Descent

Introduction

In the chapter on topologies on schemes (see Topologies, Section 020L) we introduced Zariski, étale, fppf, smooth, syntomic and fpqc coverings of schemes. In this chapter we discuss what kind of structures over schemes can be descended through such coverings. See for example [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI]. This is also meant to introduce the notions of descent, descent data, effective descent data, in the less formal setting of descent questions for quasi-coherent sheaves, schemes, etc. The formal notion, that of a stack over a site, is discussed in the chapter on stacks (see Stacks, Section 0267).

Descent data for quasi-coherent sheaves

In this chapter we will use the convention where the projection maps \(\text{pr}_i : X \times \ldots \times X \to X\) are labeled starting with \(i = 0\). Hence we have \(\text{pr}_0, \text{pr}_1 : X \times X \to X\), \(\text{pr}_0, \text{pr}_1, \text{pr}_2 : X \times X \times X \to X\), etc.

Definition

Let \(S\) be a scheme. Let \(\{f_i : S_i \to S\}_{i \in I}\) be a family of morphisms with target \(S\).

1. A descent datum \((\mathcal{F}_i, \varphi_{ij})\) for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf \(\mathcal{F}_i\) on \(S_i\) for each \(i \in I\), an isomorphism of quasi-coherent \(\mathcal{O}_{S_i \times_S S_j}\)-modules \(\varphi_{ij} : \text{pr}_0^*\mathcal{F}_i \to \text{pr}_1^*\mathcal{F}_j\) for each pair \((i, j) \in I^2\) such that for every triple of indices \((i, j, k) \in I^3\) the diagram \[\xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[rd]_{\text{pr}_{01}^*\varphi_{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi_{ik}} & & \text{pr}_2^*\mathcal{F}_k \\ & \text{pr}_1^*\mathcal{F}_j \ar[ru]_{\text{pr}_{12}^*\varphi_{jk}} & }\] of \(\mathcal{O}_{S_i \times_S S_j \times_S S_k}\)-modules commutes. This is called the cocycle condition.

2. A morphism \(\psi : (\mathcal{F}_i, \varphi_{ij}) \to (\mathcal{F}'_i, \varphi'_{ij})\) of descent data is given by a family \(\psi = (\psi_i)_{i\in I}\) of morphisms of \(\mathcal{O}_{S_i}\)-modules \(\psi_i : \mathcal{F}_i \to \mathcal{F}'_i\) such that all the diagrams \[\xymatrix{ \text{pr}_0^*\mathcal{F}_i \ar[r]_{\varphi_{ij}} \ar[d]_{\text{pr}_0^*\psi_i} & \text{pr}_1^*\mathcal{F}_j \ar[d]^{\text{pr}_1^*\psi_j} \\ \text{pr}_0^*\mathcal{F}'_i \ar[r]^{\varphi'_{ij}} & \text{pr}_1^*\mathcal{F}'_j \\ }\] commute.

A good example to keep in mind is the following. Suppose that \(S = \bigcup S_i\) is an open covering. In that case we have seen descent data for sheaves of sets in Sheaves, Section 00AK where we called them “glueing data for sheaves of sets with respect to the given covering”. Moreover, we proved that the category of glueing data is equivalent to the category of sheaves on \(S\). We will show the analogue in the setting above when \(\{S_i \to S\}_{i\in I}\) is an fpqc covering.

In the extreme case where the covering \(\{S \to S\}\) is given by \(\text{id}_S\) a descent datum is necessarily of the form \((\mathcal{F}, \text{id}_\mathcal{F})\). The cocycle condition guarantees that the identity on \(\mathcal{F}\) is the only permitted map in this case. The following lemma shows in particular that to every quasi-coherent sheaf of \(\mathcal{O}_S\)-modules there is associated a unique descent datum with respect to any given family.

Lemma

Let \(\mathcal{U} = \{U_i \to U\}_{i \in I}\) and \(\mathcal{V} = \{V_j \to V\}_{j \in J}\) be families of morphisms of schemes with fixed target. Let \((g, \alpha : I \to J, (g_i)) : \mathcal{U} \to \mathcal{V}\) be a morphism of families of maps with fixed target, see Sites, Definition 00VT. Let \((\mathcal{F}_j, \varphi_{jj'})\) be a descent datum for quasi-coherent sheaves with respect to the family \(\{V_j \to V\}_{j \in J}\). Then

1. The system \[\left(g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}\right)\] is a descent datum with respect to the family \(\{U_i \to U\}_{i \in I}\).

2. This construction is functorial in the descent datum \((\mathcal{F}_j, \varphi_{jj'})\).

3. Given a second morphism \((g', \alpha' : I \to J, (g'_i))\) of families of maps with fixed target with \(g = g'\) there exists a functorial isomorphism of descent data \[(g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}) \cong ((g'_i)^*\mathcal{F}_{\alpha'(i)}, (g'_i \times g'_{i'})^*\varphi_{\alpha'(i)\alpha'(i')}).\]

Proof

Omitted. Hint: The maps \(g_i^*\mathcal{F}_{\alpha(i)} \to (g'_i)^*\mathcal{F}_{\alpha'(i)}\) which give the isomorphism of descent data in part (3) are the pullbacks of the maps \(\varphi_{\alpha(i)\alpha'(i)}\) by the morphisms \((g_i, g'_i) : U_i \to V_{\alpha(i)} \times_V V_{\alpha'(i)}\).

Any family \(\mathcal{U} = \{S_i \to S\}_{i \in I}\) is a refinement of the trivial covering \(\{S \to S\}\) in a unique way. For a quasi-coherent sheaf \(\mathcal{F}\) on \(S\) we denote simply \((\mathcal{F}|_{S_i}, can)\) the descent datum with respect to \(\mathcal{U}\) obtained by the procedure above.

Definition

Let \(S\) be a scheme. Let \(\{S_i \to S\}_{i \in I}\) be a family of morphisms with target \(S\).

1. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_S\)-module. We call the unique descent on \(\mathcal{F}\) datum with respect to the covering \(\{S \to S\}\) the trivial descent datum.

2. The pullback of the trivial descent datum to \(\{S_i \to S\}\) is called the canonical descent datum. Notation: \((\mathcal{F}|_{S_i}, can)\).

3. A descent datum \((\mathcal{F}_i, \varphi_{ij})\) for quasi-coherent sheaves with respect to the given covering is said to be effective if there exists a quasi-coherent sheaf \(\mathcal{F}\) on \(S\) such that \((\mathcal{F}_i, \varphi_{ij})\) is isomorphic to \((\mathcal{F}|_{S_i}, can)\).

Lemma

Let \(S\) be a scheme. Let \(S = \bigcup U_i\) be an open covering. Any descent datum on quasi-coherent sheaves for the family \(\mathcal{U} = \{U_i \to S\}\) is effective. Moreover, the functor from the category of quasi-coherent \(\mathcal{O}_S\)-modules to the category of descent data with respect to \(\mathcal{U}\) is fully faithful.

Proof

This follows immediately from Sheaves, Section 00AK and the fact that being quasi-coherent is a local property, see Modules, Definition 01BE.

To prove more we first need to study the case of modules over rings.

Descent for modules

Let \(R \to A\) be a ring map. By Simplicial, Example 016N this gives rise to a cosimplicial \(R\)-algebra \[\xymatrix{ A \ar@<1ex>[r] \ar@<-1ex>[r] & A \otimes_R A \ar@<0ex>[l] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A \otimes_R A \otimes_R A \ar@<1ex>[l] \ar@<-1ex>[l] }\] Let us denote this \((A/R)_\bullet\) so that \((A/R)_n\) is the \((n + 1)\)-fold tensor product of \(A\) over \(R\). Given a map \(\varphi : [n] \to [m]\) the \(R\)-algebra map \((A/R)_\bullet(\varphi)\) is the map \[a_0 \otimes \ldots \otimes a_n \longmapsto \prod\nolimits_{\varphi(i) = 0} a_i \otimes \prod\nolimits_{\varphi(i) = 1} a_i \otimes \ldots \otimes \prod\nolimits_{\varphi(i) = m} a_i\] where we use the convention that the empty product is \(1\). Thus the first few maps, notation as in Simplicial, Section 016I, are \[\begin{matrix} \delta^1_0 & : & a_0 & \mapsto & 1 \otimes a_0 \\ \delta^1_1 & : & a_0 & \mapsto & a_0 \otimes 1 \\ \sigma^0_0 & : & a_0 \otimes a_1 & \mapsto & a_0a_1 \\ \delta^2_0 & : & a_0 \otimes a_1 & \mapsto & 1 \otimes a_0 \otimes a_1 \\ \delta^2_1 & : & a_0 \otimes a_1 & \mapsto & a_0 \otimes 1 \otimes a_1 \\ \delta^2_2 & : & a_0 \otimes a_1 & \mapsto & a_0 \otimes a_1 \otimes 1 \\ \sigma^1_0 & : & a_0 \otimes a_1 \otimes a_2 & \mapsto & a_0a_1 \otimes a_2 \\ \sigma^1_1 & : & a_0 \otimes a_1 \otimes a_2 & \mapsto & a_0 \otimes a_1a_2 \end{matrix}\] and so on.

An \(R\)-module \(M\) gives rise to a cosimplicial \((A/R)_\bullet\)-module \((A/R)_\bullet \otimes_R M\). In other words \(M_n = (A/R)_n \otimes_R M\) and using the \(R\)-algebra maps \((A/R)_n \to (A/R)_m\) to define the corresponding maps on \(M \otimes_R (A/R)_\bullet\).

The analogue to a descent datum for quasi-coherent sheaves in the setting of modules is the following.

Definition

Let \(R \to A\) be a ring map.

1. A descent datum \((N, \varphi)\) for modules with respect to \(R \to A\) is given by an \(A\)-module \(N\) and an isomorphism of \(A \otimes_R A\)-modules \[\varphi : N \otimes_R A \to A \otimes_R N\] such that the cocycle condition holds: the diagram of \(A \otimes_R A \otimes_R A\)-module maps \[\xymatrix{ N \otimes_R A \otimes_R A \ar[rr]_{\varphi_{02}} \ar[rd]_{\varphi_{01}} & & A \otimes_R A \otimes_R N \\ & A \otimes_R N \otimes_R A \ar[ru]_{\varphi_{12}} & }\] commutes (see below for notation).

2. A morphism \((N, \varphi) \to (N', \varphi')\) of descent data is a morphism of \(A\)-modules \(\psi : N \to N'\) such that the diagram \[\xymatrix{ N \otimes_R A \ar[r]_\varphi \ar[d]_{\psi \otimes \text{id}_A} & A \otimes_R N \ar[d]^{\text{id}_A \otimes \psi} \\ N' \otimes_R A \ar[r]^{\varphi'} & A \otimes_R N' }\] is commutative.

In the definition we use the notation that \(\varphi_{01} = \varphi \otimes \text{id}_A\), \(\varphi_{12} = \text{id}_A \otimes \varphi\), and \(\varphi_{02}(n \otimes 1 \otimes 1) = \sum a_i \otimes 1 \otimes n_i\) if \(\varphi(n \otimes 1) = \sum a_i \otimes n_i\). All three are \(A \otimes_R A \otimes_R A\)-module homomorphisms. Equivalently we have \[\varphi_{ij} = \varphi \otimes_{(A/R)_1, \ (A/R)_\bullet(\tau^2_{ij})} (A/R)_2\] where \(\tau^2_{ij} : [1] \to [2]\) is the map \(0 \mapsto i\), \(1 \mapsto j\). Namely, \((A/R)_{\bullet}(\tau^2_{02})(a_0 \otimes a_1) = a_0 \otimes 1 \otimes a_1\), and similarly for the others¹.

We need some more notation to be able to state the next lemma. Let \((N, \varphi)\) be a descent datum with respect to a ring map \(R \to A\). For \(n \geq 0\) and \(i \in [n]\) we set \[N_{n, i} = A \otimes_R \ldots \otimes_R A \otimes_R N \otimes_R A \otimes_R \ldots \otimes_R A\] with the factor \(N\) in the \(i\)th spot. It is an \((A/R)_n\)-module. If we introduce the maps \(\tau^n_i : [0] \to [n]\), \(0 \mapsto i\) then we see that \[N_{n, i} = N \otimes_{(A/R)_0, \ (A/R)_\bullet(\tau^n_i)} (A/R)_n\] For \(0 \leq i \leq j \leq n\) we let \(\tau^n_{ij} : [1] \to [n]\) be the map such that \(0\) maps to \(i\) and \(1\) to \(j\). Similarly to the above the homomorphism \(\varphi\) induces isomorphisms \[\varphi^n_{ij} = \varphi \otimes_{(A/R)_1, \ (A/R)_\bullet(\tau^n_{ij})} (A/R)_n : N_{n, i} \longrightarrow N_{n, j}\] of \((A/R)_n\)-modules when \(i < j\). If \(i = j\) we set \(\varphi^n_{ij} = \text{id}\). Since these are all isomorphisms they allow us to move the factor \(N\) to any spot we like. And the cocycle condition exactly means that it does not matter how we do this (e.g., as a composition of two of these or at once). Finally, for any \(\beta : [n] \to [m]\) we define the morphism \[N_{\beta, i} : N_{n, i} \to N_{m, \beta(i)}\] as the unique \((A/R)_\bullet(\beta)\)-semi linear map such that \[N_{\beta, i}(1 \otimes \ldots \otimes n \otimes \ldots \otimes 1) = 1 \otimes \ldots \otimes n \otimes \ldots \otimes 1\] for all \(n \in N\). This hints at the following lemma.

Lemma

Let \(R \to A\) be a ring map. Given a descent datum \((N, \varphi)\) we can associate to it a cosimplicial \((A/R)_\bullet\)-module \(N_\bullet\)² by the rules \(N_n = N_{n, n}\) and given \(\beta : [n] \to [m]\) setting we define \[N_\bullet(\beta) = (\varphi^m_{\beta(n)m}) \circ N_{\beta, n} : N_{n, n} \longrightarrow N_{m, m}.\] This procedure is functorial in the descent datum.

Proof

Here are the first few maps where \(\varphi(n \otimes 1) = \sum \alpha_i \otimes x_i\) \[\begin{matrix} \delta^1_0 & : & N & \to & A \otimes N & n & \mapsto & 1 \otimes n \\ \delta^1_1 & : & N & \to & A \otimes N & n & \mapsto & \sum \alpha_i \otimes x_i\\ \sigma^0_0 & : & A \otimes N & \to & N & a_0 \otimes n & \mapsto & a_0n \\ \delta^2_0 & : & A \otimes N & \to & A \otimes A \otimes N & a_0 \otimes n & \mapsto & 1 \otimes a_0 \otimes n \\ \delta^2_1 & : & A \otimes N & \to & A \otimes A \otimes N & a_0 \otimes n & \mapsto & a_0 \otimes 1 \otimes n \\ \delta^2_2 & : & A \otimes N & \to & A \otimes A \otimes N & a_0 \otimes n & \mapsto & \sum a_0 \otimes \alpha_i \otimes x_i \\ \sigma^1_0 & : & A \otimes A \otimes N & \to & A \otimes N & a_0 \otimes a_1 \otimes n & \mapsto & a_0a_1 \otimes n \\ \sigma^1_1 & : & A \otimes A \otimes N & \to & A \otimes N & a_0 \otimes a_1 \otimes n & \mapsto & a_0 \otimes a_1n \end{matrix}\] with notation as in Simplicial, Section 016I. We first verify the two properties \(\sigma^0_0 \circ \delta^1_0 = \text{id}\) and \(\sigma^0_0 \circ \delta^1_1 = \text{id}\). The first one, \(\sigma^0_0 \circ \delta^1_0 = \text{id}\), is clear from the explicit description of the morphisms above. To prove the second relation we have to use the cocycle condition (because it does not hold for an arbitrary isomorphism \(\varphi : N \otimes_R A \to A \otimes_R N\)). Write \(p = \sigma^0_0 \circ \delta^1_1 : N \to N\). By the description of the maps above we deduce that \(p\) is also equal to \[p = \varphi \otimes \text{id} : N = (N \otimes_R A) \otimes_{(A \otimes_R A)} A \longrightarrow (A \otimes_R N) \otimes_{(A \otimes_R A)} A = N\] Since \(\varphi\) is an isomorphism we see that \(p\) is an isomorphism. Write \(\varphi(n \otimes 1) = \sum \alpha_i \otimes x_i\) for certain \(\alpha_i \in A\) and \(x_i \in N\). Then \(p(n) = \sum \alpha_ix_i\). Next, write \(\varphi(x_i \otimes 1) = \sum \alpha_{ij} \otimes y_j\) for certain \(\alpha_{ij} \in A\) and \(y_j \in N\). Then the cocycle condition says that \[\sum \alpha_i \otimes \alpha_{ij} \otimes y_j = \sum \alpha_i \otimes 1 \otimes x_i.\] This means that \(p(n) = \sum \alpha_ix_i = \sum \alpha_i\alpha_{ij}y_j = \sum \alpha_i p(x_i) = p(p(n))\). Thus \(p\) is a projector, and since it is an isomorphism it is the identity.

To prove fully that \(N_\bullet\) is a cosimplicial module we have to check all 5 types of relations of Simplicial, Remark 016L. The relations on composing \(\sigma\)’s are obvious. The relations on composing \(\delta\)’s come down to the cocycle condition for \(\varphi\). In exactly the same way as above one checks the relations \(\sigma_j \circ \delta_j = \sigma_j \circ \delta_{j + 1} = \text{id}\). Finally, the other relations on compositions of \(\delta\)’s and \(\sigma\)’s hold for any \(\varphi\) whatsoever.

Note that to an \(R\)-module \(M\) we can associate a canonical descent datum, namely \((M \otimes_R A, can)\) where \(can : (M \otimes_R A) \otimes_R A \to A \otimes_R (M \otimes_R A)\) is the obvious map: \((m \otimes a) \otimes a' \mapsto a \otimes (m \otimes a')\).

Lemma

Let \(R \to A\) be a ring map. Let \(M\) be an \(R\)-module. The cosimplicial \((A/R)_\bullet\)-module associated to the canonical descent datum is isomorphic to the cosimplicial module \((A/R)_\bullet \otimes_R M\).

Proof

Omitted.

Definition

Let \(R \to A\) be a ring map. We say a descent datum \((N, \varphi)\) is effective if there exists an \(R\)-module \(M\) and an isomorphism of descent data from \((M \otimes_R A, can)\) to \((N, \varphi)\).

Let \(R \to A\) be a ring map. Let \((N, \varphi)\) be a descent datum. We may take the cochain complex \(s(N_\bullet)\) associated with \(N_\bullet\) (see Simplicial, Section 019H). It has the following shape: \[N \to A \otimes_R N \to A \otimes_R A \otimes_R N \to \ldots\] We can describe the maps. The first map is the map \[n \longmapsto 1 \otimes n - \varphi(n \otimes 1).\] The second map on pure tensors has the values \[a \otimes n \longmapsto 1 \otimes a \otimes n - a \otimes 1 \otimes n + a \otimes \varphi(n \otimes 1).\] It is clear how the pattern continues.

In the special case where \(N = A \otimes_R M\) we see that for any \(m \in M\) the element \(1 \otimes m\) is in the kernel of the first map of the cochain complex associated to the cosimplicial module \((A/R)_\bullet \otimes_R M\). Hence we get an extended cochain complex [023K]\[\begin{equation} 0 \to M \to A \otimes_R M \to A \otimes_R A \otimes_R M \to \ldots \end{equation}\] Here we think of the \(0\) as being in degree \(-2\), the module \(M\) in degree \(-1\), the module \(A \otimes_R M\) in degree \(0\), etc. Note that this complex has the shape \[0 \to R \to A \to A \otimes_R A \to A \otimes_R A \otimes_R A \to \ldots\] when \(M = R\).

Lemma

Suppose that \(R \to A\) has a section. Then for any \(R\)-module \(M\) the extended cochain complex (023K) is exact.

Proof

By Simplicial, Lemma 019Z the map \(R \to (A/R)_\bullet\) is a homotopy equivalence of cosimplicial \(R\)-algebras (here \(R\) denotes the constant cosimplicial \(R\)-algebra). Hence \(M \to (A/R)_\bullet \otimes_R M\) is a homotopy equivalence in the category of cosimplicial \(R\)-modules, because \(\otimes_R M\) is a functor from the category of \(R\)-algebras to the category of \(R\)-modules, see Simplicial, Lemma 019Y. This implies that the induced map of associated complexes is a homotopy equivalence, see Simplicial, Lemma 01A0. Since the complex associated to the constant cosimplicial \(R\)-module \(M\) is the complex \[\xymatrix{ M \ar[r]^0 & M \ar[r]^1 & M \ar[r]^0 & M \ar[r]^1 & M \ldots }\] we win (since the extended version simply puts an extra \(M\) at the beginning).

Lemma

Suppose that \(R \to A\) is faithfully flat, see Algebra, Definition 00HB. Then for any \(R\)-module \(M\) the extended cochain complex (023K) is exact.

Proof

Suppose we can show there exists a faithfully flat ring map \(R \to R'\) such that the result holds for the ring map \(R' \to A' = R' \otimes_R A\). Then the result follows for \(R \to A\). Namely, for any \(R\)-module \(M\) the cosimplicial module \((M \otimes_R R') \otimes_{R'} (A'/R')_\bullet\) is just the cosimplicial module \(R' \otimes_R (M \otimes_R (A/R)_\bullet)\). Hence the vanishing of cohomology of the complex associated to \((M \otimes_R R') \otimes_{R'} (A'/R')_\bullet\) implies the vanishing of the cohomology of the complex associated to \(M \otimes_R (A/R)_\bullet\) by faithful flatness of \(R \to R'\). Similarly for the vanishing of cohomology groups in degrees \(-1\) and \(0\) of the extended complex (proof omitted).

But we have such a faithful flat extension. Namely \(R' = A\) works because the ring map \(R' = A \to A' = A \otimes_R A\) has a section \(a \otimes a' \mapsto aa'\) and Lemma 023L applies.

Here is how the complex relates to the question of effectivity.

Lemma

Let \(R \to A\) be a faithfully flat ring map. Let \((N, \varphi)\) be a descent datum. Then \((N, \varphi)\) is effective if and only if the canonical map \[A \otimes_R H^0(s(N_\bullet)) \longrightarrow N\] is an isomorphism.

Proof

If \((N, \varphi)\) is effective, then we may write \(N = A \otimes_R M\) with \(\varphi = can\). It follows that \(H^0(s(N_\bullet)) = M\) by Lemmas 023I and 023M. Conversely, suppose the map of the lemma is an isomorphism. In this case set \(M = H^0(s(N_\bullet))\). This is an \(R\)-submodule of \(N\), namely \(M = \{n \in N \mid 1 \otimes n = \varphi(n \otimes 1)\}\). The only thing to check is that via the isomorphism \(A \otimes_R M \to N\) the canonical descent data agrees with \(\varphi\). We omit the verification.

Lemma

Let \(R \to A\) be a faithfully flat ring map, and let \(R \to R'\) be faithfully flat. Set \(A' = R' \otimes_R A\). If all descent data for \(R' \to A'\) are effective, then so are all descent data for \(R \to A\).

Proof

Let \((N, \varphi)\) be a descent datum for \(R \to A\). Set \(N' = R' \otimes_R N = A' \otimes_A N\), and denote \(\varphi' = \text{id}_{R'} \otimes \varphi\) the base change of the descent datum \(\varphi\). Then \((N', \varphi')\) is a descent datum for \(R' \to A'\) and \(H^0(s(N'_\bullet)) = R' \otimes_R H^0(s(N_\bullet))\). Moreover, the map \(A' \otimes_{R'} H^0(s(N'_\bullet)) \to N'\) is identified with the base change of the \(A\)-module map \(A \otimes_R H^0(s(N)) \to N\) via the faithfully flat map \(A \to A'\). Hence we conclude by Lemma 039W.

Here is the main result of this section. Its proof may seem a little clumsy; for a more highbrow approach see Remark 039Y below.

Proposition

Let \(R \to A\) be a faithfully flat ring map. Then

1. any descent datum on modules with respect to \(R \to A\) is effective,

2. the functor \(M \mapsto (A \otimes_R M, can)\) from \(R\)-modules to the category of descent data is an equivalence, and

3. the inverse functor is given by \((N, \varphi) \mapsto H^0(s(N_\bullet))\).

Proof

We only prove (1) and omit the proofs of (2) and (3). As \(R \to A\) is faithfully flat, there exists a faithfully flat base change \(R \to R'\) such that \(R' \to A' = R' \otimes_R A\) has a section (namely take \(R' = A\) as in the proof of Lemma 023M). Hence, using Lemma 039X we may assume that \(R \to A\) has a section, say \(\sigma : A \to R\). Let \((N, \varphi)\) be a descent datum relative to \(R \to A\). Set \[M = H^0(s(N_\bullet)) = \{n \in N \mid 1 \otimes n = \varphi(n \otimes 1)\} \subset N\] By Lemma 039W it suffices to show that \(A \otimes_R M \to N\) is an isomorphism.

Take an element \(n \in N\). Write \(\varphi(n \otimes 1) = \sum a_i \otimes x_i\) for certain \(a_i \in A\) and \(x_i \in N\). By Lemma 023H we have \(n = \sum a_i x_i\) in \(N\) (because \(\sigma^0_0 \circ \delta^1_1 = \text{id}\) in any cosimplicial object). Next, write \(\varphi(x_i \otimes 1) = \sum a_{ij} \otimes y_j\) for certain \(a_{ij} \in A\) and \(y_j \in N\). The cocycle condition means that \[\sum a_i \otimes a_{ij} \otimes y_j = \sum a_i \otimes 1 \otimes x_i\] in \(A \otimes_R A \otimes_R N\). We conclude two things from this:

1. applying \(\sigma\) to the first \(A\) we get \(\sum \sigma(a_i) \varphi(x_i \otimes 1) = \sum \sigma(a_i) \otimes x_i\),

2. applying \(\sigma\) to the middle \(A\) we get \(\sum_i a_i \otimes \sum_j \sigma(a_{ij}) y_j = \sum a_i \otimes x_i\).

Part (1) shows that \(\sum \sigma(a_i) x_i \in M\). Applying this to \(x_i\) we see that \(\sum \sigma(a_{ij})y_i \in M\) for all \(i\). Multiplying out the equation in (2) we conclude that \(\sum_i a_i (\sum_j \sigma(a_{ij}) y_j) = \sum a_i x_i = n\). Hence \(A \otimes_R M \to N\) is surjective. Finally, suppose that \(m_i \in M\) and \(\sum a_i m_i = 0\). Then we see by applying \(\varphi\) to \(\sum a_im_i \otimes 1\) that \(\sum a_i \otimes m_i = 0\). In other words \(A \otimes_R M \to N\) is injective and we win.

Remark

Let \(R\) be a ring. Let \(f_1, \ldots, f_n\in R\) generate the unit ideal. The ring \(A = \prod_i R_{f_i}\) is a faithfully flat \(R\)-algebra. We remark that the cosimplicial ring \((A/R)_\bullet\) has the following ring in degree \(n\): \[\prod\nolimits_{i_0, \ldots, i_n} R_{f_{i_0}\ldots f_{i_n}}\] Hence the results above recover Algebra, Lemmas 00EJ, 00EK and 00EQ. But the results above actually say more because of exactness in higher degrees. Namely, it implies that Čech cohomology of quasi-coherent sheaves on affines is trivial. Thus we get a second proof of Cohomology of Schemes, Lemma 01X9.

Remark

Let \(R\) be a ring. Let \(A_\bullet\) be a cosimplicial \(R\)-algebra. In this setting a descent datum corresponds to an cosimplicial \(A_\bullet\)-module \(M_\bullet\) with the property that for every \(n, m \geq 0\) and every \(\varphi : [n] \to [m]\) the map \(M(\varphi) : M_n \to M_m\) induces an isomorphism \[M_n \otimes_{A_n, A(\varphi)} A_m \longrightarrow M_m.\] Let us call such a cosimplicial module a cartesian module. In this setting, the proof of Proposition 023N can be split in the following steps

1. If \(R \to R'\) and \(R \to A\) are faithfully flat, then descent data for \(A/R\) are effective if descent data for \((R' \otimes_R A)/R'\) are effective.

2. Let \(A\) be an \(R\)-algebra. Descent data for \(A/R\) correspond to cartesian \((A/R)_\bullet\)-modules.

3. If \(R \to A\) has a section then \((A/R)_\bullet\) is homotopy equivalent to \(R\), the constant cosimplicial \(R\)-algebra with value \(R\).

4. If \(A_\bullet \to B_\bullet\) is a homotopy equivalence of cosimplicial \(R\)-algebras then the functor \(M_\bullet \mapsto M_\bullet \otimes_{A_\bullet} B_\bullet\) induces an equivalence of categories between cartesian \(A_\bullet\)-modules and cartesian \(B_\bullet\)-modules.

For (1) see Lemma 039X. Part (2) uses Lemma 023H. Part (3) we have seen in the proof of Lemma 023L (it relies on Simplicial, Lemma 019Z). Moreover, part (4) is a triviality if you think about it right!

Finite locally free descent and differential operators

For a finite locally free covering, module descent can be encoded by one endomorphism algebra. In characteristic \(p\) this gives a concrete form of Cartier descent.

Lemma

Let \(A \to B\) be a faithfully flat finite locally free ring map and set \[E = \operatorname{End}_A(B).\] View \(B\) as a subalgebra of \(E\) by left multiplication. Then

1. the functor \[\text{Mod}_A \longrightarrow \text{left }E\text{-modules}, \qquad N \longmapsto B \otimes_A N\] where \(E\) acts on the first factor is an equivalence, and

2. for a \(B\)-module \(M\), descent data on \(M\) relative to \(A \to B\) are naturally equivalent to left \(E\)-module structures on \(M\) whose restriction to \(B\) is the given \(B\)-module structure.

Proof

Write \(B^\vee = \Hom_A(B, A)\) and make it a right \(E\)-module by precomposition. An inverse in (1) is \[M \longmapsto B^\vee \otimes_E M.\] The maps which exhibit the two composites as the identity are induced by \[B^\vee \otimes_E B \longrightarrow A, \quad \lambda \otimes b \longmapsto \lambda(b)\] and \[B \otimes_A B^\vee \otimes_E M \longrightarrow M, \quad b \otimes \lambda \otimes m \longmapsto (b\lambda)m,\] where \(b\lambda \in E\) is the endomorphism \(c \mapsto b\lambda(c)\). These maps are isomorphisms. Indeed, this can be checked after localization at every prime of \(A\), where \(B\) is a nonzero finite free module and the assertion is the usual calculation with matrix units. This proves (1).

By Proposition 023N, the functor from \(A\)-modules to descent data relative to \(A \to B\) is an equivalence. This functor and the functor in (1) have the same underlying \(B\)-module \(B \otimes_A N\). Consequently they identify the extra structures on a fixed \(B\)-module, which proves (2).

Lemma

Let \(A\) be an \(\mathbf{F}_p\)-algebra and let \[B = A[x_1, \ldots, x_n]/(x_1^p-a_1, \ldots, x_n^p-a_n)\] for some \(a_i \in A\). Let \(\partial_i : B \to B\) be the \(A\)-derivation determined by \(\partial_i(x_j) = \delta_{ij}\). The \(A\)-algebra \(E = \operatorname{End}_A(B)\) is generated by the subalgebra \(B\) of left multiplication operators and by \(\partial_1, \ldots, \partial_n\), with defining relations \[\partial_i b - b\partial_i = \partial_i(b), \qquad \partial_i\partial_j = \partial_j\partial_i, \qquad \partial_i^p = 0.\] In particular, \(E\) is free as a left \(B\)-module with basis \[\partial^F = \partial_1^{f_1} \cdots \partial_n^{f_n}, \qquad 0 \leq f_i < p.\]

Proof

The displayed relations hold in \(E\) and show that the proposed generators and relations are spanned as a left \(B\)-module by the \(\partial^F\). The monomials \[x^E = x_1^{e_1} \cdots x_n^{e_n}, \qquad 0 \leq e_i < p,\] form an \(A\)-basis of \(B\). If \(F \leq E\) componentwise, then \[\partial^F(x^E) = \prod_i \frac{e_i!}{(e_i-f_i)!} x^{E-F},\] and this is zero if \(F \not\leq E\). In particular, \(\partial^E(x^E) = \prod_i e_i!\) is a unit of \(A\).

Let \(T \in \operatorname{End}_A(B)\). Recursively on \(|E| = \sum e_i\), choose \(b_E \in B\) so that \[T(x^E) = \sum_{F \leq E} b_F\partial^F(x^E).\] At the step indexed by \(E\), all terms except \(b_E\partial^E(x^E)\) are already known and the coefficient of \(b_E\) is a unit. Thus every \(A\)-linear endomorphism is a unique sum \(\sum b_F\partial^F\). The uniqueness also shows that there can be no further defining relation.

Proposition

With \(A\), \(B\), and \(\partial_i\) as in Lemma descent-lemma-height-one-differential-operators, let \(M\) be a \(B\)-module. The following kinds of data are naturally equivalent:

1. a descent datum on \(M\) relative to \(A \to B\), and

2. additive operators \(\nabla_i : M \to M\), \(i = 1, \ldots, n\), such that \[\nabla_i(bm) = \partial_i(b)m + b\nabla_i(m), \qquad [\nabla_i, \nabla_j] = 0, \qquad \nabla_i^p = 0.\]

Every datum in (2) is effective. More precisely, if \[N = \bigcap_{i = 1}^n \Ker(\nabla_i),\] then the canonical map \(B \otimes_A N \to M\) is an isomorphism.

Proof

The ring map \(A \to B\) is finite free and faithfully flat. By Lemma descent-lemma-finite-locally-free-endomorphism-descent, descent data on \(M\) are the same as \(E = \operatorname{End}_A(B)\)-module structures extending its given \(B\)-module structure. By Lemma descent-lemma-height-one-differential-operators, such a structure is exactly a collection of operators satisfying the conditions in (2).

Effectivity follows from Proposition 023N. Under an isomorphism \(M = B \otimes_A N_0\) supplied by that proposition, the operators \(\nabla_i\) become \(\partial_i \otimes 1\). Expanding in the \(A\)-basis \(x^E\) shows that their common kernel is \(1 \otimes N_0\). Thus \(N = N_0\) and the final assertion follows.

Lemma

With \(A\), \(B\), and \(\partial_i\) as in Lemma descent-lemma-height-one-differential-operators, let \(C\) be a \(B\)-algebra. Descent data on \(C\) as an algebra relative to \(A \to B\) are naturally equivalent to additive derivations \(\nabla_i : C \to C\) which extend \(\partial_i : B \to B\) and satisfy \[[\nabla_i, \nabla_j] = 0 \quad\text{and}\quad \nabla_i^p = 0.\] If \(C_0 = \bigcap \Ker(\nabla_i)\), then \(C_0\) is an \(A\)-algebra and \(B \otimes_A C_0 \to C\) is an isomorphism of \(B\)-algebras. Equivalently, the corresponding descent datum on the affine scheme \(\Spec(C)\) is effective with descended scheme \(\Spec(C_0)\).

Proof

Apply Proposition descent-proposition-cartier-descent-modules to the underlying \(B\)-module of \(C\). The multiplication and unit are morphisms of descent data if and only if each \(\nabla_i\) is a derivation extending \(\partial_i\). The common kernel is then an \(A\)-subalgebra, and the module isomorphism in the proposition respects multiplication and the unit.

Here is the coordinate-free interpretation of the proposition. The \(B\)-module \(\mathfrak g = \operatorname{Der}_A(B, B)\) is free on the \(\partial_i\) and is a restricted Lie algebra with \(D^{[p]} = D^p\). The operators in (2) determine a \(B\)-linear assignment \(D \mapsto \nabla_D\) satisfying \[\nabla_D(bm) = D(b)m + b\nabla_D(m), \qquad [\nabla_D, \nabla_{D'}] = \nabla_{[D,D']}, \qquad \nabla_D^p = \nabla_{D^{[p]}}.\] Thus Proposition descent-proposition-cartier-descent-modules identifies descent data with flat connections having zero \(p\)-curvature, and Lemma descent-lemma-cartier-descent-algebras gives the algebra and affine-scheme versions. The construction commutes with localization in \(A\), and hence gives the analogous statement for quasi-coherent modules on finite locally free morphisms which locally have the displayed height-one presentation.

Lemma

Let \(f : X \to S\) be a morphism with a section \(s : S \to X\). Assume that for every \(T \to S\) the canonical map \[\mathcal{O}_T \longrightarrow (f_T)_*\mathcal{O}_{X_T}\] is an isomorphism. Let \(S' \to S\) be an fpqc covering and set \(X' = X \times_S S'\). An invertible sheaf \(\mathcal{L}'\) on \(X'\) together with a trivialization of \(s'^*\mathcal{L}'\) is the pullback of a unique such rigidified invertible sheaf on \(X\) if and only if its two pullbacks to \(X' \times_X X'\) are isomorphic as rigidified invertible sheaves.

Without choosing a rigidification, an invertible sheaf \(\mathcal{L}'\) is Zariski locally on \(S'\) isomorphic to the pullback of an invertible sheaf on \(X\) if and only if its two pullbacks to \(X' \times_X X'\) are Zariski locally on \(S' \times_S S'\) isomorphic.

Proof

For every \(T \to S\) and every invertible sheaf \(\mathcal{L}\) on \(X_T\), we have \[\operatorname{Aut}(\mathcal{L}) = \Gamma(X_T, \mathcal{O}_{X_T}^*) = \Gamma(T, \mathcal{O}_T^*).\] It follows that an automorphism preserving a rigidification along \(s_T\) is the identity. Thus an isomorphism between two rigidified invertible sheaves, if it exists, is unique. On the triple fibre product the two composites of the isomorphism in the statement are both rigidified isomorphisms, so they agree. In other words, the cocycle condition is automatic. Effectivity now follows from Proposition 023T, and the descended module is invertible by Lemma 05B2. The rigidification descends by full faithfulness. This proves the first assertion.

