Introduction
A foundational reference is [SV].
In this chapter we only define what are called the universally integral relative cycles in [SV]. This choice makes the theory somewhat simpler to develop than in the original, but of course we also lose something.
Fix a morphism \(X \to S\) of finite type between Noetherian schemes. A family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) is simply a collection \(\alpha = (\alpha_s)_{s \in S}\) where \(\alpha_s \in Z_r(X_s)\). It is immediately clear how to base change \(g^*\alpha\) of \(\alpha\) along any morphism \(g : S' \to S\). Then we say \(\alpha\) is a relative \(r\)-cycle on \(X/S\) if \(\alpha\) is compatible with specializations, i.e., for any morphism \(g : S' \to S\) where \(S'\) is the spectrum of a discrete valuation ring, we require the generic fibre of \(g^*\alpha\) to specialize to the closed fibre of \(g^*\alpha\). See Section 0H4Z.
Conventions and notation
Please consult the chapter on Chow Homology and Chern Classes for our conventions and notation regarding cycles on schemes locally of finite type over a fixed Noetherian base, see Chow Homology, Section 02QK ff.
In particular, if \(X\) is locally of finite type over a field \(k\), then \(Z_r(X)\) denotes the group of cycles of dimension \(r\), see Chow Homology, Example 02QM and Section 02QQ. Given an integral closed subscheme \(Z \subset X\) with \(\dim(Z) = r\) we have \([Z] \in Z_r(X)\) and if \(X\) is quasi-compact, then \(Z_r(X)\) is free abelian on these classes.
Cycles relative to fields
Let \(k\) be a field. Let \(X\) be a locally algebraic scheme over \(k\). Let \(r \geq 0\) be an integer. In this setting we have the group \(Z_r(X)\) of \(r\)-cycles on \(X\), see Section 0H4D.
Base change. For any field extension \(k'/k\) there is a base change map \(Z_r(X) \to Z_r(X_{k'})\), see Chow Homology, Section 0FVF. Namely, given an integral closed subscheme \(Z \subset X\) of dimension \(r\) we send \([Z] \in Z_r(X)\) to the \(r\)-cycle \([Z_{k'}]_r \in Z_r(X_{k'})\) associated to the closed subscheme \(Z_{k'} \subset X_{k'}\) (of course in general \(Z_{k'}\) is neither irreducible nor reduced). The base change map \(Z_r(X) \to Z_r(X_{k'})\) is always injective.
Lemma
Let \(K/k\) be a field extension. Let \(Z\) be an integral locally algebraic scheme over \(k\). The multiplicity \(m_{Z', Z_K}\) of an irreducible component \(Z' \subset Z_K\) is \(1\) or a power of the characteristic of \(k\).
Proof
If the characteristic of \(k\) is zero, then \(k\) is perfect and the multiplicity is always \(1\) since \(X_K\) is reduced by Varieties, Lemma 035X. Assume the characteristic of \(k\) is \(p > 0\). Let \(L\) be the function field of \(Z\). Since \(Z\) is locally algebraic over \(k\), the field extension \(L/k\) is finitely generated. The ring \(K \otimes_k L\) is Noetherian (Algebra, Lemma 045I). Translated into algebra, we have to show that the length of the artinian local ring \((K \otimes_k L)_\mathfrak q\) is a power of \(p\) for every minimal prime ideal \(\mathfrak q\).
Let \(L'/L\) be a finite purely inseparable extension, say of degree \(p^n\). Then \(K \otimes_k L \subset K \otimes_k L'\) is a finite free ring map of degree \(p^n\) which induces a homeomorphism on spectra and purely inseparable residue field extensions. Hence for every minimal prime \(\mathfrak q\) as above there is a unique minimal prime \(\mathfrak q' \subset K \otimes_k L'\) lying over it and \[p^n \text{length}((K \otimes_k L)_\mathfrak q) = [\kappa(\mathfrak q') : \kappa(\mathfrak q)] \text{length}((K \otimes_k L')_{\mathfrak q'})\] by Algebra, Lemma 02M0 applied to \(M = (K \otimes_k L')_{\mathfrak q'} \cong (K \otimes_k L)_{\mathfrak q}^{\oplus p^n}\). Since \([\kappa(\mathfrak q') : \kappa(\mathfrak q)]\) is a power of \(p\) we conclude that it suffices to prove the statement for \(L'\) and \(\mathfrak q'\).
By the previous paragraph and Algebra, Lemma 030R we may assume that we have a subfield \(L/k'/k\) such that \(L/k'\) is separable and \(k'/k\) is finite purely inseparable. Then \(K \otimes_k k'\) is an Artinian local ring. The argument of the preceding paragraph (applied to \(L = k\) and \(L' = k'\)) shows that \(\text{length}(K \otimes_k k')\) is a power of \(p\). Since \(L/k'\) is the localization of a smooth \(k'\)-algebra (Algebra, Lemma 037X). Hence \(S = (K \otimes_k L)_\mathfrak q\) is the localization of a smooth \(R = K \otimes_k k'\)-algebra at a minimal prime. Thus \(R \to S\) is a flat local homomorphism of Artinian local rings and \(\mathfrak m_R S = \mathfrak m_S\). It follows from Algebra, Lemma 02M1 that \(\text{length}(K \otimes_k k') = \text{length}(R) = \text{length}(S) = \text{length}((K \otimes_k L)_\mathfrak q)\) and the proof is finished.
Lemma
Let \(k\) be a field of characteristic \(p > 0\) with perfect closure \(k^{perf}\). Let \(X\) be an algebraic scheme over \(k\). Let \(r \geq 0\) be an integer. The cokernel of the injective map \(Z_r(X) \to Z_r(X_{k^{perf}})\) is a \(p\)-power torsion module (More on Algebra, Definition 05E6).
Proof
Since \(X\) is quasi-compact, the abelian group \(Z_r(X)\) is free with basis given by the integral closed subschemes of dimension \(r\). Similarly for \(Z_r(X_{k^{perf}})\). Since \(X_{k^{perf}} \to X\) is a homeomorphism, it follows that \(Z_r(X) \to Z_r(X_{k^{perf}})\) is injective with torsion cokernel. Every element in the cokernel is \(p\)-power torsion by Lemma 0H4F.
Specialization of cycles
Let \(R\) be a discrete valuation ring with fraction field \(K\) and residue field \(\kappa\). Let \(X\) be a scheme locally of finite type over \(R\). Let \(r \geq 0\). There is a specialization map \[sp_{X/R} : Z_r(X_K) \longrightarrow Z_r(X_\kappa)\] defined as follows. For an integral closed subscheme \(Z \subset X_K\) of dimension \(r\) we denote \(\overline{Z}\) the scheme theoretic image of \(Z \to X\). Then we let \(sp_{X/R}\) be the unique \(\mathbf{Z}\)-linear map such that \[sp_{X/R}([Z]) = [\overline{Z}_\kappa]_r\] We briefly discuss why this is well defined. First, observe that the morphism \(X_K \to X\) is quasi-compact and hence the morphism \(Z \to X\) is quasi-compact. Thus taking the scheme theoretic image of \(Z \to X\) commutes with flat base change by Morphisms, Lemma 081I. In particular, base changing back to \(X_K\) we see that \(Z = \overline{Z}_K\). Since \(Z\) is integral, of course \(\overline{Z}\) is integral too and in fact is equal to the unique integral closed subscheme whose generic point is the (image of the) generic point of \(Z\). It follows from Varieties, Lemma 0B2J that \(Z_\kappa\) is equidimensional of dimension \(r\).
Lemma
Let \(R\) be a discrete valuation ring with fraction field \(K\) and residue field \(\kappa\). Let \(X\) be a scheme locally of finite type over \(R\). Let \(r \geq 0\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module flat over \(R\). Assume \(\dim(\text{Supp}(\mathcal{F}_K)) \leq r\). Then \(\dim(\text{Supp}(\mathcal{F}_\kappa)) \leq r\) and \[sp_{X/R}([\mathcal{F}_K]_r) = [\mathcal{F}_\kappa]_r\]
Proof
The statement on dimension follows from More on Morphisms, Lemma 0H3X. Let \(x\) be a generic point of an integral closed subscheme \(Z \subset X_\kappa\) of dimension \(r\). To finish the proof we will show that the coefficient of \([Z]\) in the left (L) and right hand side (R) of equality are the same.
Let \(A = \mathcal{O}_{X, x}\) and \(M = \mathcal{F}_x\). Observe that \(M\) is a finite \(A\)-module flat over \(R\). Let \(\pi \in R\) be a uniformizer so that \(A/\pi A = \mathcal{O}_{X_\kappa, x}\). By Chow Homology, Lemma 02QG we have \[\sum\nolimits_i \text{length}_A(A/(\pi, \mathfrak q_i)) \text{length}_{A_{\mathfrak q_i}}(M_{\mathfrak q_i}) = \text{length}_A(M/\pi M)\] where the sum is over the minimal primes \(\mathfrak q_i\) in the support of \(M\). Since \(\pi\) is a nonzerodivisor on \(M\) we see that \(\pi \not \in \mathfrak q_i\) and hence these primes correspond to those generic points \(y_i \in X_K\) of the support of \(\mathcal{F}_K\) which specialize to our chosen \(x \in X_\kappa\). Thus the left hand side is the coefficient of \([Z]\) in (L). Of course \(\text{length}_A(M/\pi M)\) is the coefficient of \([Z]\) in (R). This finishes the proof.
Lemma
Let \(R\) be a discrete valuation ring with fraction field \(K\) and residue field \(\kappa\). Let \(X\) be a scheme locally of finite type over \(R\). Let \(r \geq 0\). Let \(W \subset X\) be a closed subscheme flat over \(R\). Assume \(\dim(W_K) \leq r\). Then \(\dim(W_\kappa) \leq r\) and \[sp_{X/R}([W_K]_r) = [W_\kappa]_r\]
Proof
Taking \(\mathcal{F} = \mathcal{O}_W\) this is a special case of Lemma 0H4I. See Chow Homology, Lemma 02QY.
Lemma
Let \(R'/R\) be an extension of discrete valuation rings inducing fraction field extension \(K'/K\) and residue field extension \(\kappa'/\kappa\) (More on Algebra, Definition 09E4). Let \(X\) be locally of finite type over \(R\). Denote \(X' = X_{R'}\). Then the diagram \[\xymatrix{ Z_r(X'_{K'}) \ar[rr]_{sp_{X'/R'}} & & Z_r(X'_{\kappa'}) \\ Z_r(X_K) \ar[rr]^{sp_{X/R}} \ar[u] & & Z_r(X_\kappa) \ar[u] }\] commutes where \(r \geq 0\) and the vertical arrows are base change maps.
Proof
Observe that \(X'_{K'} = X_{K'} = X_K \times_{\Spec(K)} \Spec(K')\) and similarly for closed fibres, so that the vertical arrows indeed make sense (see Section 0H4E). Now if \(Z \subset X_K\) is an integral closed subscheme with scheme theoretic image \(\overline{Z} \subset X\), then we see that \(Z_{K'} \subset X_{K'}\) is a closed subscheme with scheme theoretic image \(\overline{Z}_{R'} \subset X_{R'}\). The base change of \([Z]\) is \([Z_{K'}]_r = [\overline{Z}_{K'}]_r\) by definition. We have \[sp_{X/R}([Z]) = [\overline{Z}_\kappa]_r \quad\text{and}\quad sp_{X'/R'}([\overline{Z}_{K'}]_r) = [(\overline{Z}_{R'})_{\kappa'}]_r\] by Lemma 0H4I. Since \((\overline{Z}_{R'})_{\kappa'} = (\overline{Z}_\kappa)_{\kappa'}\) we conclude.
Lemma
Let \(R\) be a discrete valuation ring with fraction field \(K\) and residue field \(\kappa\). Let \(X\) be a scheme locally of finite type over \(R\). Let \(f : X' \to X\) be a morphism which is locally of finite type, flat, and of relative dimension \(e\). Then the diagram \[\xymatrix{ Z_{r + e}(X'_K) \ar[rr]_{sp_{X'/R}} & & Z_{r + e}(X'_\kappa) \\ Z_r(X_K) \ar[rr]^{sp_{X/R}} \ar[u] & & Z_r(X_\kappa) \ar[u] }\] commutes where \(r \geq 0\) and the vertical arrows are given by flat pullback.
Proof
Let \(Z \subset X\) be an integral closed subscheme dominating \(R\). By the construction of \(sp_{X/R}\) we have \(sp_{X/R}([Z_K]) = [Z_\kappa]_r\) and this characterizes the specialization map. Set \(Z' = f^{-1}(Z) = X' \times_X Z\). Since \(R\) is a valuation ring, \(Z\) is flat over \(R\). Hence \(Z'\) is flat over \(R\) and \(sp_{X'/R}([Z'_K]_{r + e}) = [Z'_\kappa]_{r + e}\) by Lemma 0H4J. Since by Chow Homology, Lemma 02RE we have \(f_K^*[Z_K] = [Z'_K]_{r + e}\) and \(f_\kappa^*[Z_\kappa]_r = [Z'_\kappa]_{r + e}\) we win.
Lemma
Let \(R\) be a discrete valuation ring with fraction field \(K\) and residue field \(\kappa\). Let \(f : X \to Y\) be a proper morphism of schemes locally of finite type over \(R\). Then the diagram \[\xymatrix{ Z_r(X_K) \ar[rr]_{sp_{X/R}} \ar[d] & & Z_r(X_\kappa) \ar[d] \\ Z_r(Y_K) \ar[rr]^{sp_{Y/R}} & & Z_r(Y_\kappa) }\] commutes where \(r \geq 0\) and the vertical arrows are given by proper pushforward.
Proof
Let \(Z \subset X\) be an integral closed subscheme dominating \(R\). By the construction of \(sp_{X/R}\) we have \(sp_{X/R}([Z_K]) = [Z_\kappa]_r\) and this characterizes the specialization map. Set \(Z' = f(Z) \subset Y\). Then \(Z'\) is an integral closed subscheme of \(Y\) dominating \(R\). Thus \(sp_{Y/R}([Z'_K]) = [Z'_\kappa]_r\).
We can think of \([Z]\) as an element of \(Z_{r + 1}(X)\). By definition we have \(f_*[Z] = 0\) if \(\dim(Z') < r + 1\) and \(f_*[Z] = d[Z']\) if \(Z \to Z'\) is generically finite of degree \(d\). Since proper pushforward commutes with flat pullback by \(Y_K \to Y\) (Chow Homology, Lemma 02RG) we see that correspondingly \(f_{K, *}[Z_K] = 0\) or \(f_{K, *}[Z_K] = d[Z'_K]\). Let us apply Chow Homology, Lemma 02TA to the commutative diagram \[\xymatrix{ X_\kappa \ar[d] \ar[r]_i & X \ar[d] \\ Y_\kappa \ar[r]^j & Y }\] We obtain that \(f_{\kappa, *}[Z_\kappa]_r = 0\) or \(f_{\kappa, *}[Z_\kappa] = d[Z'_\kappa]_r\) because clearly \(i^*[Z] = [Z_k]_r\) and \(j^*[Z'] = [Z'_\kappa]_r\). Putting everything together we conclude.
Families of cycles on fibres
Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. A family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) is a family \[\alpha = (\alpha_s)_{s \in S}\] indexed by the points \(s\) of the scheme \(S\) where \(\alpha_s \in Z_r(X_s)\) is an \(r\) cycle on the scheme theoretic fibre \(X_s\) of \(f\) at \(s\). There are various constructions we can perform on families of \(r\)-cycles on fibres.
