# Descending properties of schemes and morphisms

*Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Public domain (CC0).*

Descent has two different questions. If a morphism already exists over the base, can its properties be checked after a covering? If only an object with descent data exists over the covering, does an object over the base exist at all? Properness belongs to the first question. Quasi-affine and étale descent will lead us to the second. Permanence is not automatic for an arbitrary property: each assertion needs its own argument and its specified covering topology.

We use the module descent, algebra descent and morphism descent proved in *Faithfully flat descent*, and the elementary local structure of étale morphisms. In particular, an étale morphism has an open diagonal, and an étale universally bijective morphism is an isomorphism. An fpqc covering has the finite affine refinements described in that lesson. All schemes in this lesson may be non-Noetherian unless a hypothesis explicitly says otherwise.

## 1. Descent is different from ascent

A property of schemes is **fpqc-local** if it both ascends and descends along fpqc coverings. A property that only descends should not be called fpqc-local. This distinction matters even over a field: the faithfully flat map

\[
k\longrightarrow k[\epsilon]/(\epsilon^2)
\]

starts with a reduced, normal, regular scheme and ends with one having none of those properties.

**Proposition 1.1.** Let \(\{T_i\to T\}\) be an fpqc covering.

1. If all \(T_i\) are reduced, then \(T\) is reduced.
2. If all \(T_i\) are normal, then \(T\) is normal.
3. If all \(T_i\) are locally Noetherian, then \(T\) is locally Noetherian.
4. If all \(T_i\) are regular, then \(T\) is regular. Here regular schemes are locally Noetherian and have regular local rings.

*Proof.* For a point \(t\in T\), choose a point \(u\in T_i\) above it. The flat local map

\[
R=\mathcal O_{T,t}\longrightarrow S=\mathcal O_{T_i,u}
\]

is faithfully flat and hence injective. Thus a nilpotent of \(R\) vanishes if \(S\) is reduced.

Suppose that \(S\) is a normal domain. Then \(R\) is a domain. If \(a/b\in\operatorname{Frac}(R)\) is integral over \(R\), its image is integral over \(S\), so it belongs to \(S\). Faithful flatness gives

\[
bS\cap R=bR:
\]

indeed \(R/(b)\to S/bS\) is injective. Consequently \(a=br\) for some \(r\in R\). This proves normality at every point.

For the Noetherian assertion, restrict to an affine open \(\operatorname{Spec}R\subset T\) and refine the covering to a faithfully flat map \(R\to A\), with \(A\) a finite product of Noetherian rings. Given an ideal \(I\subset R\), the ideal \(IA\) is finitely generated. Express its generators as finite sums of elements of \(I\) times elements of \(A\), and let \(I_0\subset I\) be generated by the finitely many elements of \(I\) used. Then \((I/I_0)\otimes_R A=0\), so \(I=I_0\). Thus \(R\) is Noetherian.

For regularity, we can now use Noetherian local rings \(R\to S\) as above. We use the commutative algebra criterion that a Noetherian local ring is regular precisely when its residue field has finite projective dimension, and that a regular local ring has global dimension equal to its dimension [Stacks, Tag 00OC](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-proposition-finite-gl-dim-regular). Let \(d=\dim S\). Tensor a resolution of the residue field of \(R\) by finite free \(R\)-modules with \(S\). The \(d\)-th syzygy becomes a finitely generated projective \(S\)-module. Finite projectivity descends by *Faithfully flat descent*, Section 3. The original resolution therefore has a projective syzygy and gives finite projective dimension. The criterion proves that \(R\) is regular. If \(d=0\), the same argument applies to the residue field itself. \(\square\)

Local Noetherianness also ascends along locally finitely presented maps, since a finitely presented algebra over a Noetherian ring is Noetherian. It is therefore fppf-local. It is not fpqc-local: \(k\to k[x_1,x_2,\ldots]\) is faithfully flat, and the ideal generated by all the variables is not finitely generated. Reducedness, normality and regularity do ascend along smooth maps. The precise ascent inputs are [Stacks, Tags [033B](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-reduced-goes-up), [033C](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-normal-goes-up), [07NF](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-regular-goes-up)]; together with Proposition 1.1 they make these three properties smooth-local. We distinguish those ascent theorems from the descent arguments just proved. Compare [Stacks, Sections [0347](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-properties), [034B](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-properties-fppf), [034D](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-properties-smooth)].

## 2. Checking a morphism on an fpqc covering of its target

Let \(f:Y\to S\) and let \(\{S_i\to S\}\) be an fpqc covering. A property is **fpqc-local on the target** if \(f\) has it exactly when every \(Y\times_S S_i\to S_i\) has it.

The properties considered below are stable under base change and local for the Zariski topology on the target. To prove their descent, it suffices to treat

\[
S=\operatorname{Spec}R,\qquad S_1=\operatorname{Spec}A,
\qquad R\to A\text{ faithfully flat}.
\tag{2.1}
\]

Here is the reduction. Above each affine open of \(S\), take the finite affine refinement of the fpqc covering, and combine its members into one affine scheme by taking their finite disjoint union. The property on each refined member gives the property after this combined covering. Prove the assertion in (2.1), then glue the conclusion over the affine opens of \(S\). Write \(Y_A=Y\times_R A\).

**Theorem 2.1.** Each of the following properties is fpqc-local on the target:

\[
\begin{gathered}
\text{quasi-compact, quasi-separated, universally closed, separated,}\\
\text{affine, closed immersion, locally of finite type,}\\
\text{locally of finite presentation, flat, proper, smooth, étale, finite.}
\end{gathered}
\]

*Proof of quasi-compactness.* In (2.1), \(Y_A\to Y\) is surjective. If \(Y_A\) is quasi-compact, its continuous image \(Y\) is quasi-compact. The affine-target reduction proves the statement. [Stacks, Tag 02KQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-quasi-compact)

*Proof of universal closedness.* The universally quotient property of faithfully flat affine maps was proved in *Faithfully flat descent*, Section 6. Consider any base change \(T\to S\), and a closed subset \(C\subset Y_T\). Let \(q:T_A\to T\), and let \(C_A\) be the inverse image of \(C\). Images commute with this base change as subsets:

\[
f_A(C_A)=q^{-1}(f(C)).
\tag{2.2}
\]

To check the nontrivial inclusion, a point of \(C\) over \(t\), and a point of \(T_A\) over \(t\), have a common point over them: the tensor product of their residue fields over \(\kappa(t)\) is nonzero and has a prime ideal. If \(f_A\) is universally closed, the left side of (2.2) is closed. The quotient property of \(q\) makes \(f(C)\) closed. This holds for every \(T\), so \(f\) is universally closed. Notice that this proof works for any universally submersive surjective covering, without flatness. [Stacks, Tag 02KS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-universally-closed)

*Proof of affineness.* Suppose \(Y_A\) is affine. Its canonical descent datum is effective in affine schemes by algebra descent. Let \(W\) be the descended affine \(R\)-scheme. The isomorphism \(W_A\simeq Y_A\), and its inverse, are compatible with descent. The full faithfulness for morphisms of schemes from *Faithfully flat descent*, Section 6 descends both maps. Their composites are identities because this can be checked after the covering. Thus \(Y\simeq W\) is affine. [Stacks, Tag 02L5](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-affine)

*Proof of closed immersions.* By affineness, write \(Y=\operatorname{Spec}B\). If \(Y_A\to\operatorname{Spec}A\) is a closed immersion, then \(A\to A\otimes_R B\) is surjective. Its cokernel is \(A\otimes_R\operatorname{coker}(R\to B)\), which can vanish only if \(R\to B\) is surjective. Hence \(Y\to S\) is a closed immersion. This also identifies its defining ideal through the descended ideal module. [Stacks, Tag 02L6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-closed-immersion)

*Proof of separatedness and quasi-separatedness.* Apply the closed-immersion result to the diagonal

\[
\Delta_f:Y\longrightarrow Y\times_S Y
\]

and the fpqc covering of its target induced by \(S_1\to S\). This proves separatedness. Apply quasi-compactness descent to the same diagonal to prove quasi-separatedness. [Stacks, Tags [02KU](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-separated), [02KR](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-quasi-separated)]

*Proof of local finite type and local finite presentation.* We first prove the algebra assertions. If \(A\otimes_R B\) is a finitely generated \(A\)-algebra, choose generators and express them using finitely many elements \(b_1,\ldots,b_n\in B\). Set \(B_0=R[b_1,\ldots,b_n]\). Flatness identifies \(A\otimes_R B_0\) with a subalgebra of \(A\otimes_R B\); by construction it is the whole algebra. Faithful flatness applied to \(B/B_0\) yields \(B=B_0\).