For the second assertion, replace \(\mathcal{L}'\) by \[\mathcal{L}'_0 = \mathcal{L}' \otimes (f')^*(s'^*\mathcal{L}')^{-1},\] which has a canonical rigidification. Local isomorphisms between the two pullbacks can be uniquely normalized to preserve the induced rigidifications; uniqueness makes them glue. The first assertion descends \(\mathcal{L}'_0\). Since \(s'^*\mathcal{L}'\) is Zariski locally trivial on \(S'\), this is exactly the claimed local pullback condition. The converse is immediate.

Proposition

Let \(A\), \(B\), and \(\partial_i\) be as in Lemma descent-lemma-height-one-differential-operators. Let \(f : X \to \Spec(A)\) have a section \(s\). For \(r \geq 1\), set \[S_r = \Spec(B^{\otimes_A r}), \qquad X_r = X \times_A S_r,\] write \(f_r : X_r \to S_r\) for the projection, and assume that \(\mathcal{O}_{S_r} \to (f_r)_*\mathcal{O}_{X_r}\) is an isomorphism for every \(r\). Set \(X_B = X_1\), let \(q : X_B \to X\) be the projection, and let \(\mathcal{L}\) be an invertible sheaf on \(X_B\). Normalize it along the induced section by setting \[\mathcal{L}_0 = \mathcal{L} \otimes f_B^*(s_B^*\mathcal{L})^{-1}.\] Define the it Atiyah–Cartier class \(a(\mathcal{L})\) to be the extension class of the sequence of principal parts \[0 \longrightarrow \Omega^1_{X_B/X} \otimes \mathcal{L}_0 \longrightarrow \mathcal{P}^1_{X_B/X}(\mathcal{L}_0) \longrightarrow \mathcal{L}_0 \longrightarrow 0.\] Thus \[a(\mathcal{L}) \in \Ext^1_{\mathcal{O}_{X_B}} (\mathcal{L}_0, \Omega^1_{X_B/X} \otimes \mathcal{L}_0) = H^1(X_B, \Omega^1_{X_B/X}).\] This is the ordinary first Atiyah class for \(q\). Then the following are equivalent:

1. \(a(\mathcal{L}) = 0\), and

2. there is an invertible sheaf \(\mathcal{M}\) on \(X\) such that \(\mathcal{L}\) is Zariski locally on \(\Spec(B)\) isomorphic to \(q^*\mathcal{M}\).

Proof

The sequence in the statement is the sequence of Modules, Lemma 0G3V. Its splittings are the same as connections on \(\mathcal{L}_0\) relative to \(X_B/X\). Since \[\Omega^1_{X_B/X} = \mathcal{O}_{X_B} \otimes_B \Omega^1_{B/A}\] and \(\Omega^1_{B/A}\) is free on \(\mathrm{d}x_1, \ldots, \mathrm{d}x_n\), such a connection is given by additive operators \(\nabla_i : \mathcal{L}_0 \to \mathcal{L}_0\) satisfying \[\nabla_i(gm) = \partial_i(g)m + g\nabla_i(m).\]

Assume (1). Choose a connection. Its restriction along \(s_B\) is a connection on \(s_B^*\mathcal{L}_0 = \mathcal{O}_{\Spec(B)}\). Subtracting multiplication by suitable elements of \(B\) from the \(\nabla_i\), we may and do arrange that this restricted connection is the canonical one given by the \(\partial_i\). The resulting normalized connection is unique, because the difference of two normalized connections is given by global functions on \(X_B\) and these are exactly the elements of \(B\).

The commutators \([\nabla_i, \nabla_j]\) and the operators \(\nabla_i^p\) are \(\mathcal{O}_{X_B}\)-linear. Hence they are multiplication by global functions on \(X_B\). Their restrictions along \(s_B\) vanish because the \(\partial_i\) commute and satisfy \(\partial_i^p = 0\). The hypothesis on global functions therefore gives \[[\nabla_i, \nabla_j] = 0 \quad\text{and}\quad \nabla_i^p = 0.\] On every affine open of \(X\), Proposition descent-proposition-cartier-descent-modules identifies the horizontal sections with a module whose pullback is \(\mathcal{L}_0\). These modules commute with localization and hence form the quasi-coherent sheaf \[\mathcal{M} = \bigcap_i \Ker(q_*\nabla_i)\] on \(X\). The canonical map \(q^*\mathcal{M} \to \mathcal{L}_0\) is an isomorphism, and \(\mathcal{M}\) is invertible by Lemma 05B2. The original \(\mathcal{L}\) differs from \(q^*\mathcal{M}\) by the pullback of the invertible sheaf \(s_B^*\mathcal{L}\) on \(\Spec(B)\), which is Zariski locally trivial. Thus (2) holds.

Conversely, assume (2). After normalization, the local pullback descriptions give connections which restrict to the canonical connection along \(s_B\). They agree on overlaps by the uniqueness just proved and hence give a global splitting of the sequence of principal parts. Thus \(a(\mathcal{L}) = 0\).

Descent for universally injective morphisms

Numerous constructions in algebraic geometry are made using techniques of descent, such as constructing objects over a given space by first working over a somewhat larger space which projects down to the given space, or verifying a property of a space or a morphism by pulling back along a covering map. The utility of such techniques is of course dependent on identification of a wide class of effective descent morphisms. Early in the Grothendieckian development of modern algebraic geometry, the class of morphisms which are quasi-compact and faithfully flat was shown to be effective for descending objects, morphisms, and many properties thereof.

As usual, this statement comes down to a property of rings and modules. For a homomorphism \(f: R \to S\) to be an effective descent morphism for modules, Grothendieck showed that it is sufficient for \(f\) to be faithfully flat. However, this excludes many natural examples: for instance, any split ring homomorphism is an effective descent morphism. One natural example of this even arises in the proof of faithfully flat descent: for \(f: R \to S\) any ring homomorphism, \(1_S \otimes f: S \to S \otimes_R S\) is split by the multiplication map whether or not it is flat.

One may then ask whether there is a natural ring-theoretic condition implying effective descent for modules which includes both the case of a faithfully flat morphism and that of a split ring homomorphism. It may surprise the reader (at least it surprised this author) to learn that a complete answer to this question has been known since around 1970! Namely, it is not hard to check that a necessary condition for \(f: R \to S\) to be an effective descent morphism for modules is that \(f\) must be universally injective in the category of \(R\)-modules, that is, for any \(R\)-module \(M\), the map \(1_M \otimes f: M \to M \otimes_R S\) must be injective. This then turns out to be a sufficient condition as well. For example, if \(f\) is split in the category of \(R\)-modules (but not necessarily in the category of rings), then \(f\) is an effective descent morphism for modules.

The history of this result is a bit involved: it was originally asserted by Olivier [olivier], who called universally injective morphisms pure, but without a clear indication of proof. One can extract the result from the work of Joyal and Tierney [joyal-tierney], but to the best of our knowledge, the first free-standing proof to appear in the literature is that of Mesablishvili [mesablishvili1]. The first purpose of this section is to expose Mesablishvili’s proof; this requires little modification of his original presentation aside from correcting typos, with the one exception that we make explicit the relationship between the customary definition of a descent datum in algebraic geometry and the one used in [mesablishvili1]. The proof turns out to be entirely category-theoretic, and consequently can be put in the language of monads (and thus applied in other contexts); see [janelidze-tholen].

The second purpose of this section is to collect some information about which properties of modules, algebras, and morphisms can be descended along universally injective ring homomorphisms. The cases of finite modules and flat modules were treated by Mesablishvili [mesablishvili2].

Category-theoretic preliminaries

We start by recalling a few basic notions from category theory which will simplify the exposition. In this subsection, fix an ambient category.

For two morphisms \(g_1, g_2: B \to C\), recall that an equalizer of \(g_1\) and \(g_2\) is a morphism \(f: A \to B\) which satisfies \(g_1 \circ f = g_2 \circ f\) and is universal for this property. This second statement means that any commutative diagram \[\xymatrix{A' \ar[rd]^e \ar@/^1.5pc/[rrd] \ar@{-->}[d] & & \\ A \ar[r]^f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C }\] without the dashed arrow can be uniquely completed. We also say in this situation that the diagram [08WG]\[\begin{equation} \xymatrix{ A \ar[r]^f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C } \end{equation}\] is an equalizer. Reversing arrows gives the definition of a coequalizer. See Categories, Sections 0027 and 0029.

Since it involves a universal property, the property of being an equalizer is typically not stable under applying a covariant functor. Just as for monomorphisms and epimorphisms, one can get around this in some cases by exhibiting splittings.

Definition

A split equalizer is a diagram (08WG) with \(g_1 \circ f = g_2 \circ f\) for which there exist auxiliary morphisms \(h : B \to A\) and \(i : C \to B\) such that [08WI]\[\begin{equation} h \circ f = 1_A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_B. \end{equation}\]

The point is that the equalities among arrows force (08WG) to be an equalizer: the map \(e\) factors uniquely through \(f\) by writing \(e = f \circ (h \circ e)\). Consequently, applying a covariant functor to a split equalizer gives a split equalizer; applying a contravariant functor gives a split coequalizer, whose definition is apparent.

Universally injective morphisms

Recall that \(\textit{Rings}\) denotes the category of commutative rings with \(1\). For an object \(R\) of \(\textit{Rings}\) we denote \(\text{Mod}_R\) the category of \(R\)-modules.

Remark

Any functor \(F : \mathcal{A} \to \mathcal{B}\) of abelian categories which is exact and takes nonzero objects to nonzero objects reflects injections and surjections. Namely, exactness implies that \(F\) preserves kernels and cokernels (compare with Homology, Section 010M). For example, if \(f : R \to S\) is a faithfully flat ring homomorphism, then \(\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S\) has these properties.

Let \(R\) be a ring. Recall that a morphism \(f : M \to N\) in \(\text{Mod}_R\) is universally injective if for all \(P \in \text{Mod}_R\), the morphism \(f \otimes 1_P: M \otimes_R P \to N \otimes_R P\) is injective. See Algebra, Definition 058I.

Definition

A ring map \(f: R \to S\) is universally injective if it is universally injective as a morphism in \(\text{Mod}_R\).

Example

Any split injection in \(\text{Mod}_R\) is universally injective. In particular, any split injection in \(\textit{Rings}\) is universally injective.

Example

For a ring \(R\) and \(f_1, \ldots, f_n \in R\) generating the unit ideal, the morphism \(R \to R_{f_1} \oplus \ldots \oplus R_{f_n}\) is universally injective. Although this is immediate from Lemma 08WP, it is instructive to check it directly: we immediately reduce to the case where \(R\) is local, in which case some \(f_i\) must be a unit and so the map \(R \to R_{f_i}\) is an isomorphism.

Lemma

Any faithfully flat ring map is universally injective.

Proof

This is a reformulation of Algebra, Lemma 05CK.

The key observation from [mesablishvili1] is that universal injectivity can be usefully reformulated in terms of a splitting, using the usual construction of an injective cogenerator in \(\text{Mod}_R\).

Definition

Let \(R\) be a ring. Define the contravariant functor \(C\) \(: \text{Mod}_R \to \text{Mod}_R\) by setting \[C(M) = \Hom_{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}),\] with the \(R\)-action on \(C(M)\) given by \(rf(s) = f(rs)\).

This functor was denoted \(M \mapsto M^\vee\) in More on Algebra, Section 01D8.

Lemma

For a ring \(R\), the functor \(C : \text{Mod}_R \to \text{Mod}_R\) is exact and reflects injections and surjections.

Proof

Exactness is More on Algebra, Lemma 01DA and the other properties follow from this, see Remark 08WK.

Remark

We will use frequently the standard adjunction between \(\Hom\) and tensor product, in the form of the natural isomorphism of contravariant functors [08WT]\[\begin{equation} C(\bullet_1 \otimes_R \bullet_2) \cong \Hom_R(\bullet_1, C(\bullet_2)): \text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R \end{equation}\] taking \(f: M_1 \otimes_R M_2 \to \mathbf{Q}/\mathbf{Z}\) to the map \(m_1 \mapsto (m_2 \mapsto f(m_1 \otimes m_2))\). See Algebra, Lemma 08YQ. A corollary of this observation is that if \[\xymatrix@C=9pc{ C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P) }\] is a split coequalizer diagram in \(\text{Mod}_R\), then so is \[\xymatrix@C=9pc{ C(M \otimes_R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes_R Q) \ar[r] & C(P \otimes_R Q) }\] for any \(Q \in \text{Mod}_R\).

Lemma

Let \(R\) be a ring. A morphism \(f: M \to N\) in \(\text{Mod}_R\) is universally injective if and only if \(C(f): C(N) \to C(M)\) is a split surjection.

Proof

By (08WT), for any \(P \in \text{Mod}_R\) we have a commutative diagram \[\xymatrix@C=9pc{ \Hom_R( P, C(N)) \ar[r]_{\Hom_R(P,C(f))} \ar[d]^{\cong} & \Hom_R(P,C(M)) \ar[d]^{\cong} \\ C(P \otimes_R N ) \ar[r]^{C(1_{P} \otimes f)} & C(P \otimes_R M ). }\] If \(f\) is universally injective, then \(1_{C(M)} \otimes f: C(M) \otimes_R M \to C(M) \otimes_R N\) is injective, so both rows in the above diagram are surjective for \(P = C(M)\). We may thus lift \(1_{C(M)} \in \Hom_R(C(M), C(M))\) to some \(g \in \Hom_R(C(N), C(M))\) splitting \(C(f)\). Conversely, if \(C(f)\) is a split surjection, then both rows in the above diagram are surjective, so by Lemma 08WR, \(1_{P} \otimes f\) is injective.

Remark

Let \(f: M \to N\) be a universally injective morphism in \(\text{Mod}_R\). By choosing a splitting \(g\) of \(C(f)\), we may construct a functorial splitting of \(C(1_P \otimes f)\) for each \(P \in \text{Mod}_R\). Namely, by (08WT) this amounts to splitting \(\Hom_R(P, C(f))\) functorially in \(P\), and this is achieved by the map \(g \circ \bullet\).

Descent for modules and their morphisms

Throughout this subsection, fix a ring map \(f: R \to S\). As seen in Section 023F we can use the language of cosimplicial algebras to talk about descent data for modules, but in this subsection we prefer a more down to earth terminology.

For \(i = 1, 2, 3\), let \(S_i\) be the \(i\)-fold tensor product of \(S\) over \(R\). Define the ring homomorphisms \(\delta_0^1, \delta_1^1: S_1 \to S_2\), \(\delta_{01}^1, \delta_{02}^1, \delta_{12}^1: S_1 \to S_3\), and \(\delta_0^2, \delta_1^2, \delta_2^2: S_2 \to S_3\) by the formulas \[\begin{align*} \delta^1_0 (a_0) & = 1 \otimes a_0 \\ \delta^1_1 (a_0) & = a_0 \otimes 1 \\ \delta^2_0 (a_0 \otimes a_1) & = 1 \otimes a_0 \otimes a_1 \\ \delta^2_1 (a_0 \otimes a_1) & = a_0 \otimes 1 \otimes a_1 \\ \delta^2_2 (a_0 \otimes a_1) & = a_0 \otimes a_1 \otimes 1 \\ \delta_{01}^1(a_0) & = 1 \otimes 1 \otimes a_0 \\ \delta_{02}^1(a_0) & = 1 \otimes a_0 \otimes 1 \\ \delta_{12}^1(a_0) & = a_0 \otimes 1 \otimes 1. \end{align*}\] In other words, the upper index indicates the source ring, while the lower index indicates where to insert factors of 1. (This notation is compatible with the notation introduced in Section 023F.)

Recall³ from Definition 023G that for \(M \in \text{Mod}_S\), a descent datum on \(M\) relative to \(f\) is an isomorphism \[\theta : M \otimes_{S,\delta^1_0} S_2 \longrightarrow M \otimes_{S,\delta^1_1} S_2\] of \(S_2\)-modules satisfying the cocycle condition [08WX]\[\begin{equation} (\theta \otimes \delta_2^2) \circ (\theta \otimes \delta_2^0) = (\theta \otimes \delta_2^1): M \otimes_{S, \delta^1_{01}} S_3 \to M \otimes_{S,\delta^1_{12}} S_3. \end{equation}\] Let \(DD_{S/R}\) be the category of \(S\)-modules equipped with descent data relative to \(f\).

For example, for \(M_0 \in \text{Mod}_R\) and a choice of isomorphism \(M \cong M_0 \otimes_R S\) gives rise to a descent datum by identifying \(M \otimes_{S,\delta^1_0} S_2\) and \(M \otimes_{S,\delta^1_1} S_2\) naturally with \(M_0 \otimes_R S_2\). This construction in particular defines a functor \(f^*: \text{Mod}_R \to DD_{S/R}\).

Definition

The functor \(f^*: \text{Mod}_R \to DD_{S/R}\) is called base extension along \(f\). We say that \(f\) is a descent morphism for modules if \(f^*\) is fully faithful. We say that \(f\) is an effective descent morphism for modules if \(f^*\) is an equivalence of categories.

Lemma

Assume that \[R \longrightarrow S \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} S \otimes_R S\] is an equalizer diagram, where the two arrows are \(s \mapsto s \otimes 1\) and \(s \mapsto 1 \otimes s\). If \(M\) and \(N\) are \(R\)-modules and \(N\) is flat over \(R\), then the canonical map \[\Hom_R(M, N) \longrightarrow \Hom_{DD_{S/R}}(f^*M, f^*N)\] is bijective. In particular, base extension is fully faithful on the full subcategory of flat \(R\)-modules.

Proof

Tensoring the equalizer diagram in the statement with the flat \(R\)-module \(N\) gives an equalizer diagram \[N \longrightarrow N \otimes_R S \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} N \otimes_R S \otimes_R S.\] This immediately gives injectivity of the map in the statement. Let \[u : M \otimes_R S \longrightarrow N \otimes_R S\] be a morphism between the canonical descent data. For \(m \in M\), compatibility with the descent data says that \(u(m \otimes 1)\) has the same image under the two arrows in the displayed equalizer. Thus there is a unique \(v(m) \in N\) whose image in \(N \otimes_R S\) is \(u(m \otimes 1)\). Uniqueness shows that \(m \mapsto v(m)\) is \(R\)-linear. Since \(u\) is \(S\)-linear, we have \(u = v \otimes_R S\), proving surjectivity.

Lemma

Let \((R, \mathfrak m)\) be an Artinian local ring and let \(R \to S\) be an injective ring map. Assume that

1. \(\mathfrak mS \subset R\), and

2. the diagram \[R \longrightarrow S \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} S \otimes_R S\] is an equalizer, where the two arrows are the canonical ones.

Then base extension induces an equivalence between

1. flat \(R\)-modules, and

2. descent data relative to \(R \to S\) whose underlying \(S\)-module is flat.

Proof

We first record the shape of the ring map. Since \(\mathfrak m\) is nilpotent, \(\mathfrak mS\) is a nilpotent ideal of \(S\). By (1) it is contained in \(R\), and hence in the nilradical \(\mathfrak m\) of \(R\). The reverse inclusion is automatic, so \[\mathfrak mS = \mathfrak m.\] Set \(k = R/\mathfrak m\) and \(C = S/\mathfrak m\). The map \(k \to C\) is injective and hence faithfully flat. Moreover, the diagram \[\xymatrix{ C & S \ar[l] \\ k \ar[u] & R \ar[l] \ar[u] }\] is cartesian. Indeed, if \(s \in S\) and \(a \in R\) have the same image in \(C\), then \(s - a \in \mathfrak mS = \mathfrak m\), and hence \(s \in R\).

Let \((P, \theta)\) be a descent datum relative to \(R \to S\) with \(P\) flat over \(S\). Reduction modulo \(\mathfrak m\) gives a descent datum on \(P/\mathfrak mP\) relative to the faithfully flat map \(k \to C\). By Proposition 023N, there are a \(k\)-vector space \(V\) and an isomorphism \[\beta : V \otimes_k C \longrightarrow P/\mathfrak mP\] compatible with these descent data. Apply More on Algebra, Lemma 0D2I to the cartesian diagram above and the triple \((V, P, \beta)\). We obtain a flat \(R\)-module \(M\) and an isomorphism \[\alpha : M \otimes_R S \longrightarrow P\] whose reduction modulo \(\mathfrak m\) is \(\beta\).

It remains to check that \(\alpha\) respects the descent data. Transport the canonical descent datum on \(M \otimes_R S\) across \(\alpha\) and denote it by \(\theta_0\). The reductions of \(\theta\) and \(\theta_0\) modulo \(\mathfrak m\) agree by the choice of \(\beta\). The module \(M\) is free by Algebra, Lemma 051G; fix a basis. Put \(D = S \otimes_R S\). For \(a \in \mathfrak m\) and \(s,t \in S\), we have \[a(s \otimes t) = as \otimes t = 1 \otimes ast \in \mathfrak m \subset D.\] The multiplication map \(D \to S\) shows that the resulting copy of \(\mathfrak m\) in \(D\) is embedded. Thus \(\mathfrak mD = \mathfrak m\). The same argument works in every higher tensor power of \(S\) over \(R\).

After identifying both pullbacks using the chosen basis, let \(g\) be the automorphism obtained by composing \(\theta\) with \(\theta_0^{-1}\). It is the identity modulo \(\mathfrak mD\). Hence all three pullbacks of \(g\) to \(S \otimes_R S \otimes_R S\) are the same automorphism: after subtracting the identity, its coefficients belong to the copy of \(\mathfrak m\) coming from \(R\). The cocycle condition therefore reads \(g = g^2\). Since \(g\) is invertible, \(g = 1\). Thus \(\alpha\) is compatible with the descent data, proving essential surjectivity. Full faithfulness follows from Lemma descent-lemma-equalizer-flat-module-descent and assumption (2).

Remark

The condition \(\mathfrak mS \subset R\) in Lemma descent-lemma-artinian-small-effective-descent-flat-modules cannot simply be discarded. Venken constructed a finite injective map of local Artinian rings which is a descent morphism for flat modules but which admits a non-effective descent datum on a free module of rank one; see [Venken]. Thus an equalizer statement gives full faithfulness, but not effectivity by itself.

Our goal is to show that for \(f\) universally injective, we can use \(\theta\) to locate \(M_0\) within \(M\). This process makes crucial use of some equalizer diagrams.

Lemma

For \((M,\theta) \in DD_{S/R}\), the diagram [08X0]\[\begin{equation} \xymatrix@C=8pc{ M \ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} & M \otimes_{S, \delta_1^1} S_2 \ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} & M \otimes_{S, \delta_{12}^1} S_3 } \end{equation}\] is a split equalizer.

Proof

Define the ring homomorphisms \(\sigma^0_0: S_2 \to S_1\) and \(\sigma_0^1, \sigma_1^1: S_3 \to S_2\) by the formulas \[\begin{align*} \sigma^0_0 (a_0 \otimes a_1) & = a_0a_1 \\ \sigma^1_0 (a_0 \otimes a_1 \otimes a_2) & = a_0a_1 \otimes a_2 \\ \sigma^1_1 (a_0 \otimes a_1 \otimes a_2) & = a_0 \otimes a_1a_2. \end{align*}\] We then take the auxiliary morphisms to be \(1_M \otimes \sigma_0^0: M \otimes_{S, \delta_1^1} S_2 \to M\) and \(1_M \otimes \sigma_0^1: M \otimes_{S,\delta_{12}^1} S_3 \to M \otimes_{S, \delta_1^1} S_2\). Of the compatibilities required in (08WI), the first follows from tensoring the cocycle condition (08WX) with \(\sigma_1^1\) and the others are immediate.

Lemma

For \((M, \theta) \in DD_{S/R}\), the diagram [08X2]\[\begin{equation} \xymatrix@C=8pc{ C(M \otimes_{S, \delta_{12}^1} S_3) \ar@<1ex>[r]^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))} \ar@<-1ex>[r]_{C(1_{M \otimes S_2} \otimes \delta^2_1)} & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(M). } \end{equation}\] obtained by applying \(C\) to (08X0) is a split coequalizer.

Proof

Omitted.

Lemma

The diagram [08X4]\[\begin{equation} \xymatrix@C=8pc{ S_1 \ar[r]^{\delta^1_1} & S_2 \ar@<1ex>[r]^{\delta^2_2} \ar@<-1ex>[r]_{\delta^2_1} & S_3 } \end{equation}\] is a split equalizer.

Proof

In Lemma 08WZ, take \((M, \theta) = f^*(S)\).

This suggests a definition of a potential quasi-inverse functor for \(f^*\).

Definition

Define the functor \(f_*\) \(: DD_{S/R} \to \text{Mod}_R\) by taking \(f_*(M, \theta)\) to be the \(R\)-submodule of \(M\) for which the diagram [08X6]\[\begin{equation} \xymatrix@C=8pc{f_*(M,\theta) \ar[r] & M \ar@<1ex>^{\theta \circ (1_M \otimes \delta_0^1)}[r] \ar@<-1ex>_{1_M \otimes \delta_1^1}[r] & M \otimes_{S, \delta_1^1} S_2 } \end{equation}\] is an equalizer.

Using Lemma 08WZ and the fact that the restriction functor \(\text{Mod}_S \to \text{Mod}_R\) is right adjoint to the base extension functor \(\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S\), we deduce that \(f_*\) is right adjoint to \(f^*\).

We are ready for the key lemma. In the faithfully flat case this is a triviality (see Remark 08X9), but in the general case some argument is needed.

Lemma

If \(f\) is universally injective, then the diagram [08X8]\[\begin{equation} \xymatrix@C=8pc{ f_*(M, \theta) \otimes_R S \ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} & M \otimes_{S, \delta_1^1} S_2 \ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} & M \otimes_{S, \delta_{12}^1} S_3 } \end{equation}\] obtained by tensoring (08X6) over \(R\) with \(S\) is an equalizer.

Proof

By Lemma 08WU and Remark 08WV, the map \(C(1_N \otimes f): C(N \otimes_R S) \to C(N)\) can be split functorially in \(N\). This gives the upper vertical arrows in the commutative diagram \[\xymatrix@C=8pc{ C(M \otimes_{S, \delta_1^1} S_2) \ar@<1ex>^{C(\theta \circ (1_M \otimes \delta_0^1))}[r] \ar@<-1ex>_{C(1_M \otimes \delta_1^1)}[r] \ar[d] & C(M) \ar[r]\ar[d] & C(f_*(M,\theta)) \ar@{-->}[d] \\ C(M \otimes_{S,\delta_{12}^1} S_3) \ar@<1ex>^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}[r] \ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta^2_1)}[r] \ar[d] & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} \ar[d]^{C(1_M \otimes \delta_1^1)} & C(M) \ar[d] \ar@{=}[dl] \\ C(M \otimes_{S, \delta_1^1} S_2) \ar@<1ex>[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} \ar@<-1ex>[r]_{C(1_M \otimes \delta_1^1)} & C(M) \ar[r] & C(f_*(M,\theta)) }\] in which the compositions along the columns are identity morphisms. The second row is the coequalizer diagram (08X2); this produces the dashed arrow. From the top right square, we obtain auxiliary morphisms \(C(f_*(M,\theta)) \to C(M)\) and \(C(M) \to C(M\otimes_{S,\delta_1^1} S_2)\) which imply that the first row is a split coequalizer diagram. By Remark 08WS, we may tensor with \(S\) inside \(C\) to obtain the split coequalizer diagram \[\xymatrix@C=8pc{ C(M \otimes_{S,\delta_2^2 \circ \delta_1^1} S_3) \ar@<1ex>^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}[r] \ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta^2_1)}[r] & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(f_*(M,\theta) \otimes_R S). }\] By Lemma 08WR, we conclude (08X8) must also be an equalizer.

Remark

If \(f\) is a split injection in \(\text{Mod}_R\), one can simplify the argument by splitting \(f\) directly, without using \(C\). Things are even simpler if \(f\) is faithfully flat; in this case, the conclusion of Lemma 08X7 is immediate because tensoring over \(R\) with \(S\) preserves all equalizers.

Theorem

The following conditions are equivalent.

1. The morphism \(f\) is a descent morphism for modules.

2. The morphism \(f\) is an effective descent morphism for modules.

3. The morphism \(f\) is universally injective.

Proof

It is clear that (b) implies (a). We now check that (a) implies (c). If \(f\) is not universally injective, we can find \(M \in \text{Mod}_R\) such that the map \(1_M \otimes f: M \to M \otimes_R S\) has nontrivial kernel \(N\). The natural projection \(M \to M/N\) is not an isomorphism, but its image in \(DD_{S/R}\) is an isomorphism. Hence \(f^*\) is not fully faithful.

We finally check that (c) implies (b). By Lemmas 08WZ and 08X7, for \((M, \theta) \in DD_{S/R}\), the natural map \(f^* f_*(M,\theta) \to M\) is an isomorphism of \(S\)-modules. For \(M_0 \in \text{Mod}_R\) we want to show \(M_0 \to f_*f^*M_0\) is an isomorphism. By adjunction, the composition \(f^*M_0 \to f^*f_*f^*M_0 \to f^*M_0\) is the identity, while the above shows that the second arrow is an isomorphism. Thus \(f^*M_0 \to f^*f_*f^*M_0\) is an isomorphism. By Algebra, Lemma 0H9I we conclude \(M_0 \to f_* f^* M_0\) is an isomorphism. Consequently, \(f_*\) and \(f^*\) are quasi-inverse functors, proving the claim.

Descent for properties of modules

Throughout this subsection, fix a universally injective ring map \(f : R \to S\), an object \(M \in \text{Mod}_R\), and a ring map \(R \to A\). We now investigate the question of which properties of \(M\) or \(A\) can be checked after base extension along \(f\). We start with some results from [mesablishvili2].

Lemma

If \(M \in \text{Mod}_R\) is flat, then \(C(M)\) is an injective \(R\)-module.

Proof

Let \(0 \to N \to P \to Q \to 0\) be an exact sequence in \(\text{Mod}_R\). Since \(M\) is flat, \[0 \to N \otimes_R M \to P \otimes_R M \to Q \otimes_R M \to 0\] is exact. By Lemma 08WR, \[0 \to C(Q \otimes_R M) \to C(P \otimes_R M) \to C(N \otimes_R M) \to 0\] is exact. By (08WT), this last sequence can be rewritten as \[0 \to \Hom_R(Q, C(M)) \to \Hom_R(P, C(M)) \to \Hom_R(N, C(M)) \to 0.\] Hence \(C(M)\) is an injective object of \(\text{Mod}_R\).

Theorem

If \(M \otimes_R S\) has one of the following properties as an \(S\)-module

1. finitely generated;

2. finitely presented;

3. flat;

4. faithfully flat;

5. finite projective;

then so does \(M\) as an \(R\)-module (and conversely).

Proof

To prove (a), choose a finite set \(\{n_i\}\) of generators of \(M \otimes_R S\) in \(\text{Mod}_S\). Write each \(n_i\) as \(\sum_j m_{ij} \otimes s_{ij}\) with \(m_{ij} \in M\) and \(s_{ij} \in S\). Let \(F\) be the finite free \(R\)-module with basis \(e_{ij}\) and let \(F \to M\) be the \(R\)-module map sending \(e_{ij}\) to \(m_{ij}\). Then \(F \otimes_R S\to M \otimes_R S\) is surjective, so \(\Coker(F \to M) \otimes_R S\) is zero and hence \(\Coker(F \to M)\) is zero. This proves (a).

To see (b) assume \(M \otimes_R S\) is finitely presented. Then \(M\) is finitely generated by (a). Choose a surjection \(R^{\oplus n} \to M\) with kernel \(K\). Then \(K \otimes_R S \to S^{\oplus r} \to M \otimes_R S \to 0\) is exact. By Algebra, Lemma 0519 the kernel of \(S^{\oplus r} \to M \otimes_R S\) is a finite \(S\)-module. Thus we can find finitely many elements \(k_1, \ldots, k_t \in K\) such that the images of \(k_i \otimes 1\) in \(S^{\oplus r}\) generate the kernel of \(S^{\oplus r} \to M \otimes_R S\). Let \(K' \subset K\) be the submodule generated by \(k_1, \ldots, k_t\). Then \(M' = R^{\oplus r}/K'\) is a finitely presented \(R\)-module with a morphism \(M' \to M\) such that \(M' \otimes_R S \to M \otimes_R S\) is an isomorphism. Thus \(M' \cong M\) as desired.

To prove (c), let \(0 \to M' \to M'' \to M \to 0\) be a short exact sequence in \(\text{Mod}_R\). Since \(\bullet \otimes_R S\) is a right exact functor, \(M'' \otimes_R S \to M \otimes_R S\) is surjective. So by Lemma 08WR the map \(C(M \otimes_R S) \to C(M'' \otimes_R S)\) is injective. If \(M \otimes_R S\) is flat, then Lemma 08XC shows \(C(M \otimes_R S)\) is an injective object of \(\text{Mod}_S\), so the injection \(C(M \otimes_R S) \to C(M'' \otimes_R S)\) is split in \(\text{Mod}_S\) and hence also in \(\text{Mod}_R\). Since \(C(M \otimes_R S) \to C(M)\) is a split surjection by Lemma 08WU, it follows that \(C(M) \to C(M'')\) is a split injection in \(\text{Mod}_R\). That is, the sequence \[0 \to C(M) \to C(M'') \to C(M') \to 0\] is split exact. For \(N \in \text{Mod}_R\), by (08WT) we see that \[0 \to C(M \otimes_R N) \to C(M'' \otimes_R N) \to C(M' \otimes_R N) \to 0\] is split exact. By Lemma 08WR, \[0 \to M' \otimes_R N \to M'' \otimes_R N \to M \otimes_R N \to 0\] is exact. This implies \(M\) is flat over \(R\). Namely, taking \(M'\) a free module surjecting onto \(M\) we conclude that \(\text{Tor}_1^R(M, N) = 0\) for all modules \(N\) and we can use Algebra, Lemma 00M5. This proves (c).

To deduce (d) from (c), note that if \(N \in \text{Mod}_R\) and \(M \otimes_R N\) is zero, then \(M \otimes_R S \otimes_S (N \otimes_R S) \cong (M \otimes_R N) \otimes_R S\) is zero, so \(N \otimes_R S\) is zero and hence \(N\) is zero.

To deduce (e) at this point, it suffices to recall that \(M\) is finitely generated and projective if and only if it is finitely presented and flat. See Algebra, Lemma 00NX.

Lemma

If \(f : R \to S\) is universally injective, then base extension induces an equivalence between

1. flat \(R\)-modules, and

2. descent data relative to \(f\) whose underlying \(S\)-module is flat.

Proof

Theorem 08XA says that base extension is an equivalence on all modules. It carries flat modules to flat modules by Algebra, Lemma 00HI. Conversely, if \((P, \theta)\) is a descent datum with \(P\) flat over \(S\), write \((P, \theta) \cong f^*M\) using Theorem 08XA. Then \(M \otimes_R S\) is flat over \(S\), and Theorem 08XD shows that \(M\) is flat over \(R\).

There is a variant for \(R\)-algebras.

Theorem

If \(A \otimes_R S\) has one of the following properties as an \(S\)-algebra

1. of finite type;

2. of finite presentation;

3. formally unramified;

4. unramified;

5. étale;

then so does \(A\) as an \(R\)-algebra (and of course conversely).

Proof

To prove (a), choose a finite set \(\{x_i\}\) of generators of \(A \otimes_R S\) over \(S\). Write each \(x_i\) as \(\sum_j y_{ij} \otimes s_{ij}\) with \(y_{ij} \in A\) and \(s_{ij} \in S\). Let \(F\) be the polynomial \(R\)-algebra on variables \(e_{ij}\) and let \(F \to M\) be the \(R\)-algebra map sending \(e_{ij}\) to \(y_{ij}\). Then \(F \otimes_R S\to A \otimes_R S\) is surjective, so \(\Coker(F \to A) \otimes_R S\) is zero and hence \(\Coker(F \to A)\) is zero. This proves (a).

To see (b) assume \(A \otimes_R S\) is a finitely presented \(S\)-algebra. Then \(A\) is finite type over \(R\) by (a). Choose a surjection \(R[x_1, \ldots, x_n] \to A\) with kernel \(I\). Then \(I \otimes_R S \to S[x_1, \ldots, x_n] \to A \otimes_R S \to 0\) is exact. By Algebra, Lemma 00R2 the kernel of \(S[x_1, \ldots, x_n] \to A \otimes_R S\) is a finitely generated ideal. Thus we can find finitely many elements \(y_1, \ldots, y_t \in I\) such that the images of \(y_i \otimes 1\) in \(S[x_1, \ldots, x_n]\) generate the kernel of \(S[x_1, \ldots, x_n] \to A \otimes_R S\). Let \(I' \subset I\) be the ideal generated by \(y_1, \ldots, y_t\). Then \(A' = R[x_1, \ldots, x_n]/I'\) is a finitely presented \(R\)-algebra with a morphism \(A' \to A\) such that \(A' \otimes_R S \to A \otimes_R S\) is an isomorphism. Thus \(A' \cong A\) as desired.

To prove (c), recall that \(A\) is formally unramified over \(R\) if and only if the module of relative differentials \(\Omega_{A/R}\) vanishes, see Algebra, Lemma 00UO or [EGA4, Proposition 17.2.1]. Since \(\Omega_{(A \otimes_R S)/S} = \Omega_{A/R} \otimes_R S\), the vanishing descends by Theorem 08XA.

To deduce (d) from the previous cases, recall that \(A\) is unramified over \(R\) if and only if \(A\) is formally unramified and of finite type over \(R\), see Algebra, Lemma 00UU.

To prove (e), recall that by Algebra, Lemma 08WD or [EGA4, Théorème 17.6.1] the algebra \(A\) is étale over \(R\) if and only if \(A\) is flat, unramified, and of finite presentation over \(R\).