Base change. Let \[\xymatrix{ X' \ar[r] \ar[d] & X \ar[d]^f \\ S' \ar[r]^g & S }\] be a catesian square of morphisms of schemes with \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Given a family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) we define the base change \(g^*\alpha\) of \(\alpha\) to be the family \[g^*\alpha = (\alpha'_{s'})_{s' \in S'}\] where \(\alpha'_{s'} \in Z_r(X'_{s'})\) is the base change of the cycle \(\alpha_s\) with \(s = g(s')\) as in Section 0H4E via the identitification \(X'_{s'} = X_s \times_{\Spec(\kappa(s))} \Spec(\kappa(s'))\) of scheme theoretic fibres.
Restriction. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. Let \(U \subset X\) and \(V \subset S\) be open subschemes with \(f(U) \subset V\). Given a family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) we can define the restriction \(\alpha|_U\) of \(\alpha\) to be the family of \(r\)-cycles on fibres of \(U/V\) \[\alpha|_U = (\alpha_s|_{U_s})_{s \in V}\] of restrictions to scheme theoretic fibres.
Flat pullback. Let \(X \to S\) be a morphism of schemes which is locally of finite type. Let \(r, e \geq 0\) be integers. Let \(f : X' \to X\) be a flat morphism, locally of finite type, and of relative dimension \(e\). Given a family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) we define the flat pullback \(f^*\alpha\) of \(\alpha\) to be the family of \((r + e)\)-cycles on fibres \[f^*\alpha = (f_s^*\alpha_s)_{s \in S}\] where \(f_s^*\alpha_s \in Z_{r + e}(X'_s)\) is the flat pullback of the cycle \(\alpha_s\) in \(Z_r(X_s)\) by the flat morphism \(f_s : X'_s \to X_s\) of relative dimension \(e\) of scheme theoretic fibres.
Proper pushforward. Let \[\xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & S }\] be a commutative diagram of morphisms of schemes with \(X\) and \(Y\) locally of finite type over \(S\) and \(f\) proper. Let \(r \geq 0\) be an integer. Given a family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) we define the proper pushforward \(f_*\alpha\) of \(\alpha\) to be the family of \(r\)-cycles on fibres of \(Y/S\) by \[f_*\alpha = (f_{s, *}\alpha_s)_{s \in S}\] where \(f_{s, *}\alpha_s \in Z_r(Y_s)\) is the proper pushforward of the cycle \(\alpha_s\) in \(Z_r(X_s)\) by the proper morphism \(f_s : X_s \to Y_s\) of scheme theoretic fibres.
Lemma
We have the following compatibilities between the operations above: (1) base change is functorial, (2) restriction is a combination of base change and (a special case of) flat pullback, (3) flat pullback commutes with base change, (4) flat pullback is functorial, (5) proper pushforward commutes with base change, (6) proper pushforward is functorial, and (7) proper pushforward commutes with flat pullback.
Proof
Each of these compatibilities follows directly from the corresponding results proved in the chapter on Chow homology applied to the fibres over \(S\) of the schemes in question. We omit the precise statements and the detailed proofs. Here are some references. Part (1): Chow Homology, Lemma 0FVP. Part (2): Obvious. Part (3): Chow Homology, Lemma 0FVK. Part (4): Chow Homology, Lemma 02RD. Part (5): Chow Homology, Lemma 0FVL. Part (6): Chow Homology, Lemma 02R5. Part (7): Chow Homology, Lemma 02RG.
Example
Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type. For \(s \in S\) denote \(\mathcal{F}_s\) the pullback of \(\mathcal{F}\) to \(X_s\). Assume \(\dim(\text{Supp}(\mathcal{F}_s)) \leq r\) for all \(s \in S\). Then we can associate to \(\mathcal{F}\) the family \([\mathcal{F}/X/S]_r\) of \(r\)-cycles on fibres of \(X/S\) defined by the formula \[[\mathcal{F}/X/S]_r = ([\mathcal{F}_s]_r)_{s \in S}\] where \([\mathcal{F}_s]_r\) is given by Chow Homology, Definition 02QX.
Lemma
The construction in Example 0H4Q is compatible with base change, restriction, and flat pullback.
Proof
Example
Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. Let \(Z \subset X\) be a closed subscheme. For \(s \in S\) denote \(Z_s\) the inverse image of \(Z\) in \(X_s\) or equivalently the scheme theoretic fibre of \(Z\) at \(s\) viewed as a closed subscheme of \(X_s\). Assume \(\dim(Z_s) \leq r\) for all \(s \in S\). Then we can associate to \(Z\) the family \([Z/X/S]_r\) of \(r\)-cycles on fibres of \(X/S\) defined by the formula \[[Z/X/S]_r = ([Z_s]_r)_{s \in S}\] where \([Z_s]_r\) is given by Chow Homology, Definition 02QU.
Lemma
The construction in Example 0H4S is compatible with base change, restriction, and flat pullback.
Proof
Taking \(\mathcal{F} = (Z \to X)_*\mathcal{O}_Z\) this is a special case of Lemma 0H4R. See Chow Homology, Lemma 02QY.
Remark
Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/S\). We define the support of \(\alpha\) to be \[\text{Supp}(\alpha) = \bigcup\nolimits_{s \in S} \text{Supp}(\alpha_s) \subset X\] Here \(\text{Supp}(\alpha_s) \subset X_s\) is the support of the cycle \(\alpha_s\), see Chow Homology, Definition 0H46. The support \(\text{Supp}(\alpha)\) is rarely a closed subset of \(X\).
Lemma
Taking the support as in Remark 0H4U is compatible with base change, restriction, and flat pullback.
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. Let \(g : S' \to S\) be a surjective morphism of schemes. Set \(S'' = S' \times_S S'\) and let \(f' : X' \to S'\) and \(f'' : X'' \to S''\) be the base changes of \(f\). Let \(x \in X\) with \(\text{trdeg}_{\kappa(f(x))}(\kappa(x)) = r\).
There exists an \(x' \in X'\) mapping to \(x\) with \(\text{trdeg}_{\kappa(f'(x'))}(\kappa(x')) = r\).
If \(x'_1, x'_2 \in X'\) are both as in (1), then there exists an \(x'' \in X''\) with \(\text{trdeg}_{\kappa(f''(x''))}(\kappa(x'')) = r\) and \(\text{pr}_i(x'') = x'_i\).
Proof
Part (1) is Morphisms, Lemma 02FY. Let \(x'_1, x'_2\) be as in (2). Then since \(X'' = X' \times_X X'\) we see that there exists a \(x'' \in X''\) mapping to both \(x'_1\) and \(x'_2\) (see for example Descent, Lemma 02KI). Denote \(s'' \in S''\), \(s'_i \in S'\), and \(s \in S\) the images of \(x''\), \(x'_i\), and \(x\). Denote \(k = \kappa(s)\) and let \(Z \subset X_k\) be the integral closed subscheme whose generic point is \(x\). Then \(x'_i\) is a generic point of an irreducible component of \(Z_{\kappa(s'_i)}\). Let \(Z'' \subset Z_{\kappa(s'')}\) be an irreducible component containing \(x''\). Denote \(\xi'' \in Z''\) the generic point. Since \(\xi'' \leadsto x''\) we see that \(\xi''\) must also map to \(x'_i\) under the two projections. On the other hand, we see that \(\text{trdeg}_{\kappa(s'')}(\kappa(\xi'')) = r\) because it is a generic point of an irreducible component of the base change of \(Z\).
Lemma
Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. Let \(g : S' \to S\) be a morphism of schemes and \(X' = S' \times_S X\). Assume that for every \(s \in S\) there exists a point \(s' \in S'\) with \(g(s') = s\) and such that \(\kappa(s')/\kappa(s)\) is a separable extension of fields. Then
For families \(\alpha_1\) and \(\alpha_2\) of \(r\)-cycles on fibres of \(X/S\) if \(g^*\alpha_1 = g^*\alpha_2\), then \(\alpha_1 = \alpha_2\).
Given a family \(\alpha'\) of \(r\)-cycles on fibres of \(X'/S'\) if \(\text{pr}_1^*\alpha' = \text{pr}_2^*\alpha'\) as families of \(r\)-cycles on fibres of \((S' \times_S S') \times_S X / (S' \times_S S')\), then there is a unique family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) such that \(g^*\alpha = \alpha'\).
Proof
Part (1) follows from the injectivity of the base change map discussed in Section 0H4E. (This argument works as long as \(S' \to S\) is surjective.)
Let \(\alpha'\) be as in (2). Denote \(\alpha'' = \text{pr}_1^*\alpha' = \text{pr}_2^*\alpha'\) the common value.
Let \((X/S)^{(r)}\) be the set of \(x \in X\) with \(\text{trdeg}_{\kappa(f(x))}(\kappa(x)) = r\) and similarly define \((X'/S')^{(r)}\) and \((X''/S'')^{(r)}\) Taking coefficients, we may think of \(\alpha'\) and \(\alpha''\) as functions \(\alpha' : (X'/S')^{(r)} \to \mathbf{Z}\) and \(\alpha'' : (X''/S'')^{(r)} \to \mathbf{Z}\). Given a function \[\varphi : (X/S)^{(r)} \to \mathbf{Z}\] we define \(g^*\varphi : (X'/S')^{(r)} \to \mathbf{Z}\) by analogy with our base change operation. Namely, say \(x' \in (X'/S')^{(r)}\) maps to \(x \in X\), \(s' \in S'\), and \(s \in Z\). Denote \(Z' \subset X'_{s'}\) and \(Z \subset X_s\) the integral closed subschemes with generic points \(x'\) and \(x\). Note that \(\dim(Z') = r\). If \(\dim(Z) < r\), then we set \((g^*\varphi)(x') = 0\). If \(\dim(Z) = r\), then \(Z'\) is an irreducible component of \(Z_{s'}\) and hence has a multiplicity \(m_{Z', Z_{s'}}\). Call this \(m(x', g)\). Then we define \[(g^*\varphi)(x') = m(x', g) \varphi(x)\] Note that the coefficients \(m(x', g)\) are always positive integers (see for example Lemma 0H4F). We similarly have base change maps \[\text{pr}_1^*, \text{pr}_2^* : \text{Map}((X'/S')^{(r)}, \mathbf{Z}) \longrightarrow \text{Map}((X''/S'')^{(r)}, \mathbf{Z})\] It follows from the associativity of base change that we have \(\text{pr}_1^* \circ g^* = \text{pr}_2^* \circ g^*\) (small detail omitted). To be explicitly, in terms of the maps of sets this equality just means that for \(x'' \in (X''/S'')^{(r)}\) we have \[m(x'', \text{pr}_1) m(\text{pr}_1(x''), g) = m(x'', \text{pr}_2) m(\text{pr}_2(x''), g)\] provided that \(\text{pr}_1(x'')\) and \(\text{pr}_2(x'')\) are in \((X''/S'')^{(r)}\). By Lemma 0H4W and an elementary argument1 using the previous displayed equation, it follows that there exists a unique map \[\alpha : (X/S)^{(r)} \to \mathbf{Q}\] such that \(g^*\alpha = \alpha'\). To finish the proof it suffices to show that \(\alpha\) has integer values (small detail omitted: one needs to see that \(\alpha\) determines a locally finite sum on each fibre which follows from the corresponding fact for \(\alpha'\)). Given any \(x \in (X/S)^{(r)}\) with image \(s \in S\) we can pick a point \(s' \in S'\) such that \(\kappa(s')/\kappa(s)\) is separable. Then we may choose \(x' \in (X'/S')^{(r)}\) mapping to \(s\) and \(x\) and we see that \(m(x', g) = 1\) because \(Z_{s'}\) is reduced in this case. Whence \(\alpha(x) = \alpha'(x')\) is an integer.
Lemma
Let \(g : S' \to S\) be a bijective morphism of schemes which induces isomorphisms of residue fields. Let \(f : X \to S\) be locally of finite type. Set \(X' = S' \times_S X\). Let \(r \geq 0\). Then base change by \(g\) determines a bijection between the group of families of \(r\)-cycles on fibres of \(X/S\) and the group of families of \(r\)-cycles on fibres of \(X'/S'\).
Proof
Omitted.
Relative cycles
Here is the definition we will work with; see Section 0H6T for a comparison with the definitions in [SV].
Definition
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(r \geq 0\) be an integer. A relative \(r\)-cycle on \(X/S\) is a family \(\alpha\) of \(r\)-cycles on fibres of \(X/S\) such that for every morphism \(g : S' \to S\) where \(S'\) is the spectrum of a discrete valuation ring we have \[sp_{X'/S'}(\alpha_\eta) = \alpha_0\] where \(sp_{X'/S'}\) is as in Section 0H4H and \(\alpha_\eta\) (resp. \(\alpha_0\)) is the value of the base change \(g^*\alpha\) of \(\alpha\) at the generic (resp. closed) point of \(S'\). The group of all relative \(r\)-cycles on \(X/S\) is denoted \(z(X/S, r)\).
Lemma
Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\) as in Definition 0H50. Then any restriction, base change, flat pullback, or proper pushforward of \(\alpha\) is a relative \(r\)-cycle.
Proof
For flat pullback use Lemma 0H4L. Restriction is a special case of flat pullback. To see it holds for base change use that base change is transitive. For proper pushforward use Lemma 0H4M.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/S\). Let \(\{g_i : S_i \to S\}\) be a h covering (More on Flatness, Definition 0ETS). Then \(\alpha\) is a relative \(r\)-cycle if and only if each base change \(g_i^*\alpha\) is a relative \(r\)-cycle.
Proof
If \(\alpha\) is a relative \(r\)-cycle, then each base change \(g_i^*\alpha\) is a relative \(r\)-cycle by Lemma 0H51. Assume each \(g_i^*\alpha\) is a relative \(r\)-cycle. Let \(g : S' \to S\) be a morphism where \(S'\) is the spectrum of a discrete valuation ring. After replacing \(S\) by \(S'\), \(X\) by \(X' = X \times_S S'\), and \(\alpha\) by \(\alpha' = g^*\alpha\) and using that the base change of a h covering is a h covering (More on Flatness, Lemma 0ETY) we reduce to the problem studied in the next paragraph.
Assume \(S\) is the spectrum of a discrete valuation ring with closed point \(0\) and generic point \(\eta\). We have to show that \(sp_{X/S}(\alpha_\eta) = \alpha_0\). Since a h covering is a V covering (by definition), there is an \(i\) and a specialization \(s' \leadsto s\) of points of \(S_i\) with \(g_i(s') = \eta\) and \(g_i(s) = 0\), see Topologies, Lemma 0ETN. By Properties, Lemma 054F we can find a morphism \(h : S' \to S_i\) from the spectrum \(S'\) of a discrete valuation ring which maps the generic point \(\eta'\) to \(s'\) and maps the closed point \(0'\) to \(s\). Denote \(\alpha' = h^*g_i^*\alpha\). By assumption we have \(sp_{X'/S'}(\alpha'_{\eta'}) = \alpha'_{0'}\). Since \(g = g_i \circ h : S' \to S\) is the morphism of schemes induced by an extension of discrete valuation rings we conclude that \(sp_{X/S}\) and \(sp_{X'/S'}\) are compatible with base change maps on the fibres, see Lemma 0H4K. We conclude that \(sp_{X/S}(\alpha_\eta) = \alpha_0\) because the base change map \(Z_r(X_0) \to Z_r(X'_{0'})\) is injective as discussed in Section 0H4E.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r, e \geq 0\) be integers. Let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/S\). Let \(\{f_i : X_i \to X\}\) be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension \(e\). Then \(\alpha\) is a relative \(r\)-cycle if and only if each flat pullback \(f_i^*\alpha\) is a relative \(r\)-cycle.