Now suppose \(A\otimes_R B\) is finitely presented. We have a surjection \(P=R[X_1,\ldots,X_n]\to B\), with kernel \(J\). Flatness gives kernel \(A\otimes_R J\) for \(A[X]\to A\otimes_R B\). This ideal is finitely generated. For completeness, the last assertion holds for *any* chosen finite set of algebra generators of a finitely presented algebra. Given another presentation \(A[Z]/(r_1,\ldots,r_m)\), express \(Z_j=h_j(X)\) and \(X_i=g_i(Z)\) in that algebra. The kernel for the \(X\)-presentation is generated by

\[
r_j(h(X)),\qquad X_i-g_i(h(X)).
\]

Indeed these relations construct inverse maps between the two quotient presentations. Express generators of \(A\otimes_R J\) using finitely many elements of \(J\), and let \(J_0\) be their \(P\)-span. Then \(A\otimes_R(J/J_0)=0\), so \(J=J_0\). This proves that \(B\) is finitely presented over \(R\).

For schemes, take any affine open \(U\subset Y\) over an affine open of \(S\). Its base change \(U_A\) is affine. Local finite type, or local finite presentation, of \(Y_A\to S_1\) gives the corresponding algebra property on \(U_A\). The algebra arguments give it on \(U\). Thus both properties descend. [Stacks, Tags [02KX](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-locally-finite-type), [02KY](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-locally-finite-presentation)]

*Proof of flatness.* For \(U=\operatorname{Spec}B\) as above, the \(A\)-module \(A\otimes_R B\) is flat. Given an injection \(M\to N\) of \(R\)-modules, the kernel of \(M\otimes_R B\to N\otimes_R B\) becomes zero after tensoring with \(A\), by flatness over \(A\). Faithfulness makes that kernel zero. Thus \(B\) is flat over \(R\), proving flatness of \(f\). [Stacks, Tag 02L2](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-flat)

*Proof of properness.* Finite type is quasi-compactness plus local finite type, both now proved to descend. A proper morphism is a separated, finite type, universally closed morphism. Each of its three conditions descends by the preceding arguments. [Stacks, Tag 02L1](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-proper)

*Proof of smoothness.* We use the local smoothness criterion: a morphism is smooth exactly when it is flat, locally of finite presentation, and has geometrically regular fibres; equivalently its fibres are smooth over their residue fields. The implication from the three conditions is [Stacks, Tag 01V8](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-smooth-flat-smooth-fibres); for schemes locally of finite type over a field, geometric regularity is equivalent to smoothness [Stacks, Tag 038X](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-geometrically-regular-smooth). Flatness and local finite presentation already descend.

Fix \(s\in S\) and an algebraic closure \(\Omega\) of \(\kappa(s)\). Choose \(s_1\in S_1\) over \(s\), and a field \(E\) receiving maps from both \(\Omega\) and \(\kappa(s_1)\) over \(\kappa(s)\). Such a field is obtained from a prime of their nonzero tensor product. Smoothness upstairs implies that \(Y_s\times_{\kappa(s)}E\) is regular. The faithfully flat map

\[
Y_s\times_{\kappa(s)}E\longrightarrow
Y_s\times_{\kappa(s)}\Omega
\]

descends regularity by Proposition 1.1. These schemes are locally Noetherian because their morphisms to their fields are locally of finite presentation. Thus the fibre over \(s\) is geometrically regular. The criterion proves smoothness of \(f\). [Stacks, Tag 02VL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-smooth)

*Proof of étaleness.* After smoothness descent, use the fact that an étale morphism is a smooth morphism of relative dimension zero. On an affine open \(\operatorname{Spec}B\subset Y\), base change for differentials gives

\[
\Omega_{B/R}\otimes_R A\simeq
\Omega_{(B\otimes_R A)/A}=0.
\]

Hence \(\Omega_{B/R}=0\). For a smooth morphism the rank of the differential module is the relative dimension, so every relative dimension is zero. Thus \(f\) is étale. [Stacks, Tag 02VN](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-etale)

*Proof of finiteness.* Affineness gives \(Y=\operatorname{Spec}B\) in (2.1). If \(A\otimes_R B\) is a finite \(A\)-module, finite generation of modules descends, so \(B\) is a finite \(R\)-module. This proves finiteness. [Stacks, Tag 02LA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-finite) \(\square\)

The following additional target-local results will be used with the indicated precise sources. They are not further proofs of Theorem 2.1.

| Property, fpqc-local on the target | Locator |
| --- | --- |
| Surjective | [Stacks, Tag 02KV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-surjective) |
| Universally injective | [Stacks, Tag 02KW](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-universally-injective) |
| Open immersion | [Stacks, Tag 02L3](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-open-immersion) |
| Integral | [Stacks, Tag 02L9](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-integral) |
| Quasi-finite | [Stacks, Tag 02VI](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-quasi-finite) |
| Unramified | [Stacks, Tag 02VM](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-property-unramified) |

For example, every field extension \(K/k\) is faithfully flat. Theorem 2.1 therefore proves that a \(k\)-scheme is proper, smooth or affine precisely when its base change to \(K\) has that property. No algebraicity or separability assumption on \(K/k\) is needed.

## 3. Covering the source

Let \(g:T\to Y\) be a covering and \(f:Y\to S\). Here the question is whether a property of the composite \(fg\) detects a property of \(f\). This is a different diagram from a covering of the target.

**Proposition 3.1.** Flatness is fpqc-local on the source. In fact, if \(g\) is any surjective flat morphism and \(fg\) is flat, then \(f\) is flat.

*Proof.* Choose \(t\in T\) above \(y\in Y\), and put \(s=f(y)\). The local maps are

\[
A=\mathcal O_{S,s}\longrightarrow
B=\mathcal O_{Y,y}\longrightarrow
C=\mathcal O_{T,t}.
\]

The second map is faithfully flat, and \(C\) is flat over \(A\). For an injection \(M\to N\) of \(A\)-modules, the kernel of
\(M\otimes_A B\to N\otimes_A B\) becomes zero on tensoring over \(B\) with \(C\). It is therefore zero. This proves flatness of \(B/A\), at every \(y\). Ascent follows from composition of flat maps. Apply the argument member by member for a covering family. [Stacks, Section 036F](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-properties-morphisms-local-source) \(\square\)

**Proposition 3.2.** Local finite presentation is fppf-local on the source.

We will prove this using one standard approximation input: a flat finitely presented algebra over a filtered colimit of rings is the base change of a flat finitely presented algebra at some stage. We also use the algebra characterization of finite presentation by preservation of filtered colimits of homomorphism sets. Exact locators for these inputs are listed at the end of the lesson.