Remark

It would make things easier to have a faithfully flat ring homomorphism \(g: R \to T\) for which \(T \to S \otimes_R T\) has some extra structure. For instance, if one could ensure that \(T \to S \otimes_R T\) is split in \(\textit{Rings}\), then it would follow that every property of a module or algebra which is stable under base extension and which descends along faithfully flat morphisms also descends along universally injective morphisms. An obvious guess would be to find \(g\) for which \(T\) is not only faithfully flat but also injective in \(\text{Mod}_R\), but even for \(R = \mathbf{Z}\) no such homomorphism can exist.

Fpqc descent of quasi-coherent sheaves

The main application of flat descent for modules is the corresponding descent statement for quasi-coherent sheaves with respect to fpqc-coverings.

Lemma

Let \(S\) be an affine scheme. Let \(\mathcal{U} = \{f_i : U_i \to S\}_{i = 1, \ldots, n}\) be a standard fpqc covering of \(S\), see Topologies, Definition 022F. Any descent datum on quasi-coherent sheaves for \(\mathcal{U} = \{U_i \to S\}\) is effective. Moreover, the functor from the category of quasi-coherent \(\mathcal{O}_S\)-modules to the category of descent data with respect to \(\mathcal{U}\) is fully faithful.

Proof

This is a restatement of Proposition 023N in terms of schemes. First, note that a descent datum \(\xi\) for quasi-coherent sheaves with respect to \(\mathcal{U}\) is exactly the same as a descent datum \(\xi'\) for quasi-coherent sheaves with respect to the covering \(\mathcal{U}' = \{\coprod_{i = 1, \ldots, n} U_i \to S\}\). Moreover, effectivity for \(\xi\) is the same as effectivity for \(\xi'\). Hence we may assume \(n = 1\), i.e., \(\mathcal{U} = \{U \to S\}\) where \(U\) and \(S\) are affine. In this case descent data correspond to descent data on modules with respect to the ring map \[\Gamma(S, \mathcal{O}) \longrightarrow \Gamma(U, \mathcal{O}).\] Since \(U \to S\) is surjective and flat, we see that this ring map is faithfully flat. In other words, Proposition 023N applies and we win.

Proposition

Let \(S\) be a scheme. Let \(\mathcal{U} = \{\varphi_i : U_i \to S\}\) be an fpqc covering, see Topologies, Definition 022B. Any descent datum on quasi-coherent sheaves for \(\mathcal{U} = \{U_i \to S\}\) is effective. Moreover, the functor from the category of quasi-coherent \(\mathcal{O}_S\)-modules to the category of descent data with respect to \(\mathcal{U}\) is fully faithful.

Proof

Let \(S = \bigcup_{j \in J} V_j\) be an affine open covering. For \(j, j' \in J\) we denote \(V_{jj'} = V_j \cap V_{j'}\) the intersection (which need not be affine). For \(V \subset S\) open we denote \(\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}\) which is a fpqc-covering (Topologies, Lemma 022D). By definition of an fpqc covering, we can find for each \(j \in J\) a finite set \(K_j\), a map \(\underline{i} : K_j \to I\), affine opens \(U_{\underline{i}(k), k} \subset U_{\underline{i}(k)}\), \(k \in K_j\) such that \(\mathcal{V}_j = \{U_{\underline{i}(k), k} \to V_j\}_{k \in K_j}\) is a standard fpqc covering of \(V_j\). And of course, \(\mathcal{V}_j\) is a refinement of \(\mathcal{U}_{V_j}\). Picture \[\xymatrix{ \mathcal{V}_j \ar[r] \ar@{~>}[d] & \mathcal{U}_{V_j} \ar[r] \ar@{~>}[d] & \mathcal{U} \ar@{~>}[d] \\ V_j \ar@{=}[r] & V_j \ar[r] & S }\] where the top horizontal arrows are morphisms of families of morphisms with fixed target (see Sites, Definition 00VT).

To prove the proposition you show successively the faithfulness, fullness, and essential surjectivity of the functor from quasi-coherent sheaves to descent data.

Faithfulness. Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent sheaves on \(S\) and let \(a, b : \mathcal{F} \to \mathcal{G}\) be homomorphisms of \(\mathcal{O}_S\)-modules. Suppose \(\varphi_i^*(a) = \varphi_i^*(b)\) for all \(i\). Pick \(s \in S\). Then \(s = \varphi_i(u)\) for some \(i \in I\) and \(u \in U_i\). Since \(\mathcal{O}_{S, s} \to \mathcal{O}_{U_i, u}\) is flat, hence faithfully flat (Algebra, Lemma 00HR) we see that \(a_s = b_s : \mathcal{F}_s \to \mathcal{G}_s\). Hence \(a = b\).

Fully faithfulness. Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent sheaves on \(S\) and let \(a_i : \varphi_i^*\mathcal{F} \to \varphi_i^*\mathcal{G}\) be homomorphisms of \(\mathcal{O}_{U_i}\)-modules such that \(\text{pr}_0^*a_i = \text{pr}_1^*a_j\) on \(U_i \times_U U_j\). We can pull back these morphisms to get morphisms \[a_k : \varphi_{i(k)}^*\mathcal{F}|_{U_{\underline{i}(k), k}} \longrightarrow \varphi_{i(k)}^*\mathcal{G}|_{U_{\underline{i}(k), k}}\] \(k \in K_j\) with notation as above. Moreover, Lemma 023C assures us that these define a morphism between (canonical) descent data on \(\mathcal{V}_j\). Hence, by Lemma 023S, we get correspondingly unique morphisms \(a_j : \mathcal{F}|_{V_j} \to \mathcal{G}|_{V_j}\). To see that \(a_j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}\) we use that both \(a_j\) and \(a_{j'}\) agree with the pullback of the morphism \((a_i)_{i \in I}\) of (canonical) descent data to any covering refining both \(\mathcal{V}_{j, V_{jj'}}\) and \(\mathcal{V}_{j', V_{jj'}}\), and using the faithfulness already shown. For example the covering \(\mathcal{V}_{jj'} = \{V_k \times_S V_{k'} \to V_{jj'}\}_{k \in K_j, k' \in K_{j'}}\) will do.

Essential surjectivity. Let \(\xi = (\mathcal{F}_i, \varphi_{ii'})\) be a descent datum for quasi-coherent sheaves relative to the covering \(\mathcal{U}\). Pull back this descent datum to get descent data \(\xi_j\) for quasi-coherent sheaves relative to the coverings \(\mathcal{V}_j\) of \(V_j\). By Lemma 023S once again there exist quasi-coherent sheaves \(\mathcal{F}_j\) on \(V_j\) whose associated canonical descent datum is isomorphic to \(\xi_j\). By fully faithfulness (proved above) we see there are isomorphisms \[\phi_{jj'} : \mathcal{F}_j|_{V_{jj'}} \longrightarrow \mathcal{F}_{j'}|_{V_{jj'}}\] corresponding to the isomorphism of descent data between the pullback of \(\xi_j\) and \(\xi_{j'}\) to \(\mathcal{V}_{jj'}\). To see that these maps \(\phi_{jj'}\) satisfy the cocycle condition we use faithfulness (proved above) over the triple intersections \(V_{jj'j''}\). Hence, by Lemma 023E we see that the sheaves \(\mathcal{F}_j\) glue to a quasi-coherent sheaf \(\mathcal{F}\) as desired. We still have to verify that the canonical descent datum relative to \(\mathcal{U}\) associated to \(\mathcal{F}\) is isomorphic to the descent datum we started out with. This verification is omitted.

Finite epimorphisms

In this section an effective epimorphism is meant in the category of schemes, in the sense of Sites, Definition 00WP, for a family consisting of one morphism. We first translate this categorical condition into a condition on functions for finite morphisms.

Lemma

Let \(f : X \to S\) be a finite morphism of schemes and set \(\mathcal{A} = f_*\mathcal{O}_X\). Then \(f\) is an epimorphism in the category of schemes if and only if \[\mathcal{O}_S \longrightarrow \mathcal{A}\] is injective.

Proof

Assume that \(f\) is an epimorphism. If \(\mathcal{I} = \Ker(\mathcal{O}_S \to \mathcal{A})\) is nonzero, then \(f\) factors through the proper closed subscheme \(Z \subset S\) cut out by \(\mathcal{I}\). The pushout \(S \amalg_Z S\) exists by More on Morphisms, Lemma 0B7M. The two maps \(S \to S \amalg_Z S\) are distinct and become equal after composition with \(f\), a contradiction. To see directly that the two maps are distinct, work on an affine open on which \(\mathcal{I}\) is nonzero and use the two projections \[A \times_{A/I} A \longrightarrow A.\]

Conversely, assume that \(\mathcal{O}_S \to \mathcal{A}\) is injective. The morphism \(f\) is surjective: this follows affine locally from lying over for an injective integral ring map, see Algebra, Lemma 00GQ. Let \(g, h : S \to Y\) be morphisms such that \(g \circ f = h \circ f\). The inverse image \(Z \to S\) of the diagonal of \(Y\) under \((g, h)\) is an immersion by Schemes, Lemma 01KJ, and \(f\) factors through \(Z\). Since \(f\) is surjective, the immersion \(Z \to S\) is surjective and hence is a closed immersion by Schemes, Lemma 01IQ. If \(\mathcal{J}\) is its ideal sheaf, then \(\mathcal{J}\) maps to zero in \(\mathcal{A}\). Injectivity gives \(\mathcal{J} = 0\), and hence \(Z = S\) and \(g = h\).

Lemma

Let \[\xymatrix{ S'' \ar@<0.5ex>[r]^a \ar@<-0.5ex>[r]_b & S' \ar[r]^f & S }\] be a coequalizer diagram in the category of schemes. Assume that \(S'\) and \(S''\) are finite over \(S\). Then the following statements hold.

1. The diagram of quasi-coherent \(\mathcal{O}_S\)-algebras \[\mathcal{O}_S \longrightarrow f_*\mathcal{O}_{S'} \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} (f \circ a)_*\mathcal{O}_{S''}\] is an equalizer diagram.

2. For every flat morphism \(T \to S\), the base change \[\xymatrix{ S''_T \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & S'_T \ar[r] & T }\] is a coequalizer diagram in the category of schemes.

3. If \(T\) and \(Y\) are \(S\)-schemes and \(T\) is flat over \(S\), then \[\Mor_S(T, Y) \longrightarrow \Mor_{S'}(T_{S'}, Y_{S'}) \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} \Mor_{S''}(T_{S''}, Y_{S''})\] is an equalizer diagram.

4. If \(\mathcal{F}\) and \(\mathcal{G}\) are quasi-coherent \(\mathcal{O}_S\)-modules and \(\mathcal{G}\) is flat, then the analogous diagram of sets of module maps after pullback to \(S'\) and \(S''\) is an equalizer diagram.

Proof

For a diagram of schemes finite over \(S\), the coequalizer condition is equivalent to the equalizer condition on the corresponding finite quasi-coherent algebras. Here is a reminder of the argument. The assertion is local on \(S\), where all three schemes are affine. For affine targets it is the anti-equivalence between rings and affine schemes. For an arbitrary target, finiteness shows that the structure map \(S' \to S\) is closed. The inverse image of an affine open under a map out of \(S'\) which equalizes \(a\) and \(b\) is saturated and hence descends to an open of \(S\). The affine case constructs the descended map on these opens, and uniqueness glues them. This is the same finite topological argument used in the proof of Lemma descent-lemma-finite-effective-epimorphism-functions. This proves (1) and the stated equivalence.

Formation of the finite pushforward algebras commutes with base change. Tensoring the equalizer in (1) with the flat \(\mathcal{O}_S\)-algebra \(\mathcal{O}_T\) preserves it. The equivalence just recalled gives (2), and the universal property of the coequalizer gives (3).

We prove (4) affine locally. Write the equalizer in (1) as \[A \longrightarrow B \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} C.\] Let \(M\) and \(N\) correspond to \(\mathcal{F}\) and \(\mathcal{G}\). Since \(N\) is flat, tensoring gives an equalizer \[N \longrightarrow N \otimes_A B \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} N \otimes_A C.\] A compatible \(B\)-linear map \(M \otimes_A B \to N \otimes_A B\) sends \(m \otimes 1\) into this equalizer for every \(m \in M\). It therefore comes from a unique \(A\)-linear map \(M \to N\). These maps glue by uniqueness.

Proposition

Let \(S\) be a locally Noetherian scheme, let \(s \in S\), and let \(\Omega\) be an algebraically closed extension of \(\kappa(s)\). The category of schemes finite over \(S\) has finite colimits, and the functor \[X \longmapsto X_s(\Omega)\] from this category to the category of sets commutes with finite colimits.

Proof

Finite coproducts of finite \(S\)-schemes exist and are finite over \(S\). Consider two morphisms \(X_1 \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} X_0\) of finite \(S\)-schemes. Write \[\mathcal{A}_i = (X_i \to S)_*\mathcal{O}_{X_i}\] and let \(\mathcal{A}\) be the equalizer of \(\mathcal{A}_0 \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} \mathcal{A}_1\). This is a finite quasi-coherent \(\mathcal{O}_S\)-algebra: the assertion is affine local, where it follows because a submodule of a finite module over a Noetherian ring is finite. The finite \(S\)-scheme \[X = \underline{\Spec}_S(\mathcal{A})\] is the coequalizer of \(X_1 \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} X_0\) in the category of finite \(S\)-schemes. Thus finite colimits exist by Categories, Lemma 002P.

Choose a separable closure \(\kappa(s)^{sep} \subset \Omega\) and let \(R^{sh}\) be the corresponding strict henselization of \(R = \mathcal{O}_{S, s}\). The morphism \[T = \Spec(R^{sh}) \longrightarrow S\] is flat by More on Algebra, Lemma 07QM. Consequently, the construction by equalizers above shows that base change from \(S\) to \(T\) preserves coequalizers of finite schemes.

We claim that the functor \[G : \{\text{schemes finite over }T\} \longrightarrow \{\text{finite sets}\}, \qquad Y \longmapsto Y(\Omega)\] is left adjoint to the functor \[E \longmapsto E_T = \coprod_{e \in E} T.\] To see this, write \(Y = \Spec(B)\). Since \(R^{sh}\) is henselian, Algebra, Lemma 04GG gives a product decomposition \[B = B_1 \times \ldots \times B_n\] with each \(B_i\) local. The residue field of \(B_i\) is a finite purely inseparable extension of the separably algebraically closed residue field \(\kappa(s)^{sep}\). It therefore has a unique \(\kappa(s)^{sep}\)-embedding into \(\Omega\). It follows that \(Y(\Omega)\) has exactly one element for each factor \(B_i\). On the other hand, a morphism \(Y \to E_T\) is exactly a choice of an element of \(E\) for each of these factors. We have therefore obtained, naturally in \(Y\) and \(E\), a bijection \[\Mor_T(Y, E_T) = \operatorname{Map}(Y(\Omega), E),\] which proves the claim.

As a left adjoint, \(G\) commutes with finite colimits. Finally, for every finite \(S\)-scheme \(Y\) we have \[Y_s(\Omega) = (Y \times_S T)(\Omega).\] Flat base change to \(T\) preserves the coequalizers constructed above, and \(G\) preserves them by the claim. Finite coproducts are preserved as well, which proves the proposition.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \[j : R \longrightarrow X \times_S X\] be an equivalence relation with \(R\) and \(X\) finite over \(S\). If \(Y = X/R\) is the coequalizer in the category of finite \(S\)-schemes, then the canonical morphism \[R \longrightarrow X \times_Y X\] is a surjective closed immersion. In particular, it is surjective and radicial.

Proof

The canonical morphism is a monomorphism because \(j\) is a monomorphism. It is finite, as every morphism between schemes finite over \(S\) is finite, and hence it is a closed immersion by Morphisms, Lemma 03BB.

Let \(z \in X \times_Y X\), let \(s \in S\) be its image, and choose an algebraically closed extension \(\Omega\) of \(\kappa(z)\). The point \(z\) gives two elements of \(X_s(\Omega)\) with the same image in \(Y_s(\Omega)\). By Proposition descent-proposition-finite-schemes-geometric-fibre-colimits, the latter set is the quotient of \(X_s(\Omega)\) by \(R_s(\Omega)\). Since \(R_s(\Omega)\) is already an equivalence relation, there is an element of \(R_s(\Omega)\) mapping to the given pair. Thus \(z\) is in the image of the canonical morphism, which proves surjectivity. A monomorphism is universally injective, so the final assertion follows as well.

Lemma

Let \(f : X \to S\) be a finite morphism of schemes. Let \(f_2 : X \times_S X \to S\) be the structure morphism and set \[\mathcal{A} = f_*\mathcal{O}_X, \qquad \mathcal{A}_2 = (f_2)_*\mathcal{O}_{X \times_S X}.\] The following are equivalent:

1. \(f\) is an effective epimorphism in the category of schemes, and

2. the diagram of quasi-coherent \(\mathcal{O}_S\)-algebras \[\xymatrix{ \mathcal{O}_S \ar[r] & \mathcal{A} \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{A}_2 }\] is an equalizer diagram.

These conditions are preserved by flat base change on \(S\).

Proof

Assume (1) and let \(\mathcal{B}\) be the equalizer of the two maps from \(\mathcal{A}\) to \(\mathcal{A}_2\). By Lemma descent-lemma-finite-epimorphism-injective-functions, the map \(\mathcal{O}_S \to \mathcal{A}\) is injective, and hence we have inclusions \[\mathcal{O}_S \longrightarrow \mathcal{B} \longrightarrow \mathcal{A}.\] They correspond to a factorization \[X \xrightarrow{g} T = \underline{\Spec}_S(\mathcal{B}) \xrightarrow{h} S\] of \(f\). By construction \(g \circ \text{pr}_1 = g \circ \text{pr}_2\) on \(X \times_S X\). Effectivity gives a morphism \(e : S \to T\) such that \(g = e \circ f\). Thus \[g = e \circ h \circ g, \qquad f = h \circ e \circ f.\] The maps \(\mathcal{B} \to \mathcal{A}\) and \(\mathcal{O}_S \to \mathcal{A}\) are injective. On applying the functor on functions to the displayed identities, we find that \(h^\sharp\) and \(e^\sharp\) are inverse. Hence \(T \to S\) is an isomorphism and (2) holds.

Assume (2). By Lemma descent-lemma-finite-epimorphism-injective-functions, \(f\) is an epimorphism. It remains to prove existence in the coequalizer property. Suppose that \(a : X \to Y\) satisfies \[a \circ \text{pr}_1 = a \circ \text{pr}_2\] on \(X \times_S X\). The finite surjective morphism \(f\) is closed, and hence is a quotient map on underlying topological spaces. If \(V \subset Y\) is affine open, then \(a^{-1}(V)\) is saturated for \(f\). Indeed, two points of \(X\) in the same fibre lift to a point of \(X \times_S X\) after a residue field extension; see Schemes, Lemma 01JT. Consequently, there is an open \(U \subset S\) such that \[f^{-1}(U) = a^{-1}(V).\] Explicitly, one can take \(U = S \setminus f(X \setminus a^{-1}(V))\).

The ring map defining \(a|_{f^{-1}(U)} : f^{-1}(U) \to V\) has image in the equalizer of the two maps from \(\Gamma(f^{-1}(U), \mathcal{O}_X)\) to \(\Gamma(f^{-1}(U) \times_U f^{-1}(U), \mathcal{O})\). By (2), this equalizer is \(\Gamma(U, \mathcal{O}_U)\). Schemes, Lemma 01I1 therefore gives a morphism \(U \to V\) whose pullback to \(f^{-1}(U)\) is \(a\). As \(V\) runs through an affine open covering of \(Y\), the corresponding opens \(U\) cover \(S\). The resulting morphisms agree on overlaps because \(f\) is an epimorphism, and they glue to the desired morphism \(S \to Y\).

Finally, formation of \(\mathcal{A}\) and \(\mathcal{A}_2\) commutes with base change because the morphisms involved are affine. Kernels of maps of quasi-coherent modules commute with flat base change. Thus (2), and hence (1), is preserved by flat base change.

Lemma

Let \(f : X \to S\) be a finite effective epimorphism. Let \(T\) and \(Y\) be \(S\)-schemes and assume that \(T\) is flat over \(S\). Set \(X_2 = X \times_S X\). Then \[\Mor_S(T, Y) \longrightarrow \Mor_X(T \times_S X, Y \times_S X) \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} \Mor_{X_2}(T \times_S X_2, Y \times_S X_2)\] is an equalizer diagram, where the two arrows are pullback by the projections \(X_2 \to X\).

Proof

The base change \[T \times_S X \longrightarrow T\] is an effective epimorphism by Lemma descent-lemma-finite-effective-epimorphism-functions, because \(T \to S\) is flat. The equalizer condition on a morphism \(T \times_S X \to Y \times_S X\) over \(X\) says that its underlying \(S\)-morphism to \(Y\) agrees after the two pullbacks to \[(T \times_S X) \times_T (T \times_S X) = T \times_S X_2.\] The effective epimorphism therefore gives a unique morphism \(T \to Y\). Its pullback to \(X\) is the original morphism, which proves the assertion.

Lemma

Let \(f : X \to S\) be a finite effective epimorphism. Pullback from \(S\) to \(X\) is fully faithful from the category of quasi-coherent \(\mathcal{O}_S\)-modules which are flat over \(S\) to the category of descent data relative to \(f\).

Proof

The assertion is local on \(S\). If \(S = \Spec(R)\), then \(X = \Spec(A)\) because \(f\) is finite. Lemma descent-lemma-finite-effective-epimorphism-functions says that \[R \longrightarrow A \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} A \otimes_R A\] is an equalizer. Full faithfulness now follows from Lemma descent-lemma-equalizer-flat-module-descent. The descended morphisms on affine opens agree on overlaps by uniqueness and hence glue.

Example

The equalizer condition for a finite effective epimorphism does not imply universal injectivity. Let \(k\) be a field and set \[A = k[x, y]/(xy), \qquad B = k[x] \times k[y].\] Map \(A\) to \(B\) by restriction to the two coordinate axes. We have \[B \otimes_A B \cong k[x] \times k \times k \times k[y].\] The equalizer of the two maps \(B \to B \otimes_A B\) consists of the pairs \((p(x), q(y))\) such that \(p(0) = q(0)\), and hence is \(A\). Thus \(\Spec(B) \to \Spec(A)\) is a finite effective epimorphism by Lemma descent-lemma-finite-effective-epimorphism-functions.

On the other hand, it is not universally injective. For the ideal \(I = (x + y) \subset A\), we have \[x \in IB \cap A \quad\text{but}\quad x \notin I.\] The last assertion can be checked after identifying \(A/I = k[x]/(x^2)\). Thus tensoring \(A \to B\) with \(A/I\) kills the nonzero class of \(x\). By Algebra, Definition 058I, the map \(A \to B\) is not universally injective. By Theorem 08XA, it is not an effective descent morphism for all modules.

This does not by itself answer the more restricted historical question for descent data whose underlying module is flat. Under the additional hypothesis of universal injectivity, Lemma descent-lemma-effective-descent-flat-modules gives an affirmative answer.

Theorem

Let \(S\) be a Noetherian scheme and let \(f : X \to S\) be a finite epimorphism. Put \[\mathcal{A} = \mathcal{O}_S, \qquad \mathcal{A}_0 = f_*\mathcal{O}_X,\] and define recursively \[\mathcal{A}_{i + 1} = \Ker\left( \mathcal{A}_i \longrightarrow \mathcal{A}_i \otimes_{\mathcal{A}} \mathcal{A}_i, \quad b \longmapsto b \otimes 1 - 1 \otimes b \right), \qquad X_i = \underline{\Spec}_S(\mathcal{A}_i).\] Then:

1. the inclusions \(\mathcal{A} \subset \mathcal{A}_{i + 1} \subset \mathcal{A}_i\) define finite effective epimorphisms \(X_i \to X_{i + 1}\);

2. there is an \(n \geq 0\) such that \(\mathcal{A}_n = \mathcal{A}\), and hence \[X = X_0 \longrightarrow X_1 \longrightarrow \ldots \longrightarrow X_n = S\] is a factorization of \(f\) into finite effective epimorphisms;

3. the construction commutes with flat base change on \(S\); and

4. this factorization has minimal length among factorizations of \(f\) into finite effective epimorphisms.

We call it the canonical factorization of \(f\).

Proof

Lemma descent-lemma-finite-epimorphism-injective-functions gives \(\mathcal{A} \subset \mathcal{A}_0\), and it is clear inductively that \(\mathcal{A} \subset \mathcal{A}_{i + 1} \subset \mathcal{A}_i\). All these algebras are finite as \(\mathcal{A}\)-modules: a submodule of a finite module on a Noetherian scheme is finite. The canonical surjection \[\mathcal{A}_i \otimes_{\mathcal{A}} \mathcal{A}_i \longrightarrow \mathcal{A}_i \otimes_{\mathcal{A}_{i + 1}} \mathcal{A}_i\] has kernel generated locally by \(b \otimes 1 - 1 \otimes b\) for \(b \in \mathcal{A}_{i + 1}\). These elements are already zero in the source by the definition of \(\mathcal{A}_{i + 1}\). Thus the displayed map is an isomorphism, and \[\xymatrix{ \mathcal{A}_{i + 1} \ar[r] & \mathcal{A}_i \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{A}_i \otimes_{\mathcal{A}_{i + 1}} \mathcal{A}_i }\] is an equalizer diagram. Lemma descent-lemma-finite-effective-epimorphism-functions proves (1).

The same description shows (3): tensor products commute with arbitrary base change and kernels commute with flat base change.

We prove (2). First observe that if \(\mathcal{A}_{i + 1} = \mathcal{A}_i\), then the isomorphism above says that the multiplication map \[\mathcal{A}_i \otimes_{\mathcal{A}} \mathcal{A}_i \longrightarrow \mathcal{A}_i\] is an isomorphism. Affine locally, \(A \to A_i\) is therefore a finite epimorphism of rings. It is surjective by Algebra, Lemma 04VT, and it is injective by construction. Hence \(\mathcal{A}_i = \mathcal{A}\).

It remains to show that the descending sequence \(\mathcal{A}_i\) is locally stationary. By localization and quasi-compactness, we may work over the spectrum of a Noetherian local ring \(A\) and write \(A_i\) for the corresponding finite \(A\)-algebras. We argue by induction on \(d = \dim(A)\). If \(d = 0\), then \(A_0/A\) has finite length, so its descending sequence of submodules \(A_i/A\) is stationary.

Suppose \(d > 0\) and let \(\mathfrak m\) be the maximal ideal. For every \(\mathfrak p \neq \mathfrak m\), the localized sequence is stationary by induction because \(\dim(A_{\mathfrak p}) < d\). The punctured spectrum is quasi-compact, and every \(A_i/A\) is finite. For each such \(\mathfrak p\), choose an index where the localized sequence is stationary; its stationary value is \(A_{\mathfrak p}\) by the first paragraph. This equality holds on a neighbourhood of \(\mathfrak p\). A finite subcover and the largest of the corresponding indices therefore give an \(N\) such that \((A_N/A)_{\mathfrak p} = 0\) for all \(\mathfrak p \neq \mathfrak m\). Thus \(A_i/A\) has finite length for all \(i \geq N\). The descending sequence is stationary, and the observation of the preceding paragraph shows that its stationary value is \(A\). Applying this at every point of \(S\) and using that \(S\) is quasi-compact gives one integer \(n\) with \(\mathcal{A}_n = \mathcal{A}\).

Finally, consider any factorization \[X = X'_0 \longrightarrow X'_1 \longrightarrow \ldots \longrightarrow X'_m = S\] into finite effective epimorphisms, and write \(\mathcal{B}_i\) for the finite \(\mathcal{A}\)-algebra corresponding to \(X'_i\). We show by induction that \(\mathcal{A}_i \subset \mathcal{B}_i\). This is equality for \(i = 0\). If it holds for \(i\), then the equalizer criterion of Lemma descent-lemma-finite-effective-epimorphism-functions shows that every section of \(\mathcal{A}_{i + 1}\) maps into \(\mathcal{B}_{i + 1}\). Hence the inclusion holds for \(i + 1\). Since \(\mathcal{B}_m = \mathcal{A}\), we obtain \(\mathcal{A} \subset \mathcal{A}_m \subset \mathcal{B}_m = \mathcal{A}\). Thus the canonical sequence terminates in at most \(m\) steps, proving (4).

Galois descent for quasi-coherent sheaves

Galois descent for quasi-coherent sheaves is just a special case of fpqc descent for quasi-coherent sheaves. In this section we will explain how to translate from a Galois descent to an fpqc descent and then apply earlier results to conclude.

Let \(k'/k\) be a field extension. Then \(\{\Spec(k') \to \Spec(k)\}\) is an fpqc covering. Let \(X\) be a scheme over \(k\). For a \(k\)-algebra \(A\) we set \(X_A = X \times_{\Spec(k)} \Spec(A)\). By Topologies, Lemma 022D we see that \(\{X_{k'} \to X\}\) is an fpqc covering. Observe that \[X_{k'} \times_X X_{k'} = X_{k' \otimes_k k'} \quad\text{and}\quad X_{k'} \times_X X_{k'} \times_X X_{k'} = X_{k' \otimes_k k' \otimes_k k'}\] Thus a descent datum for quasi-coherent sheaves with respect to \(\{X_{k'} \to X\}\) is given by a quasi-coherent sheaf \(\mathcal{F}\) on \(X_{k'}\), an isomorphism \(\varphi : \text{pr}_0^*\mathcal{F} \to \text{pr}_1^*\mathcal{F}\) on \(X_{k' \otimes_k k'}\) which satisfies an obvious cocycle condition on \(X_{k' \otimes_k k' \otimes_k k'}\). We will work out what this means in the case of a Galois extension below.

Let \(k'/k\) be a finite Galois extension with Galois group \(G = \text{Gal}(k'/k)\). Then there are \(k\)-algebra isomorphisms \[k' \otimes_k k' \longrightarrow \prod\nolimits_{\sigma \in G} k',\quad a \otimes b \longrightarrow \prod a\sigma(b)\] and \[k' \otimes_k k' \otimes_k k' \longrightarrow \prod\nolimits_{(\sigma, \tau) \in G \times G} k',\quad a \otimes b \otimes c \longrightarrow \prod a\sigma(b)\sigma(\tau(c))\] The reason for choosing here \(a\sigma(b)\sigma(\tau(c))\) and not \(a\sigma(b)\tau(c)\) is that the formulas below simplify but it isn’t strictly necessary. Given \(\sigma \in G\) we denote \[f_\sigma = \text{id}_X \times \Spec(\sigma) : X_{k'} \longrightarrow X_{k'}\] Please keep in mind that because \(\Spec(-)\) is a contravariant functor we have \(f_{\sigma \tau} = f_\tau \circ f_\sigma\) and not the other way around. Using the first isomorphism above we obtain an identification \[X_{k' \otimes_k k'} = \coprod\nolimits_{\sigma \in G} X_{k'}\] such that \(\text{pr}_0\) corresponds to the map \[\coprod\nolimits_{\sigma \in G} X_{k'} \xrightarrow{\coprod \text{id}} X_{k'}\] and such that \(\text{pr}_1\) corresponds to the map \[\coprod\nolimits_{\sigma \in G} X_{k'} \xrightarrow{\coprod f_\sigma} X_{k'}\] Thus we see that a descent datum \(\varphi\) on \(\mathcal{F}\) over \(X_{k'}\) corresponds to a family of isomorphisms \(\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}\). To work out the cocycle condition we use the identification \[X_{k' \otimes_k k' \otimes_k k'} = \coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'}.\] we get from our isomorphism of algebras above. Via this identification the map \(\text{pr}_{01}\) corresponds to the map \[\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'} \longrightarrow \coprod\nolimits_{\sigma \in G} X_{k'}\] which maps the summand with index \((\sigma, \tau)\) to the summand with index \(\sigma\) via the identity morphism. The map \(\text{pr}_{12}\) corresponds to the map \[\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'} \longrightarrow \coprod\nolimits_{\sigma \in G} X_{k'}\] which maps the summand with index \((\sigma, \tau)\) to the summand with index \(\tau\) via the morphism \(f_\sigma\). Finally, the map \(\text{pr}_{02}\) corresponds to the map \[\coprod\nolimits_{(\sigma, \tau) \in G \times G} X_{k'} \longrightarrow \coprod\nolimits_{\sigma \in G} X_{k'}\] which maps the summand with index \((\sigma, \tau)\) to the summand with index \(\sigma\tau\) via the identity morphism. Thus the cocycle condition \[\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi\] translates into one condition for each pair \((\sigma, \tau)\), namely \[\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma\] as maps \(\mathcal{F} \to f_{\sigma\tau}^*\mathcal{F}\). (Everything works out beautifully; for example the target of \(\varphi_\sigma\) is \(f_\sigma^*\mathcal{F}\) and the source of \(f_\sigma^*\varphi_\tau\) is \(f_\sigma^*\mathcal{F}\) as well.)

Lemma

Let \(k'/k\) be a (finite) Galois extension with Galois group \(G\). Let \(X\) be a scheme over \(k\). The category of quasi-coherent \(\mathcal{O}_X\)-modules is equivalent to the category of systems \((\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})\) where

1. \(\mathcal{F}\) is a quasi-coherent module on \(X_{k'}\),

2. \(\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}\) is an isomorphism of modules,

3. \(\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma\) for all \(\sigma, \tau \in G\).

Here \(f_\sigma = \text{id}_X \times \Spec(\sigma) : X_{k'} \to X_{k'}\).

Proof

As seen above a datum \((\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})\) as in the lemma is the same thing as a descent datum for the fpqc covering \(\{X_{k'} \to X\}\). Thus the lemma follows from Proposition 023T.

A slightly more general case of the above is the following. Suppose we have a surjective finite étale morphism \(X \to Y\) and a finite group \(G\) together with a group homomorphism \(G^{opp} \to \text{Aut}_Y(X), \sigma \mapsto f_\sigma\) such that the map \[G \times X \longrightarrow X \times_Y X,\quad (\sigma, x) \longmapsto (x, f_\sigma(x))\] is an isomorphism. Then the same result as above holds.

Lemma

Let \(X \to Y\), \(G\), and \(f_\sigma : X \to X\) be as above. The category of quasi-coherent \(\mathcal{O}_Y\)-modules is equivalent to the category of systems \((\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})\) where

1. \(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_X\)-module,

2. \(\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}\) is an isomorphism of modules,

3. \(\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma\) for all \(\sigma, \tau \in G\).

Proof

Since \(X \to Y\) is surjective finite étale \(\{X \to Y\}\) is an fpqc covering. Since \(G \times X \to X \times_Y X\), \((\sigma, x) \mapsto (x, f_\sigma(x))\) is an isomorphism, we see that \(G \times G \times X \to X \times_Y X \times_Y X\), \((\sigma, \tau, x) \mapsto (x, f_\sigma(x), f_{\sigma\tau}(x))\) is an isomorphism too. Using these identifications, the category of data as in the lemma is the same as the category of descent data for quasi-coherent sheaves for the covering \(\{x \to Y\}\). Thus the lemma follows from Proposition 023T.

Descent of finiteness properties of modules

In this section we prove that one can check quasi-coherent module has a certain finiteness conditions by checking on the members of a covering.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a finite type \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module.

Proof

Omitted. For the affine case, see Algebra, Lemma 03C4.

Lemma

Let \(f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)\) be a morphism of locally ringed spaces. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_Y\)-modules. If

1. \(f\) is open as a map of topological spaces,

2. \(f\) is surjective and flat, and

3. \(f^*\mathcal{F}\) is of finite type,

then \(\mathcal{F}\) is of finite type.

Proof

Let \(y \in Y\) be a point. Choose a point \(x \in X\) mapping to \(y\). Choose an open \(x \in U \subset X\) and elements \(s_1, \ldots, s_n\) of \(f^*\mathcal{F}(U)\) which generate \(f^*\mathcal{F}\) over \(U\). Since \(f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X\) we can after shrinking \(U\) assume \(s_i = \sum t_{ij} \otimes a_{ij}\) with \(t_{ij} \in f^{-1}\mathcal{F}(U)\) and \(a_{ij} \in \mathcal{O}_X(U)\). After shrinking \(U\) further we may assume that \(t_{ij}\) comes from a section \(s_{ij} \in \mathcal{F}(V)\) for some \(V \subset Y\) open with \(f(U) \subset V\). Let \(N\) be the number of sections \(s_{ij}\) and consider the map \[\sigma = (s_{ij}) : \mathcal{O}_V^{\oplus N} \to \mathcal{F}|_V\] By our choice of the sections we see that \(f^*\sigma|_U\) is surjective. Hence for every \(u \in U\) the map \[\sigma_{f(u)} \otimes_{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} : \mathcal{O}_{X, u}^{\oplus N} \longrightarrow \mathcal{F}_{f(u)} \otimes_{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u}\] is surjective. As \(f\) is flat, the local ring map \(\mathcal{O}_{Y, f(u)} \to \mathcal{O}_{X, u}\) is flat, hence faithfully flat (Algebra, Lemma 00HR). Hence \(\sigma_{f(u)}\) is surjective. Since \(f\) is open, \(f(U)\) is an open neighbourhood of \(y\) and the proof is done.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is an \(\mathcal{O}_{X_i}\)-module of finite presentation. Then \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite presentation.

Proof

Omitted. For the affine case, see Algebra, Lemma 03C4.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is locally generated by \(r\) sections as an \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is locally generated by \(r\) sections as an \(\mathcal{O}_X\)-module.

