Lemma 62.5.1. 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.
62.5 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 62.3 via the identitification $X'_{s'} = X_ s \times _{\mathop{\mathrm{Spec}}(\kappa (s))} \mathop{\mathrm{Spec}}(\kappa (s'))$ 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.
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 42.67.9. Part (2): Obvious. Part (3): Chow Homology, Lemma 42.67.5. Part (4): Chow Homology, Lemma 42.14.3. Part (5): Chow Homology, Lemma 42.67.6. Part (6): Chow Homology, Lemma 42.12.2. Part (7): Chow Homology, Lemma 42.15.1. $\square$
Example 62.5.2. 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 where $[\mathcal{F}_ s]_ r$ is given by Chow Homology, Definition 42.10.2.
\[ [\mathcal{F}/X/S]_ r = ([\mathcal{F}_ s]_ r)_{s \in S} \]
Lemma 62.5.3. The construction in Example 62.5.2 is compatible with base change, restriction, and flat pullback.
Proof. See Chow Homology, Lemmas 42.67.3 and 42.14.4. $\square$
Example 62.5.4. 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 where $[Z_ s]_ r$ is given by Chow Homology, Definition 42.9.2.
\[ [Z/X/S]_ r = ([Z_ s]_ r)_{s \in S} \]
Lemma 62.5.5. The construction in Example 62.5.4 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 62.5.3. See Chow Homology, Lemma 42.10.3. $\square$
Remark 62.5.6 (Support). 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 Here $\text{Supp}(\alpha _ s) \subset X_ s$ is the support of the cycle $\alpha _ s$, see Chow Homology, Definition 42.8.3. The support $\text{Supp}(\alpha )$ is rarely a closed subset of $X$.
\[ \text{Supp}(\alpha ) = \bigcup \nolimits _{s \in S} \text{Supp}(\alpha _ s) \subset X \]
Lemma 62.5.7. Taking the support as in Remark 62.5.6 is compatible with base change, restriction, and flat pullback.
Proof. Omitted. $\square$
Lemma 62.5.8. 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 29.28.3. 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 35.13.1). 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 $Z$. $\square$
Lemma 62.5.9. 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 62.3. (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 62.3.1). 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 explicity, 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 62.5.8 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. $\square$
Lemma 62.5.10. 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. $\square$
[1] 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.
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