*Proof.* First consider \(R\to B\to C\), with \(C/R\) finitely presented and \(C/B\) faithfully flat and finitely presented. Put \(D=C\otimes_B C\). The algebra \(D\) is finitely presented over \(R\), since \(C\to D\) is a base change of \(B\to C\).

Let \(L=\varinjlim L_\lambda\) be a filtered colimit of \(R\)-algebras, and let \(B\to L\) be a map. Approximate the flat finitely presented \(L\)-algebra \(C_L=L\otimes_B C\) by a flat finitely presented \(L_\lambda\)-algebra \(C_\lambda\). It can be made faithfully flat. Indeed its image in \(\operatorname{Spec}L_\lambda\) is open; write its closed complement as \(V(I)\). Images commute with base change, so \(IL=L\). An equation \(1=\sum a_i x_i\) uses finitely many elements and holds already at a later stage. At that stage the complement is empty.

Set \(D_\lambda=C_\lambda\otimes_{L_\lambda}C_\lambda\). Finite presentation of \(C/R\) and \(D/R\) descends the maps \(C\to C_L\), \(D\to C_L\otimes_L C_L\), and the two commuting squares, to one sufficiently late stage. The composite \(B\to C\to C_\lambda\) then lands in

\[
\operatorname{Eq}(C_\lambda\rightrightarrows D_\lambda)
=L_\lambda
\]

by faithfully flat module descent. It gives a factorization of \(B\to L\) through \(L_\lambda\).

To see that this factorization property proves finite presentation, express \(B\) as a filtered colimit of finitely presented \(R\)-algebras and apply it to \(\operatorname{id}_B\). Then \(B\) is a retract of one such algebra \(P\). A retract is finitely presented: write \(P=R[X_1,\ldots,X_n]/(r_1,\ldots,r_m)\), let \(e:P\to P\) be the corresponding idempotent, and represent \(e(X_i)\) by polynomials \(h_i(X)\). Its image is the quotient of \(P\) by the finitely many relations \(X_i-h_i(X)\). This quotient identifies with \(\operatorname{im}e\), since applying \(e\) kills those relations and is inverse to the induced inclusion. Thus \(B/R\) is finitely presented.

Now work over affine opens in \(S\) and \(Y\). A surjective flat locally finitely presented covering is open. Finitely many affine pieces of its members have images covering the affine open of \(Y\). Their disjoint union gives precisely the ring diagram just treated. This proves descent of local finite presentation. Ascent is composition of locally finitely presented morphisms. Compare [Stacks, Tags [02KK](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-flat-finitely-presented-permanence-algebra), [02KL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-flat-finitely-presented-permanence); Section [036J](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-fpqc-local-source)]. \(\square\)

**Corollary 3.3.** Smoothness is smooth-local on the source, and étaleness is étale-local on the source.

*Proof.* Suppose a smooth surjective covering \(g\) has smooth composites \(fg\). Propositions 3.1 and 3.2 give flatness and local finite presentation of \(f\). On every geometric fibre, the base change of \(g\) is a faithfully flat covering by regular schemes, since the composites are smooth. Proposition 1.1 makes that geometric fibre regular. The smoothness criterion proves \(f\) smooth. Composition gives the converse.

If both \(g\) and \(fg\) are étale, the same argument first proves \(f\) smooth. On a geometric fibre, choose a point upstairs above any point downstairs. The local map is flat and local, while the upstairs local ring is a field. Going down lifts any chain of primes downstairs to a chain upstairs; hence the downstairs local ring has dimension zero. A smooth geometric fibre thus has relative dimension zero everywhere, and \(f\) is étale. Again the converse is composition. [Stacks, Sections [036T](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-smooth-local-source), [036V](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-etale-local-source)] \(\square\)

Closed immersion is not even Zariski-local on the source. The fold map \(S\amalg S\to S\) restricts to an isomorphism on each of the two open components, but it is not a monomorphism when \(S\ne\varnothing\), so cannot be a closed immersion. The finite presentation condition on the covering in Proposition 3.2 was needed to approximate its algebra and its two projections; Proposition 3.1 has no such condition.

## 4. The affine hull of a quasi-affine object

A morphism is **quasi-affine** if it factors as a quasi-compact open immersion into an affine morphism. The quasi-compact condition is part of our convention.

**Lemma 4.1 (flat base change for the affine hull).** Let \(V\) be a quasi-compact separated scheme over \(\operatorname{Spec}R\). For every flat \(R\)-algebra \(A\),

\[
A\otimes_R\Gamma(V,\mathcal O_V)
\simeq\Gamma(V_A,\mathcal O_{V_A}).
\tag{4.1}
\]

*Proof.* Choose a finite affine open covering \(V=\bigcup V_i\). Because \(V/R\) is separated, every \(V_i\cap V_j\) is affine: it is the inverse image of the closed diagonal in the affine scheme \(V_i\times_R V_j\). Sections are the kernel of

\[
\prod_i\Gamma(V_i,\mathcal O)\longrightarrow
\prod_{i,j}\Gamma(V_i\cap V_j,\mathcal O),
\]

where the map takes differences of restrictions. Flat tensor product preserves that kernel and the finite products. The resulting diagram is exactly the sections diagram for \(V_A\). This proves (4.1), compatibly with all maps involved. In particular, localization gives \(\Gamma(V,\mathcal O)_r\simeq\Gamma(V_r,\mathcal O)\). \(\square\)

**Lemma 4.2.** If \(V\) is a quasi-compact open subscheme of an affine scheme, its canonical map

\[
j:V\longrightarrow\operatorname{Spec}\Gamma(V,\mathcal O_V)
\]

is a quasi-compact open immersion.

*Proof.* Write \(V\subset\operatorname{Spec}C\), and choose finitely many principal opens \(D_C(f_i)\) covering \(V\) and contained in it. Put \(B=\Gamma(V,\mathcal O_V)\). Apply Lemma 4.1 to the flat localization \(C\to C_{f_i}\). It gives

\[
B_{f_i}=\Gamma(D_C(f_i),\mathcal O)=C_{f_i}.
\]

Consequently \(j\) identifies each \(D_C(f_i)\) with \(D_B(f_i)\). These identifications agree on intersections, which are the corresponding principal opens for \(f_if_j\). They glue to an isomorphism from \(V\) onto \(\bigcup_iD_B(f_i)\). The image is quasi-compact because the union is finite. \(\square\)

**Theorem 4.3 (quasi-affine effectivity).** Quasi-affine morphisms with descent data for an fpqc covering descend effectively to quasi-affine morphisms. Quasi-affineness of an existing morphism is fpqc-local on the target. [Stacks, Tag 0247](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-quasi-affine)

*Proof for a faithfully flat affine covering.* Let \(R\to A\) be faithfully flat, and let \(V\to\operatorname{Spec}A\) be quasi-affine with a descent datum. Set \(B=\Gamma(V,\mathcal O_V)\). By (4.1), the datum induces a descent datum on the \(A\)-algebra \(B\), including its multiplication and unit. Algebra descent gives an \(R\)-algebra \(B_0\), with \(A\otimes_R B_0\simeq B\). Put \(W=\operatorname{Spec}B_0\). Lemma 4.2 identifies \(V\) with a quasi-compact open in \(W_A\), compatibly with the original datum.