Proof

By Lemma 05AZ we see that \(\mathcal{F}\) is of finite type. Hence Nakayama’s lemma (Algebra, Lemma 00DV) implies that \(\mathcal{F}\) is generated by \(r\) sections in the neighbourhood of a point \(x \in X\) if and only if \(\dim_{\kappa(x)} \mathcal{F}_x \otimes \kappa(x) \leq r\). Choose an \(i\) and a point \(x_i \in X_i\) mapping to \(x\). Then \(\dim_{\kappa(x)} \mathcal{F}_x \otimes \kappa(x) = \dim_{\kappa(x_i)} (f_i^*\mathcal{F})_{x_i} \otimes \kappa(x_i)\) which is \(\leq r\) as \(f_i^*\mathcal{F}\) is locally generated by \(r\) sections.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a flat \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a flat \(\mathcal{O}_X\)-module.

Proof

Omitted. For the affine case, see Algebra, Lemma 03C4.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a finite locally free \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a finite locally free \(\mathcal{O}_X\)-module.

Proof

This follows from the fact that a quasi-coherent sheaf is finite locally free if and only if it is of finite presentation and flat, see Algebra, Lemma 00NX. Namely, if each \(f_i^*\mathcal{F}\) is flat and of finite presentation, then so is \(\mathcal{F}\) by Lemmas 05B1 and 05B0.

The definition of a locally projective quasi-coherent sheaf can be found in Properties, Section 05JN.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fpqc covering such that each \(f_i^*\mathcal{F}\) is a locally projective \(\mathcal{O}_{X_i}\)-module. Then \(\mathcal{F}\) is a locally projective \(\mathcal{O}_X\)-module.

Proof

Omitted. For Zariski coverings this is Properties, Lemma 05JQ. For the affine case this is Algebra, Theorem 05A9.

Remark

Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology. Namely, suppose that \(R\) is a ring and that \(M\) is a projective \(R\)-module which is a countable direct sum \(M = \bigoplus L_n\) of rank 1 locally free modules, but not locally free, see Examples, Lemma 05WL. Then \(M\) becomes free on making the faithfully flat base change \[R \longrightarrow \bigoplus\nolimits_{m \geq 1} \bigoplus\nolimits_{(i_1, \ldots, i_m) \in \mathbf{Z}^{\oplus m}} L_1^{\otimes i_1} \otimes_R \ldots \otimes_R L_m^{\otimes i_m}\] But we don’t know what happens for fppf coverings. In other words, we don’t know the answer to the following question: Suppose \(A \to B\) is a faithfully flat ring map of finite presentation. Let \(M\) be an \(A\)-module such that \(M \otimes_A B\) is free. Is \(M\) a locally free \(A\)-module? It turns out that if \(A\) is Noetherian, then the answer is yes. This follows from the results of [Bass]. But in general we don’t know the answer. If you know the answer, or have a reference, please email stacks.project@gmail.com.

We also add here two results which are related to the results above, but are of a slightly different nature.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(f\) is a finite morphism. Then \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite type if and only if \(f_*\mathcal{F}\) is an \(\mathcal{O}_Y\)-module of finite type.

Proof

As \(f\) is finite it is affine. This reduces us to the case where \(f\) is the morphism \(\Spec(B) \to \Spec(A)\) given by a finite ring map \(A \to B\). Moreover, then \(\mathcal{F} = \widetilde{M}\) is the sheaf of modules associated to the \(B\)-module \(M\). Note that \(M\) is finite as a \(B\)-module if and only if \(M\) is finite as an \(A\)-module, see Algebra, Lemma 00GJ. Combined with Properties, Lemma 01PB this proves the lemma.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(f\) is finite and of finite presentation. Then \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module of finite presentation if and only if \(f_*\mathcal{F}\) is an \(\mathcal{O}_Y\)-module of finite presentation.

Proof

As \(f\) is finite it is affine. This reduces us to the case where \(f\) is the morphism \(\Spec(B) \to \Spec(A)\) given by a finite and finitely presented ring map \(A \to B\). Moreover, then \(\mathcal{F} = \widetilde{M}\) is the sheaf of modules associated to the \(B\)-module \(M\). Note that \(M\) is finitely presented as a \(B\)-module if and only if \(M\) is finitely presented as an \(A\)-module, see Algebra, Lemma 0564. Combined with Properties, Lemma 01PC this proves the lemma.

Quasi-coherent sheaves and topologies, I

The results in this section say there is a natural equivalence between the category quasi-coherent modules on a scheme \(S\) and the category of quasi-coherent modules on many of the sites associated to \(S\) in the chapter on topologies.

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_S\)-module. Consider the functor [03DS]\[\begin{equation} (\Sch/S)^{opp} \longrightarrow \textit{Ab}, \quad (f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}). \end{equation}\]

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_S\)-module. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf, \linebreak[0] fpqc\}\). The functor defined in (03DS) satisfies the sheaf condition with respect to any \(\tau\)-covering \(\{T_i \to T\}_{i \in I}\) of any scheme \(T\) over \(S\).

Proof

For \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\) a \(\tau\)-covering is also a fpqc-covering, see the results in Topologies, Lemmas 0216, 0220, 0226, 021N, and 022C. Hence it suffices to prove the theorem for a fpqc covering. Assume that \(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering where \(f : T \to S\) is given. Suppose that we have a family of sections \(s_i \in \Gamma(T_i , f_i^*f^*\mathcal{F})\) such that \(s_i|_{T_i \times_T T_j} = s_j|_{T_i \times_T T_j}\). We have to find the correspond section \(s \in \Gamma(T, f^*\mathcal{F})\). We can reinterpret the \(s_i\) as a family of maps \(\varphi_i : f_i^*\mathcal{O}_T = \mathcal{O}_{T_i} \to f_i^*f^*\mathcal{F}\) compatible with the canonical descent data associated to the quasi-coherent sheaves \(\mathcal{O}_T\) and \(f^*\mathcal{F}\) on \(T\). Hence by Proposition 023T we see that we may (uniquely) descend these to a map \(\mathcal{O}_T \to f^*\mathcal{F}\) which gives us our section \(s\).

We may in particular make the following definition.

Definition

Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\). Let \(S\) be a scheme. Let \(\Sch_\tau\) be a big site containing \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_S\)-module.

1. The structure sheaf of the big site \((\Sch/S)_\tau\) is the sheaf of rings \(T/S \mapsto \Gamma(T, \mathcal{O}_T)\) which is denoted \(\mathcal{O}\) or \(\mathcal{O}_S\).

2. If \(\tau = Zariski\) or \(\tau = \etale\) the structure sheaf of the small site \(S_{Zar}\) or \(S_\etale\) is the sheaf of rings \(T/S \mapsto \Gamma(T, \mathcal{O}_T)\) which is denoted \(\mathcal{O}\) or \(\mathcal{O}_S\).

3. The sheaf of \(\mathcal{O}\)-modules associated to \(\mathcal{F}\) on the big site \((\Sch/S)_\tau\) is the sheaf of \(\mathcal{O}\)-modules \((f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})\) which is denoted \(\mathcal{F}^a\) (and often simply \(\mathcal{F}\)).

4. If \(\tau = Zariski\) or \(\tau = \etale\) the sheaf of \(\mathcal{O}\)-modules associated to \(\mathcal{F}\) on the small site \(S_{Zar}\) or \(S_\etale\) is the sheaf of \(\mathcal{O}\)-modules \((f : T \to S) \mapsto \Gamma(T, f^*\mathcal{F})\) which is denoted \(\mathcal{F}^a\) (and often simply \(\mathcal{F}\)).

Note how we use the same notation \(\mathcal{F}^a\) in each case. No confusion can really arise from this as by definition the rule that defines the sheaf \(\mathcal{F}^a\) is independent of the site we choose to look at.

Remark

In Topologies, Lemma 020X we have seen that the small Zariski site of a scheme \(S\) is equivalent to \(S\) as a topological space in the sense that the categories of sheaves are naturally equivalent. Now that \(S_{Zar}\) is also endowed with a structure sheaf \(\mathcal{O}\) we see that sheaves of modules on the ringed site \((S_{Zar}, \mathcal{O})\) agree with sheaves of modules on the ringed space \((S, \mathcal{O}_S)\).

Remark

Let \(f : T \to S\) be a morphism of schemes. Each of the morphisms of sites \(f_{sites}\) listed in Topologies, Section 03FE becomes a morphism of ringed sites. Namely, each of these morphisms of sites \(f_{sites} : (\Sch/T)_\tau \to (\Sch/S)_{\tau'}\), or \(f_{sites} : (\Sch/S)_\tau \to S_{\tau'}\) is given by the continuous functor \(S'/S \mapsto T \times_S S'/S\). Hence, given \(S'/S\) we let \[f_{sites}^\sharp : \mathcal{O}(S'/S) \longrightarrow f_{sites, *}\mathcal{O}(S'/S) = \mathcal{O}(T \times_S S'/T)\] be the usual map \(\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times_S S')\). Similarly, the morphism \(i_f : \Sh(T_\tau) \to \Sh((\Sch/S)_\tau)\) for \(\tau \in \{Zar, \etale\}\), see Topologies, Lemmas 020Y and 021F, becomes a morphism of ringed topoi because \(i_f^{-1}\mathcal{O} = \mathcal{O}\). Here are some special cases:

1. The morphism of big sites \(f_{big} : (\Sch/X)_{fppf} \to (\Sch/Y)_{fppf}\), becomes a morphism of ringed sites \[(f_{big}, f_{big}^\sharp) : ((\Sch/X)_{fppf}, \mathcal{O}_X) \longrightarrow ((\Sch/Y)_{fppf}, \mathcal{O}_Y)\] as in Modules on Sites, Definition 03AD. Similarly for the big syntomic, smooth, étale and Zariski sites.

2. The morphism of small sites \(f_{small} : X_\etale \to Y_\etale\) becomes a morphism of ringed sites \[(f_{small}, f_{small}^\sharp) : (X_\etale, \mathcal{O}_X) \longrightarrow (Y_\etale, \mathcal{O}_Y)\] as in Modules on Sites, Definition 03AD. Similarly for the small Zariski site.

Let \(S\) be a scheme. It is clear that given an \(\mathcal{O}\)-module on (say) \((\Sch/S)_{Zar}\) the pullback to (say) \((\Sch/S)_{fppf}\) is just the fppf-sheafification. To see what happens when comparing big and small sites we have the following.

Lemma

Let \(S\) be a scheme. Denote \[\begin{matrix} \text{id}_{\tau, Zar} & : & (\Sch/S)_\tau \to S_{Zar}, & \tau \in \{Zar, \etale, smooth, syntomic, fppf\} \\ \text{id}_{\tau, \etale} & : & (\Sch/S)_\tau \to S_\etale, & \tau \in \{\etale, smooth, syntomic, fppf\} \\ \text{id}_{small, \etale, Zar} & : & S_\etale \to S_{Zar}, \end{matrix}\] the morphisms of ringed sites of Remark 070R. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_S\)-modules which we view a sheaf of \(\mathcal{O}\)-modules on \(S_{Zar}\). Then

1. \((\text{id}_{\tau, Zar})^*\mathcal{F}\) is the \(\tau\)-sheafification of the Zariski sheaf \[(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})\] on \((\Sch/S)_\tau\), and

2. \((\text{id}_{small, \etale, Zar})^*\mathcal{F}\) is the étale sheafification of the Zariski sheaf \[(f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F})\] on \(S_\etale\).

Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}\)-modules on \(S_\etale\). Then

1. \((\text{id}_{\tau, \etale})^*\mathcal{G}\) is the \(\tau\)-sheafification of the étale sheaf \[(f : T \to S) \longmapsto \Gamma(T, f_{small}^*\mathcal{G})\] where \(f_{small} : T_\etale \to S_\etale\) is the morphism of ringed small étale sites of Remark 070R.

Proof

Proof of (1). We first note that the result is true when \(\tau = Zar\) because in that case we have the morphism of topoi \(i_f : \Sh(T_{Zar}) \to \Sh((\Sch/S)_{Zar})\) such that \(\text{id}_{\tau, Zar} \circ i_f = f_{small}\) as morphisms \(T_{Zar} \to S_{Zar}\), see Topologies, Lemmas 020Y and 0211. Since pullback is transitive (see Modules on Sites, Lemma 03D8) we see that \(i_f^*(\text{id}_{\tau, Zar})^*\mathcal{F} = f_{small}^*\mathcal{F}\) as desired. Hence, by the remark preceding this lemma we see that \((\text{id}_{\tau, Zar})^*\mathcal{F}\) is the \(\tau\)-sheafification of the presheaf \(T \mapsto \Gamma(T, f^*\mathcal{F})\).

The proof of (3) is exactly the same as the proof of (1), except that it uses Topologies, Lemmas 021F and 021I. We omit the proof of (2).

Remark

Remark 070R and Lemma 070S have the following applications:

1. Let \(S\) be a scheme. The construction \(\mathcal{F} \mapsto \mathcal{F}^a\) is the pullback under the morphism of ringed sites \(\text{id}_{\tau, Zar} : ((\Sch/S)_\tau, \mathcal{O}) \to (S_{Zar}, \mathcal{O})\) or the morphism \(\text{id}_{small, \etale, Zar} : (S_\etale, \mathcal{O}) \to (S_{Zar}, \mathcal{O})\).

2. Let \(f : X \to Y\) be a morphism of schemes. For any of the morphisms \(f_{sites}\) of ringed sites of Remark 070R we have \[(f^*\mathcal{F})^a = f_{sites}^*\mathcal{F}^a.\] This follows from (1) and the fact that pullbacks are compatible with compositions of morphisms of ringed sites, see Modules on Sites, Lemma 03D8.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_S\)-module. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\).

1. The sheaf \(\mathcal{F}^a\) is a quasi-coherent \(\mathcal{O}\)-module on \((\Sch/S)_\tau\), as defined in Modules on Sites, Definition 03DL.

2. If \(\tau = Zariski\) or \(\tau = \etale\), then the sheaf \(\mathcal{F}^a\) is a quasi-coherent \(\mathcal{O}\)-module on \(S_{Zar}\) or \(S_\etale\) as defined in Modules on Sites, Definition 03DL.

Proof

Let \(\{S_i \to S\}\) be a Zariski covering such that we have exact sequences \[\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{S_i} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}_{S_i} \longrightarrow \mathcal{F}|_{S_i} \longrightarrow 0\] for some index sets \(K_i\) and \(J_i\). This is possible by the definition of a quasi-coherent sheaf on a ringed space (See Modules, Definition 01BE).

Proof of (1). Let \(\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}\). It is clear that \(\mathcal{F}^a|_{(\Sch/S_i)_\tau}\) also sits in an exact sequence \[\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow 0\] Hence \(\mathcal{F}^a\) is quasi-coherent by Modules on Sites, Lemma 03DN.

Proof of (2). Let \(\tau = \etale\). It is clear that \(\mathcal{F}^a|_{(S_i)_\etale}\) also sits in an exact sequence \[\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(S_i)_\etale} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(S_i)_\etale} \longrightarrow \mathcal{F}^a|_{(S_i)_\etale} \longrightarrow 0\] Hence \(\mathcal{F}^a\) is quasi-coherent by Modules on Sites, Lemma 03DN. The case \(\tau = Zariski\) is similar (actually, it is really tautological since the corresponding ringed topoi agree).

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\). Each of the functors \(\mathcal{F} \mapsto \mathcal{F}^a\) of Definition 03DU \[\QCoh(\mathcal{O}_S) \to \QCoh((\Sch/S)_\tau, \mathcal{O}) \quad\text{or}\quad \QCoh(\mathcal{O}_S) \to \QCoh(S_\tau, \mathcal{O})\] is fully faithful.

Proof

(By Lemma 03DV we do indeed get functors as indicated.) We may and do identify \(\mathcal{O}_S\)-modules on \(S\) with modules on \((S_{Zar}, \mathcal{O}_S)\). The functor \(\mathcal{F} \mapsto \mathcal{F}^a\) on quasi-coherent modules \(\mathcal{F}\) is given by pullback by a morphism \(f\) of ringed sites, see Remark 03FH. In each case the functor \(f_*\) is given by restriction along the inclusion functor \(S_{Zar} \to S_\tau\) or \(S_{Zar} \to (\Sch/S)_\tau\) (see discussion of how these morphisms of sites are defined in Topologies, Section 03FE). Combining this with the description of \(f^*\mathcal{F} = \mathcal{F}^a\) we see that \(f_*f^*\mathcal{F} = \mathcal{F}\) provided that \(\mathcal{F}\) is quasi-coherent. Then we see that \[\Hom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a) = \Hom_\mathcal{O}(f^*\mathcal{F}, f^*\mathcal{G}) = \Hom_{\mathcal{O}_S}(\mathcal{F}, f_*f^*\mathcal{G}) = \Hom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G})\] as desired.

Proposition

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\).

1. The functor \(\mathcal{F} \mapsto \mathcal{F}^a\) defines an equivalence of categories \[\QCoh(\mathcal{O}_S) \longrightarrow \QCoh((\Sch/S)_\tau, \mathcal{O})\] between the category of quasi-coherent sheaves on \(S\) and the category of quasi-coherent \(\mathcal{O}\)-modules on the big \(\tau\) site of \(S\).

2. Let \(\tau = Zariski\) or \(\tau = \etale\). The functor \(\mathcal{F} \mapsto \mathcal{F}^a\) defines an equivalence of categories \[\QCoh(\mathcal{O}_S) \longrightarrow \QCoh(S_\tau, \mathcal{O})\] between the category of quasi-coherent sheaves on \(S\) and the category of quasi-coherent \(\mathcal{O}\)-modules on the small \(\tau\) site of \(S\).

Proof

We have seen in Lemma 03DV that the functor is well defined. By Lemma 0GN7 the functor is fully faithful. To finish the proof we will show that a quasi-coherent \(\mathcal{O}\)-module on \((\Sch/S)_\tau\) gives rise to a descent datum for quasi-coherent sheaves relative to a \(\tau\)-covering of \(S\). Having produced this descent datum we will appeal to Proposition 023T to get the corresponding quasi-coherent sheaf on \(S\).

Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}\)-modules on the big \(\tau\) site of \(S\). By Modules on Sites, Definition 03DL there exists a \(\tau\)-covering \(\{S_i \to S\}_{i \in I}\) of \(S\) such that each of the restrictions \(\mathcal{G}|_{(\Sch/S_i)_\tau}\) has a global presentation \[\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\Sch/S_i)_\tau} \longrightarrow 0\] for some index sets \(J_i\) and \(K_i\). We claim that this implies that \(\mathcal{G}|_{(\Sch/S_i)_\tau}\) is \(\mathcal{F}_i^a\) for some quasi-coherent sheaf \(\mathcal{F}_i\) on \(S_i\). Namely, this is clear for the direct sums \(\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau}\) and \(\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau}\). Hence we see that \(\mathcal{G}|_{(\Sch/S_i)_\tau}\) is a cokernel of a map \(\varphi : \mathcal{K}_i^a \to \mathcal{L}_i^a\) for some quasi-coherent sheaves \(\mathcal{K}_i\), \(\mathcal{L}_i\) on \(S_i\). By the fully faithfulness of \((\ )^a\) we see that \(\varphi = \phi^a\) for some map of quasi-coherent sheaves \(\phi : \mathcal{K}_i \to \mathcal{L}_i\) on \(S_i\). Then it is clear that \(\mathcal{G}|_{(\Sch/S_i)_\tau} \cong \Coker(\phi)^a\) as claimed.

Since \(\mathcal{G}\) lives on all of the category \((\Sch/S)_\tau\) we see that \[(\text{pr}_0^*\mathcal{F}_i)^a \cong \mathcal{G}|_{(\Sch/(S_i \times_S S_j))_\tau} \cong (\text{pr}_1^*\mathcal{F})^a\] as \(\mathcal{O}\)-modules on \((\Sch/(S_i \times_S S_j))_\tau\). Hence, using fully faithfulness again we get canonical isomorphisms \[\phi_{ij} : \text{pr}_0^*\mathcal{F}_i \longrightarrow \text{pr}_1^*\mathcal{F}_j\] of quasi-coherent modules over \(S_i \times_S S_j\). We omit the verification that these satisfy the cocycle condition. Since they do we see by effectivity of descent for quasi-coherent sheaves and the covering \(\{S_i \to S\}\) (Proposition 023T) that there exists a quasi-coherent sheaf \(\mathcal{F}\) on \(S\) with \(\mathcal{F}|_{S_i} \cong \mathcal{F}_i\) compatible with the given descent data. In other words we are given \(\mathcal{O}\)-module isomorphisms \[\phi_i : \mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\Sch/S_i)_\tau}\] which agree over \(S_i \times_S S_j\). Hence, since \(\SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G})\) is a sheaf (Modules on Sites, Lemma 03EM), we conclude that there is a morphism of \(\mathcal{O}\)-modules \(\mathcal{F}^a \to \mathcal{G}\) recovering the isomorphisms \(\phi_i\) above. Hence this is an isomorphism and we win.

The case of the sites \(S_\etale\) and \(S_{Zar}\) is proved in the exact same manner.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\). Let \(\mathcal{P}\) be one of the properties of modules⁴ defined in Modules on Sites, Definitions 03DE, 03DL, and 03ER. The equivalences of categories \[\QCoh(\mathcal{O}_S) \longrightarrow \QCoh((\Sch/S)_\tau, \mathcal{O}) \quad\text{and}\quad \QCoh(\mathcal{O}_S) \longrightarrow \QCoh(S_\tau, \mathcal{O})\] defined by the rule \(\mathcal{F} \mapsto \mathcal{F}^a\) seen in Proposition 03DX have the property \[\mathcal{F}\text{ has }\mathcal{P} \Leftrightarrow \mathcal{F}^a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module}\] except (possibly) when \(\mathcal{P}\) is “locally free” or “coherent”. If \(\mathcal{P}=\)“coherent” the equivalence holds for \(\QCoh(\mathcal{O}_S) \to \QCoh(S_\tau, \mathcal{O})\) when \(S\) is locally Noetherian and \(\tau\) is Zariski or étale.

Proof

This is immediate for the global properties, i.e., those defined in Modules on Sites, Definition 03DE. For the local properties we can use Modules on Sites, Lemma 03DN to translate “\(\mathcal{F}^a\) has \(\mathcal{P}\)” into a property on the members of a covering of \(X\). Hence the result follows from Lemmas 05AZ, 05B0, 082U, 05B1, and 05B2. Being coherent for a quasi-coherent module is the same as being of finite type over a locally Noetherian scheme (see Cohomology of Schemes, Lemma 01XZ) hence this reduces to the case of finite type modules (details omitted).

Cohomology of quasi-coherent modules and topologies

In this section we prove that cohomology of quasi-coherent modules is independent of the choice of topology.

Lemma

Let \(S\) be a scheme. Let

1. \(\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}\) and \(\mathcal{C} = (\Sch/S)_\tau\), or

2. let \(\tau = \etale\) and \(\mathcal{C} = S_\etale\), or

3. let \(\tau = Zariski\) and \(\mathcal{C} = S_{Zar}\).

Let \(\mathcal{F}\) be an abelian sheaf on \(\mathcal{C}\). Let \(U \in \Ob(\mathcal{C})\) be affine. Let \(\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}\) be a standard affine \(\tau\)-covering in \(\mathcal{C}\). Then

1. \(\mathcal{V} = \{\coprod_{i = 1, \ldots, n} U_i \to U\}\) is a \(\tau\)-covering of \(U\),

2. \(\mathcal{U}\) is a refinement of \(\mathcal{V}\), and

3. the induced map on Čech complexes (Cohomology on Sites, Equation (03F4)) \[\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\] is an isomorphism of complexes.

Proof

This follows because \[(\coprod\nolimits_{i_0 = 1, \ldots, n} U_{i_0}) \times_U \ldots \times_U (\coprod\nolimits_{i_p = 1, \ldots, n} U_{i_p}) = \coprod\nolimits_{i_0, \ldots, i_p \in \{1, \ldots, n\}} U_{i_0} \times_U \ldots \times_U U_{i_p}\] and the fact that \(\mathcal{F}(\coprod_a V_a) = \prod_a \mathcal{F}(V_a)\) since disjoint unions are \(\tau\)-coverings.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(S\). Let \(\tau\), \(\mathcal{C}\), \(U\), \(\mathcal{U}\) be as in Lemma 03FI. Then there is an isomorphism of complexes \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a) \cong s((A/R)_\bullet \otimes_R M)\] (see Section 023F) where \(R = \Gamma(U, \mathcal{O}_U)\), \(M = \Gamma(U, \mathcal{F}^a)\) and \(R \to A\) is a faithfully flat ring map. In particular \[\check{H}^p(\mathcal{U}, \mathcal{F}^a) = 0\] for all \(p \geq 1\).

Proof

By Lemma 03FI we see that \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a)\) is isomorphic to \(\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}^a)\) where \(\mathcal{V} = \{V \to U\}\) with \(V = \coprod_{i = 1, \ldots n} U_i\) affine also. Set \(A = \Gamma(V, \mathcal{O}_V)\). Since \(\{V \to U\}\) is a \(\tau\)-covering we see that \(R \to A\) is faithfully flat. On the other hand, by definition of \(\mathcal{F}^a\) we have that the degree \(p\) term \(\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{F}^a)\) is \[\Gamma(V \times_U \ldots \times_U V, \mathcal{F}^a) = \Gamma(\Spec(A \otimes_R \ldots \otimes_R A), \mathcal{F}^a) = A \otimes_R \ldots \otimes_R A \otimes_R M\] We omit the verification that the maps of the Čech complex agree with the maps in the complex \(s((A/R)_\bullet \otimes_R M)\). The vanishing of cohomology is Lemma 023M.

Proposition

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(S\). Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\).

1. There is a canonical isomorphism \[H^q(S, \mathcal{F}) = H^q((\Sch/S)_\tau, \mathcal{F}^a).\]

2. There are canonical isomorphisms \[H^q(S, \mathcal{F}) = H^q(S_{Zar}, \mathcal{F}^a) = H^q(S_\etale, \mathcal{F}^a).\]

Proof

The result for \(q = 0\) is clear from the definition of \(\mathcal{F}^a\). Let \(\mathcal{C} = (\Sch/S)_\tau\), or \(\mathcal{C} = S_\etale\), or \(\mathcal{C} = S_{Zar}\).

We are going to apply Cohomology on Sites, Lemma 03F9 with \(\mathcal{F} = \mathcal{F}^a\), \(\mathcal{B} \subset \Ob(\mathcal{C})\) the set of affine schemes in \(\mathcal{C}\), and \(\text{Cov} \subset \text{Cov}_\mathcal{C}\) the set of standard affine \(\tau\)-coverings. Assumption (3) of the lemma is satisfied by Lemma 03FJ. Hence we conclude that \(H^p(U, \mathcal{F}^a) = 0\) for every affine object \(U\) of \(\mathcal{C}\).

Next, let \(U \in \Ob(\mathcal{C})\) be any separated object. Denote \(f : U \to S\) the structure morphism. Let \(U = \bigcup U_i\) be an affine open covering. We may also think of this as a \(\tau\)-covering \(\mathcal{U} = \{U_i \to U\}\) of \(U\) in \(\mathcal{C}\). Note that \(U_{i_0} \times_U \ldots \times_U U_{i_p} = U_{i_0} \cap \ldots \cap U_{i_p}\) is affine as we assumed \(U\) separated. By Cohomology on Sites, Lemma 03F7 and the result above we see that \[H^p(U, \mathcal{F}^a) = \check{H}^p(\mathcal{U}, \mathcal{F}^a) = H^p(U, f^*\mathcal{F})\] the last equality by Cohomology of Schemes, Lemma 01XD. In particular, if \(S\) is separated we can take \(U = S\) and \(f = \text{id}_S\) and the proposition is proved. We suggest the reader skip the rest of the proof (or rewrite it to give a clearer exposition).

Choose an injective resolution \(\mathcal{F} \to \mathcal{I}^\bullet\) on \(S\). Choose an injective resolution \(\mathcal{F}^a \to \mathcal{J}^\bullet\) on \(\mathcal{C}\). Denote \(\mathcal{J}^n|_S\) the restriction of \(\mathcal{J}^n\) to opens of \(S\); this is a sheaf on the topological space \(S\) as open coverings are \(\tau\)-coverings. We get a complex \[0 \to \mathcal{F} \to \mathcal{J}^0|_S \to \mathcal{J}^1|_S \to \ldots\] which is exact since its sections over any affine open \(U \subset S\) is exact (by the vanishing of \(H^p(U, \mathcal{F}^a)\), \(p > 0\) seen above). Hence by Derived Categories, Lemma 013P there exists map of complexes \(\mathcal{J}^\bullet|_S \to \mathcal{I}^\bullet\) which in particular induces a map \[R\Gamma(\mathcal{C}, \mathcal{F}^a) = \Gamma(S, \mathcal{J}^\bullet) \longrightarrow \Gamma(S, \mathcal{I}^\bullet) = R\Gamma(S, \mathcal{F}).\] Taking cohomology gives the map \(H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})\) which we have to prove is an isomorphism. Let \(\mathcal{U} : S = \bigcup U_i\) be an affine open covering which we may think of as a \(\tau\)-covering also. By the above we get a map of double complexes \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}|_S) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).\] This map induces a map of spectral sequences \[{}^\tau\! E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a)) \longrightarrow E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))\] The first spectral sequence converges to \(H^{p + q}(\mathcal{C}, \mathcal{F})\) and the second to \(H^{p + q}(S, \mathcal{F})\). On the other hand, we have seen that the induced maps \({}^\tau\! E_2^{p, q} \to E_2^{p, q}\) are bijections (as all the intersections are separated being opens in affines). Whence also the maps \(H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})\) are isomorphisms, and we win.

Proposition

Let \(f : T \to S\) be a morphism of schemes.

1. The equivalences of categories of Proposition 03DX are compatible with pullback. More precisely, we have \(f^*(\mathcal{G}^a) = (f^*\mathcal{G})^a\) for any quasi-coherent sheaf \(\mathcal{G}\) on \(S\).

2. The equivalences of categories of Proposition 03DX part (1) are not compatible with pushforward in general.

3. If \(f\) is quasi-compact and quasi-separated, and \(\tau \in \{Zariski, \etale\}\) then \(f_*\) and \(f_{small, *}\) preserve quasi-coherent sheaves and the diagram \[\xymatrix{ \QCoh(\mathcal{O}_T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^a} & & \QCoh(\mathcal{O}_S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^a} \\ \QCoh(T_\tau, \mathcal{O}) \ar[rr]^{f_{small, *}} & & \QCoh(S_\tau, \mathcal{O}) }\] is commutative, i.e., \(f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a\).

Proof

Part (1) follows from the discussion in Remark 03FH. Part (2) is just a warning, and can be explained in the following way: First the statement cannot be made precise since \(f_*\) does not transform quasi-coherent sheaves into quasi-coherent sheaves in general. Even if this is the case for \(f\) (and any base change of \(f\)), then the compatibility over the big sites would mean that formation of \(f_*\mathcal{F}\) commutes with any base change, which does not hold in general. An explicit example is the quasi-compact open immersion \(j : X = \mathbf{A}^2_k \setminus \{0\} \to \mathbf{A}^2_k = Y\) where \(k\) is a field. We have \(j_*\mathcal{O}_X = \mathcal{O}_Y\) but after base change to \(\Spec(k)\) by the \(0\) map we see that the pushforward is zero.

Let us prove (3) in case \(\tau = \etale\). Note that \(f\), and any base change of \(f\), transforms quasi-coherent sheaves into quasi-coherent sheaves, see Schemes, Lemma 01LC. The equality \(f_{small, *}(\mathcal{F}^a) = (f_*\mathcal{F})^a\) means that for any étale morphism \(g : U \to S\) we have \(\Gamma(U, g^*f_*\mathcal{F}) = \Gamma(U \times_S T, (g')^*\mathcal{F})\) where \(g' : U \times_S T \to T\) is the projection. This is true by Cohomology of Schemes, Lemma 02KH.

Lemma

Let \(f : T \to S\) be a quasi-compact and quasi-separated morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(T\). For either the étale or Zariski topology, there are canonical isomorphisms \(R^if_{small, *}(\mathcal{F}^a) = (R^if_*\mathcal{F})^a\).

Proof

We prove this for the étale topology; we omit the proof in the case of the Zariski topology. By Cohomology of Schemes, Lemma 01XJ the sheaves \(R^if_*\mathcal{F}\) are quasi-coherent so that the assertion makes sense. The sheaf \(R^if_{small, *}\mathcal{F}^a\) is the sheaf associated to the presheaf \[U \longmapsto H^i(U \times_S T, \mathcal{F}^a)\] where \(g : U \to S\) is an object of \(S_\etale\), see Cohomology on Sites, Lemma 072W. By our conventions the right hand side is the étale cohomology of the restriction of \(\mathcal{F}^a\) to the localization \(T_\etale/U \times_S T\) which equals \((U \times_S T)_\etale\). By Proposition 03DW this is presheaf the same as the presheaf \[U \longmapsto H^i(U \times_S T, (g')^*\mathcal{F}),\] where \(g' : U \times_S T \to T\) is the projection. If \(U\) is affine then this is the same as \(H^0(U, R^if'_*(g')^*\mathcal{F})\), see Cohomology of Schemes, Lemma 01XK. By Cohomology of Schemes, Lemma 02KH this is equal to \(H^0(U, g^*R^if_*\mathcal{F})\) which is the value of \((R^if_*\mathcal{F})^a\) on \(U\). Thus the values of the sheaves of modules \(R^if_{small, *}(\mathcal{F}^a)\) and \((R^if_*\mathcal{F})^a\) on every affine object of \(S_\etale\) are canonically isomorphic which implies they are canonically isomorphic.

Quasi-coherent sheaves and topologies, II

We continue the discussion comparing quasi-coherent modules on a scheme \(S\) with quasi-coherent modules on any of the sites associated to \(S\) in the chapter on topologies.

Lemma

In Lemma 070S the morphism of ringed sites \(\text{id}_{small, \etale, Zar} : S_\etale \to S_{Zar}\) is flat.

Proof

Let us denote \(\epsilon = \text{id}_{small, \etale, Zar}\) and \(\mathcal{O}_\etale\) and \(\mathcal{O}_{Zar}\) the structure sheaves on \(S_\etale\) and \(S_{Zar}\). We have to show that \(\mathcal{O}_\etale\) is a flat \(\epsilon^{-1}\mathcal{O}_{Zar}\)-module. Recall that étale morphisms are open, see Morphisms, Lemma 03WT. It follows (from the construction of pullback on sheaves) that \(\epsilon^{-1}\mathcal{O}_{Zar}\) is the sheafification of the presheaf \(\mathcal{O}'\) on \(S_\etale\) which sends an étale morphism \(f : V \to S\) to \(\mathcal{O}_S(f(V))\). If both \(V\) and \(U = f(V) \subset S\) are affine, then \(V \to U\) is an étale morphism of affines, hence corresponds to an étale ring map. Since étale ring maps are flat, we see that \(\mathcal{O}_S(U) = \mathcal{O}'(V) \to \mathcal{O}_\etale(V) = \mathcal{O}_V(V)\) is flat. Finally, for every étale morphism \(f : V \to S\), i.e., object of \(S_\etale\), there is an affine open covering \(V = \bigcup V_i\) such that \(f(V_i)\) is an affine open in \(S\) for all \(i\)⁵. Thus the result by Modules on Sites, Lemma 0GN1.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\). The functors \[\QCoh(\mathcal{O}_S) \longrightarrow \textit{Mod}((\Sch/S)_\tau, \mathcal{O}) \quad\text{and}\quad \QCoh(\mathcal{O}_S) \longrightarrow \textit{Mod}(S_\tau, \mathcal{O})\] defined by the rule \(\mathcal{F} \mapsto \mathcal{F}^a\) seen in Proposition 03DX are

1. fully faithful,

2. commmute with direct sums,

3. commmute with colimits,

4. right exact,

5. exact as a functor \(\QCoh(\mathcal{O}_S) \to \textit{Mod}(S_\etale, \mathcal{O})\),

6. not exact as a functor \(\QCoh(\mathcal{O}_S) \to \textit{Mod}((\Sch/S)_\tau, \mathcal{O})\) in general,

7. given two quasi-coherent \(\mathcal{O}_S\)-modules \(\mathcal{F}\), \(\mathcal{G}\) we have \((\mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G})^a = \mathcal{F}^a \otimes_\mathcal{O} \mathcal{G}^a\),

8. if \(\tau = \etale\) or \(\tau = Zariski\), given two quasi-coherent \(\mathcal{O}_S\)-modules \(\mathcal{F}\), \(\mathcal{G}\) such that \(\mathcal{F}\) is of finite presentation we have \((\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G}))^a = \SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a)\) in \(\textit{Mod}(S_\tau, \mathcal{O})\),

9. given two quasi-coherent \(\mathcal{O}_S\)-modules \(\mathcal{F}\), \(\mathcal{G}\) we do not have \((\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G}))^a = \SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a)\) in \(\textit{Mod}((\Sch/S)_\tau, \mathcal{O})\) in general even if \(\mathcal{F}\) is of finite presentation, and

10. given a short exact sequence \(0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0\) of \(\mathcal{O}\)-modules then \(\mathcal{E}\) is quasi-coherent⁶, i.e., \(\mathcal{E}\) is in the essential image of the functor.

Proof

Part (1) we saw in Proposition 03DX.

We have seen in Schemes, Section 01LA that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent sheaf. Moreover, in Remark 03FH we saw that \(\mathcal{F} \mapsto \mathcal{F}^a\) is the pullback functor for a morphism of ringed sites, hence commutes with all colimits, see Modules on Sites, Lemma 03DC. Thus (3) and its special case (2) hold.

This also shows that the functor is right exact (i.e., commutes with finite colimits), hence (4).

The functor \(\QCoh(\mathcal{O}_S) \to \textit{Mod}(S_\etale, \mathcal{O})\), \(\mathcal{F} \mapsto \mathcal{F}^a\) is left exact because an étale morphism is flat, see Morphisms, Lemma 02GS. This proves (5).