Proof
If \(\alpha\) is a relative \(r\)-cycle, then each pull back \(f_i^*\alpha\) is a relative \(r\)-cycle by Lemma 0H51. Assume each \(f_i^*\alpha\) is a relative \(r\)-cycle. Let \(g : S' \to S\) be a morphism where \(S'\) is the spectrum of a discrete valuation ring. After replacing \(S\) by \(S'\), \(X\) by \(X' = X \times_S S'\), and \(\alpha\) by \(\alpha' = g^*\alpha\) we reduce to the problem studied in the next paragraph.
Assume \(S\) is the spectrum of a discrete valuation ring with closed point \(0\) and generic point \(\eta\). We have to show that \(sp_{X/S}(\alpha_\eta) = \alpha_0\). Denote \(f_{i, 0} : X_{i, 0} \to X_0\) the base change of \(f_i\) to the closed point of \(S\). Similarly for \(f_{i, \eta}\). Observe that \[f_{i, 0}^*sp_{X/S}(\alpha_\eta) = sp_{X_i/S}(f_{i, \eta}^*\alpha_\eta) = f_{i, 0}^*\alpha_0\] Namely, the first equality holds by Lemma 0H4L and the second by assumption. Since the family of maps \(f_{i, 0}^* : Z_r(X_0) \to Z_r(X_{i, 0})\) is jointly injective (due to the fact that \(f_{i, 0}\) is jointly surjective), we conclude what we want.
Lemma
Let \(S\) be a locally Noetherian scheme. Let \(i : X \to Y\) be a closed immersion of schemes locally of finite type over \(S\). Let \(r \geq 0\). Let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/S\). Then \(\alpha\) is a relative \(r\)-cycle on \(X/S\) if and only if \(i_*\alpha\) is a relative \(r\)-cycle on \(Y/S\).
Proof
Since base change commutes with \(i_*\) (Lemma 0H4P) it suffices to prove the following: if \(S\) is the spectrum of a discrete valuation ring with generic point \(\eta\) and closed point \(0\), then \(sp_{X/S}(\alpha_\eta) = \alpha_0\) if and only if \(sp_{Y/S}(i_{\eta, *}\alpha_\eta) = i_{0, *}\alpha_0\). This is true because \(i_{0, *} : Z_r(X_0) \to Z_r(Y_0)\) is injective and because \(i_{0, *}sp_{X/S}(\alpha_\eta) = sp_{Y/S}(i_{\eta, *}\alpha_\eta)\) by Lemma 0H4M.
The following lemma will be strengthened in Lemma 0H5C.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\). Let \(\alpha\) and \(\beta\) be relative \(r\)-cycles on \(X/S\). The following are equivalent
\(\alpha = \beta\), and
\(\alpha_\eta = \beta_\eta\) for any generic point \(\eta \in S\) of an irreducible component of \(S\).
Proof
The implication (1) \(\Rightarrow\) (2) is immediate. Assume (2). For every \(s \in S\) we can find an \(\eta\) as in (2) which specializes to \(s\). By Properties, Lemma 054F we can find a morphism \(g : S' \to S\) from the spectrum \(S'\) of a discrete valuation ring which maps the generic point \(\eta'\) to \(\eta\) and maps the closed point \(0\) to \(s\). Then \(\alpha_s\) and \(\beta_s\) are elements of \(Z_r(X_s)\) which base change to the same element of \(Z_r(X_{0'})\), namely \(sp_{X_{S'}/S'}(\alpha_{\eta'})\) where \(\alpha_{\eta'}\) is the base change of \(\alpha_\eta\). Since the base change map \(Z_r(X_s) \to Z_r(X_{0'})\) is injective as discussed in Section 0H4E we conclude \(\alpha_s = \beta_s\).
Lemma
In the situation of Example 0H4Q assume \(S\) is locally Noetherian and \(\mathcal{F}\) is flat over \(S\) in dimensions \(\geq r\) (More on Flatness, Definition 0CWG). Then \([\mathcal{F}/X/S]_r\) is a relative \(r\)-cycle on \(X/S\).
Proof
By More on Flatness, Lemma 0CWF the hypothesis on \(\mathcal{F}\) is preserved by any base change. Also, formation of \([\mathcal{F}/X/S]_r\) is compatible with any base change by Lemma 0H4R. Since the condition of being compatible with specializations is checked after base change to the spectrum of a discrete valuation ring, this reduces us to the case where \(S\) is the spectrum of a valuation ring. In this case the set \(U = \{x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\}\) is open in \(X\) by More on Flatness, Lemma 05IQ. Since the complement of \(U\) in \(X\) has fibres of dimension \(< r\) over \(S\) by assumption, we see that restriction along the inclusion \(U \subset X\) induces an isomorphism on the groups of \(r\)-cycles on fibres after any base change, compatible with specialization maps and with formation of the relative cycle associated to \(\mathcal{F}\). Thus it suffices to show compatibility with specializations for \([\mathcal{F}|_U / U /S]_r\). Since \(\mathcal{F}|_U\) is flat over \(S\), this follows from Lemma 0H4I and the definitions.
Lemma
In the situation of Example 0H4S assume \(S\) is locally Noetherian and \(Z\) is flat over \(S\) in dimensions \(\geq r\). Then \([Z/X/S]_r\) is a relative \(r\)-cycle on \(X/S\).
Proof
The assumption means that \(\mathcal{O}_Z\) is flat over \(S\) in dimensions \(\geq r\). Thus applying Lemma 0H56 with \(\mathcal{F} = (Z \to X)_*\mathcal{O}_Z\) we conclude.
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a morphism which is of finite type. Let \(r \geq 0\). Denote \(Hilb(X/S, r)\) the set of closed subschemes \(Z \subset X\) such that \(Z \to S\) is flat and of relative dimension \(\leq r\). By Lemma 0H57 for each \(Z \in Hilb(X/S, r)\) we have an element \([Z/X/S]_r \in z(X/S, r)\). Thus we obtain a group homomorphism [0H58]\[\begin{equation} \text{free abelian group on }Hilb(X/S, r) \longrightarrow z(X/S, r) \end{equation}\] sending \(\sum n_i[Z_i]\) to \(\sum n_i[Z_i/X/S]_r\). A key feature of relative \(r\)-cycles is that they are locally (on \(X\) and \(S\) in suitable topologies) in the image of this map.
Lemma
Let \(f : X \to S\) be a finite type morphism of schemes with \(S\) Noetherian. Let \(r \geq 0\). Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Then there is a proper, completely decomposed (More on Morphisms, Definition 0GTI) morphism \(g : S' \to S\) such that \(g^*\alpha\) is in the image of (0H58).
Proof
By Noetherian induction, we may assume the result holds for the pullback of \(\alpha\) by any closed immersion \(g : S' \to S\) which is not an isomorphism.
Let \(S_1 \subset S\) be an irreducible component (viewed as an integral closed subscheme). Let \(S_2 \subset S\) be the closure of the complement of \(S'\) (viewed as a reduced closed subscheme). If \(S_2 \not = \emptyset\), then the result holds for the pullback of \(\alpha\) by \(S_1 \to S\) and \(S_2 \to S\). If \(g_1 : S'_1 \to S_1\) and \(g_2 : S'_2 \to S_2\) are the corresponding completely decomposed proper morphisms, then \(S' = S'_1 \amalg S'_2 \to S\) is a completely decomposed proper morphism and we see the result holds for \(S\)2 . Thus we may assume \(S' \to S\) is bijective and we reduce to the case described in the next paragraph.
Assume \(S\) is integral. Let \(\eta \in S\) be the generic point and let \(K = \kappa(\eta)\) be the function field of \(S\). Then \(\alpha_\eta\) is an \(r\)-cycle on \(X_K\). Write \(\alpha_\eta = \sum n_i[Y_i]\). Taking the closure of \(Y_i\) we obtain integral closed subschemes \(Z_i \subset X\) whose base change to \(\eta\) is \(Y_i\). By generic flatness (for example Morphisms, Proposition 052A), we see that \(Z_i\) is flat over a nonempty open \(U\) of \(S\) for each \(i\). Applying More on Flatness, Lemma 081R we can find a \(U\)-admissible blowing up \(g : S' \to S\) such that the strict transform \(Z'_i \subset X_{S'}\) of \(Z_i\) is flat over \(S'\). Then \(\beta = \sum n_i[Z'_i/X_{S'}/S']_r\) is in the image of (0H58) and \(\beta = g^*\alpha\) by Lemma 0H55.
However, this does not finish the proof as \(S' \to S\) may not be completely decomposed. This is easily fixed: denoting \(T \subset S\) the complement of \(U\) (viewed as a closed subscheme), by Noetherian induction we can find a completely decomposed proper morphism \(T' \to T\) such that \((T' \to S)^*\alpha\) is in the image of (0H58). Then \(S' \amalg T' \to S\) does the job.
Lemma
Let \(f : X \to S\) be a finite type morphism of schemes with \(S\) the spectrum of a discrete valuation ring. Let \(r \geq 0\). Then (0H58) is surjective.
Proof
This of course follows from Lemma 0H59 but we can also see it directly as follows. Say \(\alpha\) is a relative \(r\)-cycle on \(X/S\). Write \(\alpha_\eta = \sum n_i[Z_i]\) (the sum is finite). Denote \(\overline{Z}_i \subset X\) the closure of \(Z_i\) as in Section 0H4H. Then \(\alpha = \sum n_i[\overline{Z}_i/X/S]\).
Lemma
Let \(f : X \to S\) be a morphism of schemes. Let \(r \geq 0\). Assume \(S\) locally Noetherian and \(f\) smooth of relative dimension \(r\). Let \(\alpha \in z(X/S, r)\). Then the support of \(\alpha\) is open and closed in \(X\) (see proof for a more precise result).
Proof
Let \(x \in X\) with image \(s \in S\). Since \(f\) is smooth, there is a unique irreducible component \(Z(x)\) of \(X_s\) which contains \(x\). Then \(\dim(Z(x)) = r\). Let \(n_x\) be the coefficient of \(Z(x)\) in the cycle \(\alpha_s\). We will show the function \(x \mapsto n_x\) is locally constant on \(X\).
Let \(g : S' \to S\) be a morphism of locally Noetherian schemes. Let \(X'\) be the base change of \(X\) and let \(\alpha' = g^*\alpha\) be the base change of \(\alpha\). Let \(x' \in X'\) map to \(s' \in S'\), \(x \in X\), and \(s \in S\). We claim \(n_{x'} = n_x\). Namely, since \(Z(x)\) is smooth over \(\kappa(s)\) we see that \(Z(x) \times_{\Spec(\kappa(s))} \Spec(\kappa(s'))\) is reduced. Since \(Z(x')\) is an irreducible component of this scheme, we see that the coefficient \(n_{x'}\) of \(Z(x')\) in \(\alpha'_{s'}\) is the same as the coefficient \(n_x\) of \(Z(x)\) in \(\alpha_s\) by the definition of base change in Section 0H4E thereby proving the claim.
Since \(X\) is locally Noetherian, to show that \(x \mapsto n_x\) is locally constant, it suffices to show: if \(x' \leadsto x\) is a specialization in \(X\), then \(n_{x'} = n_x\). Choose a morphism \(S' \to X\) where \(S'\) is the spectrum of a discrete valuation ring mapping the generic point \(\eta\) to \(x'\) and the closed point \(0\) to \(x\). See Properties, Lemma 054F. Then the base change \(X' \to S'\) of \(f\) by \(S' \to S\) has a section \(\sigma : S' \to X'\) such that \(\sigma(\eta) \leadsto \sigma(0)\) is a specialization of points of \(X'\) mapping to \(x' \leadsto x\) in \(X\). Thus we reduce to the claim in the next paragraph.
Let \(S\) be the spectrum of a discrete valuation ring with generic point \(\eta\) and closed point \(0\) and we have a section \(\sigma : S \to X\). Claim: \(n_{\sigma(\eta)} = n_{\sigma(0)}\). By the discussion in More on Morphisms, Section 055K and especially More on Morphisms, Lemma 055R after replacing \(X\) by an open subscheme, we may assume the fibres of \(X \to S\) are connected. Since these fibres are smooth, they are irreducible. Then we see that \(\alpha_\eta = n[X_\eta]\) with \(n = n_{\sigma(\eta)}\) and the relation \(sp_{X/S}(\alpha_\eta) = \alpha_0\) implies \(\alpha_0 = n[X_0]\), i.e., \(n_{\sigma(0)} = n\) as desired.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) and \(\alpha, \beta \in z(X/S, r)\). The set \(E = \{s \in S : \alpha_s = \beta_s\}\) is closed in \(S\).
Proof
The question is local on \(S\), thus we may assume \(S\) is affine. Let \(X = \bigcup U_i\) be an affine open covering. Let \(E_i = \{s \in S : \alpha_s|_{U_{i, s}} = \beta_s|_{U_{i, s}}\}\). Then \(E = \bigcap E_i\). Hence it suffices to prove the lemma for \(U_i \to S\) and the restriction of \(\alpha\) and \(\beta\) to \(U_i\). This reduces us to the case discussed in the next paragraph.
Assume \(X\) and \(S\) are quasi-compact. Set \(\gamma = \alpha - \beta\). Then \(E = \{s \in S : \gamma_s = 0\}\). By Lemma 0H57 there exists a jointly surjective finite family of proper morphisms \(\{g_i : S_i \to S\}\) such that \(g_i^*\gamma\) is in the image of (0H58). Observe that \(E_i = g_i^{-1}(E)\) is the set of point \(t \in S_i\) such that \((g_i^*\gamma)_t = 0\). If \(E_i\) is closed for all \(i\), then \(E = \bigcup g_i(E_i)\) is closed as well. This reduces us to the case discussed in the next paragraph.
Assume \(X\) and \(S\) are quasi-compact and \(\gamma = \sum n_i[Z_i/X/S]_r\) for a finite number of closed subschemes \(Z_i \subset X\) flat and of relative dimension \(\leq r\) over \(S\). Set \(X' = \bigcup Z_i\) (scheme theoretic union). Then \(i : X' \to X\) is a closed immersion and \(X'\) has relative dimension \(\leq r\) over \(S\). Also \(\gamma = i_*\gamma'\) where \(\gamma' = \sum n_i[Z_i/X'/S]_r\). Since clearly \(E = E' = \{s \in S : \gamma'_s = 0\}\) we reduce to the case discussed in the next paragraph.
Assume \(X\) has relative dimension \(\leq r\) over \(S\). Let \(s \in S\), \(s \not \in E\). We will show that there exists an open neighbourhood \(V \subset S\) of \(s\) such that \(E \cap V\) is empty. The assumption \(s \not \in E\) means there exists an integral closed subscheme \(Z \subset X_s\) of dimension \(r\) such that the coefficient \(n\) of \([Z]\) in \(\gamma_s\) is nonzero. Let \(x \in Z\) be the generic point. Since \(\dim(Z) = r\) we see that \(x\) is a generic point of an irreducible component (namely \(Z\)) of \(X_s\). Thus after replacing \(X\) by an open neighbourhood of \(x\), we may assume that \(Z\) is the only irreducible component of \(X_s\). In particular, we have \(\gamma_s = n[Z]\).