This open is saturated under the two projections \(W_A\times_W W_A\rightrightarrows W_A\). Indeed the descent isomorphism identifies its two inverse images. For a surjective map of schemes, saturation says it is the inverse image of its image as a subset: any two points over the same point have a common point on the fibre product, by the residue-field tensor product argument used in (2.2). Let \(U\subset W\) be that image. The faithfully flat quotient property makes \(U\) open. Its topology is quasi-compact, being a continuous image of \(V\). Give it the open subscheme structure. Then \(U_A=V\) as open subschemes, with precisely the prescribed datum. The morphism \(U\to\operatorname{Spec}R\) is quasi-affine.

*Passage to arbitrary coverings.* Morphisms of type \(\mathcal P\) **satisfy descent** for a topology if every descent datum whose local objects have \(\mathcal P\) is effective as a scheme. This definition does not assert that the descended morphism has \(\mathcal P\) [Stacks, Tag 02W2](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-definition-descending-types-morphisms).

We spell out the general descent criterion used here. Let a class of morphisms \(\mathcal P\) be stable under base change and finite disjoint unions. Assume its descent data are effective for faithfully flat affine coverings of affine bases. Then every fpqc descent datum whose objects have \(\mathcal P\) is effective as a scheme. If the affine-base construction also gives \(\mathcal P\), and \(\mathcal P\) is Zariski-local on the target, the result has \(\mathcal P\).

To prove the criterion, restrict to an affine open of the base and take a finite affine refinement of the covering. Pull the given objects and their datum to that refinement, and combine them by finite disjoint union. The assumed affine result gives an object over this affine base. It represents the original datum, including members not chosen in the refinement: over any original member the refinement is an fpqc covering, and its comparison isomorphisms descend by full faithfulness for morphisms of schemes. Their inverses also descend, and compatibility on double overlaps follows by checking after the same refinement; triple-overlap equations descend by faithfulness. Finally repeat over affine opens of the base. The local objects have uniquely descended comparison isomorphisms on intersections, satisfying the cocycle. Zariski gluing produces the global scheme. This proves the criterion [Stacks, Tags [02W2](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-definition-descending-types-morphisms), [02W3](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-types-morphisms)]. It asserts existence of a scheme first; a property of that scheme still needs its own argument.

The same proof gives the version for fppf, étale, smooth and syntomic coverings: in the affine assumption use, respectively, a surjective flat finitely presented, étale, smooth or syntomic morphism of affines. Such a covering of an affine base has a finite affine refinement. Indeed its maps are flat and locally finitely presented, hence open; affine pieces upstairs have open images covering the quasi-compact base, so finitely many suffice. Their affine maps are finitely presented, and retain the specified type under restriction. Refinement full faithfulness applies since these coverings are fpqc. Thus the argument proves the full criterion in [Stacks, Tag 02W3](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-descending-types-morphisms).

Quasi-affine morphisms satisfy the fpqc conditions. A finite disjoint union over an affine base embeds as a quasi-compact open in the spectrum of the product of the corresponding algebras. For a general base, do this on affine opens. The construction above is quasi-affine on every affine open of the base. To obtain a global affine hull, Lemma 4.1 for localizations shows that \(f_*\mathcal O_U\) is quasi-coherent: on an affine base its sections localize correctly. The canonical map \(U\to\underline{\operatorname{Spec}}_S f_*\mathcal O_U\) is a quasi-compact open immersion on every affine open of \(S\), hence globally. Thus the construction proves the asserted type as well as effectivity.

Finally apply effectivity to the canonical datum of a given \(Y\to S\) whose pullbacks are quasi-affine. The descended quasi-affine scheme is isomorphic to \(Y\) by full faithfulness for schemes, just as in the affineness proof. This proves permanence. \(\square\)

## 5. Étale objects and quotient topology

For a morphism \(p:X\to S\), let \(\operatorname{Desc}_{\mathrm{\acute et}}(X/S)\) be the category of étale \(X\)-schemes with descent data on \(X\times_S X\), and compatible morphisms. Pullback gives

\[
\operatorname{\acute Et}(S)\longrightarrow
\operatorname{Desc}_{\mathrm{\acute et}}(X/S).
\tag{5.1}
\]

Here \(\operatorname{\acute Et}(S)\) includes non-separated and non-quasi-compact étale schemes. A **submersive** map is surjective and gives the target its quotient topology.

**Proposition 5.1 (faithfulness).** If \(p\) is surjective, (5.1) is faithful. [Stacks, Tag 0BTJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-lemma-faithful)

*Proof.* Let \(a,b:U_1\to U_2\) be two \(S\)-maps between étale schemes with equal pullbacks. The equalizer is an open subscheme of \(U_1\): it is the inverse image of the open diagonal of \(U_2/S\). Its pullback to \(X\) is all of \(U_{1,X}\). Surjectivity of \(U_{1,X}\to U_1\) makes its underlying open all of \(U_1\). Hence it is \(U_1\), and \(a=b\). \(\square\)

**Theorem 5.2 (full faithfulness).** Suppose that every base change
\(X\times_S T\to T\), for \(T\to S\) étale, is submersive. Then (5.1) is fully faithful. In particular this holds if \(p\) is universally submersive. [Stacks, Tags [0BTK](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-lemma-fully-faithful), [0BTL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-lemma-fully-faithful-cases)]

*Proof.* Consider a compatible map \(a_X:U_{1,X}\to U_{2,X}\) between the pullbacks of two étale \(S\)-schemes. Its graph is an open subscheme

\[
G_X\subset W_X,\qquad W=U_1\times_S U_2.
\]

Indeed the graph is a section of the étale projection \(W_X\to U_{1,X}\). Compatibility says that the two inverse images of this open in \(W_X\times_W W_X\) coincide. The residue-field tensor product argument from (2.2) shows that \(G_X\) is saturated. It is therefore the inverse image of its image \(G\subset W\). Since \(W\to S\) is étale, \(W_X\to W\) is submersive by hypothesis. Thus \(G\) is open.

The map \(G\to U_1\) is étale and becomes an isomorphism after pulling back to \(X\). We check its geometric fibres. Given a geometric point of \(U_1\), lift its image in \(S\) to \(X\), then extend the two residue fields to a common field. Over that field the fibre is a single point with no residue extension, because \(G_X\to U_{1,X}\) is an isomorphism. An étale scheme over an algebraically closed field is a disjoint union of copies of that field; extension of the field preserves the indexing set. Hence each geometric fibre of \(G\to U_1\) is one point. The map is étale and universally bijective, so is an isomorphism [Stacks, Tag 025G](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-theorem-etale-radicial-open). Its inverse, followed by \(G\hookrightarrow W\to U_2\), gives the required map \(U_1\to U_2\). Proposition 5.1 proves uniqueness. \(\square\)

Universally open surjections and universally closed surjections are universally submersive. Surjective integral maps are universally closed, so satisfy Theorem 5.2. Fpqc coverings also give quotient maps after every base change, by the quotient theorem in *Faithfully flat descent*, Section 6. Full faithfulness therefore holds in all these cases. Mere submersivity of \(X\to S\) would not justify the step with the étale scheme \(W\); the base-change hypothesis has a precise role.

**Lemma 5.3 (local effectivity).** Let \(\{S_i\}\) be an open covering of \(S\). A fixed étale datum is effective if it is effective on every \(S_i\), (5.1) is fully faithful over every \(S_i\cap S_j\), and (5.1) is faithful over every \(S_i\cap S_j\cap S_k\).

*Proof.* Let \(U_i\) represent the datum over the opens \(S_i\). On \(S_i\cap S_j\), the prescribed isomorphism of their pullbacks descends uniquely by full faithfulness. The descended maps are isomorphisms because their inverses descend too. Their cocycle equation follows from faithfulness on triple intersections. Glue the schemes \(U_i\) and their étale maps to the base. The given isomorphisms to the datum glue as well, producing the required effective object. [Stacks, Tag 0BTM](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-lemma-reduce-to-affine) \(\square\)

The following observation will keep the type of the descended object under control without assuming flatness.