To see (6), suppose that \(S = \Spec(\mathbf{Z})\). Then \(2 : \mathcal{O}_S \to \mathcal{O}_S\) is injective but the associated map of \(\mathcal{O}\)-modules on \((\Sch/S)_\tau\) isn’t injective because \(2 : \mathbf{F}_2 \to \mathbf{F}_2\) isn’t injective and \(\Spec(\mathbf{F}_2)\) is an object of \((\Sch/S)_\tau\).

Part (7) holds because, as mentioned above, the functor \(\mathcal{F} \mapsto \mathcal{F}^a\) is the pullback functor for a morphism of ringed sites and such commute with tensor products by Modules on Sites, Lemma 03EL.

Part (8) is obvious if \(\tau = Zariski\) because the category of \(\mathcal{O}\)-modules on \(S_{Zar}\) is the same as the category of \(\mathcal{O}_S\)-modules on the topological space \(S\). If \(\tau = \etale\) then (8) holds because, as mentioned above, the functor \(\mathcal{F} \mapsto \mathcal{F}^a\) is the pullback functor for the flat morphism of ringed sites \((S_\etale, \mathcal{O}) \to (S_{Zar}, \mathcal{O}_S)\), see Lemma 0GNA. Pullback by flat morphisms of ringed sites commutes with taking internal hom out of a finitely presented module by Modules on Sites, Lemma 0GN2.

To see (9), suppose that \(S = \Spec(\mathbf{Z})\). Let \(\mathcal{F} = \Coker(2 : \mathcal{O}_S \to \mathcal{O}_S)\) and \(\mathcal{G} = \mathcal{O}_S\). Then \(\mathcal{F}^a = \Coker(2 : \mathcal{O} \to \mathcal{O})\) and \(\mathcal{G}^a = \mathcal{O}\). Hence \(\SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a) = \mathcal{O}[2]\) is equal to the \(2\)-torsion in \(\mathcal{O}\), which is not zero, see proof of (6). On the other hand, the module \(\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G})\) is zero.

Proof of (10). Let \(0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0\) be a short exact sequence of \(\mathcal{O}\)-modules with \(\mathcal{F}_1\) and \(\mathcal{F}_2\) quasi-coherent on \(S\). Consider the restriction \[0 \to \mathcal{F}_1 \to \mathcal{E}|_{S_{Zar}} \to \mathcal{F}_2\] to \(S_{Zar}\). By Proposition 03DW we see that on any affine \(U \subset S\) we have \(H^1(U, \mathcal{F}_1^a) = H^1(U, \mathcal{F}_1) = 0\). Hence the sequence above is also exact on the right. By Schemes, Section 01LA we conclude that \(\mathcal{F} = \mathcal{E}|_{S_{Zar}}\) is quasi-coherent. Thus we obtain a commutative diagram \[\xymatrix{ & \mathcal{F}_1^a \ar[r] \ar[d] & \mathcal{F}^a \ar[r] \ar[d] & \mathcal{F}_2^a \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1^a \ar[r] & \mathcal{E} \ar[r] & \mathcal{F}_2^a \ar[r] & 0 }\] To finish the proof it suffices to show that the top row is also right exact. To do this, denote once more \(U = \Spec(A) \subset S\) an affine open of \(S\). We have seen above that \(0 \to \mathcal{F}_1(U) \to \mathcal{E}(U) \to \mathcal{F}_2(U) \to 0\) is exact. For any affine scheme \(V/U\), \(V = \Spec(B)\) the map \(\mathcal{F}_1^a(V) \to \mathcal{E}(V)\) is injective. We have \(\mathcal{F}_1^a(V) = \mathcal{F}_1(U) \otimes_A B\) by definition. The injection \(\mathcal{F}_1^a(V) \to \mathcal{E}(V)\) factors as \[\mathcal{F}_1(U) \otimes_A B \to \mathcal{E}(U) \otimes_A B \to \mathcal{E}(V)\] Considering \(A\)-algebras \(B\) of the form \(B = A \oplus M\) we see that \(\mathcal{F}_1(U) \to \mathcal{E}(U)\) is universally injective (see Algebra, Definition 058I). Since \(\mathcal{E}(U) = \mathcal{F}(U)\) we conclude that \(\mathcal{F}_1 \to \mathcal{F}\) remains injective after any base change, or equivalently that \(\mathcal{F}_1^a \to \mathcal{F}^a\) is injective.

Lemma

Let \(S\) be a scheme. The category \(\QCoh(S_\etale, \mathcal{O})\) of quasi-coherent modules on \(S_\etale\) has the following properties:

1. Any direct sum of quasi-coherent sheaves is quasi-coherent.

2. Any colimit of quasi-coherent sheaves is quasi-coherent.

3. The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.

4. Given a short exact sequence of \(\mathcal{O}\)-modules \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) if two out of three are quasi-coherent so is the third.

5. Given two quasi-coherent \(\mathcal{O}\)-modules the tensor product is quasi-coherent.

6. Given two quasi-coherent \(\mathcal{O}\)-modules \(\mathcal{F}\), \(\mathcal{G}\) such that \(\mathcal{F}\) is of finite presentation. then the internal hom \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) is quasi-coherent.

Proof

The corresponding facts hold for quasi-coherent modules on the scheme \(S\), see Schemes, Section 01LA. The proof will be to use Lemma 06VE to transfer these truths to \(S_\etale\).

Proof of (1). Let \(\mathcal{F}_i\), \(i \in I\) be a family of objects of \(\QCoh(S_\etale, \mathcal{O})\). Write \(\mathcal{F}_i = \mathcal{G}_i^a\) for some quasi-coherent modules \(\mathcal{G}_i\) on \(S\). Then \(\bigoplus \mathcal{F}_i = (\bigoplus \mathcal{G}_i)^a\) by the lemma cited and we conclude.

Proof of (2). Let \(\mathcal{I} \to \QCoh(S_\etale, \mathcal{O})\), \(i \mapsto \mathcal{F}_i\) be a diagram. Write \(\mathcal{F}_i = \mathcal{G}_i^a\) so we get a diagram \(\mathcal{I} \to \QCoh(\mathcal{O}_S)\). Then \(\colim \mathcal{F}_i = (\colim \mathcal{G}_i)^a\) by the lemma cited and we conclude.

Proof of (3). Let \(a : \mathcal{F} \to \mathcal{F}'\) be an arrow of \(\QCoh(S_\etale, \mathcal{O})\). Write \(a = b^a\) for some map \(b : \mathcal{G} \to \mathcal{G}'\) of quasi-coherent modules on \(S\). By the lemma cited we have \(\Ker(a) = \Ker(b)^a\) and \(\Coker(a) = \Coker(b)^a\) and we conclude.

Proof of (4). This follows from (3) except in the case when we know \(\mathcal{F}_1\) and \(\mathcal{F}_3\) are quasi-coherent. In this case write \(\mathcal{F}_1 = \mathcal{G}_1^a\) and \(\mathcal{F}_3 = \mathcal{G}_3^a\) with \(\mathcal{G}_i\) quasi-coherent on \(S\). By Lemma 06VE part (10) we conclude.

Proof of (5). Let \(\mathcal{F}\) and \(\mathcal{F}'\) be in \(\QCoh(S_\etale, \mathcal{O})\). Write \(\mathcal{F} = \mathcal{G}^a\) and \(\mathcal{F}' = (\mathcal{G}')^a\) with \(\mathcal{G}\) and \(\mathcal{G}'\) quasi-coherent on \(S\). By the lemma cited we have \(\mathcal{F} \otimes_\mathcal{O} \mathcal{F}' = (\mathcal{G} \otimes_{\mathcal{O}_S} \mathcal{G}')^a\) and we conclude.

Proof of (6). Let \(\mathcal{F}\) and \(\mathcal{G}\) be in \(\QCoh(S_\etale, \mathcal{O})\) with \(\mathcal{F}\) of finite presentation. Write \(\mathcal{F} = \mathcal{H}^a\) and \(\mathcal{G} = (\mathcal{I})^a\) with \(\mathcal{H}\) and \(\mathcal{I}\) quasi-coherent on \(S\). By Lemma 05VG we see that \(\mathcal{H}\) is of finite presentation. By Lemma 06VE part (8) we have \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) = (\SheafHom_{\mathcal{O}_S}(\mathcal{H}, \mathcal{I}))^a\) and we conclude.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\}\). The category \(\QCoh((\Sch/S)_\tau, \mathcal{O})\) of quasi-coherent modules on \((\Sch/S)_\tau\) has the following properties:

1. Any direct sum of quasi-coherent sheaves is quasi-coherent.

2. Any colimit of quasi-coherent sheaves is quasi-coherent.

3. The cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.

4. Given a short exact sequence of \(\mathcal{O}\)-modules \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) if \(\mathcal{F}_1\) and \(\mathcal{F}_3\) are quasi-coherent so is \(\mathcal{F}_2\).

5. Given two quasi-coherent \(\mathcal{O}\)-modules the tensor product is quasi-coherent.

6. Given two quasi-coherent \(\mathcal{O}\)-modules \(\mathcal{F}\), \(\mathcal{G}\) such that \(\mathcal{F}\) is finite locally free, the internal hom \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) is quasi-coherent.

Proof

The corresponding facts hold for quasi-coherent modules on the scheme \(S\), see Schemes, Section 01LA. The proof will be to use Lemma 06VE to transfer these truths to \((\Sch/S)_\tau\).

Proof of (1). Let \(\mathcal{F}_i\), \(i \in I\) be a family of objects of \(\QCoh((\Sch/S)_\tau, \mathcal{O})\). Write \(\mathcal{F}_i = \mathcal{G}_i^a\) for some quasi-coherent modules \(\mathcal{G}_i\) on \(S\). Then \(\bigoplus \mathcal{F}_i = (\bigoplus \mathcal{G}_i)^a\) by the lemma cited and we conclude.

Proof of (2). Let \(\mathcal{I} \to \QCoh((\Sch/S)_\tau, \mathcal{O})\), \(i \mapsto \mathcal{F}_i\) be a diagram. Write \(\mathcal{F}_i = \mathcal{G}_i^a\) so we get a diagram \(\mathcal{I} \to \QCoh(\mathcal{O}_S)\). Then \(\colim \mathcal{F}_i = (\colim \mathcal{G}_i)^a\) by the lemma cited and we conclude.

Proof of (3). Let \(a : \mathcal{F} \to \mathcal{F}'\) be an arrow of \(\QCoh((\Sch/S)_\tau, \mathcal{O})\). Write \(a = b^a\) for some map \(b : \mathcal{G} \to \mathcal{G}'\) of quasi-coherent modules on \(S\). By the lemma cited we have \(\Coker(a) = \Coker(b)^a\) (because a cokernel is a colimit) and we conclude.

Proof of (4). Write \(\mathcal{F}_1 = \mathcal{G}_1^a\) and \(\mathcal{F}_3 = \mathcal{G}_3^a\) with \(\mathcal{G}_i\) quasi-coherent on \(S\). By Lemma 06VE part (10) we conclude.

Proof of (5). Let \(\mathcal{F}\) and \(\mathcal{F}'\) be in \(\QCoh((\Sch/S)_\tau, \mathcal{O})\). Write \(\mathcal{F} = \mathcal{G}^a\) and \(\mathcal{F}' = (\mathcal{G}')^a\) with \(\mathcal{G}\) and \(\mathcal{G}'\) quasi-coherent on \(S\). By the lemma cited we have \(\mathcal{F} \otimes_\mathcal{O} \mathcal{F}' = (\mathcal{G} \otimes_{\mathcal{O}_S} \mathcal{G}')^a\) and we conclude.

Proof of (6). Write \(\mathcal{F} = \mathcal{H}^a\) for some quasi-coherent \(\mathcal{O}_S\)-module. By Lemma 05VG we see that \(\mathcal{H}\) is finite locally free. The problem is Zariski local on \(S\) (details omitted) hence we may assume \(\mathcal{H} = \mathcal{O}_S^{\oplus n}\) is finite free. Then \(\mathcal{F} = \mathcal{O}^{\oplus n}\) and \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) = \mathcal{G}^{\oplus n}\) is quasi-coherent.

Example

Let \(S\) be a scheme. Let \(\mathcal{F}\) and \(\mathcal{G}\) be quasi-coherent modules on \((\Sch/S)_\tau\) for one of the topologies \(\tau\) considered in Lemma 0GNC. In general it is not the case that \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) is quasi-coherent even if \(\mathcal{F}\) is of finite presentation. Namely, say \(S = \Spec(\mathbf{Z})\), \(\mathcal{F} = \Coker(2 : \mathcal{O} \to \mathcal{O})\), and \(\mathcal{G} = \mathcal{O}\). Then \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G}) = \mathcal{O}[2]\) is equal to the \(2\)-torsion in \(\mathcal{O}\), which is not quasi-coherent.

Lemma

Let \(S\) be a scheme.

1. The category \(\QCoh((\Sch/S)_{fppf}, \mathcal{O})\) has colimits and they agree with colimits in the categories \(\textit{Mod}((\Sch/S)_{Zar}, \mathcal{O})\), \(\textit{Mod}((\Sch/S)_\etale, \mathcal{O})\), and \(\textit{Mod}((\Sch/S)_{fppf}, \mathcal{O})\).

2. Given \(\mathcal{F}, \mathcal{G}\) in \(\QCoh((\Sch/S)_{fppf}, \mathcal{O})\) the tensor products \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G}\) computed in \(\textit{Mod}((\Sch/S)_{Zar}, \mathcal{O})\), \(\textit{Mod}((\Sch/S)_\etale, \mathcal{O})\), or \(\textit{Mod}((\Sch/S)_{fppf}, \mathcal{O})\) agree and the common value is an object of \(\QCoh((\Sch/S)_{fppf}, \mathcal{O})\).

3. Given \(\mathcal{F}, \mathcal{G}\) in \(\QCoh((\Sch/S)_{fppf}, \mathcal{O})\) with \(\mathcal{F}\) finite locally free (in fppf, or equivalently étale, or equivalently Zariski topology) the internal homs \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) computed in \(\textit{Mod}((\Sch/S)_{Zar}, \mathcal{O})\), \(\textit{Mod}((\Sch/S)_\etale, \mathcal{O})\), or \(\textit{Mod}((\Sch/S)_{fppf}, \mathcal{O})\) agree and the common value is an object of \(\QCoh((\Sch/S)_{fppf}, \mathcal{O})\).

Proof

This lemma collects the results shown above in a slightly different manner. First of all, by Lemma 0GNC we already know the output of the construction in (1), (2), or (3) ends up in \(\QCoh((\Sch/S)_\tau, \mathcal{O})\). It remains to show in each case that the result is independent of the topology used. The key to this is that the equivalence \(\QCoh(\mathcal{O}_S) \to \QCoh((\Sch/S)_\tau, \mathcal{O})\), \(\mathcal{F} \mapsto \mathcal{F}^a\) of Proposition 03DX is given by the same formula independent of the choice of the topology \(\tau \in \{Zariski, \etale, fppf\}\).

Proof of (1). Let \(\mathcal{I} \to \QCoh((\Sch/S)_{fppf}, \mathcal{O})\), \(i \mapsto \mathcal{F}_i\) be a diagram. Write \(\mathcal{F}_i = \mathcal{G}_i^a\) so we get a diagram \(\mathcal{I} \to \QCoh(\mathcal{O}_S)\). Then \(\colim \mathcal{F}_i = (\colim \mathcal{G}_i)^a\) in \(\textit{Mod}((\Sch/S)_\tau, \mathcal{O})\) for \(\tau \in \{Zariski, \etale, fppf\}\) by Lemma 06VE. This proves (1).

Proof of (2). Write \(\mathcal{F} = \mathcal{H}^a\) and \(\mathcal{G} = (\mathcal{I})^a\) with \(\mathcal{H}\) and \(\mathcal{I}\) quasi-coherent on \(S\). Then \(\mathcal{F} \otimes_\mathcal{O} \mathcal{G} = (\mathcal{H} \otimes_\mathcal{O} \mathcal{I})^a\) in \(\textit{Mod}((\Sch/S)_\tau, \mathcal{O})\) for \(\tau \in \{Zariski, \etale, fppf\}\) by Lemma 06VE. This proves (2).

Proof of (3). Let \(\mathcal{F}\) and \(\mathcal{G}\) be in \(\QCoh((\Sch/S)_{fppf}, \mathcal{O})\). Write \(\mathcal{F} = \mathcal{H}^a\) with \(\mathcal{H}\) quasi-coherent on \(S\). By Lemma 05VG we have \[\begin{align*} \mathcal{F}\text{ finite locally free in fppf topology} & \Leftrightarrow \mathcal{H}\text{ finite locally free on }S \\ & \Leftrightarrow \mathcal{F}\text{ finite locally free in \'etale topology} \\ & \Leftrightarrow \mathcal{H}\text{ finite locally free on }S \\ & \Leftrightarrow \mathcal{F}\text{ finite locally free in Zariski topology} \end{align*}\] This explains the parenthetical statement of part (3). Now, if these equivalent conditions hold, then \(\mathcal{H}\) is finite locally free. The construction of \(\SheafHom_\mathcal{O}(\mathcal{F}, \mathcal{G})\) in Modules on Sites, Section 04TT depends only on \(\mathcal{F}\) and \(\mathcal{G}\) as presheaves of modules (only whether the output \(\SheafHom\) is a sheaf depends on whether \(\mathcal{F}\) and \(\mathcal{G}\) are sheaves).

Quasi-coherent modules and affines

Let \(S\) be a scheme⁷. Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Recall that \((\textit{Aff}/S)_\tau\) is the full subcategory of \((\Sch/S)_\tau\) whose objects are affine turned into a site by declaring the coverings to be the standard \(\tau\)-coverings. By Topologies, Lemmas 020W, 021E, 06VC, 06VD, and 021V we have an equivalence of topoi \(g : \Sh((\textit{Aff}/S)_\tau) \to \Sh((\Sch/S)_\tau)\) whose pullback functor is given by restriction. Recalling that \(\mathcal{O}\) denotes the structure sheaf on \((\Sch/S)_\tau\), let us temporarily and pedantically denote \(\mathcal{O}_{\textit{Aff}}\) the restriction of \(\mathcal{O}\) to \((\textit{Aff}/S)_\tau\). Then we obtain an equivalence [0GZU]\[\begin{equation} (\Sh((\textit{Aff}/S)_\tau), \mathcal{O}_{\textit{Aff}}) \longrightarrow (\Sh((\Sch/S)_\tau), \mathcal{O}) \end{equation}\] of ringed topoi. Having said this we can compare quasi-coherent modules as well.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}_{\textit{Aff}}\)-modules on \((\textit{Aff}/S)_{fppf}\). The following are equivalent

1. for every morphism \(U \to U'\) of \((\textit{Aff}/S)_{fppf}\) the map \(\mathcal{F}(U') \otimes_{\mathcal{O}(U')} \mathcal{O}(U) \to \mathcal{F}(U)\) is an isomorphism,

2. \(\mathcal{F}\) is a sheaf on \((\textit{Aff}/S)_{Zar}\) and a quasi-coherent module on the ringed site \(((\textit{Aff}/S)_{Zar}, \mathcal{O}_{\textit{Aff}})\) in the sense of Modules on Sites, Definition 03DL,

3. same as in (2) for the étale topology,

4. same as in (2) for the smooth topology,

5. same as in (2) for the syntomic topology,

6. same as in (2) for the fppf topology,

7. \(\mathcal{F}\) corresponds to a quasi-coherent module on \((\Sch/S)_{Zar}\), \((\Sch/S)_\etale\), \((\Sch/S)_{smooth}\), \((\Sch/S)_{syntomic}\), or \((\Sch/S)_{fppf}\) via the equivalence (0GZU),

8. \(\mathcal{F}\) comes from a unique quasi-coherent \(\mathcal{O}_S\)-module \(\mathcal{G}\) by the procedure described in Section 03DR.

Proof

Since the notion of a quasi-coherent module is intrinsic (Modules on Sites, Lemma 03DM) we see that the equivalence (0GZU) induces an equivalence between categories of quasi-coherent modules. Proposition 03DX says the topology we use to study quasi-coherent modules on \(\Sch/S\) does not matter and it also tells us that (8) is the same as (7). Hence we see that (2) – (8) are all equivalent.

Assume the equivalent conditions (2) – (8) hold and let \(\mathcal{G}\) be as in (8). Let \(h : U \to U' \to S\) be a morphism of \(\textit{Aff}/S\). Denote \(f : U \to S\) and \(f' : U' \to S\) the structure morphisms, so that \(f = f' \circ h\). We have \(\mathcal{F}(U') = \Gamma(U', (f')^*\mathcal{G})\) and \(\mathcal{F}(U) = \Gamma(U, f^*\mathcal{G}) = \Gamma(U, h^*(f')^*\mathcal{G})\). Hence (1) holds by Schemes, Lemma 01I9.

Assume (1) holds. To finish the proof it suffices to prove (2). Let \(U\) be an object of \((\textit{Aff}/S)_{Zar}\). Say \(U = \Spec(R)\). A standard open covering \(U = U_1 \cup \ldots \cup U_n\) is given by \(U_i = D(f_i)\) for some elements \(f_1, \ldots, f_n \in R\) generating the unit ideal of \(R\). By property (1) we see that \[\mathcal{F}(U_i) = \mathcal{F}(U) \otimes_R R_{f_i} = \mathcal{F}(U)_{f_i}\] and \[\mathcal{F}(U_i \cap U_j) = \mathcal{F}(U) \otimes_R R_{f_if_j} = \mathcal{F}(U)_{f_if_j}\] Thus we conclude from Algebra, Lemma 00EK that \(\mathcal{F}\) is a sheaf on \((\textit{Aff}/S)_{Zar}\). Choose a presentation \[\bigoplus\nolimits_{k \in K} R \longrightarrow \bigoplus\nolimits_{l \in L} R \longrightarrow \mathcal{F}(U) \longrightarrow 0\] by free \(R\)-modules. By property (1) and the right exactness of tensor product we see that for every morphism \(U' \to U\) in \((\textit{Aff}/S)_{Zar}\) we obtain a presentation \[\bigoplus\nolimits_{k \in K} \mathcal{O}_{Aff}(U') \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}_{Aff}(U') \longrightarrow \mathcal{F}(U') \longrightarrow 0\] In other words, we see that the restriction of \(\mathcal{F}\) to the localized category \((\textit{Aff}/S)_{Zar}/U\) has a presentation \[\bigoplus\nolimits_{k \in K} \mathcal{O}_{Aff}|_{(\textit{Aff}/S)_{Zar}/U} \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}_{Aff}|_{(\textit{Aff}/S)_{Zar}/U} \longrightarrow \mathcal{F}|_{(\textit{Aff}/S)_{Zar}/U} \longrightarrow 0\] With apologies for the horrible notation, this finishes the proof.

We continue the discussion started in the introduction to this section. Let \(\tau \in \{Zariski, \etale\}\). Recall that \(S_{affine, \tau}\) is the full subcategory of \(S_\tau\) whose objects are affine turned into a site by declaring the coverings to be the standard \(\tau\) coverings. See Topologies, Definitions 020T and 021B. By Topologies, Lemmas 0F1B, resp. 04HR we have an equivalence of topoi \(g : \Sh(S_{affine, \tau}) \to \Sh(S_\tau)\), whose pullback functor is given by restriction. Recalling that \(\mathcal{O}\) denotes the structure sheaf on \(S_\tau\) let us temporarily and pedantically denote \(\mathcal{O}_{affine}\) the restriction of \(\mathcal{O}\) to \(S_{affine, \tau}\). Then we obtain an equivalence [0GZW]\[\begin{equation} (\Sh(S_{affine, \tau}), \mathcal{O}_{affine}) \longrightarrow (\Sh(S_\tau), \mathcal{O}) \end{equation}\] of ringed topoi. Having said this we can compare quasi-coherent modules as well.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zariski, \etale\}\). Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}_{affine}\)-modules on \(S_{affine, \tau}\). The following are equivalent

1. for every morphism \(U \to U'\) of \(S_{affine, \tau}\) the map \(\mathcal{F}(U') \otimes_{\mathcal{O}(U')} \mathcal{O}(U) \to \mathcal{F}(U)\) is an isomorphism,

2. \(\mathcal{F}\) is a sheaf on \(S_{affine, \tau}\) and a quasi-coherent module on the ringed site \((S_{affine, \tau}, \mathcal{O}_{affine})\) in the sense of Modules on Sites, Definition 03DL,

3. \(\mathcal{F}\) corresponds to a quasi-coherent module on \(S_\tau\) via the equivalence (0GZW),

4. \(\mathcal{F}\) comes from a unique quasi-coherent \(\mathcal{O}_S\)-module \(\mathcal{G}\) by the procedure described in Section 03DR.

Proof

Let us prove this in the case of the étale topology.

Assume (1) holds. To show that \(\mathcal{F}\) is a sheaf, let \(\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}\) be a covering of \(S_{affine, \etale}\). The sheaf condition for \(\mathcal{F}\) and \(\mathcal{U}\), by our assumption on \(\mathcal{F}\). reduces to showing that \[0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U) \otimes_{\mathcal{O}(U)} \mathcal{O}(U_i) \to \prod \mathcal{F}(U) \otimes_{\mathcal{O}(U)} \mathcal{O}(U_i \times_U U_j)\] is exact. This is true because \(\mathcal{O}(U) \to \prod \mathcal{O}(U_i)\) is faithfully flat (by Lemma 03FI and the fact that coverings in \(S_{affine, \etale}\) are standard étale coverings) and we may apply Lemma 023M. Next, we show that \(\mathcal{F}\) is quasi-coherent on \(S_{affine, \etale}\). Namely, for \(U\) in \(S_{affine, \etale}\), set \(R = \mathcal{O}(U)\) and choose a presentation \[\bigoplus\nolimits_{k \in K} R \longrightarrow \bigoplus\nolimits_{l \in L} R \longrightarrow \mathcal{F}(U) \longrightarrow 0\] by free \(R\)-modules. By property (1) and the right exactness of tensor product we see that for every morphism \(U' \to U\) in \(S_{affine, \etale}\) we obtain a presentation \[\bigoplus\nolimits_{k \in K} \mathcal{O}(U') \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}(U') \longrightarrow \mathcal{F}(U') \longrightarrow 0\] In other words, we see that the restriction of \(\mathcal{F}\) to the localized category \(S_{affine, etale}/U\) has a presentation \[\bigoplus\nolimits_{k \in K} \mathcal{O}_{affine}|_{S_{affine, \etale}/U} \longrightarrow \bigoplus\nolimits_{l \in L} \mathcal{O}_{affine}|_{S_{affine, \etale}/U} \longrightarrow \mathcal{F}|_{S_{affine, \etale}/U} \longrightarrow 0\] as required to show that \(\mathcal{F}\) is quasi-coherent. With apologies for the horrible notation, this finishes the proof that (1) implies (2).

Since the notion of a quasi-coherent module is intrinsic (Modules on Sites, Lemma 03DM) we see that the equivalence (0GZW) induces an equivalence between categories of quasi-coherent modules. Thus we have the equivalence of (2) and (3).

The equivalence of (3) and (4) follows from Proposition 03DX.

Let us assume (4) and prove (1). Namely, let \(\mathcal{G}\) be as in (4). Let \(h : U \to U' \to S\) be a morphism of \(S_{affine, \etale}\). Denote \(f : U \to S\) and \(f' : U' \to S\) the structure morphisms, so that \(f = f' \circ h\). We have \(\mathcal{F}(U') = \Gamma(U', (f')^*\mathcal{G})\) and \(\mathcal{F}(U) = \Gamma(U, f^*\mathcal{G}) = \Gamma(U, h^*(f')^*\mathcal{G})\). Hence (1) holds by Schemes, Lemma 01I9.

We omit the proof in the case of the Zariski topology.

Parasitic modules

Parasitic modules are those which are zero when restricted to schemes flat over the base scheme. Here is the formal definition.

Definition

Let \(S\) be a scheme. Let \(\tau \in \{Zar, \etale, smooth, syntomic, fppf\}\). Let \(\mathcal{F}\) be a presheaf of \(\mathcal{O}\)-modules on \((\Sch/S)_\tau\).

1. \(\mathcal{F}\) is called parasitic⁸ if for every flat morphism \(U \to S\) we have \(\mathcal{F}(U) = 0\).

2. \(\mathcal{F}\) is called parasitic for the \(\tau\)-topology if for every \(\tau\)-covering \(\{U_i \to S\}_{i \in I}\) we have \(\mathcal{F}(U_i) = 0\) for all \(i\).

If \(\tau = fppf\) this means that \(\mathcal{F}|_{U_{Zar}} = 0\) whenever \(U \to S\) is flat and locally of finite presentation; similar for the other cases.

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zar, \etale, smooth, syntomic, fppf\}\). Let \(\mathcal{G}\) be a presheaf of \(\mathcal{O}\)-modules on \((\Sch/S)_\tau\).

1. If \(\mathcal{G}\) is parasitic for the \(\tau\)-topology, then \(H^p_\tau(U, \mathcal{G}) = 0\) for every \(U\) open in \(S\), resp. étale over \(S\), resp. smooth over \(S\), resp. syntomic over \(S\), resp. flat and locally of finite presentation over \(S\).

2. If \(\mathcal{G}\) is parasitic then \(H^p_\tau(U, \mathcal{G}) = 0\) for every \(U\) flat over \(S\).

Proof

Proof in case \(\tau = fppf\); the other cases are proved in the exact same way. The assumption means that \(\mathcal{G}(U) = 0\) for any \(U \to S\) flat and locally of finite presentation. Apply Cohomology on Sites, Lemma 03F9 to the subset \(\mathcal{B} \subset \Ob((\Sch/S)_{fppf})\) consisting of \(U \to S\) flat and locally of finite presentation and the collection \(\text{Cov}\) of all fppf coverings of elements of \(\mathcal{B}\).

Lemma

Let \(f : T \to S\) be a morphism of schemes. For any parasitic \(\mathcal{O}\)-module on \((\Sch/T)_\tau\) the pushforward \(f_*\mathcal{F}\) and the higher direct images \(R^if_*\mathcal{F}\) are parasitic \(\mathcal{O}\)-modules on \((\Sch/S)_\tau\).

Proof

Recall that \(R^if_*\mathcal{F}\) is the sheaf associated to the presheaf \[U \mapsto H^i((\Sch/U \times_S T)_\tau, \mathcal{F})\] see Cohomology on Sites, Lemma 072W. If \(U \to S\) is flat, then \(U \times_S T \to T\) is flat as a base change. Hence the displayed group is zero by Lemma 0755. If \(\{U_i \to U\}\) is a \(\tau\)-covering then \(U_i \times_S T \to T\) is also flat. Hence it is clear that the sheafification of the displayed presheaf is zero on schemes \(U\) flat over \(S\).

Lemma

Let \(S\) be a scheme. Let \(\tau \in \{Zar, \etale\}\). Let \(\mathcal{G}\) be a sheaf of \(\mathcal{O}\)-modules on \((\Sch/S)_{fppf}\) such that

1. \(\mathcal{G}|_{S_\tau}\) is quasi-coherent, and

2. for every flat, locally finitely presented morphism \(g : U \to S\) the canonical map \(g_{\tau, small}^*(\mathcal{G}|_{S_\tau}) \to \mathcal{G}|_{U_\tau}\) is an isomorphism.

Then \(H^p(U, \mathcal{G}) = H^p(U, \mathcal{G}|_{U_\tau})\) for every \(U\) flat and locally of finite presentation over \(S\).

Proof

Let \(\mathcal{F}\) be the pullback of \(\mathcal{G}|_{S_\tau}\) to the big fppf site \((\Sch/S)_{fppf}\). Note that \(\mathcal{F}\) is quasi-coherent. There is a canonical comparison map \(\varphi : \mathcal{F} \to \mathcal{G}\) which by assumptions (1) and (2) induces an isomorphism \(\mathcal{F}|_{U_\tau} \to \mathcal{G}|_{U_\tau}\) for all \(g : U \to S\) flat and locally of finite presentation. Hence in the short exact sequences \[0 \to \Ker(\varphi) \to \mathcal{F} \to \Im(\varphi) \to 0\] and \[0 \to \Im(\varphi) \to \mathcal{G} \to \Coker(\varphi) \to 0\] the sheaves \(\Ker(\varphi)\) and \(\Coker(\varphi)\) are parasitic for the fppf topology. By Lemma 0755 we conclude that \(H^p(U, \mathcal{F}) \to H^p(U, \mathcal{G})\) is an isomorphism for \(g : U \to S\) flat and locally of finite presentation. Since the result holds for \(\mathcal{F}\) by Proposition 03DW we win.

Fpqc coverings are universal effective epimorphisms

We apply the material above to prove an interesting result, namely Lemma 023Q. By Sites, Section 00WO this lemma implies that the representable presheaves on any of the sites \((\Sch/S)_\tau\) are sheaves for \(\tau \in \{Zariski, fppf, \etale, smooth, syntomic\}\). First we prove a helper lemma.

Lemma

For a scheme \(X\) denote \(|X|\) the underlying set. Let \(f : X \to S\) be a morphism of schemes. Then \[|X \times_S X| \to |X| \times_{|S|} |X|\] is surjective.

Proof

Follows immediately from the description of points on the fibre product in Schemes, Lemma 01JT.

Lemma

Let \(\{f_i : X_i \to X\}_{i \in I}\) be a family of morphisms of affine schemes. The following are equivalent

1. for any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \[\Gamma(X, \mathcal{F}) = \text{Equalizer}\left( \xymatrix{ \prod\nolimits_{i \in I} \Gamma(X_i, f_i^*\mathcal{F}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{i, j \in I} \Gamma(X_i \times_X X_j, (f_i \times f_j)^*\mathcal{F}) } \right)\]

2. \(\{f_i : X_i \to X\}_{i \in I}\) is a universal effective epimorphism (Sites, Definition 00WP) in the category of affine schemes.

Proof

Assume (2) holds and let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Consider the scheme (Constructions, Section 01LQ) \[X' = \underline{\Spec}_X(\mathcal{O}_X \oplus \mathcal{F})\] where \(\mathcal{O}_X \oplus \mathcal{F}\) is an \(\mathcal{O}_X\)-algebra with multiplication \((f, s)(f', s') = (ff', fs' + f's)\). If \(s_i \in \Gamma(X_i, f_i^*\mathcal{F})\) is a section, then \(s_i\) determines a unique element of \[\Gamma(X' \times_X X_i, \mathcal{O}_{X' \times_X X_i}) = \Gamma(X_i, \mathcal{O}_{X_i}) \oplus \Gamma(X_i, f_i^*\mathcal{F})\] Proof of equality omitted. If \((s_i)_{i \in I}\) is in the equalizer of (1), then, using the equality \[\Mor(T, \mathbf{A}^1_\mathbf{Z}) = \Gamma(T, \mathcal{O}_T)\] which holds for any scheme \(T\), we see that these sections define a family of morphisms \(h_i : X' \times_X X_i \to \mathbf{A}^1_\mathbf{Z}\) with \(h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2\) as morphisms \((X' \times_X X_i) \times_{X'} (X' \times_X X_j) \to \mathbf{A}^1_\mathbf{Z}\). Since we’ve assume (2) we obtain a morphism \(h : X' \to \mathbf{A}^1_\mathbf{Z}\) compatible with the morphisms \(h_i\) which in turn determines an element \(s \in \Gamma(X, \mathcal{F})\). We omit the verification that \(s\) maps to \(s_i\) in \(\Gamma(X_i, f_i^*\mathcal{F})\).

Assume (1). Let \(T\) be an affine scheme and let \(h_i : X_i \to T\) be a family of morphisms such that \(h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2\) on \(X_i \times_X X_j\) for all \(i, j \in I\). Then \[\prod h_i^\sharp : \Gamma(T, \mathcal{O}_T) \to \prod \Gamma(X_i, \mathcal{O}_{X_i})\] maps into the equalizer and we find that we get a ring map \(\Gamma(T, \mathcal{O}_T) \to \Gamma(X, \mathcal{O}_X)\) by the assumption of the lemma for \(\mathcal{F} = \mathcal{O}_X\). This ring map corresponds to a morphism \(h : X \to T\) such that \(h_i = h \circ f_i\). Hence our family is an effective epimorphism.

Let \(p : Y \to X\) be a morphism of affines. We will show the base changes \(g_i : Y_i \to Y\) of \(f_i\) form an effective epimorphism by applying the result of the previous paragraph. Namely, if \(\mathcal{G}\) is a quasi-coherent \(\mathcal{O}_Y\)-module, then \[\Gamma(Y, \mathcal{G}) = \Gamma(X, p_*\mathcal{G}),\quad \Gamma(Y_i, g_i^*\mathcal{G}) = \Gamma(X, f_i^*p_*\mathcal{G}),\] and \[\Gamma(Y_i \times_Y Y_j, (g_i \times g_j)^*\mathcal{G}) = \Gamma(X, (f_i \times f_j)^*p_*\mathcal{G})\] by the trivial base change formula (Cohomology of Schemes, Lemma 02KG). Thus we see property (1) lemma holds for the family \(g_i\).

Lemma

Let \(\{f_i : X_i \to X\}_{i \in I}\) be a family of morphisms of schemes.

1. If the family is universal effective epimorphism in the category of schemes, then \(\coprod f_i\) is surjective.

2. If \(X\) and \(X_i\) are affine and the family is a universal effective epimorphism in the category of affine schemes, then \(\coprod f_i\) is surjective.

Proof

Omitted. Hint: perform base change by \(\Spec(\kappa(x)) \to X\) to see that any \(x \in X\) has to be in the image.

Lemma

Let \(\{f_i : X_i \to X\}_{i \in I}\) be a family of morphisms of schemes. If for every morphism \(Y \to X\) with \(Y\) affine the family of base changes \(g_i : Y_i \to Y\) forms an effective epimorphism, then the family of \(f_i\) forms a universal effective epimorphism in the category of schemes.