At this point we apply More on Morphisms, Lemma 052E and we obtain a diagram \[\xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] }\] with all the properties listed there. Let \(\gamma' = g^*\gamma\) be the flat pullback. Note that \(E \subset E' = \{s \in S: \gamma'_s = 0\}\) and that \(s \not \in E'\) because the coefficient of \(Z'\) in \(\gamma'_s\) is nonzero, where \(Z' \subset X'_s\) is the closure of \(x'\). Similarly, set \(\gamma'' = \pi_*\gamma'\). Then we have \(E' \subset E'' = \{s \in S: \gamma''_s = 0\}\) and \(s \not \in E''\) because the coefficient of \(Z''\) in \(\gamma''_s\) is nonzero, where \(Z'' \subset Y_s\) is the closure of \(y\). By Lemma 0H5B and openness of \(Y \to S\) we see that an open neighbourhood of \(s\) is disjoint from \(E''\) and the proof is complete.
Lemma
Let \(S = \lim_{i \in I} S_i\) be the limit of a directed inverse system of Noetherian schemes with affine transition morphisms. Let \(0 \in I\) and let \(X_0 \to S_0\) be a finite type morphism of schemes. For \(i \geq 0\) set \(X_i = S_i \times_{S_0} X_0\) and set \(X = S \times_{S_0} X_0\). If \(S\) is Noetherian too, then \[z(X/S, r) = \colim_{i \geq 0} z(X_i/S_i, r)\] where the transition maps are given by base change of relative \(r\)-cycles.
Proof
Suppose that \(i \geq 0\) and \(\alpha_i, \beta_i \in z(X_i/S_i, r)\) map to the same element of \(z(X/S, r)\). Then \(S \to S_i\) maps into the closed subset \(E \subset S_i\) of Lemma 0H5C. Hence for some \(j \geq i\) the morphism \(S_j \to S_i\) maps into \(E\), see Limits, Lemma 05F4. It follows that the base change of \(\alpha_i\) and \(\beta_i\) to \(S_j\) agree. Thus the map is injective.
Let \(\alpha \in z(X/S, r)\). Applying Lemma 0H59 a completely decomposed proper morphism \(g : S' \to S\) such that \(g^*\alpha\) is in the image of (0H58). Set \(X' = S' \times_S X\). We write \(g^*\alpha = \sum n_a [Z_a/X'/S']_r\) for some \(Z_a \subset X'\) closed subscheme flat and of relative dimension \(\leq r\) over \(S'\).
Now we bring the machinery of Limits, Section 01ZL ff to bear. We can find an \(i \geq 0\) such that there exist
a completely decomposed proper morphism \(g_i : S'_i \to S_i\) whose base change to \(S\) is \(g : S' \to S\),
setting \(X'_i = S'_i \times_{S_i} X_i\) closed subschemes \(Z_{ai} \subset X'_i\) flat and of relative dimension \(\leq r\) over \(S'_i\) whose base change to \(S'\) is \(Z_a\).
To do this one uses Limits, Lemmas 01ZM, 01ZP, 04AI, 081F, and 05M5 and More on Morphisms, Lemma 0GTM. Consider \(\alpha'_i = \sum n_a [Z_{ai}/X'_i/S'_i]_r \in z(X'_i/S'_i, r)\). The image of \(\alpha'_i\) in \(z(X'/S', r)\) agrees with the base change \(g^*\alpha\) by construction.
Set \(S''_i = S'_i \times_{S_i} S'_i\) and \(X''_i = S''_i \times_{S_i} X_i\) and set \(S'' = S' \times_S S'\) and \(X'' = S'' \times_S X\). We denote \(\text{pr}_1, \text{pr}_2 : S'' \to S'\) and \(\text{pr}_1, \text{pr}_2 : S''_i \to S'_i\) the projections. The two base changes \(\text{pr}_1^*\alpha'_i\) and \(\text{pr}_1^*\alpha'_i\) map to the same element of \(z(X''/S'', r)\) because \(\text{pr}_1^*g^*\alpha = \text{pr}_1^*g^*\alpha\). Hence after increasing \(i\) we may assume that \(\text{pr}_1^*\alpha'_i = \text{pr}_1^*\alpha'_i\) by the first paragraph of the proof. By Lemma 0H4X we obtain a unique family \(\alpha_i\) of \(r\)-cycles on fibres of \(X_i/S_i\) with \(g_i^*\alpha_i = \alpha'_i\) (this uses that \(S'_i \to S_i\) is completely decomposed). By Lemma 0H52 we see that \(\alpha_i \in z(X_i/S_i, r)\). The uniqueness in Lemma 0H4X implies that the image of \(\alpha_i\) in \(z(X/S, r)\) is \(\alpha\) and the proof is complete.
Lemma
Let \(S\) be a locally Noetherian scheme. Let \(i : X \to X'\) be a thickening of schemes locally of finite type over \(S\). Let \(r \geq 0\). Then \(i_* : z(X/S, r) \to z(X'/S, r)\) is a bijection.
Proof
Since \(i_s : X_s \to X'_s\) is a thickening it is clear that \(i_*\) induces a bijection between families of \(r\)-cycles on the fibres of \(X/S\) and families of \(r\)-cycles on the fibres of \(X'/S\). Also, given a family \(\alpha\) of \(r\)-cycles on the fibres of \(X/S\) \(\alpha \in z(X/S, r) \Leftrightarrow i_*\alpha \in z(X'/S, r)\) by Lemma 0H54. The lemma follows.
Lemma
Let \(S\) be a locally Noetherian scheme. Let \(X\) be a scheme locally of finite type over \(S\). Let \(r \geq 0\). Let \(U \subset X\) be an open such that \(X \setminus U\) has relative dimension \(< r\) over \(S\), i.e., \(\dim(X_s \setminus U_s) < r\) for all \(s \in S\). Then restriction defines a bijection \(z(X/S, r) \to z(U/S, r)\).
Proof
Since \(Z_r(X_s) \to Z_r(U_s)\) is a bijection by the dimension assumption, we see that restriction induces a bijection between families of \(r\)-cycles on the fibres of \(X/S\) and families of \(r\)-cycles on the fibres of \(U/S\). These restriction maps \(Z_r(X_s) \to Z_r(U_s)\) are compatible with base change and with specializations, see Lemma 0H4P and 0H4L. The lemma follows easily from this; details omitted.
Lemma
Let \(g : S' \to S\) be a universal homeomorphism of locally Noetherian schemes which induces isomorphisms of residue fields. Let \(f : X \to S\) be locally of finite type. Set \(X' = S' \times_S X\). Let \(r \geq 0\). Then base change by \(g\) determines a bijection \(z(X/S, r) \to z(X'/S', r)\).
Proof
By Lemma 0H4Y we have a bijection between the group of families of \(r\)-cycles on fibres of \(X/S\) and the group of families of \(r\)-cycles on fibres of \(X'/S'\). Say \(\alpha\) is a families of \(r\)-cycles on fibres of \(X/S\) and \(\alpha' = g^*\alpha\) is the base change. If \(R\) is a discrete valuation ring, then any morphism \(h : \Spec(R) \to S\) factors as \(g \circ h'\) for some unique morphism \(h' : \Spec(R) \to S'\). Namely, the morphism \(S' \times_S \Spec(R) \to \Spec(R)\) is a universal homomorphism inducing bijections on residue fields, and hence has a section (for example because \(R\) is a seminormal ring, see Morphisms, Section 0EUK). Thus the condition that \(\alpha\) is compatible with specializations (i.e., is a relative \(r\)-cycle) is equivalent to the condition that \(\alpha'\) is compatible with specializations.
Equidimensional relative cycles
Here is the definition.
Definition
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. We say a relative \(r\)-cycle \(\alpha\) on \(X/S\) equidimensional if the support of \(\alpha\) (Remark 0H4U) is contained in a closed subset \(W \subset X\) whose relative dimension over \(S\) is \(\leq r\). The group of all equidimensional relative \(r\)-cycles on \(X/S\) is denoted \(z_{equi}(X/S, r)\).
Example
There exist relative \(r\)-cycles which are not equidimensional. Namely, let \(k\) be a field and let \(X = \Spec(k[x, y, t])\) over \(S = \Spec(k[x, y])\). Let \(s\) be a point of \(S\) and denote \(a, b \in \kappa(s)\) the images of \(x\) and \(y\). Consider the family \(\alpha\) of \(0\)-cycles on \(X/S\) defined by
\(\alpha_s = 0\) if \(b = 0\) and otherwise
\(\alpha_s = [p] - [q]\) where \(p\), resp. \(q\) is the \(\kappa(s)\)-rational point of \(\Spec(\kappa(s)[t])\) with \(t = a/b\), resp. \(t = (a + b^2)/b\).
We leave it to the reader to show that this is compatible with specializations; the idea is that \(a/b\) and \((a + b^2)/b = a/b + b\) limit to the same point in \(\mathbf{P}^1\) over the residue field of any valuation \(v\) on \(\kappa(s)\) with \(v(b) > 0\). On the other hand, the closure of the support of \(\alpha\) contains the whole fibre over \((0, 0)\).
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). If \(\alpha\) is equidimensional, then any restriction, base change, or flat pullback of \(\alpha\) is equidimensional.
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Then to check that \(\alpha\) is equidimensional we may work Zariski locally on \(X\) and \(S\).
Proof
Namely, the condition that \(\alpha\) is equidimensional just means that the closure of the support of \(\alpha\) has relative dimension \(\leq r\) over \(S\). Since taking closures commutes with restriction to opens, the lemma follows (small detail omitted).
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Let \(\{g_i : S_i \to S\}\) be an fppf covering. Then \(\alpha\) is equidimensional if and only if each base change \(g_i^*\alpha\) is equidimensional.
Proof
If \(\alpha\) is equidimensional, then each \(g_i^*\alpha\) is too by Lemma 0H5K. Assume each \(g_i^*\alpha\) is equidimensional. Denote \(W\) the closure of \(\text{Supp}(\alpha)\) in \(X\). Since \(g_i : S_i \to S\) is universally open (being flat and locally of finite presentation), so is the morphism \(f_i : X_i = S_i \times_S X \to X\). Denote \(\alpha_i = g_i^*\alpha\). We have \(\text{Supp}(\alpha_i) = f_i^{-1}(\text{Supp}(\alpha))\) by Lemma 0H4V. Since \(f_i\) is open, we see that \(W_i = f_i^{-1}(W)\) is the closure of \(\text{Supp}(\alpha_i)\). Hence by assumption the morphism \(W_i \to S_i\) has relative dimension \(\leq r\). By Morphisms, Lemma 02FY (and the fact that the morphisms \(S_i \to S\) are jointly surjective) we conclude that \(W \to S\) has relative dimension \(\leq r\).
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r, e \geq 0\) be integers. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Let \(\{f_i : X_i \to X\}\) be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension \(e\). Then \(\alpha\) is equidimensional if and only if each flat pullback \(f_i^*\alpha\) is equidimensional.
Proof
Omitted. Hint: As in the proof of Lemma 0H5M one shows that the inverse image by \(f_i\) of the closure \(W\) of the support of \(\alpha\) is the closure \(W_i\) of the support of \(f_i^*\alpha\). Then \(W \to S\) has relative dimension \(\leq r\) holds if \(W_i \to S\) has relative dimension \(\leq r + e\) for all \(i\).
Weightings and relative zero cycles
In this section we show that for a quasi-finite morphism, relative zero cycles are closely related to weighthings, as defined in More on Morphisms, Section 0F38.
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a locally quasi-finite morphism of schemes. Then we have \(z(X/S, 0) = z_{equi}(X/S, 0)\) and \(z(X/S, r) = 0\) for \(r > 0\). Given \(\alpha \in z(X/S, 0)\) let us define a map \[w_\alpha : X \longrightarrow \mathbf{Z},\quad x \mapsto \alpha(x) [\kappa(x) : \kappa(s)]_i \quad\text{where }s = f(x)\] Here \(\alpha(x)\) denotes the coefficient of \(x\) in the \(0\)-cycle \(\alpha_s\) on the fibre \(X_s\) and \([K : k]_i\) denotes the inseparable degree of a finite field extension.
Lemma
In the situation above, if \(g : S' \to S\) is a morphism of locally Noetherian schemes, then \(w_{g^*\alpha} = w_\alpha \circ g'\) where \(g' : X' \to X\) is the projection \(X' = S' \times_S X \to X\).
Proof
Let \(x' \in X'\) with images \(s', s, x\) in \(S', S, X\). Then the coefficient of \([x']\) in the base change of \([x]\) by \(\kappa(s')/\kappa(s)\) is the length of the local ring \((\kappa(s') \otimes_{\kappa(s)} \kappa(x))_\mathfrak q\). Here \(\mathfrak q\) is the prime ideal corresponding to \(x'\). Thus compatibility with base change follows if \[[\kappa(x) : \kappa(s)]_i = \text{length}((\kappa(s') \otimes_{\kappa(s)} \kappa(x))_\mathfrak q) [\kappa(x') : \kappa(s')]_i\] Let \(k/\kappa(s')\) be an algebraically closure. Choose a prime \(\mathfrak p \subset k \otimes_{\kappa(s)} \kappa(x)\) lying over \(\mathfrak q\). Suppose we can show that \[[\kappa(x) : \kappa(s)]_i = \text{length}((k \otimes_{\kappa(s)} \kappa(x))_\mathfrak p) \quad\text{and}\quad [\kappa(x') : \kappa(s')]_i = \text{length}((k \otimes_{\kappa(s')} \kappa(x'))_\mathfrak p)\] Then we win because \[\text{length}((\kappa(s') \otimes_{\kappa(s)} \kappa(x))_\mathfrak q) \text{length}((k \otimes_{\kappa(s')} \kappa(x'))_\mathfrak p) = \text{length}((k \otimes_{\kappa(s)} \kappa(x))_\mathfrak p)\] by Algebra, Lemma 02M1 and flatness of \(\kappa(s') \otimes_{\kappa(s)} \kappa(x) \to k \otimes_{\kappa(s)} \kappa(x)\). To show the two equalities, it suffices to prove the first. Let \(\kappa(x)/\kappa/\kappa(s)\) be the subfield constructed in Fields, Lemma 030K. Then we see that \[k \otimes_{\kappa(s)} \kappa(x) = \prod\nolimits_{\sigma : \kappa \to k} k \otimes_{\sigma, \kappa} \kappa(x)\] and each of the factors is local of degree \([\kappa(x) : \kappa] = [\kappa(x) : \kappa(s)]_i\) as desired.
Lemma
Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a locally quasi-finite morphism of schemes.
For \(\alpha \in z(X/S, 0)\) the map \(w_\alpha : X \to \mathbf{Z}\) constructed above is a weighting.
If \(X\) is quasi-compact, then given a weighting \(w : X \to \mathbf{Z}\) there exists an integer \(n > 0\) such that \(nw = w_\alpha\) for some \(\alpha \in z(X/S, 0)\).
The integer \(n\) in (2) may be chosen to be a power of the prime \(p\) if \(S\) is a scheme over \(\mathbf{F}_p\).
Proof
Let \(\alpha \in z(X/S, 0)\) and choose a diagram \[\xymatrix{ X \ar[d]_f & U \ar[l]^h \ar[d]^\pi \\ Y & V \ar[l]_g }\] as in More on Morphisms, Definition 0F3A. Denote \(\beta \in z(U/V, 0)\) the restriction of the base change \(g^*\alpha\). By the compatibility with base change (Lemma relative-cycles-lemma-weightings-pre) we have \(w_\beta = w_\alpha \circ h\) and it suffices to show that \(\int_\pi w_\beta\) is locally constant on \(V\). Next, note that \[\begin{align*} \left( \int_\pi w_\beta \right)(v) & = \sum\nolimits_{u \in U, \pi(u) = v} \beta(u) [\kappa(u) : \kappa(v)]_i [\kappa(u) : \kappa(v)]_s \\ & = \sum\nolimits_{u \in U, \pi(u) = v} \beta(u)[\kappa(u) : \kappa(v)] \end{align*}\] This last expression is the coefficient of \(v\) in \(\pi_*\beta \in z(V/V, 0)\). By Lemma 0H5B this function is locally constant on \(V\).