**Lemma 5.4.** Let \(p:X\to S\) be surjective and integral, and let \(U\to S\) already be étale. If \(U_X\to X\) is quasi-compact and separated, then \(U\to S\) is quasi-compact and separated. If \(U_X\to X\) is finite, then \(U\to S\) is finite.

*Proof.* Restrict to an affine open of \(S\). Then \(X\) is affine, \(U_X\) is quasi-compact, and its continuous surjective image \(U\) is quasi-compact. For separatedness, the diagonal of \(U/S\) is an open immersion. Its image becomes closed after the integral surjective map

\[
(U\times_S U)_X\longrightarrow U\times_S U,
\]

because \(U_X/X\) is separated. A closed surjection is a quotient map, so this image is closed downstairs. An open immersion with closed image is a closed immersion: the image is open and closed, and its complementary open defines the required ideal sheaf. Thus \(U/S\) is separated.

If \(U_X/X\) is finite, it is universally closed. The proof using (2.2) and quotient topology descends universal closedness through the integral surjection. The étale map \(U/S\) is locally of finite presentation and quasi-finite; quasi-compactness makes it of finite type. Hence it is proper and quasi-finite. To see that this makes it finite, apply Zariski's Main Theorem locally on an affine base: \(U\) is an open in a finite \(S\)-scheme \(T\). Since \(U/S\) is proper and \(T/S\) separated, the open immersion \(U\to T\) is proper. Its image is therefore open and closed, and \(U\) is finite over \(S\). \(\square\)

## 6. Effective descent along an arbitrary integral surjection

We now prove the result needed for singular curves.

**Theorem 6.1.** Quasi-compact separated étale morphisms have effective fpqc descent.

*Proof.* On an affine base, a quasi-compact separated étale scheme is quasi-finite and separated. Zariski's Main Theorem factors it as a quasi-compact open in a finite scheme [Stacks, Tag 05K0](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-quasi-finite-separated-pass-through-finite). It is therefore quasi-affine. Apply Theorem 4.3 to a standard faithfully flat affine covering. Its descended morphism is étale, quasi-compact and separated by Theorem 2.1. The refinement and gluing criterion proved in Section 4 gives effectivity for arbitrary fpqc coverings, with the same properties. \(\square\)

Zariski's Main Theorem, as used here, says that a quasi-finite separated map to a quasi-compact quasi-separated base factors as a quasi-compact open immersion followed by a finite map. It is a prerequisite, rather than an integral-descent result.

For the non-flat case we use three precise facts about henselian pairs. A pair \((A,I)\) is henselian if \(I\) lies in the Jacobson radical and coprime monic factorizations modulo \(I\) lift. The facts are:

- Every pair has a henselization \((A^h,IA^h)\), with \(A\to A^h\) flat and \(A/I\simeq A^h/IA^h\) [Stacks, Tag 0AGU](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-henselization-flat).
- For a henselian pair, idempotents lift uniquely in every integral \(A\)-algebra from its quotient modulo \(I\) [Stacks, Tag 09XI](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-characterize-henselian-pair).
- Reduction induces an equivalence between finite étale \(A\)-algebras and finite étale \(A/I\)-algebras [Stacks, Tag 09ZL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-finite-etale-equivalence).

These are the henselian-pair prerequisites; we will prove the splitting, gluing and approximation arguments that use them.

**Lemma 6.2 (splitting off the finite part).** Let \((A,I)\) be henselian and \(V\to\operatorname{Spec}A\) quasi-compact, separated and étale. Suppose \(V_{A/I}\) is finite. There is a unique decomposition

\[
V=V_{\mathrm{fin}}\amalg V_{\mathrm{away}},
\tag{6.1}
\]

where \(V_{\mathrm{fin}}/A\) is finite and \(V_{\mathrm{away}}\) has empty inverse image of \(V(I)\). This decomposition commutes with any integral base change of henselian pairs.

*Proof.* Embed \(V\) as a quasi-compact open in a finite \(A\)-scheme \(T\), using Zariski's Main Theorem. The open immersion \(V_{A/I}\to T_{A/I}\) is also closed, because its source is finite over \(A/I\) and its target is separated. It is therefore a union of connected components, specified by an idempotent of the finite algebra of \(T_{A/I}\). Lift this idempotent uniquely. It splits \(T=T_0\amalg T_1\), with \((T_0)_{A/I}=V_{A/I}\).

Every closed point of \(T_0\) maps to a closed point of \(\operatorname{Spec}A\), since \(T_0/A\) is integral. Every such base point lies in \(V(I)\), because \(I\) is in the Jacobson radical. Thus every closed point of \(T_0\) belongs to the open \(V\). The closed subset \(T_0\setminus V\), if nonempty, would contain a closed point: \(T_0\) is affine and any proper ideal is contained in a maximal ideal. Therefore \(T_0\subset V\). Set \(V_{\mathrm{fin}}=T_0\) and \(V_{\mathrm{away}}=V\cap T_1\).

To prove uniqueness, compare two finite open-and-closed parts \(F,G\subset V\) with the prescribed closed fibre. The open-and-closed subscheme \(F\setminus G\) is finite over \(A\) and has zero reduction modulo \(I\). Its finite algebra is zero by Nakayama's lemma, so \(F\subset G\). Interchanging them proves equality. This also proves compatibility with integral base changes: the pulled-back decomposition still has a finite part and an away part, and uniqueness identifies it with the new decomposition. The base-changed pair is henselian, because any integral algebra over it is integral over \(A\), so the idempotent characterization applies. Compare [Stacks, Tags [0BTN](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-lemma-split-henselian), [09XK](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-integral-over-henselian-pair)]. \(\square\)

**Lemma 6.3 (a finite neighbourhood of the generic point).** Let \(Z\) be integral and \(U/Z\) quasi-compact, separated and étale. Some nonempty open of \(Z\) has \(U\) finite étale over it.

*Proof.* Work on an affine open \(Z=\operatorname{Spec}R\), with fraction field \(K\). Choose a finite affine covering \(U=\bigcup\operatorname{Spec}B_i\). Each \(B_i\) is a finitely generated \(R\)-algebra and \(B_i\otimes_R K\) is a finite étale \(K\)-algebra. Its generators satisfy monic equations over \(K\). Localize \(R\) at one nonzero element to put all coefficients in \(R\), and then at another to make the finitely many equations hold in each \(B_i\). These equations hold over \(K\), so their errors are torsion; finitely many errors can be killed by one localization. Now every \(B_i\) is finite over \(R\).

Each affine open \(U_i=\operatorname{Spec}B_i\) is consequently also closed in \(U\): a map from a proper \(R\)-scheme to a separated \(R\)-scheme is proper, so the open immersion \(U_i\to U\) has closed image. A finite covering by open-and-closed finite schemes can be made disjoint by successively removing the earlier members. Each resulting part is an open-and-closed subscheme of one \(U_i\), hence finite. Their finite disjoint union is \(U\), proving finiteness after this localization. Compare [Stacks, Tag 02NW](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-generically-finite). \(\square\)

**Theorem 6.4 (integral descent).** For any surjective integral morphism \(p:X\to S\), pullback is an equivalence

\[
\left\{\begin{array}{c}
\text{quasi-compact separated}\\
\text{étale }S\text{-schemes}
\end{array}\right\}
\simeq
\left\{\begin{array}{c}
\text{quasi-compact separated étale }X\text{-schemes}\\
\text{with descent data for }X/S
\end{array}\right\}.
\tag{6.2}
\]

No finite type or Noetherian condition is imposed on \(X\), \(S\), or \(p\). [Stacks, Tag 0BTP](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-proposition-effective)

*Proof.* Full faithfulness follows from Theorem 5.2, since an integral surjection is universally closed. It holds after every base change. Let \((V,\theta)\) be a datum on the right. Lemma 5.3 reduces existence to affine opens in \(S\). After constructing an étale object, Lemma 5.4 will give its quasi-compactness and separatedness. We prove existence in three steps.