Proof

Let \(Y \to X\) be a morphism of schemes. We have to show that the base changes \(g_i : Y_i \to Y\) form an effective epimorphism. To do this, assume given a scheme \(T\) and morphisms \(h_i : Y_i \to T\) with \(h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2\) on \(Y_i \times_Y Y_j\). Choose an affine open covering \(Y = \bigcup V_\alpha\). Set \(V_{\alpha, i}\) equal to the inverse image of \(V_\alpha\) in \(Y_i\). Then we see that \(V_{\alpha, i} \to V_\alpha\) is the base change of \(f_i\) by \(V_\alpha \to X\). Thus by assumption the family of restrictions \(h_i|_{V_{\alpha, i}}\) come from a morphism of schemes \(h_\alpha : V_\alpha \to T\). We leave it to the reader to show that these agree on overlaps and define the desired morphism \(Y \to T\). See discussion in Schemes, Section 01JA.

Lemma

Let \(\{f_i : X_i \to X\}_{i \in I}\) be a family of morphisms of affine schemes. Assume the equivalent assumption of Lemma 0EUA hold and that moreover for any morphism of affines \(Y \to X\) the map \[\coprod X_i \times_X Y \longrightarrow Y\] is a submersive map of topological spaces (Topology, Definition 0406). Then our family of morphisms is a universal effective epimorphism in the category of schemes.

Proof

By Lemma 0EUC it suffices to base change our family of morphisms by \(Y \to X\) with \(Y\) affine. Set \(Y_i = X_i \times_X Y\). Let \(T\) be a scheme and let \(h_i : Y_i \to T\) be a family of morphisms such that \(h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2\) on \(Y_i \times_Y Y_j\). Note that \(Y\) as a set is the coequalizer of the two maps from \(\coprod Y_i \times_Y Y_j\) to \(\coprod Y_i\). Namely, surjectivity by the affine case of Lemma 0EUB and injectivity by Lemma 02KI. Hence there is a set map of underlying sets \(h : Y \to T\) compatible with the maps \(h_i\). By the second condition of the lemma we see that \(h\) is continuous! Thus if \(y \in Y\) and \(U \subset T\) is an affine open neighbourhood of \(h(y)\), then we can find an affine open \(V \subset Y\) such that \(h(V) \subset U\). Setting \(V_i = Y_i \times_Y V = X_i \times_X V\) we can use the result proved in Lemma 0EUA to see that \(h|_V : V \to U \subset T\) comes from a unique morphism of affine schemes \(h_V : V \to U\) agreeing with \(h_i|_{V_i}\) as morphisms of schemes for all \(i\). Glueing these \(h_V\) (see Schemes, Section 01JA) gives a morphism \(Y \to T\) as desired.

Lemma

Let \(\{f_i : T_i \to T\}_{i \in I}\) be a fpqc covering. Suppose that for each \(i\) we have an open subset \(W_i \subset T_i\) such that for all \(i, j \in I\) we have \(\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)\) as open subsets of \(T_i \times_T T_j\). Then there exists a unique open subset \(W \subset T\) such that \(W_i = f_i^{-1}(W)\) for each \(i\).

Proof

Apply Lemma 02KI to the map \(\coprod_{i \in I} T_i \to T\). It implies there exists a subset \(W \subset T\) such that \(W_i = f_i^{-1}(W)\) for each \(i\), namely \(W = \bigcup f_i(W_i)\). To see that \(W\) is open we may work Zariski locally on \(T\). Hence we may assume that \(T\) is affine. Using Topologies, Definition 022B we may choose a standard fpqc covering \(\{g_j : V_j \to T\}_{j \in J}\) which refines \(\{T_i \to T\}_{i \in I}\). Let \(\alpha : J \to I\) and \(h_j : V_j \to T_{\alpha(j)}\) be as in Sites, Definition 00VT. Then \(g_j^{-1}(W) = h_j^{-1}(W_{\alpha(j)})\). Thus we may assume \(\{f_i : T_i \to T\}\) is a standard fpqc covering. In this case we may apply Morphisms, Lemma 02JY to the morphism \(\coprod T_i \to T\) to conclude that \(W\) is open.

Lemma

Let \(\{T_i \to T\}\) be an fpqc covering, see Topologies, Definition 022B. Then \(\{T_i \to T\}\) is a universal effective epimorphism in the category of schemes, see Sites, Definition 00WP. In other words, every representable functor on the category of schemes satisfies the sheaf condition for the fpqc topology, see Topologies, Definition 022G.

Proof

Let \(S\) be a scheme. We have to show the following: Given morphisms \(\varphi_i : T_i \to S\) such that \(\varphi_i|_{T_i \times_T T_j} = \varphi_j|_{T_i \times_T T_j}\) there exists a unique morphism \(T \to S\) which restricts to \(\varphi_i\) on each \(T_i\). In other words, we have to show that the functor \(h_S = \Mor_{\Sch}( - , S)\) satisfies the sheaf property for the fpqc topology.

If \(\{T_i \to T\}\) is a Zariski covering, then this follows from Schemes, Lemma 01JB. Thus Topologies, Lemma 022H reduces us to the case of a covering \(\{X \to Y\}\) given by a single surjective flat morphism of affines.

First proof. By Lemma 03DT we have the sheaf condition for quasi-coherent modules for \(\{X \to Y\}\). By Lemma 03N0 the morphism \(X \to Y\) is universally submersive. Hence we may apply Lemma 0EUD to see that \(\{X \to Y\}\) is a universal effective epimorphism.

Second proof. Let \(R \to A\) be the faithfully flat ring map corresponding to our surjective flat morphism \(\pi : X \to Y\). Let \(f : X \to S\) be a morphism such that \(f \circ \text{pr}_1 = f \circ \text{pr}_2\) as morphisms \(X \times_Y X = \Spec(A \otimes_R A) \to S\). By Lemma 02KI we see that as a map on the underlying sets \(f\) is of the form \(f = g \circ \pi\) for some (set theoretic) map \(g : \Spec(R) \to S\). By Morphisms, Lemma 02JY and the fact that \(f\) is continuous we see that \(g\) is continuous.

Pick \(y \in Y = \Spec(R)\). Choose \(U \subset S\) affine open containing \(g(y)\). Say \(U = \Spec(B)\). By the above we may choose an \(r \in R\) such that \(y \in D(r) \subset g^{-1}(U)\). The restriction of \(f\) to \(\pi^{-1}(D(r))\) into \(U\) corresponds to a ring map \(B \to A_r\). The two induced ring maps \(B \to A_r \otimes_{R_r} A_r = (A \otimes_R A)_r\) are equal by assumption on \(f\). Note that \(R_r \to A_r\) is faithfully flat. By Lemma 023M the equalizer of the two arrows \(A_r \to A_r \otimes_{R_r} A_r\) is \(R_r\). We conclude that \(B \to A_r\) factors uniquely through a map \(B \to R_r\). This map in turn gives a morphism of schemes \(D(r) \to U \to S\), see Schemes, Lemma 01I1.

What have we proved so far? We have shown that for any prime \(\mathfrak p \subset R\), there exists a standard affine open \(D(r) \subset \Spec(R)\) such that the morphism \(f|_{\pi^{-1}(D(r))} : \pi^{-1}(D(r)) \to S\) factors uniquely through some morphism of schemes \(D(r) \to S\). We omit the verification that these morphisms glue to the desired morphism \(\Spec(R) \to S\).

Lemma

Consider schemes \(X, Y, Z\) and morphisms \(a, b : X \to Y\) and a morphism \(c : Y \to Z\) with \(c \circ a = c \circ b\). Set \(d = c \circ a = c \circ b\). If there exists an fpqc covering \(\{Z_i \to Z\}\) such that

1. for all \(i\) the morphism \(Y \times_{c, Z} Z_i \to Z_i\) is the coequalizer of \((a, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i\) and \((b, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i\), and

2. for all \(i\) and \(i'\) the morphism \(Y \times_{c, Z} (Z_i \times_Z Z_{i'}) \to (Z_i \times_Z Z_{i'})\) is the coequalizer of \((a, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to Y \times_{c, Z} (Z_i \times_Z Z_{i'})\) and \((b, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to Y \times_{c, Z} (Z_i \times_Z Z_{i'})\)

then \(c\) is the coequalizer of \(a\) and \(b\).

Proof

Namely, for a scheme \(T\) a morphism \(Z \to T\) is the same thing as a collection of morphism \(Z_i \to T\) which agree on overlaps by Lemma 023Q.

Descent of finiteness and smoothness properties of morphisms

In this section we show that several properties of morphisms (being smooth, locally of finite presentation, and so on) descend under faithfully flat morphisms. We start with an algebraic version. (The “Noetherian” reader should consult Lemma 0367 instead of the next lemma.)

Lemma

Let \(R \to A \to B\) be ring maps. Assume \(R \to B\) is of finite presentation and \(A \to B\) faithfully flat and of finite presentation. Then \(R \to A\) is of finite presentation.

Proof

Consider the algebra \(C = B \otimes_A B\) together with the pair of maps \(p, q : B \to C\) given by \(p(b) = b \otimes 1\) and \(q(b) = 1 \otimes b\). Of course the two compositions \(A \to B \to C\) are the same. Note that as \(p : B \to C\) is flat and of finite presentation (base change of \(A \to B\)), the ring map \(R \to C\) is of finite presentation (as the composite of \(R \to B \to C\)).

We are going to use the criterion Algebra, Lemma 00QO to show that \(R \to A\) is of finite presentation. Let \(S\) be any \(R\)-algebra, and suppose that \(S = \colim_{\lambda \in \Lambda} S_\lambda\) is written as a directed colimit of \(R\)-algebras. Let \(A \to S\) be an \(R\)-algebra homomorphism. We have to show that \(A \to S\) factors through one of the \(S_\lambda\). Consider the rings \(B' = S \otimes_A B\) and \(C' = S \otimes_A C = B' \otimes_S B'\). As \(B\) is faithfully flat of finite presentation over \(A\), also \(B'\) is faithfully flat of finite presentation over \(S\). By Algebra, Lemma 02JO part (2) applied to the pair \((S \to B', B')\) and the system \((S_\lambda)\) there exists a \(\lambda_0 \in \Lambda\) and a flat, finitely presented \(S_{\lambda_0}\)-algebra \(B_{\lambda_0}\) such that \(B' = S \otimes_{S_{\lambda_0}} B_{\lambda_0}\). For \(\lambda \geq \lambda_0\) set \(B_\lambda = S_\lambda \otimes_{S_{\lambda_0}} B_{\lambda_0}\) and \(C_\lambda = B_\lambda \otimes_{S_\lambda} B_\lambda\).

We interrupt the flow of the argument to show that \(S_\lambda \to B_\lambda\) is faithfully flat for \(\lambda\) large enough. (This should really be a separate lemma somewhere else, maybe in the chapter on limits.) Since \(\Spec(B_{\lambda_0}) \to \Spec(S_{\lambda_0})\) is flat and of finite presentation it is open (see Morphisms, Lemma 01UA). Let \(I \subset S_{\lambda_0}\) be an ideal such that \(V(I) \subset \Spec(S_{\lambda_0})\) is the complement of the image. Note that formation of the image commutes with base change. Hence, since \(\Spec(B') \to \Spec(S)\) is surjective, and \(B' = B_{\lambda_0} \otimes_{S_{\lambda_0}} S\) we see that \(IS = S\). Thus for some \(\lambda \geq \lambda_0\) we have \(IS_{\lambda} = S_\lambda\). For this and all greater \(\lambda\) the morphism \(\Spec(B_\lambda) \to \Spec(S_\lambda)\) is surjective.

By analogy with the notation in the first paragraph of the proof denote \(p_\lambda, q_\lambda : B_\lambda \to C_\lambda\) the two canonical maps. Then \(B' = \colim_{\lambda \geq \lambda_0} B_\lambda\) and \(C' = \colim_{\lambda \geq \lambda_0} C_\lambda\). Since \(B\) and \(C\) are finitely presented over \(R\) there exist (by Algebra, Lemma 00QO applied several times) a \(\lambda \geq \lambda_0\) and an \(R\)-algebra maps \(B \to B_\lambda\), \(C \to C_\lambda\) such that the diagram \[\xymatrix{ C \ar[rr] & & C_\lambda \\ B \ar[rr] \ar@<1ex>[u]^-p \ar@<-1ex>[u]_-q & & B_\lambda \ar@<1ex>[u]^-{p_\lambda} \ar@<-1ex>[u]_-{q_\lambda} }\] is commutative. OK, and this means that \(A \to B \to B_\lambda\) maps into the equalizer of \(p_\lambda\) and \(q_\lambda\). By Lemma 023M we see that \(S_\lambda\) is the equalizer of \(p_\lambda\) and \(q_\lambda\). Thus we get the desired ring map \(A \to S_\lambda\) and we win.

Here is an easier version of this dealing with the property of being of finite type.

Lemma

Let \(R \to A \to B\) be ring maps. Assume \(R \to B\) is of finite type and \(A \to B\) faithfully flat and of finite presentation. Then \(R \to A\) is of finite type.

Proof

By Algebra, Lemma 034Y there exists a commutative diagram \[\xymatrix{ R \ar[r] \ar@{=}[d] & A_0 \ar[d] \ar[r] & B_0 \ar[d] \\ R \ar[r] & A \ar[r] & B }\] with \(R \to A_0\) of finite presentation, \(A_0 \to B_0\) faithfully flat of finite presentation and \(B = A \otimes_{A_0} B_0\). Since \(R \to B\) is of finite type by assumption, we may add some elements to \(A_0\) and assume that the map \(B_0 \to B\) is surjective! In this case, since \(A_0 \to B_0\) is faithfully flat, we see that as \[(A_0 \to A) \otimes_{A_0} B_0 \cong (B_0 \to B)\] is surjective, also \(A_0 \to A\) is surjective. Hence we win.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S }\] be a commutative diagram of morphisms of schemes. Assume that \(f\) is surjective, flat and locally of finite presentation and assume that \(p\) is locally of finite presentation (resp. locally of finite type). Then \(q\) is locally of finite presentation (resp. locally of finite type).

Proof

The problem is local on \(S\) and \(Y\). Hence we may assume that \(S\) and \(Y\) are affine. Since \(f\) is flat and locally of finite presentation, we see that \(f\) is open (Morphisms, Lemma 01UA). Hence, since \(Y\) is quasi-compact, there exist finitely many affine opens \(X_i \subset X\) such that \(Y = \bigcup f(X_i)\). Clearly we may replace \(X\) by \(\coprod X_i\), and hence we may assume \(X\) is affine as well. In this case the lemma is equivalent to Lemma 02KK (resp. Lemma 0367) above.

We use this to improve some of the results on morphisms obtained earlier.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S }\] be a commutative diagram of morphisms of schemes. Assume that

1. \(f\) is surjective, and syntomic (resp. smooth, resp. étale),

2. \(p\) is syntomic (resp. smooth, resp. étale).

Then \(q\) is syntomic (resp. smooth, resp. étale).

Proof

Combine Morphisms, Lemmas 02K3, 02K5, and 02K6 with Lemma 02KL above.

Actually we can strengthen this result as follows.

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S }\] be a commutative diagram of morphisms of schemes. Assume that

1. \(f\) is surjective, flat, and locally of finite presentation,

2. \(p\) is smooth (resp. étale).

Then \(q\) is smooth (resp. étale).

Proof

Assume (1) and that \(p\) is smooth. By Lemma 02KL we see that \(q\) is locally of finite presentation. By Morphisms, Lemma 02JZ we see that \(q\) is flat. Hence now it suffices to show that the fibres of \(q\) are smooth, see Morphisms, Lemma 01V8. Apply Varieties, Lemma 05AX to the flat surjective morphisms \(X_s \to Y_s\) for \(s \in S\) to conclude. We omit the proof of the étale case.

Remark

With the assumptions (1) and \(p\) smooth in Lemma 05B5 it is not automatically the case that \(X \to Y\) is smooth. A counter example is \(S = \Spec(k)\), \(X = \Spec(k[s])\), \(Y = \Spec(k[t])\) and \(f\) given by \(t \mapsto s^2\). But see also Lemma 05B7 for some information on the structure of \(f\).

Lemma

Let \[\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S }\] be a commutative diagram of morphisms of schemes. Assume that

1. \(f\) is surjective, flat, and locally of finite presentation,

2. \(p\) is syntomic.

Then both \(q\) and \(f\) are syntomic.

Proof

By Lemma 02KL we see that \(q\) is of finite presentation. By Morphisms, Lemma 02JZ we see that \(q\) is flat. By Morphisms, Lemma 01UE it now suffices to show that the local rings of the fibres of \(Y \to S\) and the fibres of \(X \to Y\) are local complete intersection rings. To do this we may take the fibre of \(X \to Y \to S\) at a point \(s \in S\), i.e., we may assume \(S\) is the spectrum of a field. Pick a point \(x \in X\) with image \(y \in Y\) and consider the ring map \[\mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x}\] This is a flat local homomorphism of local Noetherian rings. The local ring \(\mathcal{O}_{X, x}\) is a complete intersection. Thus may use Avramov’s result, see Divided Power Algebra, Lemma 09Q7, to conclude that both \(\mathcal{O}_{Y, y}\) and \(\mathcal{O}_{X, x}/\mathfrak m_y\mathcal{O}_{X, x}\) are complete intersection rings.

The following type of lemma is occasionally useful.

Lemma

Let \(X \to Y \to Z\) be morphism of schemes. Let \(P\) be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that \(X \to Z\) has \(P\) and that \(\{X \to Y\}\) can be refined by an fppf covering of \(Y\). Then \(Y \to Z\) is \(P\).

Proof

Let \(\Spec(C) \subset Z\) be an affine open and let \(\Spec(B) \subset Y\) be an affine open which maps into \(\Spec(C)\). The assumption on \(X \to Y\) implies we can find a standard affine fppf covering \(\{\Spec(B_j) \to \Spec(B)\}\) and lifts \(x_j : \Spec(B_j) \to X\). Since \(\Spec(B_j)\) is quasi-compact we can find finitely many affine opens \(\Spec(A_i) \subset X\) lying over \(\Spec(B)\) such that the image of each \(x_j\) is contained in the union \(\bigcup \Spec(A_i)\). Hence after replacing each \(\Spec(B_j)\) by a standard affine Zariski coverings of itself we may assume we have a standard affine fppf covering \(\{\Spec(B_i) \to \Spec(B)\}\) such that each \(\Spec(B_i) \to Y\) factors through an affine open \(\Spec(A_i) \subset X\) lying over \(\Spec(B)\). In other words, we have ring maps \(C \to B \to A_i \to B_i\) for each \(i\). Note that we can also consider \[C \to B \to A = \prod A_i \to B' = \prod B_i\] and that the ring map \(B \to \prod B_i\) is faithfully flat and of finite presentation.

The case \(P = flat\). In this case we know that \(C \to A\) is flat and we have to prove that \(C \to B\) is flat. Suppose that \(N \to N' \to N''\) is an exact sequence of \(C\)-modules. We want to show that \(N \otimes_C B \to N' \otimes_C B \to N'' \otimes_C B\) is exact. Let \(H\) be its cohomology and let \(H'\) be the cohomology of \(N \otimes_C B' \to N' \otimes_C B' \to N'' \otimes_C B'\). As \(B \to B'\) is flat we know that \(H' = H \otimes_B B'\). On the other hand \(N \otimes_C A \to N' \otimes_C A \to N'' \otimes_C A\) is exact hence has zero cohomology. Hence the map \(H \to H'\) is zero (as it factors through the zero module). Thus \(H' = 0\). As \(B \to B'\) is faithfully flat we conclude that \(H = 0\) as desired.

The case \(P = locally\ finite\ type\). In this case we know that \(C \to A\) is of finite type and we have to prove that \(C \to B\) is of finite type. Because \(B \to B'\) is of finite presentation (hence of finite type) we see that \(A \to B'\) is of finite type, see Algebra, Lemma 00F4. Therefore \(C \to B'\) is of finite type and we conclude by Lemma 0367.

The case \(P = locally\ finite\ presentation\). In this case we know that \(C \to A\) is of finite presentation and we have to prove that \(C \to B\) is of finite presentation. Because \(B \to B'\) is of finite presentation and \(B \to A\) of finite type we see that \(A \to B'\) is of finite presentation, see Algebra, Lemma 00F4. Therefore \(C \to B'\) is of finite presentation and we conclude by Lemma 02KK.

Local properties of schemes

It often happens one can prove the members of a covering of a scheme have a certain property. In many cases this implies the scheme has the property too. For example, if \(S\) is a scheme, and \(f : S' \to S\) is a surjective flat morphism such that \(S'\) is a reduced scheme, then \(S\) is reduced. You can prove this by looking at local rings and using Algebra, Lemma 033F. We say that the property of being reduced descends through flat surjective morphisms. Some results of this type are collected in Algebra, Section 033D and for schemes in Section 06QL. Some analogous results on descending properties of morphisms are in Section 02KJ.

On the other hand, there are examples of surjective flat morphisms \(f : S' \to S\) with \(S\) reduced and \(S'\) not, for example the morphism \(\Spec(k[x]/(x^2)) \to \Spec(k)\). Hence the property of being reduced does not ascend along flat morphisms. Having infinite residue fields is a property which does ascend along flat morphisms (but does not descend along surjective flat morphisms of course). Some results of this type are collected in Algebra, Section 0336.

Finally, we say that a property is local for the flat topology if it ascends along flat morphisms and descends along flat surjective morphisms. A somewhat silly example is the property of having residue fields of a given characteristic. To be more precise, and to tie this in with the various topologies on schemes, we make the following formal definition.

Definition

Let \(\mathcal{P}\) be a property of schemes. Let \(\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale, \linebreak[0] Zariski\}\). We say \(\mathcal{P}\) is local in the \(\tau\)-topology if for any \(\tau\)-covering \(\{S_i \to S\}_{i \in I}\) (see Topologies, Section 020M) we have \[S \text{ has }\mathcal{P} \Leftrightarrow \text{each }S_i \text{ has }\mathcal{P}.\]

To be sure, since isomorphisms are always coverings we see (or require) that property \(\mathcal{P}\) holds for \(S\) if and only if it holds for any scheme \(S'\) isomorphic to \(S\). In fact, if \(\tau = fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] \etale\), or \(Zariski\), then if \(S\) has \(\mathcal{P}\) and \(S' \to S\) is flat, flat and locally of finite presentation, syntomic, smooth, étale, or an open immersion, then \(S'\) has \(\mathcal{P}\). This is true because we can always extend \(\{S' \to S\}\) to a \(\tau\)-covering.

We have the following implications: \(\mathcal{P}\) is local in the fpqc topology \(\Rightarrow\) \(\mathcal{P}\) is local in the fppf topology \(\Rightarrow\) \(\mathcal{P}\) is local in the syntomic topology \(\Rightarrow\) \(\mathcal{P}\) is local in the smooth topology \(\Rightarrow\) \(\mathcal{P}\) is local in the étale topology \(\Rightarrow\) \(\mathcal{P}\) is local in the Zariski topology. This follows from Topologies, Lemmas 0216, 0220, 0226, 021N, and 022C.

Lemma

Let \(\mathcal{P}\) be a property of schemes. Let \(\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}\). Assume that

1. the property is local in the Zariski topology,

2. for any morphism of affine schemes \(S' \to S\) which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether \(\tau\) is fpqc, fppf, étale, smooth, or syntomic, property \(\mathcal{P}\) holds for \(S'\) if property \(\mathcal{P}\) holds for \(S\), and

3. for any surjective morphism of affine schemes \(S' \to S\) which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether \(\tau\) is fpqc, fppf, étale, smooth, or syntomic, property \(\mathcal{P}\) holds for \(S\) if property \(\mathcal{P}\) holds for \(S'\).

Then \(\mathcal{P}\) is \(\tau\) local on the base.

Proof

This follows almost immediately from the definition of a \(\tau\)-covering, see Topologies, Definition 022B 021M 0215 021Z, or 0225 and Topologies, Lemma 022E, 021P, 0218, 0222, or 0228. Details omitted.

Remark

In Lemma 0349 above if \(\tau = smooth\) then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when \(\tau = syntomic\) or \(\tau = \etale\).

Properties of schemes local in the fppf topology

In this section we find some properties of schemes which are local on the base in the fppf topology.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is locally Noetherian” is local in the fppf topology.

Proof

We will use Lemma 0349. First we note that “being locally Noetherian” is local in the Zariski topology. This is clear from the definition, see Properties, Definition 01OV. Next, we show that if \(S' \to S\) is a flat, finitely presented morphism of affines and \(S\) is locally Noetherian, then \(S'\) is locally Noetherian. This is Morphisms, Lemma 01T6. Finally, we have to show that if \(S' \to S\) is a surjective flat, finitely presented morphism of affines and \(S'\) is locally Noetherian, then \(S\) is locally Noetherian. This follows from Algebra, Lemma 033E. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is Jacobson” is local in the fppf topology.

Proof

We will use Lemma 0349. First we note that “being Jacobson” is local in the Zariski topology. This is Properties, Lemma 01P4. Next, we show that if \(S' \to S\) is a flat, finitely presented morphism of affines and \(S\) is Jacobson, then \(S'\) is Jacobson. This is Morphisms, Lemma 02J5. Finally, we have to show that if \(f : S' \to S\) is a surjective flat, finitely presented morphism of affines and \(S'\) is Jacobson, then \(S\) is Jacobson. Say \(S = \Spec(A)\) and \(S' = \Spec(B)\) and \(S' \to S\) given by \(A \to B\). Then \(A \to B\) is finitely presented and faithfully flat. Moreover, the ring \(B\) is Jacobson, see Properties, Lemma 01P4.

By Algebra, Lemma 034Z there exists a diagram \[\xymatrix{ B \ar[rr] & & B' \\ & A \ar[ru] \ar[lu] & }\] with \(A \to B'\) finitely presented, faithfully flat and quasi-finite. In particular, \(B \to B'\) is finite type, and we see from Algebra, Proposition 00GB that \(B'\) is Jacobson. Hence we may assume that \(A \to B\) is quasi-finite as well as faithfully flat and of finite presentation.

Assume \(A\) is not Jacobson to get a contradiction. According to Algebra, Lemma 034J there exists a nonmaximal prime \(\mathfrak p \subset A\) and an element \(f \in A\), \(f \not \in \mathfrak p\) such that \(V(\mathfrak p) \cap D(f) = \{\mathfrak p\}\).

This leads to a contradiction as follows. First let \(\mathfrak p \subset \mathfrak m\) be a maximal ideal of \(A\). Pick a prime \(\mathfrak m' \subset B\) lying over \(\mathfrak m\) (exists because \(A \to B\) is faithfully flat, see Algebra, Lemma 00HQ). As \(A \to B\) is flat, by going down see Algebra, Lemma 00HS, we can find a prime \(\mathfrak q \subset \mathfrak m'\) lying over \(\mathfrak p\). In particular we see that \(\mathfrak q\) is not maximal. Hence according to Algebra, Lemma 034J again the set \(V(\mathfrak q) \cap D(f)\) is infinite (here we finally use that \(B\) is Jacobson). All points of \(V(\mathfrak q) \cap D(f)\) map to \(V(\mathfrak p) \cap D(f) = \{\mathfrak p\}\). Hence the fibre over \(\mathfrak p\) is infinite. This contradicts the fact that \(A \to B\) is quasi-finite (see Algebra, Lemma 00PM or more explicitly Morphisms, Lemma 02NH). Thus the lemma is proved.

Lemma

The property \(\mathcal{P}(S) =\)“every quasi-compact open of \(S\) has a finite number of irreducible components” is local in the fppf topology.

Proof

We will use Lemma 0349. First we note that \(\mathcal{P}\) is local in the Zariski topology. Next, we show that if \(T \to S\) is a flat, finitely presented morphism of affines and \(S\) has a finite number of irreducible components, then so does \(T\). Namely, since \(T \to S\) is flat, the generic points of \(T\) map to the generic points of \(S\), see Morphisms, Lemma 03HV. Hence it suffices to show that for \(s \in S\) the fibre \(T_s\) has a finite number of generic points. Note that \(T_s\) is an affine scheme of finite type over \(\kappa(s)\), see Morphisms, Lemma 01T4. Hence \(T_s\) is Noetherian and has a finite number of irreducible components (Morphisms, Lemma 01T6 and Properties, Lemma 0BA8). Finally, we have to show that if \(T \to S\) is a surjective flat, finitely presented morphism of affines and \(T\) has a finite number of irreducible components, then so does \(S\). This follows from Topology, Lemma 0GM2. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Properties of schemes local in the syntomic topology

In this section we find some properties of schemes which are local on the base in the syntomic topology.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is locally Noetherian and \((S_k)\)” is local in the syntomic topology.

Proof

We will check (1), (2) and (3) of Lemma 0349. As a syntomic morphism is flat of finite presentation (Morphisms, Lemmas 01UL and 01UK) we have already checked this for “being locally Noetherian” in the proof of Lemma 034C. We will use this without further mention in the proof. First we note that \(\mathcal{P}\) is local in the Zariski topology. This is clear from the definition, see Cohomology of Schemes, Definition 0341. Next, we show that if \(S' \to S\) is a syntomic morphism of affines and \(S\) has \(\mathcal{P}\), then \(S'\) has \(\mathcal{P}\). This is Algebra, Lemma 0339 (use Morphisms, Lemma 01UD and Algebra, Definition 00SL and Lemma 00SB). Finally, we show that if \(S' \to S\) is a surjective syntomic morphism of affines and \(S'\) has \(\mathcal{P}\), then \(S\) has \(\mathcal{P}\). This is Algebra, Lemma 0352. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is Cohen-Macaulay” is local in the syntomic topology.

Proof

This is clear from Lemma 036A above since a scheme is Cohen-Macaulay if and only if it is locally Noetherian and \((S_k)\) for all \(k \geq 0\), see Properties, Lemma 0342.

Properties of schemes local in the smooth topology

In this section we find some properties of schemes which are local on the base in the smooth topology.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is reduced” is local in the smooth topology.

Proof

We will use Lemma 0349. First we note that “being reduced” is local in the Zariski topology. This is clear from the definition, see Schemes, Definition 01J0. Next, we show that if \(S' \to S\) is a smooth morphism of affines and \(S\) is reduced, then \(S'\) is reduced. This is Algebra, Lemma 033B. Finally, we show that if \(S' \to S\) is a surjective smooth morphism of affines and \(S'\) is reduced, then \(S\) is reduced. This is Algebra, Lemma 033F. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is normal” is local in the smooth topology.

Proof

We will use Lemma 0349. First we show “being normal” is local in the Zariski topology. This is clear from the definition, see Properties, Definition 033I. Next, we show that if \(S' \to S\) is a smooth morphism of affines and \(S\) is normal, then \(S'\) is normal. This is Algebra, Lemma 033C. Finally, we show that if \(S' \to S\) is a surjective smooth morphism of affines and \(S'\) is normal, then \(S\) is normal. This is Algebra, Lemma 033G. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is locally Noetherian and \((R_k)\)” is local in the smooth topology.

Proof

We will check (1), (2) and (3) of Lemma 0349. As a smooth morphism is flat of finite presentation (Morphisms, Lemmas 01VF and 01VE) we have already checked this for “being locally Noetherian” in the proof of Lemma 034C. We will use this without further mention in the proof. First we note that \(\mathcal{P}\) is local in the Zariski topology. This is clear from the definition, see Properties, Definition 033Q. Next, we show that if \(S' \to S\) is a smooth morphism of affines and \(S\) has \(\mathcal{P}\), then \(S'\) has \(\mathcal{P}\). This is Algebra, Lemmas 033A (use Morphisms, Lemma 01V6, Algebra, Lemmas 00T4 and 00TT). Finally, we show that if \(S' \to S\) is a surjective smooth morphism of affines and \(S'\) has \(\mathcal{P}\), then \(S\) has \(\mathcal{P}\). This is Algebra, Lemma 0353. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is regular” is local in the smooth topology.

Proof

This is clear from Lemma 036C above since a locally Noetherian scheme is regular if and only if it is locally Noetherian and \((R_k)\) for all \(k \geq 0\).

Lemma

The property \(\mathcal{P}(S) =\)“\(S\) is Nagata” is local in the smooth topology.

Proof

We will check (1), (2) and (3) of Lemma 0349. First we note that being Nagata is local in the Zariski topology. This is Properties, Lemma 033X. Next, we show that if \(S' \to S\) is a smooth morphism of affines and \(S\) is Nagata, then \(S'\) is Nagata. This is Morphisms, Lemma 035A. Finally, we show that if \(S' \to S\) is a surjective smooth morphism of affines and \(S'\) is Nagata, then \(S\) is Nagata. This is Algebra, Lemma 0354. Thus (1), (2) and (3) of Lemma 0349 hold and we win.

Variants on descending properties

Sometimes one can descend properties, which are not local. We put results of this kind in this section. See also Section 02KJ on descending properties of morphisms, such as smoothness.

Lemma

If \(f : X \to Y\) is a flat and surjective morphism of schemes and \(X\) is reduced, then \(Y\) is reduced.

Proof

The result follows by looking at local rings (Schemes, Definition 01J0) and Algebra, Lemma 033F.

Lemma

Let \(f : X \to Y\) be a morphism of algebraic spaces. If \(f\) is locally of finite presentation, flat, and surjective and \(X\) is regular, then \(Y\) is regular.

Proof

This lemma reduces to the following algebra statement: If \(A \to B\) is a faithfully flat, finitely presented ring homomorphism with \(B\) Noetherian and regular, then \(A\) is Noetherian and regular. We see that \(A\) is Noetherian by Algebra, Lemma 033E and regular by Algebra, Lemma 00OF.

Germs of schemes

Definition

Germs of schemes.

1. A pair \((X, x)\) consisting of a scheme \(X\) and a point \(x \in X\) is called the germ of \(X\) at \(x\).

2. A morphism of germs \(f : (X, x) \to (S, s)\) is an equivalence class of morphisms of schemes \(f : U \to S\) with \(f(x) = s\) where \(U \subset X\) is an open neighbourhood of \(x\). Two such \(f\), \(f'\) are said to be equivalent if and only if \(f\) and \(f'\) agree in some open neighbourhood of \(x\).

3. We define the composition of morphisms of germs by composing representatives (this is well defined).

Before we continue we need one more definition.

Definition

Let \(f : (X, x) \to (S, s)\) be a morphism of germs. We say \(f\) is étale (resp. smooth) if there exists a representative \(f : U \to S\) of \(f\) which is an étale morphism (resp. a smooth morphism) of schemes.

Local properties of germs

Definition

Let \(\mathcal{P}\) be a property of germs of schemes. We say that \(\mathcal{P}\) is étale local (resp. smooth local) if for any étale (resp. smooth) morphism of germs \((U', u') \to (U, u)\) we have \(\mathcal{P}(U, u) \Leftrightarrow \mathcal{P}(U', u')\).

Let \((X, x)\) be a germ of a scheme. The dimension of \(X\) at \(x\) is the minimum of the dimensions of open neighbourhoods of \(x\) in \(X\), and any small enough open neighbourhood has this dimension. Hence this is an invariant of the isomorphism class of the germ. We denote this simply \(\dim_x(X)\). The following lemma tells us that the assertion \(\dim_x(X) = d\) is an étale local property of germs.

Lemma

Let \(f : U \to V\) be an étale morphism of schemes. Let \(u \in U\) and \(v = f(u)\). Then \(\dim_u(U) = \dim_v(V)\).

Proof

In the statement \(\dim_u(U)\) is the dimension of \(U\) at \(u\) as defined in Topology, Definition 0055 as the minimum of the Krull dimensions of open neighbourhoods of \(u\) in \(U\). Similarly for \(\dim_v(V)\).

Let us show that \(\dim_v(V) \geq \dim_u(U)\). Let \(V'\) be an open neighbourhood of \(v\) in \(V\). Then there exists an open neighbourhood \(U'\) of \(u\) in \(U\) contained in \(f^{-1}(V')\) such that \(\dim_u(U) = \dim(U')\). Suppose that \(Z_0 \subset Z_1 \subset \ldots \subset Z_n\) is a chain of irreducible closed subschemes of \(U'\). If \(\xi_i \in Z_i\) is the generic point then we have specializations \(\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0\). This gives specializations \(f(\xi_n) \leadsto f(\xi_{n - 1}) \leadsto \ldots \leadsto f(\xi_0)\) in \(V'\). Note that \(f(\xi_j) \not = f(\xi_i)\) if \(i \not = j\) as the fibres of \(f\) are discrete (see Morphisms, Lemma 02GL). Hence we see that \(\dim(V') \geq n\). The inequality \(\dim_v(V) \geq \dim_u(U)\) follows formally.

Let us show that \(\dim_u(U) \geq \dim_v(V)\). Let \(U'\) be an open neighbourhood of \(u\) in \(U\). Note that \(V' = f(U')\) is an open neighbourhood of \(v\) by Morphisms, Lemma 01UA. Hence \(\dim(V') \geq \dim_v(V)\). Pick a chain \(Z_0 \subset Z_1 \subset \ldots \subset Z_n\) of irreducible closed subschemes of \(V'\). Let \(\xi_i \in Z_i\) be the generic point, so we have specializations \(\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0\). Since \(\xi_0 \in f(U')\) we can find a point \(\eta_0 \in U'\) with \(f(\eta_0) = \xi_0\). Consider the map of local rings \[\mathcal{O}_{V', \xi_0} \longrightarrow \mathcal{O}_{U', \eta_0}\] which is a flat local ring map by Morphisms, Lemma 02GS. Note that the points \(\xi_i\) correspond to primes of the ring on the left by Schemes, Lemma 01J7. Hence by going down (see Algebra, Section 00HU) for the displayed ring map we can find a sequence of specializations \(\eta_n \leadsto \eta_{n - 1} \leadsto \ldots \leadsto \eta_0\) in \(U'\) mapping to the sequence \(\xi_n \leadsto \xi_{n - 1} \leadsto \ldots \leadsto \xi_0\) under \(f\). This implies that \(\dim_u(U) \geq \dim_v(V)\).