Conversely, let \(w : X \to S\) be a weighting and \(X\) quasi-compact. Choose a sufficiently divisible integer \(n\). Let \(\alpha\) be the family of \(0\)-cycles on fibres of \(X/S\) such that for \(s \in S\) we have \[\alpha_s = \sum\nolimits_{f(x) = s} \frac{n w(x)}{[\kappa(x) : \kappa(s)]_i} [x]\] as a zero cycle on \(X_s\). This makes sense since the fibres of \(f\) are universally bounded (Morphisms, Lemma 03JA) hence we can find \(n\) such that the right hand side is an integer for all \(s \in S\). The final statement of the lemma also follows, provided we show \(\alpha\) is a relative \(0\)-cycle. To do this we have to show that \(\alpha\) is compatible with specializations along discrete valuation rings. Note that neither the construction of \(w_\alpha\) nor the proof of Lemma relative-cycles-lemma-weightings-pre uses that \(\alpha\) is a relative cycle. Hence we have \(w_\alpha = nw\) and this remains true after base change. As well, the base change of a weighting is a weighting, see More on Morphisms, Lemma 0F3B. Thus we reduce to the problem studied in the next paragraph.
Assume \(S\) is the spectrum of a discrete valuation ring with generic point \(\eta\) and closed point \(0\). Let \(w : X \to S\) be a weighting with \(X\) quasi-finite over \(S\). Let \(\alpha\) be the family of \(0\)-cycles on fibres of \(X/S\) constructed in the previous paragraph (for a suitable \(n\)). We have to show that \(sp_{X/S}(\alpha_\eta) = \alpha_0\). Let \(\beta \in z(X/S, 0)\) be the relative \(0\)-cycle on \(X/S\) with \(\beta_\eta = \alpha_\eta\) and \(\beta_0 = sp_{X/S}(\alpha_\eta)\). Then \(w' = w_\beta - nw : X \to \mathbf{Z}\) is a weighting (using the result above) and zero in the points of \(X\) which map to \(\eta\). It follows from More on Morphisms, Lemma more-morphisms-lemma-weighting-specialization that \(w' = 0\). This means \(w_\beta = nw\), hence \(\alpha = \beta\) as desired.
Example
Let \(p\) be a prime number. Let \(k'/k\) be a purely inseparable extension of fields of degree \(p^f\). Set \(X = \Spec(k')\) and \(S = \Spec(k)\). The map \(w : X \to \mathbf{Z}\) sending the unique point \(x\) of \(X\) to \(1\) is a weighting of \(X \to S\). On the other hand, the group \(z(X/S, 0)\) is freely generated by the cycle \(\alpha = [x]\). Thus we see that the smallest integer \(n\) that works in Lemma 0H5P is \(n = p^f\) in this case.
Example
The discussion above can be used to “explain” an example due to A.S. Merkurjev [SV, Example 3.5.10]. Let \(p\) be a prime number and let \(k = \mathbf{F}_p(a, b)\) be the purely transcendental extension of \(\mathbf{F}_p\) on \(a\) and \(b\). Set \[A = k[x, y, z]/(a x^p + b y^p - z^p)\] This is a normal domain of finite type over \(k\). The spectrum \(S = \Spec(A)\) is regular, except for a unique singular point \(s\) corresponding to the maximal ideal \((x, y, z)\) of \(A\) with residue field \(k\). Let \(k' = k(a^{1/p}, b^{1/p})\) and \(B = k'[x, y]\). Consider the map \(A \to B\) using the inclusion \(k \to k'\), sending \(x\) to \(x\), \(y\) to \(y\), and \(z\) to \(a^{1/p}x + b^{1/p}y\). Set \(X = \Spec(B)\) and consider the morphism \(f : X \to S\) corresponding to the ring map \(A \to B\). The weighting \(w : X \to \mathbf{Z}\) of \(f\) produced by More on Morphisms, Lemma 0F3E has constant value \(1\) since the fraction field of \(B\) is purely inseparable over \(A\)! Set \(V = f^{-1}(U) \subset X\). By miracle flatness we see that \(V \to U\) is flat of degree \(p\). It follows that the cycle \(\alpha = [V/V/U]_0 \in z(V/U, 0)\) satisfies \(w_\alpha = pw|_V\). However, since the unique point \(x\) of \(X\) lying over \(s\) has residue field \(k'\) of degree \(p^2\) over \(k\), we see that there cannot be an element \(\beta\) of \(z(X/S, 0)\) restricting to \(\alpha\). Namely, the weighting \(pw - w_\beta\) would be zero by More on Morphisms, Lemma more-morphisms-lemma-weighting-specialization which would imply (by the formula in the proof of the lemma above) that \[\beta_s = \frac{pw(x)}{[\kappa(x) : \kappa(s)]_i} [x] = \frac{1}{p}[x]\] in \(Z_0(X_s)\) and this is impossible3. Of course, the cycle \(p\alpha\) is the restriction of a unique element of \(z(X/S, 0)\).
Effective relative cycles
Here is the definition.
Definition
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. We say a relative \(r\)-cycle \(\alpha\) on \(X/S\) effective if \(\alpha_s\) is an effective cycle (Chow Homology, Definition 0H47) for all \(s \in S\). The monoid of all effective relative \(r\)-cycles on \(X/S\) is denoted \(z^{eff}(X/S, r)\).
Below we will show that an effective relative cycle is equidimensional, see Lemma 0H5X.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). If \(\alpha\) is effective, then any restriction, base change, flat pullback, or proper pushforward of \(\alpha\) is effective.
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Then to check that \(\alpha\) is effective we may work Zariski locally on \(X\) and \(S\).
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Let \(g : S' \to S\) be a surjective morphism. Then \(\alpha\) is effective if and only if the base change \(g^*\alpha\) is effective.
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r, e \geq 0\) be integers. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Let \(\{f_i : X_i \to X\}\) be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension \(e\). Then \(\alpha\) is effective if and only if each flat pullback \(f_i^*\alpha\) is effective.
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r, e \geq 0\) be integers. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). If \(\alpha\) is effective, then \(\text{Supp}(\alpha)\) is closed in \(X\).
Proof
Let \(g : S' \to S\) be the inclusion of an irreducible component viewed as an integral closed subscheme. By Lemmas 0H5S and 0H4V it suffices to show that the support of the base change \(g^*\alpha\) is closed in \(S' \times_S S\). Thus we may assume \(S\) is an integral scheme with generic point \(\eta\). We will show that \(\text{Supp}(\alpha)\) is the closure of \(\text{Supp}(\alpha_\eta)\). To do this, pick any \(s \in S\). We can find a morphism \(g : S' \to S\) where \(S'\) is the spectrum of a discrete valuation ring mapping the generic point \(\eta' \in S'\) to \(\eta\) and the closed point \(0 \in S'\) to \(s\), see Properties, Lemma 054F. Then it suffices to prove that the support of \(g^*\alpha\) is equal to the closure of \(\text{Supp}((g^\alpha)_{\eta'})\). This reduces us to the case discussed in the next paragraph.
Here \(S\) is the spectrum of a discrete valuation ring with generic point \(\eta\) and closed point \(0\). We have to show that \(\text{Supp}(\alpha)\) is the closure of \(\text{Supp}(\alpha_\eta)\). Since \(\alpha\) is effective we may write \(\alpha_\eta = \sum n_i[Z_i]\) with \(n_i > 0\) and \(Z_i \subset X_\eta\) integral closed of dimension \(r\). Since \(\alpha_0 = sp_{X/S}(\alpha_\eta)\) we know that \(\alpha_0 = \sum n_i [\overline{Z}_{i, 0}]_r\) where \(\overline{Z}_i\) is the closure of \(Z_i\). By Varieties, Lemma 0B2J we see that \(\overline{Z}_{i, 0}\) is equidimensional of dimension \(r\). Since \(n_i > 0\) we conclude that \(\text{Supp}(\alpha_0)\) is equal to the union of the \(\overline{Z}_{i, 0}\) which is the fibre over \(0\) of \(\bigcup \overline{Z}_i\) which in turn is the closure of \(\bigcup Z_i\) as desired.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r, e \geq 0\) be integers. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). If \(\alpha\) is effective, then \(\alpha\) is equidimensional.
Proof
Assume \(\alpha\) is effective. By Lemma 0H5W the support \(\text{Supp}(\alpha)\) is closed in \(X\). Thus \(\alpha\) is equidimensional as the fibres of \(\text{Supp}(\alpha) \to S\) are the supports of the cycles \(\alpha_s\) and hence have dimension \(r\).
Remark
Let \(f : X \to S\) be a morphism of schemes with \(S\) locally Noetherian and \(f\) locally of finite type. We can ask if the contravariant functor \[\begin{matrix} \text{schemes }S'\text{ locally} \\ \text{of finite type over }S \end{matrix} \longrightarrow z^{eff}(X'/S', r)\text{ where }X' = S' \times_S X\] is representable. Since \(z(X'/S', r) = z(X'_{red}/S'_{red}, r)\) this cannot be true (we leave it to the reader to make an actual counter example). A better question would be if we can find a subcategory of the left hand side on which the functor is representable. Lemma 0H5G suggests we should restrict at least to the category of seminormal schemes over \(S\).
If \(S/\Spec(\mathbf{Q})\) is Nagata and \(f\) is a projective morphism, then it turns out that \(S' \mapsto z^{eff}(X'/S', r)\) is representable on the category of seminormal \(S'\). Roughly speaking this is the content of [KRC, Theorem 3.21].
If \(S\) has points of positive characteristic, then this no longer works even if we replace seminormality with weak normality; a locally Noetherian scheme \(T\) is weakly normal if any birational universal homeomorphism \(T' \to T\) has a section. An example is to consider \(0\)-cycles of degree \(2\) on \(X = \mathbf{A}^2_k\) over \(S = \Spec(k)\) where \(k\) is a field of characteristic \(2\). Namely, over \(W = X \times_S X\) we have a canonical relative \(0\)-cycle \(\alpha \in z^{eff}(X_W/W, 0)\): for \(w = (x_1, x_2) \in W = X^2\) we have the cycle \(\alpha_w = [x_1] + [x_2]\). This cycle is invariant under the involution \(\sigma : W \to W\) switching the factors. Since \(W\) is smooth (hence normal, hence weakly normal), if \(z(-/-, r)\) was representable by \(M\) on the category of weakly normal schemes of finite type over \(k\) we would get a \(\sigma\)-invariant morphism from \(W\) to \(M\). This in turn would define a morphism from the quotient scheme \(\text{Sym}^2_S(X) = W/\langle \sigma \rangle\) to \(M\). Since \(\text{Sym}^2_S(X)\) is normal, we would by the moduli property of \(M\) obtain a relative \(0\)-cycle \(\beta\) on \(X \times_S \text{Sym}^2_S(X) / \text{Sym}^2_S(X)\) whose pullback to \(W\) is \(\alpha\). However, there is no such cycle \(\beta\). Namely, writing \(X = \Spec(k[u, v])\) the scheme \(\text{Sym}^2_S(X)\) is the spectrum of \[k[u_1 + u_2, u_1u_2, v_1 + v_2, v_1v_2, u_1v_1 + u_2v_2] \subset k[u_1, u_2, v_1, v_2]\] The image of the diagonal \(u_1 = u_2, v_1 = v_2\) in \(\text{Sym}^2_S(X)\) is the closed subscheme \(V = \Spec(k[u_1^2, v_1^2])\); here we use that the characteristic of \(k\) is \(2\). Looking at the generic point \(\eta\) of \(V\), the cycle \(\beta_\eta\) would be a zero cycle of degree \(2\) on \(\mathbf{A}^2_{k(u_1^2, v_1^2)}\) whose pullback to \(\mathbf{A}^2_{k(u_1, u_2)}\) would be \(2[\text{the point with coordinates} (u_1, v_2)]\). This is clearly impossible.
The discussion above does not contradict [KRC, Theorem 4.13] as the Chow variety in that theorem only coarsely represents a functor (in fact 2 distinct functors, only one of which agrees with ours for projective \(X\) as one can see with some work). Similarly, in [SV, Section 4.4] it is shown that for projective \(X/S\) the \(h\)-sheafification of the presheaf \(S' \mapsto z^{eff}(S' \times_S X/S', r)\) is equal to the \(h\)-sheafification of a representable functor.
Remark
Let \(f : X \to S\) be a morphism of schemes. Let \(r \geq 0\). Let \(Z \subset X\) be a closed subscheme. Assume
\(S\) is Noetherian and geometrically unibranch,
\(f\) is of finite type, and
\(Z \to S\) has relative dimension \(\leq r\).
Then for all sufficiently divisible integers \(n \geq 1\) there exists a unique effective relative \(r\)-cycle \(\alpha\) on \(X/S\) such that \(\alpha_\eta = n[Z_\eta]_r\) for every generic point \(\eta\) of \(S\). This is a reformulation of [SV, Theorem 3.4.2]. If we ever need this result, we will precisely state and prove it here.
Proper relative cycles
In our setting, the following is probably the correct definition.
Definition
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. We say a relative \(r\)-cycle \(\alpha\) on \(X/S\) is a proper relative cycle if the support of \(\alpha\) (Remark 0H4U) is contained in a closed subset \(W \subset X\) proper over \(S\) (Cohomology of Schemes, Definition 0CYM). The group of all proper relative \(r\)-cycles on \(X/S\) is denoted \(c(X/S, r)\).
By Cohomology of Schemes, Lemma 0CYN this just means that the closure of the support is proper over the base. To see that these form a group, use Cohomology of Schemes, Lemma 0CYR.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). If \(\alpha\) is proper, then any base change \(\alpha\) is proper.
Proof
Omitted.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) locally Noetherian and \(f\) locally of finite type. Let \(r \geq 0\) be an integer. Let \(\alpha\) be a relative \(r\)-cycle on \(X/S\). Let \(\{g_i : S_i \to S\}\) be a h covering. Then \(\alpha\) is proper if and only if each base change \(g_i^*\alpha\) is proper.
Proof
If \(\alpha\) is proper, then each \(g_i^*\alpha\) is too by Lemma 0H62. Assume each \(g_i^*\alpha\) is proper. To prove that \(\alpha\) is proper, it clearly suffices to work affine locally on \(S\). Thus we may and do assume that \(S\) is affine. Then we can refine our covering \(\{S_i \to S\}\) by a family \(\{T_j \to S\}\) where \(g : T \to S\) is a proper surjective morphism and \(T = \bigcup T_j\) is an open covering. It follows that \(\beta = g^*\alpha\) is proper on \(Y = T \times_S X\) over \(T\). By Lemma 0H4V we find that the support of \(\beta\) is the inverse image of the support of \(\alpha\) by the morphism \(f : Y \to X\). Hence the closure \(W \subset Y\) of \(f^{-1}\text{Supp}(\alpha)\) is proper over \(T\). Since the morphism \(T \to S\) is proper, it follows that \(W\) is proper over \(S\). Then by Cohomology of Schemes, Lemma 0CYQ the image \(f(W) \subset X\) is a closed subset proper over \(S\). Since \(f(W)\) contains \(\text{Supp}(\alpha)\) we conclude \(\alpha\) is proper.