**Step A: the henselian extension step.** Suppose

\[
S=\operatorname{Spec}A,\quad X=\operatorname{Spec}B,\quad
Z=V(I),\quad S^\circ=S\setminus Z,
\]

with \((A,I)\) henselian and \(B/A\) integral and surjective on spectra. Assume the datum is already effective over \(S^\circ\), represented by \(U^\circ\), and its restriction to \(Z\) is represented by a finite étale \(U_Z/Z\).

Lift \(U_Z\) to a finite étale \(U_{\mathrm{fin}}/S\) by the henselian equivalence. The pair \((B,IB)\) is henselian: the idempotent characterization in integral algebras is preserved by transitivity of integrality. Apply Lemma 6.2 to \(V/B\), whose closed fibre is the pullback of \(U_Z\). It gives (6.1). Over \(B\otimes_A B\), both pulled-back decompositions have the same finite closed fibre. Uniqueness makes \(\theta\) preserve their finite and away parts.

The finite étale equivalence for \((B,IB)\) gives a unique isomorphism

\[
(U_{\mathrm{fin}})_X\simeq V_{\mathrm{fin}}
\]

lifting the prescribed isomorphism on the closed fibre. Applying the same uniqueness over \(B\otimes_A B\) makes this isomorphism compatible with the datum. Full faithfulness over \(S^\circ\) now descends the inclusion of this part to a map

\[
(U_{\mathrm{fin}})_{S^\circ}\longrightarrow U^\circ.
\tag{6.3}
\]

This is an open-and-closed immersion. To verify that assertion without flatness, its upstairs image is open and closed and saturated, so the integral quotient map on \(U^\circ\) descends that image to an open-and-closed subset. The map to the corresponding open subscheme is étale and becomes an isomorphism upstairs. The geometric-fibre argument in Theorem 5.2 makes it an isomorphism.

Glue \(U_{\mathrm{fin}}\) and \(U^\circ\) along (6.3). This is gluing along open subschemes, hence produces an étale \(U/S\). After pullback to \(X\), the finite part covers \(V_{\mathrm{fin}}\), and the other piece covers \(V\) off \(V(IB)\). Together they cover \(V\), because \(V_{\mathrm{away}}\) is away from that closed set. The comparison maps agree on their intersection and glue to an isomorphism \(U_X\simeq V\) respecting the datum. This completes the henselian extension step.

**Step B: finite surjections over a Noetherian affine base.** Suppose \(A\) is Noetherian and \(B\) is finite over \(A\). The empty open is an effectivity locus. Since a Noetherian space has no infinite strictly ascending chain of opens, there is a maximal open \(S^\circ\subset S\) over which the datum is effective. The union of two effectivity opens is an effectivity open by Lemma 5.3.

If \(S^\circ\ne S\), choose a generic point \(\xi\) of an irreducible component of \(S\setminus S^\circ\). Restrict to an affine neighbourhood of \(\xi\) avoiding the other components. Give the remaining closed complement \(Z\) its reduced structure, so \(Z\) is integral. We may shrink this neighbourhood further whenever necessary.

The finite map \(X_Z\to Z\) becomes flat on a nonempty open of \(Z\). Here an elementary generic freeness argument suffices. For a finite module \(M\) over a domain \(R\), choose lifts of a basis of \(M\otimes_R\operatorname{Frac}(R)\). The resulting map \(R^d\to M\) is injective, and its cokernel is finitely generated torsion; one nonzero element kills that cokernel after localization. Thus \(M\) becomes free. Apply this to the finite algebra of \(X_Z\). Its generic rank is positive by surjectivity, so the resulting finite flat map is faithfully flat.

Theorem 6.1 now descends \(V_Z\) to a quasi-compact separated étale \(U_Z/Z\). Lemma 6.3 lets us shrink once more to make \(U_Z/Z\) finite. A shrinking inside \(Z\) can be realized by shrinking the ambient affine neighbourhood of \(\xi\), so all these reductions preserve our extension problem.

Write \(Z=V(I)\). Replace \(A\) by its henselization \(A^h\) along \(I\). The closed fibre is still \(Z\), and the datum remains effective on the inverse image of \(S^\circ\). Step A produces an effective object over \(\operatorname{Spec}A^h\). It is quasi-compact and separated by Lemma 5.4.

The family

\[
\{\operatorname{Spec}A^h\to S,\ S^\circ\to S\}
\tag{6.4}
\]

is fpqc. Both maps are flat. The first covers \(Z\), since its reduction is the identity on \(Z\); the second covers the complement. The open \(S^\circ\) is quasi-compact because \(S\) is Noetherian, so it can be refined by finitely many affine principal opens. Together with the affine henselization this gives a finite affine faithfully flat refinement.

The objects over the two members of (6.4) have uniquely descended comparison isomorphisms on all double overlaps, by full faithfulness for the base-changed integral surjection. Their cocycle follows by faithfulness on triple overlaps. Theorem 6.1 descends this fpqc datum to an étale scheme on the ambient neighbourhood. Thus we have an effectivity open containing \(\xi\); its union with \(S^\circ\) is strictly larger. This contradicts maximality, proving effectivity over all of \(S\). Notice that the henselized ring was not assumed Noetherian: Step A did not require it.

**Step C: arbitrary integral surjections.** Now let \(A\to B\) be any integral map with surjective spectrum. Construct a directed system of subrings

\[
A_\lambda\subset A,\qquad B_\lambda\subset B
\]

as follows. Choose finitely many elements of \(A\), finitely many of \(B\), and coefficients of monic equations satisfied by the chosen elements of \(B\). Let \(A_\lambda\) be the ring generated by the selected elements and coefficients over \(\mathbf Z\), and \(B_\lambda\) the \(A_\lambda\)-algebra generated by the selected elements of \(B\). Enlarging the finite selections gives

\[
(A\to B)=\varinjlim_\lambda(A_\lambda\to B_\lambda),
\tag{6.5}
\]

with every \(A_\lambda\) of finite type over \(\mathbf Z\) and every \(B_\lambda\) finite over \(A_\lambda\) [Stacks, Tag 0BTG](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-limit-integral). Every stage is already surjective on spectra. Indeed \(\ker(A\to B)\) consists of nilpotents, since the spectrum map is surjective. The kernel of \(A_\lambda\to B_\lambda\) is its intersection with \(A_\lambda\), and consists of nilpotents as well. The image of an integral spectrum map is the vanishing set of its kernel, so this image is all of \(\operatorname{Spec}A_\lambda\).

The morphism \(V\to\operatorname{Spec}B\) is of finite presentation: it is étale, quasi-compact and separated. Finite-presentation approximation descends \(V\) to a finitely presented \(V_\lambda/B_\lambda\). It also descends \(\theta\), its inverse, and their equations. This is applied over the systems

\[
B_\lambda,\quad B_\lambda\otimes_{A_\lambda}B_\lambda,\quad
B_\lambda\otimes_{A_\lambda}B_\lambda\otimes_{A_\lambda}B_\lambda,
\]

whose colimits are the required single, double and triple bases. The isomorphism and cocycle use only finitely many maps and equations, so they all hold at a single sufficiently late stage. The exact approximation input is [Stacks, Tag 01ZM](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/limits.html#limits-lemma-descend-finite-presentation). At a later stage \(V_\lambda/B_\lambda\) is separated and étale, by [Stacks, Tags [01ZQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/limits.html#limits-lemma-descend-separated-finite-presentation), [07RP](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/limits.html#limits-lemma-descend-etale)]; quasi-compactness is already part of finite presentation.