Let \((X, x)\) be a germ of a scheme. The isomorphism class of the local ring \(\mathcal{O}_{X, x}\) is an invariant of the germ. The following lemma says that the property \(\dim(\mathcal{O}_{X, x}) = d\) is an étale local property of germs.

Lemma

Let \(f : U \to V\) be an étale morphism of schemes. Let \(u \in U\) and \(v = f(u)\). Then \(\dim(\mathcal{O}_{U, u}) = \dim(\mathcal{O}_{V, v})\).

Proof

The algebraic statement we are asked to prove is the following: If \(A \to B\) is an étale ring map and \(\mathfrak q\) is a prime of \(B\) lying over \(\mathfrak p \subset A\), then \(\dim(A_{\mathfrak p}) = \dim(B_{\mathfrak q})\). This is More on Algebra, Lemma 07QP.

Let \((X, x)\) be a germ of a scheme. The isomorphism class of the local ring \(\mathcal{O}_{X, x}\) is an invariant of the germ. The following lemma says that the property “\(\mathcal{O}_{X, x}\) is regular” is an étale local property of germs.

Lemma

Let \(f : U \to V\) be an étale morphism of schemes. Let \(u \in U\) and \(v = f(u)\). Then \(\mathcal{O}_{U, u}\) is a regular local ring if and only if \(\mathcal{O}_{V, v}\) is a regular local ring.

Proof

The algebraic statement we are asked to prove is the following: If \(A \to B\) is an étale ring map and \(\mathfrak q\) is a prime of \(B\) lying over \(\mathfrak p \subset A\), then \(A_{\mathfrak p}\) is regular if and only if \(B_{\mathfrak q}\) is regular. This is More on Algebra, Lemma 0AH0.

Properties of morphisms local on the target

Suppose that \(f : X \to Y\) is a morphism of schemes. Let \(g : Y' \to Y\) be a morphism of schemes. Let \(f' : X' \to Y'\) be the base change of \(f\) by \(g\): \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] Let \(\mathcal{P}\) be a property of morphisms of schemes. Then we can wonder if (a) \(\mathcal{P}(f) \Rightarrow \mathcal{P}(f')\), and also whether the converse (b) \(\mathcal{P}(f') \Rightarrow \mathcal{P}(f)\) is true. If (a) holds whenever \(g\) is flat, then we say \(\mathcal{P}\) is preserved under flat base change. If (b) holds whenever \(g\) is surjective and flat, then we say \(\mathcal{P}\) descends through flat surjective base changes. If \(\mathcal{P}\) is preserved under flat base changes and descends through flat surjective base changes, then we say \(\mathcal{P}\) is flat local on the target. Compare with the discussion in Section 0347. This turns out to be a very important notion which we formalize in the following definition.

Definition

Let \(\mathcal{P}\) be a property of morphisms of schemes over a base. Let \(\tau \in \{fpqc, fppf, syntomic, smooth, \etale, Zariski\}\). We say \(\mathcal{P}\) is \(\tau\) local on the base, or \(\tau\) local on the target, or local on the base for the \(\tau\)-topology if for any \(\tau\)-covering \(\{Y_i \to Y\}_{i \in I}\) (see Topologies, Section 020M) and any morphism of schemes \(f : X \to Y\) over \(S\) we have \[f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_i \times_Y X \to Y_i\text{ has }\mathcal{P}.\]

To be sure, since isomorphisms are always coverings we see (or require) that property \(\mathcal{P}\) holds for \(X \to Y\) if and only if it holds for any arrow \(X' \to Y'\) isomorphic to \(X \to Y\). If a property is \(\tau\)-local on the target then it is preserved by base changes by morphisms which occur in \(\tau\)-coverings. Here is a formal statement.

Lemma

Let \(\tau \in \{fpqc, fppf, syntomic, smooth, \etale, Zariski\}\). Let \(\mathcal{P}\) be a property of morphisms which is \(\tau\) local on the target. Let \(f : X \to Y\) have property \(\mathcal{P}\). For any morphism \(Y' \to Y\) which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, resp. an open immersion, the base change \(f' : Y' \times_Y X \to Y'\) of \(f\) has property \(\mathcal{P}\).

Proof

This is true because we can fit \(Y' \to Y\) into a family of morphisms which forms a \(\tau\)-covering.

A simple often used consequence of the above is that if \(f : X \to Y\) has property \(\mathcal{P}\) which is \(\tau\)-local on the target and \(f(X) \subset V\) for some open subscheme \(V \subset Y\), then also the induced morphism \(X \to V\) has \(\mathcal{P}\). Proof: The base change \(f\) by \(V \to Y\) gives \(X \to V\).

Lemma

Let \(\tau \in \{fppf, syntomic, smooth, \etale\}\). Let \(\mathcal{P}\) be a property of morphisms which is \(\tau\) local on the target. For any morphism of schemes \(f : X \to Y\) there exists a largest open \(W(f) \subset Y\) such that the restriction \(X_{W(f)} \to W(f)\) has \(\mathcal{P}\). Moreover,

1. if \(g : Y' \to Y\) is flat and locally of finite presentation, syntomic, smooth, or étale and the base change \(f' : X_{Y'} \to Y'\) has \(\mathcal{P}\), then \(g(Y') \subset W(f)\),

2. if \(g : Y' \to Y\) is flat and locally of finite presentation, syntomic, smooth, or étale, then \(W(f') = g^{-1}(W(f))\), and

3. if \(\{g_i : Y_i \to Y\}\) is a \(\tau\)-covering, then \(g_i^{-1}(W(f)) = W(f_i)\), where \(f_i\) is the base change of \(f\) by \(Y_i \to Y\).

Proof

Consider the union \(W\) of the images \(g(Y') \subset Y\) of morphisms \(g : Y' \to Y\) with the properties:

1. \(g\) is flat and locally of finite presentation, syntomic, smooth, or étale, and

2. the base change \(Y' \times_{g, Y} X \to Y'\) has property \(\mathcal{P}\).

Since such a morphism \(g\) is open (see Morphisms, Lemma 01UA) we see that \(W \subset Y\) is an open subset of \(Y\). Since \(\mathcal{P}\) is local in the \(\tau\) topology the restriction \(X_W \to W\) has property \(\mathcal{P}\) because we are given a covering \(\{Y' \to W\}\) of \(W\) such that the pullbacks have \(\mathcal{P}\). This proves the existence and proves that \(W(f)\) has property (1). To see property (2) note that \(W(f') \supset g^{-1}(W(f))\) because \(\mathcal{P}\) is stable under base change by flat and locally of finite presentation, syntomic, smooth, or étale morphisms, see Lemma 04QU. On the other hand, if \(Y'' \subset Y'\) is an open such that \(X_{Y''} \to Y''\) has property \(\mathcal{P}\), then \(Y'' \to Y\) factors through \(W\) by construction, i.e., \(Y'' \subset g^{-1}(W(f))\). This proves (2). Assertion (3) follows from (2) because each morphism \(Y_i \to Y\) is flat and locally of finite presentation, syntomic, smooth, or étale by our definition of a \(\tau\)-covering.

Lemma

Let \(\mathcal{P}\) be a property of morphisms of schemes over a base. Let \(\tau \in \{fpqc, fppf, \etale, smooth, syntomic\}\). Assume that

1. the property is preserved under flat, flat and locally of finite presentation, étale, smooth, or syntomic base change depending on whether \(\tau\) is fpqc, fppf, étale, smooth, or syntomic (compare with Schemes, Definition 01JZ),

2. the property is Zariski local on the base.

3. for any surjective morphism of affine schemes \(S' \to S\) which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether \(\tau\) is fpqc, fppf, étale, smooth, or syntomic, and any morphism of schemes \(f : X \to S\) property \(\mathcal{P}\) holds for \(f\) if property \(\mathcal{P}\) holds for the base change \(f' : X' = S' \times_S X \to S'\).

Then \(\mathcal{P}\) is \(\tau\) local on the base.

Proof

This follows almost immediately from the definition of a \(\tau\)-covering, see Topologies, Definition 022B 021M 0215 021Z, or 0225 and Topologies, Lemma 022E, 021P, 0218, 0222, or 0228. Details omitted.

Remark

(This is a repeat of Remark 034A above.) In Lemma 02KP above if \(\tau = smooth\) then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when \(\tau = syntomic\) or \(\tau = \etale\).

Properties of morphisms local in the fpqc topology on the target

In this section we find a large number of properties of morphisms of schemes which are local on the base in the fpqc topology. By contrast, in Examples, Section 08J0 we will show that the properties “projective” and “quasi-projective” are not local on the base even in the Zariski topology.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is quasi-compact” is fpqc local on the base.

Proof

A base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 01K5. Being quasi-compact is Zariski local on the base, see Schemes, Lemma 01K4. Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is quasi-compact. Then \(X'\) is quasi-compact, and \(X' \to X\) is surjective. Hence \(X\) is quasi-compact. This implies that \(f\) is quasi-compact. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is quasi-separated” is fpqc local on the base.

Proof

Any base change of a quasi-separated morphism is quasi-separated, see Schemes, Lemma 01KU. Being quasi-separated is Zariski local on the base (from the definition or by Schemes, Lemma 01KO). Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is quasi-separated. This means that \(\Delta' : X' \to X'\times_{S'} X'\) is quasi-compact. Note that \(\Delta'\) is the base change of \(\Delta : X \to X \times_S X\) via \(S' \to S\). By Lemma 02KQ this implies \(\Delta\) is quasi-compact, and hence \(f\) is quasi-separated. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is universally closed” is fpqc local on the base.

Proof

A base change of a universally closed morphism is universally closed by definition. Being universally closed is Zariski local on the base (from the definition or by Morphisms, Lemma 02K7). Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is universally closed. Let \(T \to S\) be any morphism. Consider the diagram \[\xymatrix{ X' \ar[d] & S' \times_S T \times_S X \ar[d] \ar[r] \ar[l] & T \times_S X \ar[d] \\ S' & S' \times_S T \ar[r] \ar[l] & T }\] in which both squares are cartesian. Thus the assumption implies that the middle vertical arrow is closed. The right horizontal arrows are flat, quasi-compact and surjective (as base changes of \(S' \to S\)). Hence a subset of \(T\) is closed if and only if its inverse image in \(S' \times_S T\) is closed, see Morphisms, Lemma 02JY. An easy diagram chase shows that the right vertical arrow is closed too, and we conclude \(X \to S\) is universally closed. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is universally open” is fpqc local on the base.

Proof

The proof is the same as the proof of Lemma 02KS.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is universally submersive” is fpqc local on the base.

Proof

The proof is the same as the proof of Lemma 02KS using that a quasi-compact flat surjective morphism is universally submersive by Morphisms, Lemma 02JY.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is separated” is fpqc local on the base.

Proof

A base change of a separated morphism is separated, see Schemes, Lemma 01KU. Being separated is Zariski local on the base (from the definition or by Schemes, Lemma 01KP). Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is separated. This means that \(\Delta' : X' \to X'\times_{S'} X'\) is a closed immersion, hence universally closed. Note that \(\Delta'\) is the base change of \(\Delta : X \to X \times_S X\) via \(S' \to S\). By Lemma 02KS this implies \(\Delta\) is universally closed. Since it is an immersion (Schemes, Lemma 01KJ) we conclude \(\Delta\) is a closed immersion. Hence \(f\) is separated. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is surjective” is fpqc local on the base.

Proof

This is clear.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is quasi-compact and dominant” is fpqc local on the base.

Proof

By Morphisms, Lemma 0H3F, quasi-compact dominant morphisms are preserved by flat pullback. The other direction is easier. Indeed, quasi-compactness is fpqc local on the base by Lemma 02KQ. Being dominant is clearly Zariski local on the base. Finally, let \(S' \to S\) be a surjective morphism of schemes (not necessarily flat), and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is dominant. Then \(X' \to S'\to S\) is a composite of two dominant morphisms, hence dominant. Since this is also the composite \(X' \to X \to S\), it follows that \(X\to S\) is dominant.

Dominant morphisms are not preserved by flat pullback in full generality, but the following special case is worth mentioning. See also Lemma 0H8J.

Lemma

Let \(E/k\) be a field extension. Then a morphism \(X \to Y\) over \(k\) is dominant if and only if the pullback \(X_E \to Y_E\) is dominant.

Proof

By Morphisms, Lemma 0383, the morphism \(\Spec(E) \to \Spec(k)\) is universally open. So \(Y_E \to Y\) is open. Therefore, if \(X \to Y\) is dominant, Morphisms, Lemma 0H8F gives that \(X_E \to Y_E\) is dominant. Conversely, suppose that \(X_E \to Y_E\) is dominant. The morphism \(Y_E \to Y\) is surjective by Morphisms, Lemma 01S1, and so the composition \(X_E \to Y_E \to Y\) is dominant. This is also the composition \(X_E \to X \to Y\), and so \(X \to Y\) is dominant.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is universally injective” is fpqc local on the base.

Proof

A base change of a universally injective morphism is universally injective (this is formal). Being universally injective is Zariski local on the base; this is clear from the definition. Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is universally injective. Let \(K\) be a field, and let \(a, b : \Spec(K) \to X\) be two morphisms such that \(f \circ a = f \circ b\). As \(S' \to S\) is surjective and by the discussion in Schemes, Section 01J5 there exists a field extension \(K'/K\) and a morphism \(\Spec(K') \to S'\) such that the following solid diagram commutes \[\xymatrix{ \Spec(K') \ar[rrd] \ar@{-->}[rd]_{a', b'} \ar[dd] \\ & X' \ar[r] \ar[d] & S' \ar[d] \\ \Spec(K) \ar[r]^{a, b} & X \ar[r] & S }\] As the square is cartesian we get the two dotted arrows \(a'\), \(b'\) making the diagram commute. Since \(X' \to S'\) is universally injective we get \(a' = b'\), by Morphisms, Lemma 01S4. Clearly this forces \(a = b\) (by the discussion in Schemes, Section 01J5). Therefore Lemma 02KP applies and we win.

An alternative proof would be to use the characterization of a universally injective morphism as one whose diagonal is surjective, see Morphisms, Lemma 01S4. The lemma then follows from the fact that the property of being surjective is fpqc local on the base, see Lemma 02KV. (Hint: use that the base change of the diagonal is the diagonal of the base change.)

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a universal homeomorphism” is fpqc local on the base.

Proof

This can be proved in exactly the same manner as Lemma 02KS. Alternatively, one can use that a map of topological spaces is a homeomorphism if and only if it is injective, surjective, and open. Thus a universal homeomorphism is the same thing as a surjective, universally injective, and universally open morphism. Thus the lemma follows from Lemmas 02KV, 02KW, and 02KT.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is locally of finite type” is fpqc local on the base.

Proof

Being locally of finite type is preserved under base change, see Morphisms, Lemma 01T4. Being locally of finite type is Zariski local on the base, see Morphisms, Lemma 01T2. Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is locally of finite type. Let \(U \subset X\) be an affine open. Then \(U' = S' \times_S U\) is affine and of finite type over \(S'\). Write \(S = \Spec(R)\), \(S' = \Spec(R')\), \(U = \Spec(A)\), and \(U' = \Spec(A')\). We know that \(R \to R'\) is faithfully flat, \(A' = R' \otimes_R A\) and \(R' \to A'\) is of finite type. We have to show that \(R \to A\) is of finite type. This is the result of Algebra, Lemma 00QP. It follows that \(f\) is locally of finite type. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is locally of finite presentation” is fpqc local on the base.

Proof

Being locally of finite presentation is preserved under base change, see Morphisms, Lemma 01TS. Being locally of finite type is Zariski local on the base, see Morphisms, Lemma 01TQ. Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is locally of finite presentation. Let \(U \subset X\) be an affine open. Then \(U' = S' \times_S U\) is affine and of finite type over \(S'\). Write \(S = \Spec(R)\), \(S' = \Spec(R')\), \(U = \Spec(A)\), and \(U' = \Spec(A')\). We know that \(R \to R'\) is faithfully flat, \(A' = R' \otimes_R A\) and \(R' \to A'\) is of finite presentation. We have to show that \(R \to A\) is of finite presentation. This is the result of Algebra, Lemma 00QQ. It follows that \(f\) is locally of finite presentation. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is of finite type” is fpqc local on the base.

Proof

Combine Lemmas 02KQ and 02KX.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is of finite presentation” is fpqc local on the base.

Proof

Combine Lemmas 02KQ, 02KR and 02KY.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is proper” is fpqc local on the base.

Proof

The lemma follows by combining Lemmas 02KS, 02KU and 02KZ.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is flat” is fpqc local on the base.

Proof

Being flat is preserved under arbitrary base change, see Morphisms, Lemma 01U9. Being flat is Zariski local on the base by definition. Finally, let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is flat. Let \(U \subset X\) be an affine open. Then \(U' = S' \times_S U\) is affine. Write \(S = \Spec(R)\), \(S' = \Spec(R')\), \(U = \Spec(A)\), and \(U' = \Spec(A')\). We know that \(R \to R'\) is faithfully flat, \(A' = R' \otimes_R A\) and \(R' \to A'\) is flat. Goal: Show that \(R \to A\) is flat. This follows immediately from Algebra, Lemma 00HJ. Hence \(f\) is flat. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is an open immersion” is fpqc local on the base.

Proof

The property of being an open immersion is stable under base change, see Schemes, Lemma 01JY. The property of being an open immersion is Zariski local on the base (this is obvious).

Let \(S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is an open immersion. We claim that \(f\) is an open immersion. Then \(f'\) is universally open, and universally injective. Hence we conclude that \(f\) is universally open by Lemma 02KT, and universally injective by Lemma 02KW. In particular \(f(X) \subset S\) is open. If for every affine open \(U \subset f(X)\) we can prove that \(f^{-1}(U) \to U\) is an isomorphism, then \(f\) is an open immersion and we’re done. If \(U' \subset S'\) denotes the inverse image of \(U\), then \(U' \to U\) is a faithfully flat morphism of affines and \((f')^{-1}(U') \to U'\) is an isomorphism (as \(f'(X')\) contains \(U'\) by our choice of \(U\)). Thus we reduce to the case discussed in the next paragraph.

Let \(S' \to S\) be a flat surjective morphism of affine schemes, let \(f : X \to S\) be a morphism, and assume that the base change \(f' : X' \to S'\) is an isomorphism. We have to show that \(f\) is an isomorphism also. It is clear that \(f\) is surjective, universally injective, and universally open (see arguments above for the last two). Hence \(f\) is bijective, i.e., \(f\) is a homeomorphism. Thus \(f\) is affine by Morphisms, Lemma 04DE. Since \[\mathcal{O}(S') \to \mathcal{O}(X') = \mathcal{O}(S') \otimes_{\mathcal{O}(S)} \mathcal{O}(X)\] is an isomorphism and since \(\mathcal{O}(S) \to \mathcal{O}(S')\) is faithfully flat this implies that \(\mathcal{O}(S) \to \mathcal{O}(X)\) is an isomorphism. Thus \(f\) is an isomorphism. This finishes the proof of the claim above. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is an isomorphism” is fpqc local on the base.

Proof

Combine Lemmas 02KV and 02L3.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is affine” is fpqc local on the base.

Proof

A base change of an affine morphism is affine, see Morphisms, Lemma 01SD. Being affine is Zariski local on the base, see Morphisms, Lemma 01S8. Finally, let \(g : S' \to S\) be a flat surjective morphism of affine schemes, and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is affine. In other words, \(X'\) is affine, say \(X' = \Spec(A')\). Also write \(S = \Spec(R)\) and \(S' = \Spec(R')\). We have to show that \(X\) is affine.

By Lemmas 02KQ and 02KU we see that \(X \to S\) is separated and quasi-compact. Thus \(f_*\mathcal{O}_X\) is a quasi-coherent sheaf of \(\mathcal{O}_S\)-algebras, see Schemes, Lemma 01LC. Hence \(f_*\mathcal{O}_X = \widetilde{A}\) for some \(R\)-algebra \(A\). In fact \(A = \Gamma(X, \mathcal{O}_X)\) of course. Also, by flat base change (see for example Cohomology of Schemes, Lemma 02KH) we have \(g^*f_*\mathcal{O}_X = f'_*\mathcal{O}_{X'}\). In other words, we have \(A' = R' \otimes_R A\). Consider the canonical morphism \[X \longrightarrow \Spec(A)\] over \(S\) from Schemes, Lemma 01I1. By the above the base change of this morphism to \(S'\) is an isomorphism. Hence it is an isomorphism by Lemma 02L4. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a closed immersion” is fpqc local on the base.

Proof

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{Y_i \to Y\}\) be an fpqc covering. Assume that each \(f_i : Y_i \times_Y X \to Y_i\) is a closed immersion. This implies that each \(f_i\) is affine, see Morphisms, Lemma 01SE. By Lemma 02L5 we conclude that \(f\) is affine. It remains to show that \(\mathcal{O}_Y \to f_*\mathcal{O}_X\) is surjective. For every \(y \in Y\) there exists an \(i\) and a point \(y_i \in Y_i\) mapping to \(y\). By Cohomology of Schemes, Lemma 02KH the sheaf \(f_{i, *}(\mathcal{O}_{Y_i \times_Y X})\) is the pullback of \(f_*\mathcal{O}_X\). By assumption it is a quotient of \(\mathcal{O}_{Y_i}\). Hence we see that \[\Big( \mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y \Big) \otimes_{\mathcal{O}_{Y, y}} \mathcal{O}_{Y_i, y_i}\] is surjective. Since \(\mathcal{O}_{Y_i, y_i}\) is faithfully flat over \(\mathcal{O}_{Y, y}\) this implies the surjectivity of \(\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_X)_y\) as desired.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is quasi-affine” is fpqc local on the base.

Proof

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{g_i : Y_i \to Y\}\) be an fpqc covering. Assume that each \(f_i : Y_i \times_Y X \to Y_i\) is quasi-affine. This implies that each \(f_i\) is quasi-compact and separated. By Lemmas 02KQ and 02KU this implies that \(f\) is quasi-compact and separated. Consider the sheaf of \(\mathcal{O}_Y\)-algebras \(\mathcal{A} = f_*\mathcal{O}_X\). By Schemes, Lemma 01LC it is a quasi-coherent \(\mathcal{O}_Y\)-algebra. Consider the canonical morphism \[j : X \longrightarrow \underline{\Spec}_Y(\mathcal{A})\] see Constructions, Lemma 01LY. By flat base change (see for example Cohomology of Schemes, Lemma 02KH) we have \(g_i^*f_*\mathcal{O}_X = f_{i, *}\mathcal{O}_{X'}\) where \(g_i : Y_i \to Y\) are the given flat maps. Hence the base change \(j_i\) of \(j\) by \(g_i\) is the canonical morphism of Constructions, Lemma 01LY for the morphism \(f_i\). By assumption and Morphisms, Lemma 01SM all of these morphisms \(j_i\) are quasi-compact open immersions. Hence, by Lemmas 02KQ and 02L3 we see that \(j\) is a quasi-compact open immersion. Hence by Morphisms, Lemma 01SM again we conclude that \(f\) is quasi-affine.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a quasi-compact immersion” is fpqc local on the base.

Proof

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{Y_i \to Y\}\) be an fpqc covering. Write \(X_i = Y_i \times_Y X\) and \(f_i : X_i \to Y_i\) the base change of \(f\). Also denote \(q_i : Y_i \to Y\) the given flat morphisms. Assume each \(f_i\) is a quasi-compact immersion. By Schemes, Lemma 01L7 each \(f_i\) is separated. By Lemmas 02KQ and 02KU this implies that \(f\) is quasi-compact and separated. Let \(X \to Z \to Y\) be the factorization of \(f\) through its scheme theoretic image. By Morphisms, Lemma 01R8 the closed subscheme \(Z \subset Y\) is cut out by the quasi-coherent sheaf of ideals \(\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)\) as \(f\) is quasi-compact. By flat base change (see for example Cohomology of Schemes, Lemma 02KH; here we use \(f\) is separated) we see \(f_{i, *}\mathcal{O}_{X_i}\) is the pullback \(q_i^*f_*\mathcal{O}_X\). Hence \(Y_i \times_Y Z\) is cut out by the quasi-coherent sheaf of ideals \(q_i^*\mathcal{I} = \Ker(\mathcal{O}_{Y_i} \to f_{i, *}\mathcal{O}_{X_i})\). By Morphisms, Lemma 01RG the morphisms \(X_i \to Y_i \times_Y Z\) are open immersions. Hence by Lemma 02L3 we see that \(X \to Z\) is an open immersion and hence \(f\) is an immersion as desired (we already saw it was quasi-compact).

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is integral” is fpqc local on the base.

Proof

An integral morphism is the same thing as an affine, universally closed morphism. See Morphisms, Lemma 01WM. Hence the lemma follows on combining Lemmas 02KS and 02L5.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is finite” is fpqc local on the base.

Proof

An finite morphism is the same thing as an integral morphism which is locally of finite type. See Morphisms, Lemma 01WJ. Hence the lemma follows on combining Lemmas 02KX and 02L9.

Lemma

The properties \(\mathcal{P}(f) =\)“\(f\) is locally quasi-finite” and \(\mathcal{P}(f) =\)“\(f\) is quasi-finite” are fpqc local on the base.

Proof

Let \(f : X \to S\) be a morphism of schemes, and let \(\{S_i \to S\}\) be an fpqc covering such that each base change \(f_i : X_i \to S_i\) is locally quasi-finite. We have already seen (Lemma 02KX) that “locally of finite type” is fpqc local on the base, and hence we see that \(f\) is locally of finite type. Then it follows from Morphisms, Lemma 01TM that \(f\) is locally quasi-finite. The quasi-finite case follows as we have already seen that “quasi-compact” is fpqc local on the base (Lemma 02KQ).

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is locally of finite type of relative dimension \(d\)” is fpqc local on the base.

Proof

This follows immediately from the fact that being locally of finite type is fpqc local on the base and Morphisms, Lemma 02FY.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is syntomic” is fpqc local on the base.

Proof

A morphism is syntomic if and only if it is locally of finite presentation, flat, and has locally complete intersections as fibres. We have seen already that being flat and locally of finite presentation are fpqc local on the base (Lemmas 02L2, and 02KY). Hence the result follows for syntomic from Morphisms, Lemma 02V3.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is smooth” is fpqc local on the base.

Proof

A morphism is smooth if and only if it is locally of finite presentation, flat, and has smooth fibres. We have seen already that being flat and locally of finite presentation are fpqc local on the base (Lemmas 02L2, and 02KY). Hence the result follows for smooth from Morphisms, Lemma 02V4.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is unramified” is fpqc local on the base. The property \(\mathcal{P}(f) =\)“\(f\) is G-unramified” is fpqc local on the base.

Proof

A morphism is unramified (resp. G-unramified) if and only if it is locally of finite type (resp. finite presentation) and its diagonal morphism is an open immersion (see Morphisms, Lemma 02GE). We have seen already that being locally of finite type (resp. locally of finite presentation) and an open immersion is fpqc local on the base (Lemmas 02KY, 02KX, and 02L3). Hence the result follows formally.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is étale” is fpqc local on the base.

Proof

A morphism is étale if and only if it flat and G-unramified. See Morphisms, Lemma 02GV. We have seen already that being flat and G-unramified are fpqc local on the base (Lemmas 02L2, and 02VM). Hence the result follows.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is finite locally free” is fpqc local on the base. Let \(d \geq 0\). The property \(\mathcal{P}(f) =\)“\(f\) is finite locally free of degree \(d\)” is fpqc local on the base.

Proof

Being finite locally free is equivalent to being finite, flat and locally of finite presentation (Morphisms, Lemma 02KB). Hence this follows from Lemmas 02LA, 02L2, and 02KY. If \(f : Z \to U\) is finite locally free, and \(\{U_i \to U\}\) is a surjective family of morphisms such that each pullback \(Z \times_U U_i \to U_i\) has degree \(d\), then \(Z \to U\) has degree \(d\), for example because we can read off the degree in a point \(u \in U\) from the fibre \((f_*\mathcal{O}_Z)_u \otimes_{\mathcal{O}_{U, u}} \kappa(u)\).

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is a monomorphism” is fpqc local on the base.

Proof

Let \(f : X \to S\) be a morphism of schemes. Let \(\{S_i \to S\}\) be an fpqc covering, and assume each of the base changes \(f_i : X_i \to S_i\) of \(f\) is a monomorphism. Let \(a, b : T \to X\) be two morphisms such that \(f \circ a = f \circ b\). We have to show that \(a = b\). Since \(f_i\) is a monomorphism we see that \(a_i = b_i\), where \(a_i, b_i : S_i \times_S T \to X_i\) are the base changes. In particular the compositions \(S_i \times_S T \to T \to X\) are equal. Since \(\coprod S_i \times_S T \to T\) is an epimorphism (see e.g. Lemma 023Q) we conclude \(a = b\).

Lemma

The properties

1. \(\mathcal{P}(f) =\)“\(f\) is a Koszul-regular immersion”,

2. \(\mathcal{P}(f) =\)“\(f\) is an \(H_1\)-regular immersion”, and

3. \(\mathcal{P}(f) =\)“\(f\) is a quasi-regular immersion”

are fpqc local on the base.

Proof

We will use the criterion of Lemma 02KP to prove this. By Divisors, Definition 063J being a Koszul-regular (resp. \(H_1\)-regular, quasi-regular) immersion is Zariski local on the base. By Divisors, Lemma 067P being a Koszul-regular (resp. \(H_1\)-regular, quasi-regular) immersion is preserved under flat base change. The final hypothesis (3) of Lemma 02KP translates into the following algebra statement: Let \(A \to B\) be a faithfully flat ring map. Let \(I \subset A\) be an ideal. If \(IB\) is locally on \(\Spec(B)\) generated by a Koszul-regular (resp. \(H_1\)-regular, quasi-regular) sequence in \(B\), then \(I \subset A\) is locally on \(\Spec(A)\) generated by a Koszul-regular (resp. \(H_1\)-regular, quasi-regular) sequence in \(A\). This is More on Algebra, Lemma 068N.

Properties of morphisms local in the fppf topology on the target

In this section we find some properties of morphisms of schemes for which we could not (yet) show they are local on the base in the fpqc topology which, however, are local on the base in the fppf topology.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is an immersion” is fppf local on the base.

Proof

The property of being an immersion is stable under base change, see Schemes, Lemma 01JY. The property of being an immersion is Zariski local on the base. Finally, let \(\pi : S' \to S\) be a surjective morphism of affine schemes, which is flat and locally of finite presentation. Note that \(\pi : S' \to S\) is open by Morphisms, Lemma 01UA. Let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is an immersion. In particular we see that \(f'(X') = \pi^{-1}(f(X))\) is locally closed. Hence by Topology, Lemma 02YB we see that \(f(X) \subset S\) is locally closed. Let \(Z \subset S\) be the closed subset \(Z = \overline{f(X)} \setminus f(X)\). By Topology, Lemma 02YB again we see that \(f'(X')\) is closed in \(S' \setminus Z'\). Hence we may apply Lemma 02L6 to the fpqc covering \(\{S' \setminus Z' \to S \setminus Z\}\) and conclude that \(f : X \to S \setminus Z\) is a closed immersion. In other words, \(f\) is an immersion. Therefore Lemma 02KP applies and we win.

Lemma

The property \(\mathcal{P}(f) =\)“\(f\) is dominant” is fppf local on the base.

Proof

By Morphisms, Lemma 0H8F, dominant morphisms are preserved under pullback by open morphisms, hence by flat morphisms locally of finite presentation (Morphisms, Lemma 01UA). The other direction is easier. Indeed, being dominant is clearly Zariski local on the base. Next, let \(S' \to S\) be a surjective morphism of schemes (not necessarily flat or locally of finite presentation), and let \(f : X \to S\) be a morphism. Assume that the base change \(f' : X' \to S'\) is dominant. Then \(X' \to S'\to S\) is a composition of two dominant morphisms, hence dominant. Since this is also the composition \(X' \to X \to S\), it follows that \(X\to S\) is dominant.

Application of fpqc descent of properties of morphisms

The following lemma may seem a bit frivolous but turns out is a useful tool in studying étale and unramified morphisms.

Lemma

Let \(f : X \to Y\) be a flat, quasi-compact, surjective monomorphism. Then f is an isomorphism.

Proof

As \(f\) is a flat, quasi-compact, surjective morphism we see \(\{X \to Y\}\) is an fpqc covering of \(Y\). The diagonal \(\Delta : X \to X \times_Y X\) is an isomorphism (Schemes, Lemma 01L3). This implies that the base change of \(f\) by \(f\) is an isomorphism. Hence we see \(f\) is an isomorphism by Lemma 02L4.

We can use this lemma to show the following important result; we also give a proof avoiding fpqc descent. We will discuss this and related results in more detail in Étale Morphisms, Section 025F.

Lemma

A universally injective étale morphism is an open immersion.

Proof

Let \(f : X \to Y\) be an étale morphism which is universally injective. Then \(f\) is open (Morphisms, Lemma 03WT) hence we can replace \(Y\) by \(f(X)\) and we may assume that \(f\) is surjective. Then \(f\) is bijective and open hence a homeomorphism. Hence \(f\) is quasi-compact. Thus by Lemma 06NC it suffices to show that \(f\) is a monomorphism. As \(X \to Y\) is étale the morphism \(\Delta_{X/Y} : X \to X \times_Y X\) is an open immersion by Morphisms, Lemma 02GE (and Morphisms, Lemma 02GV). As \(f\) is universally injective \(\Delta_{X/Y}\) is also surjective, see Morphisms, Lemma 01S4. Hence \(\Delta_{X/Y}\) is an isomorphism, i.e., \(X \to Y\) is a monomorphism.

Proof

Let \(f : X \to Y\) be an étale morphism which is universally injective. Then \(f\) is open (Morphisms, Lemma 03WT) hence we can replace \(Y\) by \(f(X)\) and we may assume that \(f\) is surjective. Since the hypotheses remain satisfied after any base change, we conclude that \(f\) is a universal homeomorphism. Therefore \(f\) is integral, see Morphisms, Lemma 04DF. It follows that \(f\) is finite by Morphisms, Lemma 01WJ. It follows that \(f\) is finite locally free by Morphisms, Lemma 02KB. To finish the proof, it suffices that \(f\) is finite locally free of degree \(1\) (a finite locally free morphism of degree \(1\) is an isomorphism). There is decomposition of \(Y\) into open and closed subschemes \(V_d\) such that \(f^{-1}(V_d) \to V_d\) is finite locally free of degree \(d\), see Morphisms, Lemma 04MH. If \(V_d\) is not empty, we can pick a morphism \(\Spec(k) \to V_d \subset Y\) where \(k\) is an algebraically closed field (just take the algebraic closure of the residue field of some point of \(V_d\)). Then \(\Spec(k) \times_Y X \to \Spec(k)\) is a disjoint union of copies of \(\Spec(k)\), by Morphisms, Lemma 02GL and the fact that \(k\) is algebraically closed. However, since \(f\) is universally injective, there can only be one copy and hence \(d = 1\) as desired.

We can reformulate the hypotheses in the lemma above a bit by using the following characterization of flat universally injective morphisms.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(X^0\) denote the set of generic points of irreducible components of \(X\). If

1. \(f\) is flat and separated,

2. for \(\xi \in X^0\) we have \(\kappa(f(\xi)) = \kappa(\xi)\), and

3. if \(\xi, \xi' \in X^0\), \(\xi \not = \xi'\), then \(f(\xi) \not = f(\xi')\),

then \(f\) is universally injective.

Proof

We have to show that \(\Delta : X \to X \times_Y X\) is surjective, see Morphisms, Lemma 01S4. As \(X \to Y\) is separated, the image of \(\Delta\) is closed. Thus if \(\Delta\) is not surjective, we can find a generic point \(\eta \in X \times_S X\) of an irreducible component of \(X \times_S X\) which is not in the image of \(\Delta\). The projection \(\text{pr}_1 : X \times_Y X \to X\) is flat as a base change of the flat morphism \(X \to Y\), see Morphisms, Lemma 01U9. Hence generalizations lift along \(\text{pr}_1\), see Morphisms, Lemma 03HV. We conclude that \(\xi = \text{pr}_1(\eta) \in X^0\). However, assumptions (2) and (3) guarantee that the scheme \((X \times_Y X)_{f(\xi)}\) has at most one point for every \(\xi \in X^0\). In other words, we have \(\Delta(\xi) = \eta\) a contradiction.

Thus we can reformulate Lemma 02LC as follows.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(X^0\) denote the set of generic points of irreducible components of \(X\). If

1. \(f\) is étale and separated,

2. for \(\xi \in X^0\) we have \(\kappa(f(\xi)) = \kappa(\xi)\), and

3. if \(\xi, \xi' \in X^0\), \(\xi \not = \xi'\), then \(f(\xi) \not = f(\xi')\),

then \(f\) is an open immersion.

Proof

Immediate from Lemmas 09NP and 02LC.

Lemma

Let \(f : X \to Y\) be a morphism of schemes which is locally of finite type. Let \(Z\) be a closed subset of \(X\). If there exists an fpqc covering \(\{Y_i \to Y\}\) such that the inverse image \(Z_i \subset Y_i \times_Y X\) is proper over \(Y_i\) (Cohomology of Schemes, Definition 0CYM) then \(Z\) is proper over \(Y\).

Proof

Endow \(Z\) with the reduced induced closed subscheme structure, see Schemes, Definition 01J4. For every \(i\) the base change \(Y_i \times_Y Z\) is a closed subscheme of \(Y_i \times_Y X\) whose underlying closed subset is \(Z_i\). By definition (via Cohomology of Schemes, Lemma 0CYL) we conclude that the projections \(Y_i \times_Y Z \to Y_i\) are proper morphisms. Hence \(Z \to Y\) is a proper morphism by Lemma 02L1. Thus \(Z\) is proper over \(Y\) by definition.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(\{g_i : S_i \to S\}_{i \in I}\) be an fpqc covering. Let \(f_i : X_i \to S_i\) be the base change of \(f\) and let \(\mathcal{L}_i\) be the pullback of \(\mathcal{L}\) to \(X_i\). The following are equivalent

1. \(\mathcal{L}\) is ample on \(X/S\), and

2. \(\mathcal{L}_i\) is ample on \(X_i/S_i\) for every \(i \in I\).

