42.33 Bivariant intersection theory

In order to intelligently talk about higher Chern classes of vector bundles we introduce bivariant chow classes as in [F]. Our definition differs from [F] in two respects: (1) we work in a different setting, and (2) we only require our bivariant classes commute with the gysin homomorphisms for zero schemes of sections of invertible modules (Section 42.29). We will see later, in Lemma 42.54.8, that our bivariant classes commute with all higher codimension gysin homomorphisms and hence satisfy all properties required of them in [F]; see also [Theorem 17.1, F].

Definition 42.33.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. A bivariant class $c$ of degree $p$ for $f$ is given by a rule which assigns to every locally of finite type morphism $Y' \to Y$ and every $k$ a map

$c \cap - : \mathop{\mathrm{CH}}\nolimits _ k(Y') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - p}(X')$

where $X' = Y' \times _ Y X$, satisfying the following conditions

1. if $Y'' \to Y'$ is a proper, then $c \cap (Y'' \to Y')_*\alpha '' = (X'' \to X')_*(c \cap \alpha '')$ for all $\alpha ''$ on $Y''$ where $X'' = Y'' \times _ Y X$,

2. if $Y'' \to Y'$ is flat locally of finite type of fixed relative dimension, then $c \cap (Y'' \to Y')^*\alpha ' = (X'' \to X')^*(c \cap \alpha ')$ for all $\alpha '$ on $Y'$, and

3. if $(\mathcal{L}', s', i' : D' \to Y')$ is as in Definition 42.29.1 with pullback $(\mathcal{N}', t', j' : E' \to X')$ to $X'$, then we have $c \cap (i')^*\alpha ' = (j')^*(c \cap \alpha ')$ for all $\alpha '$ on $Y'$.

The collection of all bivariant classes of degree $p$ for $f$ is denoted $A^ p(X \to Y)$.

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X \to Y$ and $Y \to Z$ be morphisms of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. It is clear that $A^ p(X \to Y)$ is an abelian group. Moreover, it is clear that we have a bilinear composition

$A^ p(X \to Y) \times A^ q(Y \to Z) \to A^{p + q}(X \to Z)$

which is associative.

Lemma 42.33.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a flat morphism of relative dimension $r$ between schemes locally of finite type over $S$. Then the rule that to $Y' \to Y$ assigns $(f')^* : \mathop{\mathrm{CH}}\nolimits _ k(Y') \to \mathop{\mathrm{CH}}\nolimits _{k + r}(X')$ where $X' = X \times _ Y Y'$ is a bivariant class of degree $-r$.

Lemma 42.33.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 42.29.1. Then the rule that to $f : X' \to X$ assigns $(i')^* : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(D')$ where $D' = D \times _ X X'$ is a bivariant class of degree $1$.

Lemma 42.33.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes locally of finite type over $S$. Let $c \in A^ p(X \to Z)$ and assume $f$ is proper. Then the rule that to $Z' \to Z$ assigns $\alpha \longmapsto f'_*(c \cap \alpha )$ is a bivariant class denoted $f_* \circ c \in A^ p(Y \to Z)$.

Remark 42.33.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X \to Y$ and $Y' \to Y$ be morphisms of schemes locally of finite type over $S$. Let $X' = Y' \times _ Y X$. Then there is an obvious restriction map

$A^ p(X \to Y) \longrightarrow A^ p(X' \to Y'),\quad c \longmapsto res(c)$

obtained by viewing a scheme $Y''$ locally of finite type over $Y'$ as a scheme locally of finite type over $Y$ and settting $res(c) \cap \alpha '' = c \cap \alpha ''$ for any $\alpha '' \in \mathop{\mathrm{CH}}\nolimits _ k(Y'')$. This restriction operation is compatible with compositions in an obvious manner.

Remark 42.33.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. For $i = 1, 2$ let $Z_ i \to X$ be a morphism of schemes locally of finite type. Let $c_ i \in A^{p_ i}(Z_ i \to X)$, $i = 1, 2$ be bivariant classes. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we can ask whether

$c_1 \cap c_2 \cap \alpha = c_2 \cap c_1 \cap \alpha$

in $\mathop{\mathrm{CH}}\nolimits _{k - p_1 - p_2}(Z_1 \times _ X Z_2)$. If this is true and if it holds after any base change by $X' \to X$ locally of finite type, then we say $c_1$ and $c_2$ commute. Of course this is the same thing as saying that

$res(c_1) \circ c_2 = res(c_2) \circ c_1$

in $A^{p_1 + p_2}(Z_1 \times _ X Z_2 \to X)$. Here $res(c_1) \in A^{p_1}(Z_1 \times _ X Z_2 \to Z_2)$ is the restriction of $c_1$ as in Remark 42.33.5; similarly for $res(c_2)$.

Example 42.33.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ a triple as in Definition 42.29.1. Let $Z \to X$ be a morphism of schemes locally of finite type and let $c \in A^ p(Z \to X)$ be a bivariant class. Then the bivariant gysin class $c' \in A^1(D \to X)$ of Lemma 42.33.3 commutes with $c$ in the sense of Remark 42.33.6. Namely, this is a restatement of condition (3) of Definition 42.33.1.

Remark 42.33.8. There is a more general type of bivariant class that doesn't seem to be considered in the literature. Namely, suppose we are given a diagram

$X \longrightarrow Z \longleftarrow Y$

of schemes locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Let $p \in \mathbf{Z}$. Then we can consider a rule $c$ which assigns to every $Z' \to Z$ locally of finite type maps

$c \cap - : \mathop{\mathrm{CH}}\nolimits _ k(Y') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - p}(X')$

for all $k \in \mathbf{Z}$ where $X' = X \times _ Z Z'$ and $Y' = Z' \times _ Z Y$ compatible with

1. proper pushforward if given $Z'' \to Z'$ proper,

2. flat pullback if given $Z'' \to Z'$ flat of fixed relative dimension, and

3. gysin maps if given $D' \subset Z'$ as in Definition 42.29.1.

We omit the detailed formulations. Suppose we denote the collection of all such operations $A^ p(X \to Z \leftarrow Y)$. A simple example of the utility of this concept is when we have a proper morphism $f : X_2 \to X_1$. Then $f_*$ isn't a bivariant operation in the sense of Definition 42.33.1 but it is in the above generalized sense, namely, $f_* \in A^0(X_1 \to X_1 \leftarrow X_2)$.

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