Step B applies to this finite surjection over the Noetherian ring \(A_\lambda\). Let \(U_\lambda/A_\lambda\) be its effective object. Base change to \(A\) gives an étale \(U/S\) whose pullback to \(B\), including its datum, is \((V,\theta)\). Lemma 5.4 supplies the required quasi-compactness and separatedness. Finally the affine solutions glue by Lemma 5.3 over an arbitrary \(S\). This proves (6.2) in its full stated generality. \(\square\)

## 7. A nodal cubic: the datum is a permutation

Let \(k\) be algebraically closed, of any characteristic. Consider the cubic

\[
C:\quad y^2z+xyz=x^3\subset\mathbf P^2_k.
\tag{7.1}
\]

Its node is \(c=[0:0:1]\). In the chart \(z=1\), the quadratic tangent cone is \(y(y+x)\), so its two tangent directions are distinct even in characteristic \(2\). The only other point with \(z=0\) is \([0:1:0]\), which is smooth since the derivative with respect to \(z\) is \(y^2+xy\).

The normalization is

\[
\nu:\mathbf P^1_k\longrightarrow C,\qquad
[P:Q]\longmapsto
[P(P+Q)Q:P^2(P+Q):Q^3].
\tag{7.2}
\]

The three coordinates have no simultaneous zero. On \(Q\ne0\), with \(t=P/Q\), the map reads

\[
x=t(t+1),\qquad y=t^2(t+1).
\tag{7.3}
\]

Substitution verifies (7.1). The affine polynomial \(y^2+xy-x^3\) is irreducible: a factorization of this monic quadratic in \(y\) would have polynomial roots \(a(x),b(x)\) with \(a+b=-x\) and \(ab=-x^3\). Each root would be a nonzero scalar times a power of \(x\), and no pair of such powers with exponents summing to \(3\) has sum \(-x\). Thus \(C\) is integral. Its function field contains \(t=y/x\), and (7.3) identifies it with \(k(t)\).

The affine map is finite, since \(t\) satisfies \(t^2+t-x=0\). Near \([0:1:0]\), put \(a=x/y\), \(b=z/y\). The equation is \((1+a)b=a^3\); on \(1+a\ne0\), this identifies the curve with an open of the \(a\)-line. Under (7.2), \(a=Q/P\), so \(\nu\) is an isomorphism there. These charts show that \(\nu\) is finite and birational. Since \(\mathbf P^1\) is normal, it is the normalization. The two inverse images of the node are

\[
a_0=0,\qquad a_1=-1.
\]

We need the *scheme* fibre product, not just its points. The relative differentials on the affine chart are the quotient of \(k[t]\,dt\) by

\[
(2t+1)\,dt,\qquad (3t^2+2t)\,dt.
\]

These coefficients generate the unit ideal. In characteristic \(2\) the first is \(1\); otherwise its possible zero is \(-1/2\), where the second is \(-1/4\ne0\). Together with the isomorphism near infinity, this proves that \(\nu\) is unramified. Its diagonal is therefore open, and it is also closed because \(\nu\) is separated.

Off the diagonal in the affine fibre product, invert \(t-u\). Equality of the \(x\)-coordinates gives \(t+u+1=0\). Equality of the \(y\)-coordinates, divided by \(t-u\), then gives \(t(t+1)=0\). Since \(t-u=2t+1\) is a unit at both roots, the off-diagonal algebra is \(k\times k\), with points \((a_0,a_1)\) and \((a_1,a_0)\). There are no off-diagonal points near infinity. Consequently

\[
\mathbf P^1\times_C\mathbf P^1
=
\mathbf P^1_{\mathrm{diag}}
\amalg\{(a_0,a_1)\}
\amalg\{(a_1,a_0)\},
\tag{7.4}
\]

with both isolated points reduced. The triple fibre product consists of the diagonal \(\mathbf P^1\) and the six reduced mixed triples from \(\{a_0,a_1\}^3\). For example, fixing an off-diagonal pair leaves the third coordinate in the node fibre
\(k[t]/(t(t+1))=k\times k\), so produces no hidden nilpotents.

**Lemma 7.1.** Every finite étale cover of \(\mathbf P^1_k\) is a finite disjoint union of copies of \(\mathbf P^1_k\).

*Proof.* A connected component \(D\) of such a cover is a smooth proper connected curve. It is integral: irreducible components of a regular curve are disjoint, so connectedness leaves one. Since \(k\) is algebraically closed, \(H^0(D,\mathcal O_D)=k\). Let its finite étale degree be \(d\ge1\). Étaleness gives

\[
\Omega_{D/k}\simeq f^*\Omega_{\mathbf P^1/k}.
\]

The degree of a pulled-back line bundle is multiplied by \(d\); this follows by pulling back divisors, since every fibre here has \(d\) points with multiplicity \(1\). The canonical degree formula for a smooth proper curve [Stacks, Tag 0C1A](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/curves.html#curves-lemma-genus-smooth) gives

\[
2g(D)-2=\deg\Omega_{D/k}=-2d.
\]

As \(g(D)\ge0\), this forces \(d=1\). A finite étale algebra of rank one is its base ring: the unit is a basis on every residue fibre and hence locally a basis by Nakayama. Therefore \(D\simeq\mathbf P^1\). The finite cover has only finitely many components, proving the statement. \(\square\)

Choose a finite set \(E\), and consider \(E\times\mathbf P^1\). On the diagonal part of (7.4), a descent isomorphism must be the identity: the cocycle on equal triples says that its square equals itself, and it is invertible. On \((a_0,a_1)\), it gives a bijection between the two labelled fibres, hence a permutation

\[
\sigma:E\longrightarrow E.
\]

The reverse pair must carry \(\sigma^{-1}\). These data satisfy every cocycle equation. The only triples with a change of branch compose \(\sigma\) with its inverse or an identity; no third branch creates an additional relation. Thus an arbitrary permutation determines a descent datum.

By Theorem 6.4 this datum has an effective quasi-compact separated étale \(C\)-scheme. Lemma 5.4 makes it finite, because its pullback is finite. Morphisms of trivial covers of \(\mathbf P^1\) are exactly maps of their finite indexing sets. Compatibility at the two off-diagonal points is exactly

\[
h\sigma=\tau h.
\]

We have proved the equivalence

\[
\operatorname{F\acute Et}(C)
\simeq
\{\text{finite sets }E\text{ with a permutation }\sigma\}.
\tag{7.5}
\]

Changing the labelling of \(E\) conjugates the permutation; it does not change the cover. A cycle of length \(n\) gives a connected cover built from \(n\) copies of \(\mathbf P^1\), with the \(a_0\)-point of each copy identified with the \(a_1\)-point of the next according to \(\sigma\). Its local node structure is the one supplied by étale descent. Connectedness can also be checked without assuming this picture: the orbits give open-and-closed subcovers by descent. If a single orbit split into two nonempty open-and-closed subcovers, their inverse images would give a partition of the sheets preserved by \(\sigma\), contradicting transitivity.

The Galois-category lesson will identify finite sets with a permutation with continuous finite \(\widehat{\mathbf Z}\)-sets. Thus (7.5) already contains the computation of the nodal cubic's fundamental group. The integral descent theorem is what turns the combinatorial datum into an actual cover of the singular scheme.