Proof

The implication (1) \(\Rightarrow\) (2) follows from Morphisms, Lemma 0893. Assume \(\mathcal{L}_i\) is ample on \(X_i/S_i\) for every \(i \in I\). By Morphisms, Definition 01VH this implies that \(X_i \to S_i\) is quasi-compact and by Morphisms, Lemma 01VI this implies \(X_i \to S\) is separated. Hence \(f\) is quasi-compact and separated by Lemmas 02KQ and 02KU.

This means that \(\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}\) is a quasi-coherent graded \(\mathcal{O}_S\)-algebra (Schemes, Lemma 01LC). Moreover, the formation of \(\mathcal{A}\) commutes with flat base change by Cohomology of Schemes, Lemma 02KH. In particular, if we set \(\mathcal{A}_i = \bigoplus_{d \geq 0} f_{i, *}\mathcal{L}_i^{\otimes d}\) then we have \(\mathcal{A}_i = g_i^*\mathcal{A}\). It follows that the natural maps \(\psi_d : f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}\) of \(\mathcal{O}_X\) pullback to give the natural maps \(\psi_{i, d} : f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}\) of \(\mathcal{O}_{X_i}\)-modules. Since \(\mathcal{L}_i\) is ample on \(X_i/S_i\) we see that for any point \(x_i \in X_i\), there exists a \(d \geq 1\) such that \(f_i^*(\mathcal{A}_i)_d \to \mathcal{L}_i^{\otimes d}\) is surjective on stalks at \(x_i\). This follows either directly from the definition of a relatively ample module or from Morphisms, Lemma 01VJ. If \(x \in X\), then we can choose an \(i\) and an \(x_i \in X_i\) mapping to \(x\). Since \(\mathcal{O}_{X, x} \to \mathcal{O}_{X_i, x_i}\) is flat hence faithfully flat, we conclude that for every \(x \in X\) there exists a \(d \geq 1\) such that \(f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d}\) is surjective on stalks at \(x\). This implies that the open subset \(U(\psi) \subset X\) of Constructions, Lemma 01O9 corresponding to the map \(\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}\) of graded \(\mathcal{O}_X\)-algebras is equal to \(X\). Consider the corresponding morphism \[r_{\mathcal{L}, \psi} : X \longrightarrow \underline{\text{Proj}}_S(\mathcal{A})\] It is clear from the above that the base change of \(r_{\mathcal{L}, \psi}\) to \(S_i\) is the morphism \(r_{\mathcal{L}_i, \psi_i}\) which is an open immersion by Morphisms, Lemma 01VJ. Hence \(r_{\mathcal{L}, \psi}\) is an open immersion by Lemma 02L3 and we conclude \(\mathcal{L}\) is ample on \(X/S\) by Morphisms, Lemma 01VJ.

Properties of morphisms local on the source

It often happens one can prove a morphism has a certain property after precomposing with some other morphism. In many cases this implies the morphism has the property too. We formalize this in the following definition.

Definition

Let \(\mathcal{P}\) be a property of morphisms of schemes. Let \(\tau \in \{Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}\). We say \(\mathcal{P}\) is \(\tau\) local on the source, or local on the source for the \(\tau\)-topology if for any morphism of schemes \(f : X \to Y\) over \(S\), and any \(\tau\)-covering \(\{X_i \to X\}_{i \in I}\) we have \[f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_i \to Y\text{ has }\mathcal{P}.\]

To be sure, since isomorphisms are always coverings we see (or require) that property \(\mathcal{P}\) holds for \(X \to Y\) if and only if it holds for any arrow \(X' \to Y'\) isomorphic to \(X \to Y\). If a property is \(\tau\)-local on the source then it is preserved by precomposing with morphisms which occur in \(\tau\)-coverings. Here is a formal statement.

Lemma

Let \(\tau \in \{fpqc, fppf, syntomic, smooth, \etale, Zariski\}\). Let \(\mathcal{P}\) be a property of morphisms which is \(\tau\) local on the source. Let \(f : X \to Y\) have property \(\mathcal{P}\). For any morphism \(a : X' \to X\) which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. smooth, resp. étale, resp. an open immersion, the composition \(f \circ a : X' \to Y\) has property \(\mathcal{P}\).

Proof

This is true because we can fit \(X' \to X\) into a family of morphisms which forms a \(\tau\)-covering.

Lemma

Let \(\tau \in \{fppf, syntomic, smooth, \etale\}\). Let \(\mathcal{P}\) be a property of morphisms which is \(\tau\) local on the source. For any morphism of schemes \(f : X \to Y\) there exists a largest open \(W(f) \subset X\) such that the restriction \(f|_{W(f)} : W(f) \to Y\) has \(\mathcal{P}\). Moreover, if \(g : X' \to X\) is flat and locally of finite presentation, syntomic, smooth, or étale and \(f' = f \circ g : X' \to Y\), then \(g^{-1}(W(f)) = W(f')\).

Proof

Consider the union \(W\) of the images \(g(X') \subset X\) of morphisms \(g : X' \to X\) with the properties:

1. \(g\) is flat and locally of finite presentation, syntomic, smooth, or étale, and

2. the composition \(X' \to X \to Y\) has property \(\mathcal{P}\).

Since such a morphism \(g\) is open (see Morphisms, Lemma 01UA) we see that \(W \subset X\) is an open subset of \(X\). Since \(\mathcal{P}\) is local in the \(\tau\) topology the restriction \(f|_W : W \to Y\) has property \(\mathcal{P}\) because we are given a \(\tau\) covering \(\{X' \to W\}\) of \(W\) such that the pullbacks have \(\mathcal{P}\). This proves the existence of \(W(f)\). The compatibility stated in the last sentence follows immediately from the construction of \(W(f)\).

Lemma

Let \(\mathcal{P}\) be a property of morphisms of schemes. Let \(\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}\). Assume that

1. the property is preserved under precomposing with flat, flat locally of finite presentation, étale, smooth or syntomic morphisms depending on whether \(\tau\) is fpqc, fppf, étale, smooth, or syntomic,

2. the property is Zariski local on the source,

3. the property is Zariski local on the target,

4. for any morphism of affine schemes \(f : X \to Y\), and any surjective morphism of affine schemes \(X' \to X\) which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether \(\tau\) is fpqc, fppf, étale, smooth, or syntomic, property \(\mathcal{P}\) holds for \(f\) if property \(\mathcal{P}\) holds for the composition \(f' : X' \to Y\).

Then \(\mathcal{P}\) is \(\tau\) local on the source.

Proof

This follows almost immediately from the definition of a \(\tau\)-covering, see Topologies, Definition 022B 021M 0215 021Z, or 0225 and Topologies, Lemma 022E, 021P, 0218, 0222, or 0228. Details omitted. (Hint: Use locality on the source and target to reduce the verification of property \(\mathcal{P}\) to the case of a morphism between affines. Then apply (1) and (4).)

Remark

(This is a repeat of Remarks 034A and 034G above.) In Lemma 036H above if \(\tau = smooth\) then in condition (4) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when \(\tau = syntomic\) or \(\tau = \etale\).

Properties of morphisms local in the fpqc topology on the source

Here are some properties of morphisms that are fpqc local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is flat” is fpqc local on the source.

Proof

Since flatness is defined in terms of the maps of local rings (Morphisms, Definition 01U3) what has to be shown is the following algebraic fact: Suppose \(A \to B \to C\) are local homomorphisms of local rings, and assume \(B \to C\) is flat. Then \(A \to B\) is flat if and only if \(A \to C\) is flat. If \(A \to B\) is flat, then \(A \to C\) is flat by Algebra, Lemma 00HC. Conversely, assume \(A \to C\) is flat. Note that \(B \to C\) is faithfully flat, see Algebra, Lemma 00HR. Hence \(A \to B\) is flat by Algebra, Lemma 039V. (Also see Morphisms, Lemma 02JZ for a direct proof.)

Lemma

The property \(\mathcal{P}(f : X \to Y)=\)“for every \(x \in X\) the map of local rings \(\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}\) is injective” is fpqc local on the source.

Proof

Omitted. This is just a (probably misguided) attempt to be playful.

Properties of morphisms local in the fppf topology on the source

Here are some properties of morphisms that are fppf local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is locally of finite presentation” is fppf local on the source.

Proof

Being locally of finite presentation is Zariski local on the source and the target, see Morphisms, Lemma 01TQ. It is a property which is preserved under composition, see Morphisms, Lemma 01TR. This proves (1), (2) and (3) of Lemma 036H. The final condition (4) is Lemma 02KK. Hence we win.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is locally of finite type” is fppf local on the source.

Proof

Being locally of finite type is Zariski local on the source and the target, see Morphisms, Lemma 01T2. It is a property which is preserved under composition, see Morphisms, Lemma 01T3, and a flat morphism locally of finite presentation is locally of finite type, see Morphisms, Lemma 01TW. This proves (1), (2) and (3) of Lemma 036H. The final condition (4) is Lemma 0367. Hence we win.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is open” is fppf local on the source.

Proof

Being an open morphism is clearly Zariski local on the source and the target. It is a property which is preserved under composition, see Morphisms, Lemma 02V2, and a flat morphism of finite presentation is open, see Morphisms, Lemma 01UA This proves (1), (2) and (3) of Lemma 036H. The final condition (4) follows from Morphisms, Lemma 02JY. Hence we win.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is universally open” is fppf local on the source.

Proof

Let \(f : X \to Y\) be a morphism of schemes. Let \(\{X_i \to X\}_{i \in I}\) be an fppf covering. Denote by \(f_i : X_i \to Y\) the compositions. We have to show that \(f\) is universally open if and only if each \(f_i\) is universally open. If \(f\) is universally open, then also each \(f_i\) is universally open since the maps \(X_i \to X\) are universally open and compositions of universally open morphisms are universally open (Morphisms, Lemmas 01UA and 02V2). Conversely, assume each \(f_i\) is universally open. Let \(Y' \to Y\) be a morphism of schemes. Denote \(X' = Y' \times_Y X\) and \(X'_i = Y' \times_Y X_i\). Note that \(\{X_i' \to X'\}_{i \in I}\) is an fppf covering also. The morphisms \(f'_i : X_i' \to Y'\) are open by assumption. Hence by the Lemma 036P above we conclude that \(f' : X' \to Y'\) is open as desired.

Properties of morphisms local in the syntomic topology on the source

Here are some properties of morphisms that are syntomic local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is syntomic” is syntomic local on the source.

Proof

Combine Lemma 036H with Morphisms, Lemma 01UD (local for Zariski on source and target), Morphisms, Lemma 01UH (pre-composing), and Lemma 02KM (part (4)).

Properties of morphisms local in the smooth topology on the source

Here are some properties of morphisms that are smooth local on the source. Note also the (in some respects stronger) result on descending smoothness via flat morphisms, Lemma 05B5.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is smooth” is smooth local on the source.

Proof

Combine Lemma 036H with Morphisms, Lemma 01V6 (local for Zariski on source and target), Morphisms, Lemma 01VA (pre-composing), and Lemma 02KM (part (4)).

Properties of morphisms local in the étale topology on the source

Here are some properties of morphisms that are étale local on the source.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is étale” is étale local on the source.

Proof

Combine Lemma 036H with Morphisms, Lemma 02GJ (local for Zariski on source and target), Morphisms, Lemma 02GN (pre-composing), and Lemma 02KM (part (4)).

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is locally quasi-finite” is étale local on the source.

Proof

We are going to use Lemma 036H. By Morphisms, Lemma 01TK the property of being locally quasi-finite is local for Zariski on source and target. By Morphisms, Lemmas 01TL and 03WS we see the precomposition of a locally quasi-finite morphism by an étale morphism is locally quasi-finite. Finally, suppose that \(X \to Y\) is a morphism of affine schemes and that \(X' \to X\) is a surjective étale morphism of affine schemes such that \(X' \to Y\) is locally quasi-finite. Then \(X' \to Y\) is of finite type, and by Lemma 0367 we see that \(X \to Y\) is of finite type also. Moreover, by assumption \(X' \to Y\) has finite fibres, and hence \(X \to Y\) has finite fibres also. We conclude that \(X \to Y\) is quasi-finite by Morphisms, Lemma 02NH. This proves the last assumption of Lemma 036H and finishes the proof.

Lemma

The property \(\mathcal{P}(f)=\)“\(f\) is unramified” is étale local on the source. The property \(\mathcal{P}(f)=\)“\(f\) is G-unramified” is étale local on the source.

Proof

We are going to use Lemma 036H. By Morphisms, Lemma 02G6 the property of being unramified (resp. G-unramified) is local for Zariski on source and target. By Morphisms, Lemmas 02G9 and 02GK we see the precomposition of an unramified (resp. G-unramified) morphism by an étale morphism is unramified (resp. G-unramified). Finally, suppose that \(X \to Y\) is a morphism of affine schemes and that \(f : X' \to X\) is a surjective étale morphism of affine schemes such that \(X' \to Y\) is unramified (resp. G-unramified). Then \(X' \to Y\) is of finite type (resp. finite presentation), and by Lemma 0367 (resp. Lemma 02KK) we see that \(X \to Y\) is of finite type (resp. finite presentation) also. By Morphisms, Lemma 02K4 we have a short exact sequence \[0 \to f^*\Omega_{X/Y} \to \Omega_{X'/Y} \to \Omega_{X'/X} \to 0.\] As \(X' \to Y\) is unramified we see that the middle term is zero. Hence, as \(f\) is faithfully flat we see that \(\Omega_{X/Y} = 0\). Hence \(X \to Y\) is unramified (resp. G-unramified), see Morphisms, Lemma 02G5. This proves the last assumption of Lemma 036H and finishes the proof.

Properties of morphisms étale local on source-and-target

Let \(\mathcal{P}\) be a property of morphisms of schemes. There is an intuitive meaning to the phrase “\(\mathcal{P}\) is étale local on the source and target”. However, it turns out that this notion is not the same as asking \(\mathcal{P}\) to be both étale local on the source and étale local on the target. Before we discuss this further we give two silly examples.

Example

Consider the property \(\mathcal{P}\) of morphisms of schemes defined by the rule \(\mathcal{P}(X \to Y) =\)“\(Y\) is locally Noetherian”. The reader can verify that this is étale local on the source and étale local on the target (omitted, see Lemma 034C). But it is not true that if \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is étale, then \(g \circ f\) has \(\mathcal{P}\). Namely, \(f\) could be the identity on \(Y\) and \(g\) could be an open immersion of a locally Noetherian scheme \(Y\) into a non locally Noetherian scheme \(Z\).

The following example is in some sense worse.

Example

Consider the property \(\mathcal{P}\) of morphisms of schemes defined by the rule \(\mathcal{P}(f : X \to Y) =\)“for every \(y \in Y\) which is a specialization of some \(f(x)\), \(x \in X\) the local ring \(\mathcal{O}_{Y, y}\) is Noetherian”. Let us verify that this is étale local on the source and étale local on the target. We will freely use Schemes, Lemma 01J7.

Local on the target: Let \(\{g_i : Y_i \to Y\}\) be an étale covering. Let \(f_i : X_i \to Y_i\) be the base change of \(f\), and denote \(h_i : X_i \to X\) the projection. Assume \(\mathcal{P}(f)\). Let \(f_i(x_i) \leadsto y_i\) be a specialization. Then \(f(h_i(x_i)) \leadsto g_i(y_i)\) so \(\mathcal{P}(f)\) implies \(\mathcal{O}_{Y, g_i(y_i)}\) is Noetherian. Also \(\mathcal{O}_{Y, g_i(y_i)} \to \mathcal{O}_{Y_i, y_i}\) is a localization of an étale ring map. Hence \(\mathcal{O}_{Y_i, y_i}\) is Noetherian by Algebra, Lemma 00FN. Conversely, assume \(\mathcal{P}(f_i)\) for all \(i\). Let \(f(x) \leadsto y\) be a specialization. Choose an \(i\) and \(y_i \in Y_i\) mapping to \(y\). Since \(x\) can be viewed as a point of \(\Spec(\mathcal{O}_{Y, y}) \times_Y X\) and \(\mathcal{O}_{Y, y} \to \mathcal{O}_{Y_i, y_i}\) is faithfully flat, there exists a point \(x_i \in \Spec(\mathcal{O}_{Y_i, y_i}) \times_Y X\) mapping to \(x\). Then \(x_i \in X_i\), and \(f_i(x_i)\) specializes to \(y_i\). Thus we see that \(\mathcal{O}_{Y_i, y_i}\) is Noetherian by \(\mathcal{P}(f_i)\) which implies that \(\mathcal{O}_{Y, y}\) is Noetherian by Algebra, Lemma 033E.

Local on the source: Let \(\{h_i : X_i \to X\}\) be an étale covering. Let \(f_i : X_i \to Y\) be the composition \(f \circ h_i\). Assume \(\mathcal{P}(f)\). Let \(f_i(x_i) \leadsto y\) be a specialization. Then \(f(h_i(x_i)) \leadsto y\) so \(\mathcal{P}(f)\) implies \(\mathcal{O}_{Y, y}\) is Noetherian. Thus \(\mathcal{P}(f_i)\) holds. Conversely, assume \(\mathcal{P}(f_i)\) for all \(i\). Let \(f(x) \leadsto y\) be a specialization. Choose an \(i\) and \(x_i \in X_i\) mapping to \(x\). Then \(y\) is a specialization of \(f_i(x_i) = f(x)\). Hence \(\mathcal{P}(f_i)\) implies \(\mathcal{O}_{Y, y}\) is Noetherian as desired.

We claim that there exists a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with surjective étale vertical arrows, such that \(h\) has \(\mathcal{P}\) and \(f\) does not have \(\mathcal{P}\). Namely, let \[Y = \Spec\Big( \mathbf{C}[x_n; n \in \mathbf{Z}]/(x_n x_m; n \not = m) \Big)\] and let \(X \subset Y\) be the open subscheme which is the complement of the point all of whose coordinates \(x_n = 0\). Let \(U = X\), let \(V = X \amalg Y\), let \(a, b\) be the obvious maps, and let \(h : U \to V\) be the inclusion of \(U = X\) into the first summand of \(V\). The claim above holds because \(U\) is locally Noetherian, but \(Y\) is not.

What should be the correct notion of a property which is étale local on the source-and-target? We think that, by analogy with Morphisms, Definition 01SR it should be the following.

Definition

Let \(\mathcal{P}\) be a property of morphisms of schemes. We say \(\mathcal{P}\) is étale local on source-and-target if

1. (stable under precomposing with étale maps) if \(f : X \to Y\) is étale and \(g : Y \to Z\) has \(\mathcal{P}\), then \(g \circ f\) has \(\mathcal{P}\),

2. (stable under étale base change) if \(f : X \to Y\) has \(\mathcal{P}\) and \(Y' \to Y\) is étale, then the base change \(f' : Y' \times_Y X \to Y'\) has \(\mathcal{P}\), and

3. (locality) given a morphism \(f : X \to Y\) the following are equivalent

   1. \(f\) has \(\mathcal{P}\),

   2. for every \(x \in X\) there exists a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with étale vertical arrows and \(u \in U\) with \(a(u) = x\) such that \(h\) has \(\mathcal{P}\).

It turns out this definition excludes the behavior seen in Examples 04QX and 04QY. We will compare this to the definition in the paper [DM] by Deligne and Mumford in Remark 04R4. Moreover, a property which is étale local on the source-and-target is étale local on the source and étale local on the target. Finally, the converse is almost true as we will see in Lemma 04R2.

Lemma

Let \(\mathcal{P}\) be a property of morphisms of schemes which is étale local on source-and-target. Then

1. \(\mathcal{P}\) is étale local on the source,

2. \(\mathcal{P}\) is étale local on the target,

3. \(\mathcal{P}\) is stable under postcomposing with étale morphisms: if \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is étale, then \(g \circ f\) has \(\mathcal{P}\), and

4. \(\mathcal{P}\) has a permanence property: given \(f : X \to Y\) and \(g : Y \to Z\) étale such that \(g \circ f\) has \(\mathcal{P}\), then \(f\) has \(\mathcal{P}\).

Proof

We write everything out completely.

Proof of (1). Let \(f : X \to Y\) be a morphism of schemes. Let \(\{X_i \to X\}_{i \in I}\) be an étale covering of \(X\). If each composition \(h_i : X_i \to Y\) has \(\mathcal{P}\), then for each \(x \in X\) we can find an \(i \in I\) and a point \(x_i \in X_i\) mapping to \(x\). Then \((X_i, x_i) \to (X, x)\) is an étale morphism of germs, and \(\text{id}_Y : Y \to Y\) is an étale morphism, and \(h_i\) is as in part (3) of Definition 04QZ. Thus we see that \(f\) has \(\mathcal{P}\). Conversely, if \(f\) has \(\mathcal{P}\) then each \(X_i \to Y\) has \(\mathcal{P}\) by Definition 04QZ part (1).

Proof of (2). Let \(f : X \to Y\) be a morphism of schemes. Let \(\{Y_i \to Y\}_{i \in I}\) be an étale covering of \(Y\). Write \(X_i = Y_i \times_Y X\) and \(h_i : X_i \to Y_i\) for the base change of \(f\). If each \(h_i : X_i \to Y_i\) has \(\mathcal{P}\), then for each \(x \in X\) we pick an \(i \in I\) and a point \(x_i \in X_i\) mapping to \(x\). Then \((X_i, x_i) \to (X, x)\) is an étale morphism of germs, \(Y_i \to Y\) is étale, and \(h_i\) is as in part (3) of Definition 04QZ. Thus we see that \(f\) has \(\mathcal{P}\). Conversely, if \(f\) has \(\mathcal{P}\), then each \(X_i \to Y_i\) has \(\mathcal{P}\) by Definition 04QZ part (2).

Proof of (3). Assume \(f : X \to Y\) has \(\mathcal{P}\) and \(g : Y \to Z\) is étale. For every \(x \in X\) we can think of \((X, x) \to (X, x)\) as an étale morphism of germs, \(Y \to Z\) is an étale morphism, and \(h = f\) is as in part (3) of Definition 04QZ. Thus we see that \(g \circ f\) has \(\mathcal{P}\).

Proof of (4). Let \(f : X \to Y\) be a morphism and \(g : Y \to Z\) étale such that \(g \circ f\) has \(\mathcal{P}\). Then by Definition 04QZ part (2) we see that \(\text{pr}_Y : Y \times_Z X \to Y\) has \(\mathcal{P}\). But the morphism \((f, 1) : X \to Y \times_Z X\) is étale as a section to the étale projection \(\text{pr}_X : Y \times_Z X \to X\), see Morphisms, Lemma 02GW. Hence \(f = \text{pr}_Y \circ (f, 1)\) has \(\mathcal{P}\) by Definition 04QZ part (1).

The following lemma is the analogue of Morphisms, Lemma 01SU.

Lemma

Let \(\mathcal{P}\) be a property of morphisms of schemes which is étale local on source-and-target. Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent:

1. \(f\) has property \(\mathcal{P}\),

2. for every \(x \in X\) there exists an étale morphism of germs \(a : (U, u) \to (X, x)\), an étale morphism \(b : V \to Y\), and a morphism \(h : U \to V\) such that \(f \circ a = b \circ h\) and \(h\) has \(\mathcal{P}\),

3. for any commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(a\), \(b\) étale the morphism \(h\) has \(\mathcal{P}\),

4. for some diagram as in (c) with \(a : U \to X\) surjective \(h\) has \(\mathcal{P}\),

5. there exists an étale covering \(\{Y_i \to Y\}_{i \in I}\) such that each base change \(Y_i \times_Y X \to Y_i\) has \(\mathcal{P}\),

6. there exists an étale covering \(\{X_i \to X\}_{i \in I}\) such that each composition \(X_i \to Y\) has \(\mathcal{P}\),

7. there exists an étale covering \(\{Y_i \to Y\}_{i \in I}\) and for each \(i \in I\) an étale covering \(\{X_{ij} \to Y_i \times_Y X\}_{j \in J_i}\) such that each morphism \(X_{ij} \to Y_i\) has \(\mathcal{P}\).

Proof

The equivalence of (a) and (b) is part of Definition 04QZ. The equivalence of (a) and (e) is Lemma 04R0 part (2). The equivalence of (a) and (f) is Lemma 04R0 part (1). As (a) is now equivalent to (e) and (f) it follows that (a) is equivalent to (g).

It is clear that (c) implies (a). If (a) holds, then for any diagram as in (c) the morphism \(f \circ a\) has \(\mathcal{P}\) by Definition 04QZ part (1), whereupon \(h\) has \(\mathcal{P}\) by Lemma 04R0 part (4). Thus (a) and (c) are equivalent. It is clear that (c) implies (d). To see that (d) implies (a) assume we have a diagram as in (c) with \(a : U \to X\) surjective and \(h\) having \(\mathcal{P}\). Then \(b \circ h\) has \(\mathcal{P}\) by Lemma 04R0 part (3). Since \(\{a : U \to X\}\) is an étale covering we conclude that \(f\) has \(\mathcal{P}\) by Lemma 04R0 part (1).

It seems that the result of the following lemma is not a formality, i.e., it actually uses something about the geometry of étale morphisms.

Lemma

Let \(\mathcal{P}\) be a property of morphisms of schemes. Assume

1. \(\mathcal{P}\) is étale local on the source,

2. \(\mathcal{P}\) is étale local on the target, and

3. \(\mathcal{P}\) is stable under postcomposing with open immersions: if \(f : X \to Y\) has \(\mathcal{P}\) and \(Y \subset Z\) is an open subscheme then \(X \to Z\) has \(\mathcal{P}\).

Then \(\mathcal{P}\) is étale local on the source-and-target.

Proof

Let \(\mathcal{P}\) be a property of morphisms of schemes which satisfies conditions (1), (2) and (3) of the lemma. By Lemma 04QV we see that \(\mathcal{P}\) is stable under precomposing with étale morphisms. By Lemma 04QU we see that \(\mathcal{P}\) is stable under étale base change. Hence it suffices to prove part (3) of Definition 04QZ holds.

More precisely, suppose that \(f : X \to Y\) is a morphism of schemes which satisfies Definition 04QZ part (3)(b). In other words, for every \(x \in X\) there exists an étale morphism \(a_x : U_x \to X\), a point \(u_x \in U_x\) mapping to \(x\), an étale morphism \(b_x : V_x \to Y\), and a morphism \(h_x : U_x \to V_x\) such that \(f \circ a_x = b_x \circ h_x\) and \(h_x\) has \(\mathcal{P}\). The proof of the lemma is complete once we show that \(f\) has \(\mathcal{P}\). Set \(U = \coprod U_x\), \(a = \coprod a_x\), \(V = \coprod V_x\), \(b = \coprod b_x\), and \(h = \coprod h_x\). We obtain a commutative diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with \(a\), \(b\) étale, \(a\) surjective. Note that \(h\) has \(\mathcal{P}\) as each \(h_x\) does and \(\mathcal{P}\) is étale local on the target. Because \(a\) is surjective and \(\mathcal{P}\) is étale local on the source, it suffices to prove that \(b \circ h\) has \(\mathcal{P}\). This reduces the lemma to proving that \(\mathcal{P}\) is stable under postcomposing with an étale morphism.

During the rest of the proof we let \(f : X \to Y\) be a morphism with property \(\mathcal{P}\) and \(g : Y \to Z\) is an étale morphism. Consider the following statements:

1. With no additional assumptions \(g \circ f\) has property \(\mathcal{P}\).

2. Whenever \(Z\) is affine \(g \circ f\) has property \(\mathcal{P}\).

3. Whenever \(X\) and \(Z\) are affine \(g \circ f\) has property \(\mathcal{P}\).

4. Whenever \(X\), \(Y\), and \(Z\) are affine \(g \circ f\) has property \(\mathcal{P}\).

Once we have proved (-) the proof of the lemma will be complete.

Claim 1: (AAA) \(\Rightarrow\) (AA). Namely, let \(f : X \to Y\), \(g : Y \to Z\) be as above with \(X\), \(Z\) affine. As \(X\) is affine hence quasi-compact we can find finitely many affine open \(Y_i \subset Y\), \(i = 1, \ldots, n\) such that \(X = \bigcup_{i = 1, \ldots, n} f^{-1}(Y_i)\). Set \(X_i = f^{-1}(Y_i)\). By Lemma 04QU each of the morphisms \(X_i \to Y_i\) has \(\mathcal{P}\). Hence \(\coprod_{i = 1, \ldots, n} X_i \to \coprod_{i = 1, \ldots, n} Y_i\) has \(\mathcal{P}\) as \(\mathcal{P}\) is étale local on the target. By (AAA) applied to \(\coprod_{i = 1, \ldots, n} X_i \to \coprod_{i = 1, \ldots, n} Y_i\) and the étale morphism \(\coprod_{i = 1, \ldots, n} Y_i \to Z\) we see that \(\coprod_{i = 1, \ldots, n} X_i \to Z\) has \(\mathcal{P}\). Now \(\{\coprod_{i = 1, \ldots, n} X_i \to X\}\) is an étale covering, hence as \(\mathcal{P}\) is étale local on the source we conclude that \(X \to Z\) has \(\mathcal{P}\) as desired.

Claim 2: (AAA) \(\Rightarrow\) (A). Namely, let \(f : X \to Y\), \(g : Y \to Z\) be as above with \(Z\) affine. Choose an affine open covering \(X = \bigcup X_i\). As \(\mathcal{P}\) is étale local on the source we see that each \(f|_{X_i} : X_i \to Y\) has \(\mathcal{P}\). By (AA), which follows from (AAA) according to Claim 1, we see that \(X_i \to Z\) has \(\mathcal{P}\) for each \(i\). Since \(\{X_i \to X\}\) is an étale covering and \(\mathcal{P}\) is étale local on the source we conclude that \(X \to Z\) has \(\mathcal{P}\).

Claim 3: (AAA) \(\Rightarrow\) (-). Namely, let \(f : X \to Y\), \(g : Y \to Z\) be as above. Choose an affine open covering \(Z = \bigcup Z_i\). Set \(Y_i = g^{-1}(Z_i)\) and \(X_i = f^{-1}(Y_i)\). By Lemma 04QU each of the morphisms \(X_i \to Y_i\) has \(\mathcal{P}\). By (A), which follows from (AAA) according to Claim 2, we see that \(X_i \to Z_i\) has \(\mathcal{P}\) for each \(i\). Since \(\mathcal{P}\) is local on the target and \(X_i = (g \circ f)^{-1}(Z_i)\) we conclude that \(X \to Z\) has \(\mathcal{P}\).

Thus to prove the lemma it suffices to prove (AAA). Let \(f : X \to Y\) and \(g : Y \to Z\) be as above \(X, Y, Z\) affine. Note that an étale morphism of affines has universally bounded fibres, see Morphisms, Lemma 03WS and Lemma 03JA. Hence we can do induction on the integer \(n\) bounding the degree of the fibres of \(Y \to Z\). See Morphisms, Lemma 03WU for a description of this integer in the case of an étale morphism. If \(n = 1\), then \(Y \to Z\) is an open immersion, see Lemma 02LC, and the result follows from assumption (3) of the lemma. Assume \(n > 1\).

Consider the following commutative diagram \[\xymatrix{ X \times_Z Y \ar[d] \ar[r]_{f_Y} & Y \times_Z Y \ar[d] \ar[r]_-{\text{pr}} & Y \ar[d] \\ X \ar[r]^f & Y \ar[r]^g & Z }\] Note that we have a decomposition into open and closed subschemes \(Y \times_Z Y = \Delta_{Y/Z}(Y) \amalg Y'\), see Morphisms, Lemma 02GE. As a base change the degrees of the fibres of the second projection \(\text{pr} : Y \times_Z Y \to Y\) are bounded by \(n\), see Morphisms, Lemma 03J7. On the other hand, \(\text{pr}|_{\Delta(Y)} : \Delta(Y) \to Y\) is an isomorphism and every fibre has exactly one point. Thus, on applying Morphisms, Lemma 03WU we conclude the degrees of the fibres of the restriction \(\text{pr}|_{Y'} : Y' \to Y\) are bounded by \(n - 1\). Set \(X' = f_Y^{-1}(Y')\). Picture \[\xymatrix{ X \amalg X' \ar@{=}[d] \ar[r]_-{f \amalg f'} & \Delta(Y) \amalg Y' \ar@{=}[d] \ar[r] & Y \ar@{=}[d] \\ X \times_Z Y \ar[r]^{f_Y} & Y \times_Z Y \ar[r]^-{\text{pr}} & Y }\] As \(\mathcal{P}\) is étale local on the target and hence stable under étale base change (see Lemma 04QU) we see that \(f_Y\) has \(\mathcal{P}\). Hence, as \(\mathcal{P}\) is étale local on the source, \(f' = f_Y|_{X'}\) has \(\mathcal{P}\). By induction hypothesis we see that \(X' \to Y\) has \(\mathcal{P}\). As \(\mathcal{P}\) is local on the source, and \(\{X \to X \times_Z Y, X' \to X \times_Z Y\}\) is an étale covering, we conclude that \(\text{pr} \circ f_Y\) has \(\mathcal{P}\). Note that \(g \circ f\) can be viewed as a morphism \(g \circ f : X \to g(Y)\). As \(\text{pr} \circ f_Y\) is the pullback of \(g \circ f : X \to g(Y)\) via the étale covering \(\{Y \to g(Y)\}\), and as \(\mathcal{P}\) is étale local on the target, we conclude that \(g \circ f : X \to g(Y)\) has property \(\mathcal{P}\). Finally, applying assumption (3) of the lemma once more we conclude that \(g \circ f : X \to Z\) has property \(\mathcal{P}\).

Remark

Using Lemma 04R2 and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are étale local on the source-and-target. In each case we list the lemma which implies the property is étale local on the source and the lemma which implies the property is étale local on the target. In each case the third assumption of Lemma 04R2 is trivial to check, and we omit it. Here is the list:

1. flat, see Lemmas 036K and 02L2,

2. locally of finite presentation, see Lemmas 036N and 02KY,

3. locally of finite type, see Lemmas 036O and 02KX,

4. universally open, see Lemmas 036Q and 02KT,

5. syntomic, see Lemmas 036S and 02VK,

6. smooth, see Lemmas 036U and 02VL,

7. étale, see Lemmas 036W and 02VN,

8. locally quasi-finite, see Lemmas 03X4 and 02VI,

9. unramified, see Lemmas 03YV and 02VM,

10. G-unramified, see Lemmas 03YV and 02VM, and

11. add more here as needed.

Remark

At this point we have three possible definitions of what it means for a property \(\mathcal{P}\) of morphisms to be “étale local on the source and target”:

1. \(\mathcal{P}\) is étale local on the source and \(\mathcal{P}\) is étale local on the target,

2. (the definition in the paper [DM, Page 100] by Deligne and Mumford) for every diagram \[\xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y }\] with surjective étale vertical arrows we have \(\mathcal{P}(h) \Leftrightarrow \mathcal{P}(f)\), and

3. \(\mathcal{P}\) is étale local on the source-and-target.

In this section we have seen that (SP) \(\Rightarrow\) (DM) \(\Rightarrow\) (ST). The Examples 04QX and 04QY show that neither implication can be reversed. Finally, Lemma 04R2 shows that the difference disappears when looking at properties of morphisms which are stable under postcomposing with open immersions, which in practice will always be the case.

Lemma

Let \(\mathcal{P}\) be a property of morphisms of schemes which is étale local on the source-and-target. Given a commutative diagram of schemes \[\vcenter{ \xymatrix{ X' \ar[d]_{g'} \ar[r]_{f'} & Y' \ar[d]^g \\ X \ar[r]^f & Y } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x' \ar[d] \ar[r] & y' \ar[d] \\ x \ar[r] & y } }\] such that \(g'\) is étale at \(x'\) and \(g\) is étale at \(y'\), then \(x \in W(f) \Leftrightarrow x' \in W(f')\) where \(W(-)\) is as in Lemma 0CEY.

Proof

Lemma 0CEY applies since \(\mathcal{P}\) is étale local on the source by Lemma 04R0.

Assume \(x \in W(f)\). Let \(U' \subset X'\) and \(V' \subset Y'\) be open neighbourhoods of \(x'\) and \(y'\) such that \(f'(U') \subset V'\), \(g'(U') \subset W(f)\) and \(g'|_{U'}\) and \(g|_{V'}\) are étale. Then \(f \circ g'|_{U'} = g \circ f'|_{U'}\) has \(\mathcal{P}\) by property (1) of Definition 04QZ. Then \(f'|_{U'} : U' \to V'\) has property \(\mathcal{P}\) by (4) of Lemma 04R0. Then by (3) of Lemma 04R0 we conclude that \(f'|_{U'} : U' \to Y'\) has \(\mathcal{P}\). Hence \(U' \subset W(f')\) by definition. Hence \(x' \in W(f')\).

Assume \(x' \in W(f')\). Let \(U' \subset X'\) and \(V' \subset Y'\) be open neighbourhoods of \(x'\) and \(y'\) such that \(f'(U') \subset V'\), \(U' \subset W(f')\) and \(g'|_{U'}\) and \(g|_{V'}\) are étale. Then \(U' \to Y'\) has \(\mathcal{P}\) by definition of \(W(f')\). Then \(U' \to V'\) has \(\mathcal{P}\) by (4) of Lemma 04R0. Then \(U' \to Y\) has \(\mathcal{P}\) by (3) of Lemma 04R0. Let \(U \subset X\) be the image of the étale (hence open) morphism \(g'|_{U'} : U' \to X\). Then \(\{U' \to U\}\) is an étale covering and we conclude that \(U \to Y\) has \(\mathcal{P}\) by (1) of Lemma 04R0. Thus \(U \subset W(f)\) by definition. Hence \(x \in W(f)\).