## 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].

reference
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

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 $X'' = Y'' \times _ Y X$,

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

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 $-r$.

**Proof.**
This follows from Lemmas 42.20.2, 42.14.3, 42.15.1, and 42.29.9.
$\square$

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 $1$.

**Proof.**
This follows from Lemmas 42.30.2, 42.29.8, 42.29.9, and 42.30.5.
$\square$

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 $f_* \circ c \in A^ p(Y \to Z)$.

**Proof.**
This follows from Lemmas 42.12.2, 42.15.1, and 42.29.8.
$\square$

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.

## Comments (0)