## 8. Exercises and complete solutions

**Exercise 8.1 (easy).** Let \(\{S_i\to S\}\) be an fpqc covering. Show directly that quasi-compactness of \(Y\to S\) follows from quasi-compactness of every \(Y_{S_i}\to S_i\).

*Solution.* Work over an affine open \(T\subset S\). Choose a finite affine refinement \(T_j\to T\) of the covering. Each \(Y_{T_j}\) is quasi-compact: it is the restriction of a quasi-compact morphism to an affine target. The finite disjoint union \(\coprod_jY_{T_j}\) is quasi-compact and maps surjectively to \(Y_T\). Its image is quasi-compact. Since this holds over every affine open of \(S\), \(Y\to S\) is quasi-compact. Surjectivity and the finite refinement are enough for this argument; the quotient-topology theorem is not needed.

**Exercise 8.2 (medium).** For a field extension \(K/k\), prove that \(Y/k\) is proper, smooth or affine exactly when \(Y_K/K\) has the corresponding property.

*Solution.* The map \(k\to K\) is flat because every vector space is free. It is faithful because a nonzero \(k\)-vector space remains nonzero after extending scalars. Thus it is an fpqc covering. Base change preserves each of the three properties. For descent, affineness follows from descending the affine algebra and its comparison isomorphism. Properness follows from descent of quasi-compactness, local finite type, the closed diagonal and universal closedness. Smoothness follows from flatness, local finite presentation and regularity of geometric fibres as in Theorem 2.1. These arguments apply to arbitrary schemes, and to arbitrary extensions \(K/k\).

**Exercise 8.3 (medium).** Descend a quasi-affine object over a faithfully flat affine covering using its affine hull. Explain both why the hull has descent data and why the open subset descends.

*Solution.* Let \(R\to A\) be faithfully flat and \(V/A\) quasi-affine with datum \(\theta\). Put \(B=\Gamma(V,\mathcal O_V)\). The two projections from \(\operatorname{Spec}(A\otimes_R A)\) are flat. Lemma 4.1 therefore identifies the two pulled-back section algebras with the section algebras of the pulled-back schemes. Applying sections to \(\theta\) supplies an algebra isomorphism, and functoriality transfers its cocycle. Descend this algebra to \(B_0/R\).

The canonical map \(V\to\operatorname{Spec}B\) is a quasi-compact open immersion by Lemma 4.2. It is preserved by the induced datum, so its two inverse images in the double overlap coincide. The open is saturated for \(\operatorname{Spec}(A\otimes_R B_0)\to\operatorname{Spec}B_0\). Its image is open by quotient topology and quasi-compact by the continuous-image property. That image, with its open subscheme structure, pulls back to \(V\) with its original datum. For a general fpqc covering, apply the finite refinement and Zariski gluing argument in Theorem 4.3.

**Exercise 8.4 (hard).** Use integral descent along the normalization of the nodal cubic (7.1) to prove (7.5), including its statement about morphisms.

*Solution.* Pull a finite étale \(C\)-scheme back to \(\mathbf P^1\). Lemma 7.1 makes it \(E\times\mathbf P^1\) for a finite set \(E\). Formula (7.4) shows that a datum is the identity on the diagonal and one bijection \(\sigma\) between the two node fibres, with inverse on the reverse pair. The six mixed triples impose precisely the inverse and identity equations, so every permutation is allowed.

Conversely, any finite \(E\) and permutation \(\sigma\) define such a datum. Theorem 6.4 descends it to a quasi-compact separated étale cover of \(C\); Lemma 5.4 makes this cover finite. A map of trivial covers upstairs is a single map \(h:E\to F\), because \(\mathbf P^1\) is connected. It respects the datum precisely when \(h\sigma=\tau h\). Full faithfulness in Theorem 5.2 descends this map uniquely. Thus the two constructions are mutually inverse on objects and morphisms, as required.

## 9. What this lesson does not prove

The module, algebra and morphism descent used here was proved in *Faithfully flat descent*. The elementary structure theorems for étale morphisms belong to *Étale morphisms and their local structure*: an étale diagonal is open, fibres over an algebraically closed field are disjoint unions of points, and an étale universally bijective map is an isomorphism [Stacks, Tag 025G](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-theorem-etale-radicial-open). The smoothness criterion uses flatness, local finite presentation and geometrically regular fibres [Stacks, Tags [01V8](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-smooth-flat-smooth-fibres), [038X](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-geometrically-regular-smooth)]. We do not reprove those prerequisites.

The other inputs are stated here to specify the boundaries of the proofs:

- A Noetherian local ring is regular precisely when its residue field has finite projective dimension; a regular local ring has global dimension its Krull dimension [Stacks, Tag 00OC](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-proposition-finite-gl-dim-regular). Smooth morphisms preserve reducedness, normality and regularity [Stacks, Tags [033B](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-reduced-goes-up), [033C](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-normal-goes-up), [07NF](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-regular-goes-up)].
- Flat finitely presented algebras can be approximated over a filtered colimit by flat finitely presented algebras [Stacks, Tag 02JO](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-flat-finite-presentation-limit-flat). Finite presentation is characterized by preservation of filtered colimits of algebra homomorphisms [Stacks, Tag 00QO](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-characterize-finite-presentation).
- Finite-presentation schemes and their morphisms descend to a stage of an affine-transition inverse system of quasi-compact quasi-separated schemes, and equality of two morphisms holds at some stage if it holds in the limit [Stacks, Tag 01ZM](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/limits.html#limits-lemma-descend-finite-presentation). Separatedness and étaleness of a finitely presented morphism hold at some stage when they hold in the limit [Stacks, Tags [01ZQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/limits.html#limits-lemma-descend-separated-finite-presentation), [07RP](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/limits.html#limits-lemma-descend-etale)].
- A quasi-finite separated morphism to a quasi-compact quasi-separated scheme factors as a quasi-compact open immersion followed by a finite morphism: Zariski's Main Theorem [Stacks, Tag 05K0](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-quasi-finite-separated-pass-through-finite).
- Henselization is flat and preserves the quotient defining the pair; idempotents in integral algebras over a henselian pair lift uniquely; finite étale algebras are equivalent to their reductions modulo that pair [Stacks, Tags [0AGU](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-henselization-flat), [09XI](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-characterize-henselian-pair), [09ZL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-finite-etale-equivalence)].
- A smooth proper curve with constant field \(k\) has \(\deg\Omega=2g-2\), where \(g\ge0\) is its genus [Stacks, Tag 0C1A](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/curves.html#curves-lemma-genus-smooth). This is the curve-theory input to Lemma 7.1.
- The six additional fpqc-local target properties in the table following Theorem 2.1 are used as cited results.

Projectivity deserves a separate argument from properness. Local projective embeddings supply relatively ample line bundles, and those line bundles need not come with compatible descent isomorphisms. If a relatively ample invertible sheaf does have such a datum, quasi-coherent descent first supplies its descended invertible sheaf; one must then verify relative ampleness. The proof of properness in this lesson does not supply that additional datum. See [Stacks, Section 02YJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-properties-morphisms-fpqc) for the distinction between projective and fpqc-local properties.

The principal descent references are [Stacks, Sections [02KN](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-properties-morphisms), [02YJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-properties-morphisms-fpqc), [02W1](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-descending-types-morphisms), [0246](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-section-quasi-affine), [0BTH](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-section-descending-etale)]. The proofs above separate their mechanisms: faithful flatness detects algebraic equations and finite presentations; quotient topology descends invariant open subsets; henselian lifting extends finite étale pieces across a closed set. The next lesson uses these results to construct the Galois category of finite étale covers.