Proper and equidimensional relative cycles
Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian and \(f\) is locally of finite type. Let \(r \geq 0\) be an integer. We say a relative \(r\)-cycle \(\alpha\) on \(X/S\) is a proper and equidimensional relative cycle if \(\alpha\) is both equidimensional (Definition 0H5I) and proper (Definition 0H61). The group of all proper, equidimensional relative \(r\)-cycles on \(X/S\) is denoted \(c_{equi}(X/S, r)\).
Similarly we say a relative \(r\)-cycle \(\alpha\) on \(X/S\) is a proper and effective relative cycle if \(\alpha\) is both effective (Definition 0H5R) and proper (Definition 0H61). The monoid of all proper, effective relative \(r\)-cycles on \(X/S\) is denoted \(c^{eff}(X/S, r)\). Observe that these are equidimensional by Lemma 0H5X.
Thus we have the following diagram of inclusion maps \[\xymatrix{ c^{eff}(X/S, r) \ar[r] \ar[d] & c_{equi}(X/S, r) \ar[r] \ar[d] & c(X/S, r) \ar[d] \\ z^{eff}(X/S, r) \ar[r] & z_{equi}(X/S, r) \ar[r] & z(X/S, r) }\]
Action on cycles
Let \(S\) be a locally Noetherian, universally catenary scheme endowed with a dimension function \(\delta\), see Chow Homology, Section 02QK. Let \(X \to Y\) be a morphism of schemes over \(S\), both locally of finite type over \(S\). Let \(r \geq 0\). Finally, let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/Y\). For \(e \in \mathbf{Z}\) we will construct an operation \[\alpha \cap - : Z_e(Y) \longrightarrow Z_{r + e}(X)\] Namely, given \(\beta \in Z_e(Y)\) write \(\beta = \sum n_i[Z_i]\) where \(Z_i \subset Y\) is an integral closed subscheme of \(\delta\)-dimension \(e\) and the family \(Z_i\) is locally finite in the scheme \(Y\). Let \(y_i \in Z_i\) be the generic point. Write \(\alpha_{y_i} = \sum m_{ij} [V_{ij}]\). Thus \(V_{ij} \subset X_{y_i}\) is an integral closed subscheme of dimension \(r\) and the family \(V_{ij}\) is locally finite in the scheme \(X_{y_i}\). Then we set \[\alpha \cap \beta = \sum n_i m_{ij} [\overline{V}_{ij}] \quad\in\quad Z_{r + e}(X)\] Here \(\overline{V}_{ij} \subset X\) is the scheme theoretic image of the morphism \(V_{ij} \to X_{y_i} \to X\) or equivalently, \(\overline{V}_{ij} \subset X\) is an integral closed subscheme mapping dominantly to \(Z_i \subset Y\) whose generic fibre is \(V_{ij}\). It follows readily that \(\dim_\delta(\overline{V}_{ij}) = r + e\) and that the family of closed subschemes \(\overline{V}_{ij} \subset X\) is locally finite (we omit the verifications). Hence \(\alpha \cap \beta\) is indeed an element of \(Z_{r + e}(X)\).
Lemma
The construction above is bilinear, i.e., we have \((\alpha_1 + \alpha_2) \cap \beta = \alpha_1 \cap \beta + \alpha_2 \cap \beta\) and \(\alpha \cap (\beta_1 + \beta_2) = \alpha \cap \beta_1 + \alpha \cap \beta_2\).
Proof
Omitted.
Lemma
If \(U \subset X\) and \(V \subset Y\) are open and \(f(U) \subset V\), then \((\alpha \cap \beta)|_U\) is equal to \(\alpha|_U \cap \beta|_V\).
Proof
Immediate from the explicit description of \(\alpha \cap \beta\) given above.
Lemma
Forming \(\alpha \cap \beta\) is compatible with flat base change and flat pullback (see proof for elucidation).
Proof
Let \((S, \delta)\), \((S', \delta')\), \(g : S' \to S\), and \(c \in \mathbf{Z}\) be as in Chow Homology, Situation 0FVG. Let \(X \to Y\) be a morphism of schemes locally of finite type over \(S\). Denote \(X' \to Y'\) the base change of \(X \to Y\) by \(g\). Let \(\alpha\) be a family of \(r\)-cycles on the fibres of \(X/Y\). Let \(\beta \in Z_e(Y)\). Denote \(\alpha'\) the base change of \(\alpha\) by \(Y' \to Y\). Denote \(\beta' = g^*\beta \in Z_{e + c}(Y')\) the pullback of \(\beta\) by \(g\), see Chow Homology, Section 0FVF. Compatibility with base change means \(\alpha' \cap \beta'\) is the base change of \(\alpha \cap \beta\).
Proof of compatibility with base change. Since we are proving an equality of cycles on \(X'\), we may work locally on \(Y\), see Lemma 0H67. Thus we may assume \(Y\) is affine. In particular \(\beta\) is a finite linear combination of prime cycles. Since \(- \cap -\) is linear in the second variable (Lemma 0H66), it suffices to prove the equality when \(\beta = [Z]\) for some integral closed subscheme \(Z \subset Y\) of \(\delta\)-dimension \(e\).
Let \(y \in Z\) be the generic point. Write \(\alpha_y = \sum m_j [V_j]\). Let \(\overline{V}_j\) be the closure of \(V_j\) in \(X\). Then we have \[\alpha \cap \beta = \sum m_j[\overline{V}_j]\] The base change of \(\beta\) is \(\beta' = \sum [Z \times_S S']_{e + c}\) as a cycle on \(Y' = Y \times_S S'\). Let \(Z'_a \subset Z \times_S S'\) be the irreducible components, denote \(y'_a \in Z'_a\) their generic points, and denote \(n_a\) the multiplicity of \(Z'_a\) in \(Z \times_S S'\). We have \[\beta' = \sum [Z \times_S S']_{e + c} = \sum n_a[Z'_a]\] We have \(\alpha'_{y'_a} = \sum m_j [V_{j, \kappa(y'_a)}]_r\) because \(\alpha'\) is the base change of \(\alpha\) by \(Y' \to Y\). Let \(V'_{jab} \subset V_{j, \kappa(y'_a)}\) be the irreducible components and denote \(m_{jab}\) the multiplicity of \(V'_{jab}\) in \(V_{j, \kappa(y'_a)}\). We have \[\alpha'_{y'_a} = \sum m_j [V_{j, \kappa(y'_a)}]_r = \sum m_j m_{jab} [V'_{jab}]\] Thus we we have \[\alpha' \cap \beta' = \sum n_a m_j m_{jab} [\overline{V}'_{jab}]\] where \(\overline{V}'_{jab}\) is the closure of \(V'_{jab}\) in \(X'\). Thus to prove the desired equality it suffices to prove
the irreducible components of \(\overline{V}_j \times_S S'\) are the schemes \(\overline{V}'_{jab}\) and
the multiplicity of \(\overline{V}'_{jab}\) in \(\overline{V}_j \times_S S'\) is equal to \(n_a m_{jab}\).
Note that \(V_j \to \overline{V}_j\) is a birational morphism of integral schemes. The morphisms \(V_j \times_S S' \to V_j\) and \(\overline{V}_j \times_S S' \to \overline{V}_j\) are flat and hence map generic points of irreducible components to the (unique) generic points of \(V_j\) and \(\overline{V}_j\). It follows that \(V_j \times_S S' \to \overline{V}_j \times_S S'\) is a birational morphisms hence induces a bijection on irreducible components and identifies their multiplicities. This means that it suffices to prove that the irreducible components of \(V_j \times_S S'\) are the schemes \(V'_{jab}\) and the multiplicity of \(V'_{jab}\) in \(V_j \times_S S'\) is equal to \(n_a m_{jab}\). However, then we are just saying that the diagram \[\xymatrix{ Z_r(V_j) \ar[r] & Z_{r + c}(V_j \times_S S') \\ Z_0(\Spec(\kappa(y))) \ar[r] \ar[u] & Z_c(\Spec(\kappa(y)) \times_S S') \ar[u] }\] is commutative where the horizontal arrows are base change by \(\Spec(\kappa(y)) \times_S S' \to \Spec(\kappa(y))\) and the vertical arrows are flat pullback. This was shown in Chow Homology, Lemma 0FVK.
The statement in the lemma on flat pullback means the following. Let \((S, \delta)\), \(X \to Y\), \(\alpha\), and \(\beta\) be as in the construction of \(\alpha \cap \beta\) above. Let \(Y' \to Y\) be a flat morphism, locally of finite type, and of relative dimension \(c\). Then we can let \(\alpha'\) be the base change of \(\alpha\) by \(Y' \to Y\) and \(\beta'\) the flat pullback of \(\beta\). Compatibility with flat pullback means \(\alpha' \cap \beta'\) is the flat pullback of \(\alpha \cap \beta\) by \(X \times_Y Y' \to Y\). This is actually a special case of the discussion above if we set \(S = Y\) and \(S' = Y'\).
Lemma
Let \((S, \delta)\) and \(f : X \to Y\) be as above. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module with \(\dim(\text{Supp}(\mathcal{F}_y)) \leq r\) for all \(y \in Y\). Let \(\mathcal{G}\) be a coherent \(\mathcal{O}_Y\)-module with \(\dim_\delta(\text{Supp}(\mathcal{G})) \leq e\). Set \(\alpha = [\mathcal{F}/X/Y]_r\) (Example 0H4Q) and \(\beta = [\mathcal{G}]_e\) (Chow Homology, Definition 02QX). If \(\mathcal{F}\) is flat over \(Y\), then \(\alpha \cap \beta = [\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}]_{r + e}\).
Proof
Observe that \[\text{Supp}(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \text{Supp}(\mathcal{F}) \cap f^{-1}\text{Supp}(\mathcal{G}) = \bigcup\nolimits_{y \in \text{Supp}(\mathcal{G})} \text{Supp}(\mathcal{F}_y)\] It follows that this is a closed subset of \(\delta\)-dimension \(\leq r + e\). Whence the expression \([\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}]_{r + e}\) makes sense.
We will use the notation \(\beta = \sum n_i[Z_i]\), \(y_i \in Z_i\), \(\alpha_{y_i} = \sum m_{ij} [V_{ij}]\), and \(\overline{V}_{ij}\) introduced in the construction of \(\alpha \cap \beta\). Since \(\beta = [\mathcal{G}]_e\) we see that the \(Z_i\) are the irreducible components of \(\text{Supp}(\mathcal{G})\) which have \(\delta\)-dimension \(e\). Similarly, the \(V_{ij}\) are the irreducible components of \(\text{Supp}(\mathcal{F}_{y_i})\) having dimension \(r\). It follows from this and the equation in the first paragraph that \(\overline{V}_{ij}\) are the irreducible components of \(\text{Supp}(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})\) having \(\delta\)-dimension \(r + e\). Thus to prove the lemma it now suffices to show that \[\text{length}_{\mathcal{O}_{X, \xi_{ij}}}( (\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})_{\xi_{ij}}) = \text{length}_{\mathcal{O}_{X_{y_i}, \xi_{ij}}}((\mathcal{F}_{y_i})_{\xi_{ij}}) \cdot \text{length}_{\mathcal{O}_{Y, y_i}}(\mathcal{G}_{y_i})\] By the first paragraph of the proof the left hand side is equal to the length of the \(B = \mathcal{O}_{X, \xi_{ij}}\)-module \[\mathcal{G}_{y_i} \otimes_{\mathcal{O}_{Y, y_i}} \mathcal{F}_{\xi_{ij}} = M \otimes_A N\] Here \(M = \mathcal{G}_{y_i}\) is a finite length \(A = \mathcal{O}_{Y, y_i}\)-module and \(N = \mathcal{F}_{\xi_{ij}}\) is a finite \(B\)-module such that \(N/\mathfrak m_AN\) has finite length. Since \(\mathcal{F}\) is flat over \(Y\) the module \(N\) is \(A\)-flat. The right hand side of the formula is equal to \[\text{length}_B(N/\mathfrak m_A N) \cdot \text{length}_A(M)\] Thus the right and left hand side of the formula are additive in \(M\) (use flatness of \(N\) over \(A\)). Thus it suffices to prove the formula with \(M = \kappa_A\) is the residue field in which case it is immediate.
Lemma
Let \((S, \delta)\) and \(f : X \to Y\) be as above. Let \(Z \subset X\) be a closed subscheme of relative dimension \(\leq r\) over \(Y\). Set \(\alpha = [Z/X/Y]_r\) (Example 0H4S). Let \(W \subset Y\) be a closed subscheme of \(\delta\)-dimension \(\leq e\). Set \(\beta = [W]_e\) (Chow Homology, Definition 02QU). If \(Z\) is flat over \(Y\), then \(\alpha \cap \beta = [Z \times_Y W]_{r + e}\).
Proof
This is a special case of Lemma 0H69 if we take \(\mathcal{F} = \mathcal{O}_Z\) and \(\mathcal{F} = \mathcal{O}_W\).
Lemma
Let \((S, \delta)\) and \(f : X \to Y\) be as above. Assume \(f\) is flat of relative dimension \(e\). For \(\beta \in Z_r(Y)\) we have \(f^*\beta = [X/X/Y]_e \cap \beta\) in \(Z_{e + r}(X)\).
Proof
It suffices to prove the equality locally on \(X\). Thus we may assume \(Y\) is affine. In this case \(\beta\) is a finite integer linear combination of prime cycles. Since both sides of the equality are additive in \(\beta\) we may assume \(\beta = [W]\) where \(W\) is an integral closed subscheme of \(Y\) of \(\delta\)-dimension \(r\). Let \(f^{-1}(W) = W \times_Y X\) be the scheme theoretic inverse image of \(W\) in \(X\). By Lemma 0H6A we have \([X/X/Y]_e \cap \beta = [W \times_Y X]_{r + e}\) and by Chow Homology, Definition 02RB we have \(f^*\beta = [f^{-1}(W)]_{r + e}\). Thus we have the desired equality.
Lemma
Let \((S, \delta)\) be as above. Let \[\xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of schemes locally of finite type over \(S\) with \(g\) proper. Let \(r, e \geq 0\). Let \(\alpha\) be a family of \(r\)-cycles on the fibres of \(X/Y\). Let \(\beta' \in Z_e(Y')\). Then we have \(f_*(g^*\alpha \cap \beta') = \alpha \cap g_*\beta'\).
Proof
Since we are proving an equality of cycles on \(X\), we may work locally on \(Y\), see Lemma 0H67. Thus we may assume \(Y\) is affine. Thus \(Y'\) is quasi-compact. In particular \(\beta'\) is a finite linear combination of prime cycles. Since \(- \cap -\) is linear in the second variable (Lemma 0H66), it suffices to prove the equality when \(\beta' = [Z']\) for some integral closed subscheme \(Z' \subset Y'\) of \(\delta\)-dimension \(e\). Set \(Z = g(Z')\). This is an integral closed subscheme of \(Y\) of \(\delta\)-dimension \(\leq e\). For simplicity we are going to assume \(Z\) has \(\delta\)-dimension equal to \(e\) and leave the other case (which is easier) to the reader. Let \(y \in Z\) and \(y' \in Z'\) be the generic points. Write \(\alpha_y = \sum m_j[V_j]\) with \(V_j \subset X_y\) integral closed subschemes of dimension \(r\).
Assume first \(g\) is a closed immersion. Then \(g_*\beta' = [Z]\) and \((g^*\alpha)_{y'} = \sum n_j[V_j]\); this makes sense because \(V_j\) is contained in the closed subscheme \(X'_{y'}\) of \(X_y\). Thus in this case the equality is obvious: in both cases we obtain \(\sum m_j[\overline{V}_j]\) where \(\overline{V}_j\) is the closure of \(V_j\) in the closed subscheme \(X' \subset X\).
Back to the general case with \(\beta' = [Z']\) as above. Set \(W = Z \times_Y X\) and \(W' = Z' \times_{Y'} X'\). Consider the cartesian squares \[\xymatrix{ W \ar[r] \ar[d] & X \ar[d] \\ Z \ar[r] & Y } \quad \xymatrix{ W' \ar[r] \ar[d] & X' \ar[d] \\ Z' \ar[r] & Y' } \quad \xymatrix{ W' \ar[r] \ar[d] & W \ar[d] \\ Z' \ar[r] & Z }\] Since we know the result for the first two squares with by the previous paragraph, a formal argument shows that it suffices to prove the result for the last square and the element \(\beta' = [Z'] \in Z_e(Z')\). This reduces us to the case discussed in the next paragraph.
Assume \(Y' \to Y\) is a generically finite morphism of integral schemes of \(\delta\)-dimension \(e\) and \(\beta' = [Y']\). In this case both \(f_*(g^*\alpha \cap \beta')\) and \(\alpha \cap g_*\beta'\) are cycles which can be written as a sum of prime cycles dominant over \(Y\). Thus we may replace \(Y\) by a nonempty open subscheme in order to check the equality. After such a replacement we may assume \(g\) is finite and flat, say of degree \(d \geq 1\). Of course, this means that \(g_*\beta' = g_*[Y'] = d[Y]\). Also \(\beta' = [Y'] = g^*[Y]\). Hence \[f_*(g^*\alpha \cap \beta') = f_*(g^*\alpha \cap g^*[Y]) = f_*f^*(\alpha \cap [Y]) = d (\alpha \cap [Y]) = \alpha \cap g_*\beta'\] as desired. The second equality is Lemma 0H68 and the third equality is Chow Homology, Lemma 02RH.
Action on chow groups
When \(\alpha\) is a relative \(r\)-cycle, the operation \(\alpha \cap -\) of Section 0H65 factors through rational equivalence and defines a bivariant class.
Lemma
Let \((S, \delta)\) be as in Section 0H65. Let \(f : X' \to X\) be a proper morphism of schemes locally of finite type over \(S\). Let \((\mathcal{L}, s, i : D \to X)\) be as in Chow Homology, Definition 02T8. Form the diagram \[\xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X }\] as in Chow Homology, Remark 0B6Y. If \(\mathcal{L}|_D \cong \mathcal{O}_D\), then \(i^*f_*\alpha' = g_*(i')^*\alpha'\) in \(Z_k(D)\) for any \(\alpha' \in Z_{k + 1}(X')\).
Proof
The statement makes sense as all operations are defined on the level of cycles, see Chow Homology, Remark 0B6Z for the Gysin maps. Suppose \(\alpha = [W']\) for some integral closed subscheme \(W' \subset X'\). Let \(W = f(W') \subset X\). In case \(W' \not \subset D'\), then \(W \not \subset D\) and we see that \[[W' \cap D']_k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad\text{and}\quad [W \cap D]_k = \text{div}_{\mathcal{L}|_W}(s|_W)\] and hence \(f_*\) of the first cycle equals the second cycle by Chow Homology, Lemma 02ST. Hence the equality holds as cycles. In case \(W' \subset D'\), then \(W \subset D\) and both sides are zero by construction.
Lemma
Let \((S, \delta)\) be as in Section 0H65. Let \(X \to Y\) be a morphism of schemes locally of finite type over \(S\). Let \(r \geq 0\) and let \(\alpha \in z(X/Y, r)\) be a relative \(r\)-cycle on \(X/Y\). Let \((\mathcal{L}, s, i : D \to Y)\) be as in Chow Homology, Definition 02T8. Form the cartesian diagram \[\xymatrix{ E \ar[d] \ar[r]_j & X \ar[d] \\ D \ar[r]^i & Y }\] See Chow Homology, Remark 0B6Y. If \(\mathcal{L}|_D \cong \mathcal{O}_D\), then for \(e \in \mathbf{Z}\) the diagram \[\xymatrix{ Z_e(D) \ar[rr]_{i^*\alpha \cap -} & & Z_{e + r}(E) \\ Z_{e + 1}(Y) \ar[u]^{i^*} \ar[rr]^{\alpha \cap -} & & Z_{r + e + 1}(X) \ar[u]_{j^*} }\] commutes where the vertical arrows \(i^*\) and \(j^*\) are the Gysin maps on cycles as in Chow Homology, Remark 0B6Z.
Proof
Preliminary remark. Suppose that \(g : Y' \to Y\) is an envelope (Chow Homology, Definition 0GU5). Denote \(D', i', E', j', X', \alpha'\) the base changes of \(D, i, E, j, X, \alpha\) by \(g\) and denote \(f : X' \to X\) the projection. Assume the lemma holds for \(D', i', E', j', X', Y', \alpha'\). Then, if \(\beta' \in Z_{e + 1}(Y')\), we have \[\begin{align*} i^*\alpha \cap i^*g_*\beta' & = i^*\alpha \cap f_*(i')^*\beta' \\ & = f_*(f^*i^*\alpha \cap (i')^*\beta') \\ & = f_*((i')^*\alpha' \cap (i')^*\beta') \\ & = f_*((j')^*(\alpha' \cap \beta')) \\ & = j^*(f_*(f^*\alpha \cap \beta')) \\ & = j^*(\alpha \cap g_*\beta') \end{align*}\] Here the first equality is Lemma 0H6D, the second equality is Lemma 0H6B, the third equality is the definition of \(\alpha'\), the fourth equality is the assumption that our lemma holds for \(D', i', E', j', X', \alpha'\), the fifth equality is Lemma 0H6D, and the sixth equality is Lemma 0H6B. Thus we see that our lemma holds for the image of \(g_* : Z_{e + 1}(Y') \to Z_e(Y)\). However, since \(g\) is completely decomposed this map is surjective and we conclude the lemma holds for \(D, i, E, j, X, Y, \alpha\).
Let \(\beta \in Z_{e + 1}(Y)\). We have to show that \((D \to Y)^*\alpha \cap i^*\beta = j^*(\alpha \cap \beta)\) as cycles on \(E\). This question is local on \(E\) hence we can replace \(X\) and \(Y\) by open subschemes. (This uses that formation of the operators \(i^*\), \(j^*\), \(\alpha \cap -\) and \((D \to Y)^*\alpha \cap -\) commute with localization. This is obvious for the Gysin maps and follows from Lemma 0H67 for the others.) Thus we may assume that \(X\) and \(Y\) are affine and we reduce to the case discussed in the next paragraph.
Assume \(X\) and \(Y\) are quasi-compact. By the first paragraph of the proof and Lemma 0H59 we may in addition assume that \(\alpha\) is in the image of (0H58). By linearity of the operations in question, we may assume that \(\alpha = [Z/X/Y]_r\) for some closed subscheme \(Z \subset X\) which is flat and of relative dimension \(\leq r\) over \(Y\). Also, as \(Y\) is quasi-compact, the cycle \(\beta\) is a finite linear combination of prime cycles. Since the operations in question are linear, it suffices to prove the equality when \(\beta = [W]\) for some integral closed subscheme \(W \subset Y\) of \(\delta\)-dimension \(e + 1\).
If \(W \subset D\), then on the one hand \(i^*[W] = 0\) and on the other hand \(\alpha \cap [W]\) is supported on \(E\) so also \(j^*(\alpha \cap [W]) = 0\). Thus the equality holds in this case.
Say \(W \not \subset D\). Then \(i^*[W] = [D \cap W]_e\). Note that the pullback \(i^*\alpha\) of \(\alpha = [Z/X/Y]_r\) by \(i\) is \([(E \cap Z)/E/D]_r\) and that \((E \cap Z) = E \times_Y Z = D \times_Y Z\) is flat over \(D\). Hence by Lemma 0H6A used twice we have \[i^*\alpha \cap i^*[W] = [(E \cap Z) \times_D (D \cap W)]_{r + e} = [E \cap (Z \times_Y W)]_{r + e} = j^*(\alpha \cap [W])\] as desired.
Proposition
Let \((S, \delta)\) be as in Section 0H65. Let \(X \to Y\) be a morphism of schemes locally of finite type over \(S\). Let \(r \geq 0\) and let \(\alpha \in z(X/Y, r)\) be a relative \(r\)-cycle on \(X/Y\). The rule that to every morphism \(g : Y' \to Y\) locally of finite type and every \(e \in \mathbf{Z}\) associates the operation \[g^*\alpha \cap - : Z_e(Y') \to Z_{r + e}(X')\] where \(X' = Y' \times_Y X\) factors through rational equivalence to define a bivariant class \(c(\alpha) \in A^{-r}(X \to Y)\).
Proof
The operation factors through rational equivalence by Lemma 0H6E and Chow Homology, Lemma 0B7A. The resulting operation on chow groups is a bivariant class by Chow Homology, Lemma 0F9A and Lemmas 0H6B, 0H68, and 0H6E.
Remark
Let \((S, \delta)\) be as in Section 0H65. Let \(X \to Y\) be a morphism of schemes locally of finite type over \(S\). Let \(r \geq 0\). Let \(c\) be a rule that to every morphism \(g : Y' \to Y\) locally of finite type and every \(e \in \mathbf{Z}\) associates an operation \[c \cap - : Z_e(Y') \to Z_{r + e}(X')\] compatible with proper pushforward, flat pullback, and Gysin maps as in Lemma 0H6E. Then we claim there is a relative \(r\)-cycle \(\alpha\) on \(X/Y\) such that \(c \cap = g^*\alpha \cap -\) for every \(g\) as above. If we ever need this, we will carefully state and prove this here.
Composition of families of cycles on fibres
Let \(X \to Y \to S\) be morphisms of schemes, both locally of finite type. Let \(r, e \geq 0\). Let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/Y\) and let \(\beta\) be a family of \(e\)-cycles on fibres of \(Y/S\). Then we obtain a family of of \((r + e)\)-cycles \(\alpha \circ \beta\) on the fibres of \(X/S\) by setting \[(\alpha \circ \beta)_s = (Y_s \to Y)^*\alpha \cap \beta_s\] More precisely, the expression \((Y_s \to Y)^*\alpha\) denotes the base change of \(\alpha\) by \(Y_s \to Y\) to a family of \(r\)-cycles on the fibres of \(X_s/Y_s\) and the operation \(- \cap -\) was defined and studied in Section 0H654.
Lemma
The construction above is bilinear, i.e., we have \((\alpha_1 + \alpha_2) \circ \beta \alpha_1 \circ \beta + \alpha_1 \circ \beta\) and \(\alpha \circ (\beta_1 + \beta_2) = \alpha \circ \beta_1 + \alpha \circ \beta_2\).
Proof
Omitted. Hint: on fibres the construction is bilinear by Lemma 0H66.
Lemma
If \(U \subset X\) and \(V \subset Y\) are open and \(f(U) \subset V\), then \((\alpha \circ \beta)|_U\) is equal to \(\alpha|_U \circ \beta|_V\).
Proof
Omitted. Hint: on fibres use Lemma 0H67.
Lemma
The formation of \(\alpha \circ \beta\) is compatible with base change.
Proof
Let \(g : S' \to S\) be a morphism of schemes. Denote \(X' \to Y'\) the base change of \(X \to Y\) by \(g\). Denote \(\alpha'\) the base change of \(\alpha\) with respect to \(Y' \to Y\). Denote \(\beta'\) the base change of \(\beta\) with respect to \(S' \to S\). The assertion means that \(\alpha' \circ \beta'\) is the base change of \(\alpha \circ \beta\) by \(g : S' \to S\).
Let \(s' \in S'\) be a point with image \(s \in S\). Then \[(\alpha' \circ \beta')_{s'} = (Y'_{s'} \to Y')^*\alpha' \cap \beta'_{s'}\] We observe that \[(Y'_{s'} \to Y')^*\alpha' = (Y'_{s'} \to Y')^*(Y' \to Y)^*\alpha = (Y'_{s'} \to Y_s)^*(Y_s \to Y)^*\alpha\] and that \(\beta'_{s'}\) is the base change of \(\beta_s\) by \(s' = \Spec(\kappa(s')) \to \Spec(\kappa(s)) = s\). Hence the result follows from Lemma 0H68 applied to \((Y_s \to Y)^*\alpha\), \(\beta_s\), \(X_s \to Y_s \to s\), and base change by \(s' \to s\).
Lemma
Let \(f : X \to Y\) and \(Y \to S\) be morphisms of schemes, both locally of finite type. Let \(r, e \geq 0\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite type, with \(\dim(\text{Supp}(\mathcal{F}_y)) \leq r\) for all \(y \in Y\). Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module of finite type, with \(\dim(\text{Supp}(\mathcal{G}_s)) \leq e\) for all \(s \in S\). If \(\alpha = [\mathcal{F}/X/Y]_r\) and \(\beta = [\mathcal{G}/Y/S]_e\) (Example 0H4Q) and \(\mathcal{F}\) is flat over \(Y\), then \(\alpha \circ \beta = [\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}/X/S]_{r + e}\).
Proof
First we observe that \(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}\) is a quasi-coherent \(\mathcal{O}_X\)-module of finite type. Let \(s \in S\). Observe that \[(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})_s = \mathcal{F}_s \otimes_{\mathcal{O}_{X_s}} f_s^*\mathcal{G}_s\] by right exactness of tensor products. Moreover \(\mathcal{F}_s\) is flat over \(Y_s\) as a base change of a flat module. Thus the equality \((\alpha \circ \beta)_s = [(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})_s]_{r + e}\) follows from Lemma 0H69.
Lemma
Let \(f : X \to Y\) and \(Y \to S\) be morphisms of schemes, both locally of finite type. Let \(r, e \geq 0\). Let \(Z \subset X\) be a closed subscheme of relative dimension \(\leq r\) over \(Y\). Let \(W \subset Y\) be a closed subscheme of relative dimension \(\leq e\) over \(S\). If \(\alpha = [Z/X/Y]_r\) and \(\beta = [W/Y/S]_e\) (Example 0H4S) and \(Z\) is flat over \(Y\), then \(\alpha \circ \beta = [Z \times_Y W/X/S]_{r + e}\).
Proof
This is a special case of Lemma 0H6L if we take \(\mathcal{F} = \mathcal{O}_Z\) and \(\mathcal{F} = \mathcal{O}_W\).
Lemma
Let \(f : X \to Y\) and \(Y \to S\) be morphisms of schemes, both locally of finite type. Assume \(f\) is flat of relative dimension \(e\). Let \(\beta\) be a family of \(r\)-cycles on the fibres of \(Y/S\). Then \(f^*\beta = [X/X/Y]_e \circ \beta\) as families of \((e + r)\)-cycles on \(X/S\).
Proof
After unwinding the definitions and using that formation of \([X/X/Y]_e\) is compatible with base change (Lemma 0H4T) this follows from Lemma relative-cycles-lemma-flat-pullback-as-action.
Lemma
Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of schemes locally of finite type over \(S\) with \(g\) proper. Let \(r, e \geq 0\). Let \(\alpha\) be a family of \(r\)-cycles on the fibres of \(X/Y\). Let \(\beta'\) be a family of \(e\)-cycles on the fibres of \(Y'/S\). Then we have \(f_*(g^*(\alpha) \circ \beta') = \alpha \circ g_*\beta'\).
Proof
Unwinding the definitions, this follows from Lemma 0H6B.
Lemma
Let \((S, \delta)\) be as in Chow Homology, Situation 02QL. Let \(X \to Y \to Z\) be morphisms of schemes locally of finite type over \(S\). Let \(r, s, e \geq 0\). Then \[(\alpha \circ \beta) \cap \gamma = \alpha \cap (\beta \cap \gamma) \quad\text{in}\quad Z_{r + s + e}(X)\] where \(\alpha\) is a family of \(r\)-cycles on fibres of \(X/Y\), \(\beta\) is a family of \(s\)-cycles on fibres of \(Y/Z\), and \(\gamma \in Z_e(Z)\).
Proof
Since we are proving an equality of cycles on \(X\), we may work locally on \(Z\), see Lemma 0H67. Thus we may assume \(Z\) is affine. In particular \(\gamma\) is a finite linear combination of prime cycles. Since \(- \cap -\) is linear in the second variable (Lemma 0H66), it suffices to prove the equality when \(\gamma = [W]\) for some integral closed subscheme \(W \subset Z\) of \(\delta\)-dimension \(e\).
Let \(z \in W\) be the generic point. Write \(\beta_z = \sum m_j[V_j]\) in \(Z_s(Y_z)\). Then \(\beta \cap \gamma\) is equal to \(\sum m_j[\overline{V}_j]\) where \(\overline{V}_j \subset Y\) is an integral closed subscheme mapped by \(Y \to Z\) into \(W\) with generic fibre \(V_j\). Let \(y_j \in V_j\) be the generic point. We may and do view also as the generic point of \(\overline{V}_j\) (mapping to \(z\) in \(W\)). Write \(\alpha_{y_j} = \sum n_{jk} [W_{jk}]\) in \(Z_r(X_{y_j})\). Then \(\alpha \cap (\beta \cap \gamma)\) is equal to \[\sum m_j n_{jk} [\overline{W}_{jk}]\] where \(\overline{W}_{jk} \subset X\) is an integral closed subscheme mapped by \(X \to Y\) into \(\overline{V}_j\) with generic fibre \(W_{jk}\).
On the other hand, let us consider \[(\alpha \circ \beta)_z = (Y_z \to Y)^*\alpha \cap \beta_z = (Y_z \to Y)^*\alpha \cap (\sum m_j [V_j])\] By the construction of \(- \cap -\) this is equal to the cycle \[\sum m_j n_{jk} [(\overline{W}_{jk})_z]\] on \(X_z\). Thus by definition we obtain \[(\alpha \circ \beta) \cap [W] = \sum m_j n_{jk} [\widetilde{W}_{jk}]\] where \(\widetilde{W}_{jk} \subset X\) is an integral closed subscheme which is mapped by \(X \to Z\) into \(W\) with generic fibre \((\overline{W}_{jk})_z\). Clearly, we must have \(\widetilde{W}_{jk} = \overline{W}_{jk}\) and the proof is complete.
Composition of relative cycles
Let \(S\) be a locally Noetherian scheme. Let \(X \to Y\) be a morphism of schemes locally of finite type over \(S\). We are going to define a map \[z(X/Y, r) \otimes_\mathbf{Z} z(Y/S, e) \longrightarrow z(X/S, r + e),\quad \alpha \otimes \beta \longmapsto \alpha \circ \beta\] using the construction in Section 0H6H. We already know the construction is bilinear (Lemma 0H6I) hence we obtain the displayed arrow once we show the following.
Lemma
If \(\alpha\) and \(\beta\) are relative cycles, then so is \(\alpha \circ \beta\).
Proof
The formation of \(\alpha \circ \beta\) is compatible with base change by Lemma 0H6K. Thus we may assume \(S\) is the spectrum of a discrete valuation ring with generic point \(\eta\) and closed point \(0\) and we have to show that \(sp_{X/S}((\alpha \circ \beta)_\eta) = (\alpha \circ \beta)_0\). Since we are trying to prove an equality of cycles, we may work locally on \(Y\) and \(X\) (this uses Lemmas 0H6J and 0H4L to see that the constructions commute with restriction). Thus we may assume \(X\) and \(Y\) are affine. By Lemma 0H59 we can find a completely decomposed proper morphism \(g : Y' \to Y\) such that \(g^*\alpha\) is in the image of (0H58).
Since the family of morphisms \(g_\eta : Y'_\eta \to Y_\eta\) is completely decomposed, we can find \(\beta'_\eta \in Z_e(Y'_\eta)\) such that \(\beta_\eta = \sum g_{\eta, *}\beta'_\eta\), see Chow Homology, Lemma 0GU8. Set \(\beta'_0 = sp_{Y'/S}(\beta'_\eta)\) so that \(\beta' = (\beta'_\eta, \beta'_0)\) is a relative \(e\)-cycle on \(Y'/S\). Then \(g_*\beta'\) and \(\beta\) are relative \(e\)-cycles on \(Y/S\) (Lemma 0H51) which have the same value at \(\eta\) and hence are equal (Lemma 0H55). By linearity (Lemma 0H6I) it suffices to show that \(\alpha \circ g_*\beta'\) is a relative \((r + e)\)-cycle.
Set \(X' = X \times_Y Y'\) and denote \(f : X' \to X\) the projection. By Lemma 0H6N we see that \(\alpha \circ g_*\beta' = f_*(g^*\alpha \circ \beta')\). By Lemma 0H51 it suffices to show that \(g^*\alpha \circ \beta'\) is a relative \((r + e)\)-cycle. Using Lemma 0H5A and bilinearity this reduces us to the case discussed in the next paragraph.
Assume \(\alpha = [Z/X/Y]_r\) and \(\beta = [W/Y/S]\) where \(Z \subset X\) is a closed subscheme flat and of relative dimension \(\leq r\) over \(Y\) and \(W \subset Y\) is a closed subscheme flat and of relative dimension \(\leq e\) over \(S\). By Lemma 0H6M we see that \[\alpha \circ \beta = [Z \times_X W/X/S]_{r + e}\] and \(Z \times_X W \subset X\) is a closed subscheme flat over \(S\) of relative dimension \(\leq r + e\). This is a relative \((r + e)\)-cycle by Lemma 0H57.
Lemma
Let \(f : X \to Y\) and \(g : Y \to S\) be a morphisms of schemes. Assume \(S\) locally Noetherian, \(g\) locally of finite type and flat of relative dimension \(e \ge 0\), and \(f\) locally of finite type and flat of relative dimension \(r \geq 0\). Then \([X/X/Y]_r \circ [Y/Y/S]_e = [X/X/S]_{r + e}\) in \(z(X/S, r + e)\).
Proof
Special case of Lemma 0H6M.
Comparison with Suslin and Voevodsky
We have tried to use the same notation as in [SV], except that our notation for cycles is taken from Chow Homology, Section 02QQ ff. Here is a comparison:
In [SV, Section 3.1] there is a notion of a “relative cycle”, of a “relative cycle of dimension \(r\)”, and of a “equidimensional relative cycle of dimension \(r\)”. There is no corresponding notion in this chapter. Consequently, the groups \(Cycl(X/S, r)\), \(Cycl_{equi}(X/S, r)\), \(PropCycl(X/S, r)\), and \(PropCycl_{equi}(X/S, r)\), have no counter parts in this chapter.
On the bottom of [SV, page 36] the groups \(z(X/S, r)\), \(c(X/S, r)\), \(z_{equi}(X/S, r)\), \(c_{equi}(X/S, r)\) are defined. These agree with our notions when \(S\) is separated Noetherian and \(X \to S\) is separated and of finite type.
In [SV] the symbol \(z(X/S, r)\) is sometimes used for the presheaf \(S' \mapsto z(S' \times_S X/S', r)\) on the category of schemes of finite type over \(S\). Similarly for \(c(X/S, r)\), \(z_{equi}(X/S, r)\), and \(c_{equi}(X/S, r)\).
Base change, flat pullback, and proper pushforward as defined in [SV] agrees with ours when both apply.
For \(\alpha \in z(X/S, r)\) the operation \(\alpha \cap - : Z_e(S) \to Z_{e + r}(X)\) defined in Section 0H65 agrees with the operation \(Cor(\alpha, -)\) in [SV, Section 3.7] when both are defined.
For \(X \to Y \to S\) the composition law \(z(X/Y, r) \otimes_\mathbf{Z} z(Y/S, e) \longrightarrow z(X/S, r + e)\) defined in Section 0H6Q agrees with the operation \(Cor_{X/Y}(-, -)\) in [SV, Corollary 3.7.5].
Relative cycles in the non-Noetherian case
We urge the reader to skip this section.
Let \(f : X \to S\) be a morphism of schemes of finite presentation. Let \(r \geq 0\). Denote \(Hilb(X/S, r)\) the set of closed subschemes \(Z \subset X\) such that \(Z \to S\) is flat, of finite presentation, and of relative dimension \(\leq r\). We consider the group homomorphism [0H6V]\[\begin{equation} \begin{matrix} \text{free abelian group} \\ \text{on }Hilb(X/S, r) \end{matrix} \longrightarrow \begin{matrix} \text{families of }r\text{-cycles}\\ \text{on fibres of }X/S \end{matrix} \end{equation}\] sending \(\sum n_i[Z_i]\) to \(\sum n_i[Z_i/X/S]_r\).
Lemma
Let \(S\) be a quasi-compact and quasi-separated scheme. Let \(f : X \to S\) be a morphism of finite presentation. Let \(r \geq 0\) and let \(\alpha\) be a family of \(r\)-cycles on fibres of \(X/S\). The following are equivalent
there exists a cartesian diagram \[\xymatrix{ X \ar[r] \ar[d] & X_0 \ar[d] \\ S \ar[r] & S_0 }\] where \(X_0 \to S_0\) is a finite type morphism of Noetherian schemes and \(\alpha_0 \in z(X_0/S_0, r)\) such that \(\alpha\) is the base change of \(\alpha_0\) by \(S \to S_0\)
there exists a completely decomposed proper morphism \(g : S' \to S\) of finite presentation such that \(g^*\alpha\) is in the image of (0H6V).
Proof
Let a diagram and \(\alpha_0 \in z(X_0/S_0, r)\) as in (1) be given. By Lemma 0H59 there exists a completely decomposed proper morphism \(g_0 : S'_0 \to S_0\) such that \(g_0^*\alpha_0\) is in the image of (0H6V). Namely, since \(S'_0\) is Noetherian, every closed subscheme of \(S'_0 \times_{S_0} X_0\) is of finite presentation over \(S'_0\). Setting \(S' = S \times_{S_0} S'_0\) and using base change by \(S' \to S'_0\) we see that (2) holds.
Conversely, assume that (2) holds. Choose a completely decomposed proper morphism \(g : S' \to S\) of finite presentation such that \(g^*\alpha\) is in the image of (0H6V). Set \(X' = S' \times_S X\). Write \(g^*\alpha = \sum n_a [Z_a/X'/S']_r\) for some \(Z_a \subset X'\) closed subscheme flat, of finite presentation, and of relative dimension \(\leq r\) over \(S'\).
Write \(S = \lim S_i\) as a directed limit with affine transition morphisms with \(S_i\) of finite type over \(\mathbf{Z}\), see Limits, Proposition 01ZA. We can find an \(i\) large enough such that there exist
a completely decomposed proper morphism \(g_i : S'_i \to S_i\) whose base change to \(S\) is \(g : S' \to S\),
setting \(X'_i = S'_i \times_{S_i} X_i\) closed subschemes \(Z_{ai} \subset X'_i\) flat and of relative dimension \(\leq r\) over \(S'_i\) whose base change to \(S'\) is \(Z_a\).
To do this one uses Limits, Lemmas 01ZM, 01ZP, 04AI, 07RR, 081F, and 05M5 and and More on Morphisms, Lemma 0GTM. Consider \(\alpha'_i = \sum n_a [Z_{ai}/X'_i/S_i]_r \in z(X'_i/S'_i, r)\). The base change of \(\alpha'_i\) to a family of \(r\)-cycles on fibres of \(X'/S'\) agrees with the base change \(g^*\alpha\) by construction.
Set \(S''_i = S'_i \times_{S_i} S'_i\) and \(X''_i = S''_i \times_{S_i} X_i\) and set \(S'' = S' \times_S S'\) and \(X'' = S'' \times_S X\). We denote \(\text{pr}_1, \text{pr}_2 : S'' \to S'\) and \(\text{pr}_1, \text{pr}_2 : S''_i \to S'_i\) the projections. The relative \(r\)-cycles \(\text{pr}_1^*\alpha'_i\) and \(\text{pr}_1^*\alpha'_i\) on \(X''_i/S''_i\) base change to the same family of \(r\)-cycles on fibres of \(X''/S''\) because \(\text{pr}_1^*g^*\alpha = \text{pr}_1^*g^*\alpha\). Hence the morphism \(S'' \to S''_i\) maps into \(E = \{s \in S''_i : (\text{pr}_1^*\alpha'_i)_s = (\text{pr}_1^*\alpha'_i)_s\}\). By Lemma 0H5C this is a closed subset. Since \(S'' = \lim_{i' \geq i} S''_{i'}\) we see from Limits, Lemma 05F4 that for some \(i' \geq i\) the morphism \(S''_{i'} \to S''_i\) maps into \(E\). Therefore, after replacing \(i\) by \(i'\), we may assume that \(\text{pr}_1^*\alpha'_i = \text{pr}_1^*\alpha'_i\). By Lemma 0H4X we obtain a unique family \(\alpha_i\) of \(r\)-cycles on fibres of \(X_i/S_i\) with \(g_i^*\alpha_i = \alpha'_i\) (this uses that \(S'_i \to S_i\) is completely decomposed). By Lemma 0H52 we see that \(\alpha_i \in z(X_i/S_i, r)\). The uniqueness in Lemma 0H4X implies that the base change of \(\alpha_i\) is \(\alpha\) and we see (1) holds.
Discussion. If \(f : X \to S\), \(r\), and \(\alpha\) are as in Lemma 0H6W, then it makes sense to say that \(\alpha\) is a relative \(r\)-cycle on \(X/S\) if the equivalent conditions (1) and (2) of Lemma 0H6W hold. This definition has many good properties; for example it doesn’t conflict with the earlier definition in case \(S\) is Noetherian and most of the results of Section 0H4Z generalize to this setting.
We may still generalize further as follows. Assume \(S\) is arbitrary and \(f : X \to S\) is locally of finite presentation. Let \(r \geq 0\) and let \(\alpha\) be a family of \(r\)-cycles \(\alpha\) on fibres of \(X/S\). Then \(\alpha\) is an relative \(r\)-cycle on \(X/S\) if for \(U \subset X\) and \(V \subset S\) affine open with \(f(U) \subset V\) the restriction \(\alpha|_U\) is a relative \(r\)-cycle on \(U/V\) as defined in the previous paragraph. Again many of the earlier results generalize to this setting.
If we ever need these generalizations we will carefully state and prove them here.
Given \(x \in (X/S)^{(r)}\) pick \(x' \in (X'/S')^{(r)}\) mapping to \(x\) and set \(\alpha(x) = \alpha'(x')/m(x', g)\). This is well defined by the formula and the lemma.↩︎
Namely, any closed subscheme of \(S'_1 \times_S X\) flat and of relative dimension \(\leq r\) over \(S'_1\) may be viewed as a closed subscheme of \(S' \times_S X\) flat and of relative dimension \(\leq r\) over \(S'\).↩︎
In [SV] there does exist an element of the group of relative cycles \(Cycl(X/S, 0)\) corresponding to \(pw\) and this cycle does specialize to a cycle on \(X_s\) with non-integer coefficients.↩︎
To be sure, we use \(s = \Spec(\kappa(s))\) as the base scheme with \(\delta(s) = 0\).↩︎