Definition 42.34.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. The Chow cohomology of $X$ is the graded $\mathbf{Z}$-algebra $A^*(X)$ whose degree $p$ component is $A^ p(X \to X)$.
42.34 Chow cohomology and the first Chern class
We will be most interested in $A^ p(X) = A^ p(X \to X)$, which will always mean the bivariant cohomology classes for $\text{id}_ X$. Namely, that is where Chern classes will live.
Warning: It is not clear that the $\mathbf{Z}$-algebra structure on $A^*(X)$ is commutative, but we will see that Chern classes live in its center.
Remark 42.34.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : Y' \to Y$ be a morphism of schemes locally of finite type over $S$. As a special case of Remark 42.33.5 there is a canonical $\mathbf{Z}$-algebra map $res : A^*(Y) \to A^*(Y')$. This map is often denoted $f^*$ in the literature.
Lemma 42.34.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then the rule that to $f : X' \to X$ assigns $c_1(f^*\mathcal{L}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(X')$ is a bivariant class of degree $1$.
Proof. This follows from Lemmas 42.28.2, 42.26.4, 42.26.2, and 42.30.4. $\square$
The lemma above finally allows us to make the following definition.
Definition 42.34.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The first Chern class $c_1(\mathcal{L}) \in A^1(X)$ of $\mathcal{L}$ is the bivariant class of Lemma 42.34.3.
For finite locally free modules we construct the Chern classes in Section 42.38. Let us prove that $c_1(\mathcal{L})$ is in the center of $A^*(X)$.
Lemma 42.34.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then
$c_1(\mathcal{L}) \in A^1(X)$ is in the center of $A^*(X)$ and
if $f : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_1(\mathcal{L}) = c_1(f^*\mathcal{L}) \circ c$.
Proof. Of course (2) implies (1). Let $p : L \to X$ be as in Lemma 42.32.2 and let $o : X \to L$ be the zero section. Denote $p' : L' \to X'$ and $o' : X' \to L'$ their base changes. By Lemma 42.32.4 we have
\[ p^*(c_1(\mathcal{L}) \cap \alpha ) = - o_* \alpha \quad \text{and}\quad (p')^*(c_1(f^*\mathcal{L}) \cap \alpha ') = - o'_* \alpha ' \]
Since $c$ is a bivariant class we have
\begin{align*} (p')^*(c \cap c_1(\mathcal{L}) \cap \alpha ) & = c \cap p^*(c_1(\mathcal{L}) \cap \alpha ) \\ & = - c \cap o_* \alpha \\ & = - o'_*(c \cap \alpha ) \\ & = (p')^*(c_1(f^*\mathcal{L}) \cap c \cap \alpha ) \end{align*}
Since $(p')^*$ is injective by one of the lemmas cited above we obtain $c \cap c_1(\mathcal{L}) \cap \alpha = c_1(f^*\mathcal{L}) \cap c \cap \alpha $. The same is true after any base change by $Y \to X$ locally of finite type and hence we have the equality of bivariant classes stated in (2). $\square$
Lemma 42.34.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a finite type scheme over $S$ which has an ample invertible sheaf. Assume $d = \dim (X) < \infty $ (here we really mean dimension and not $\delta $-dimension). Then for any invertible sheaves $\mathcal{L}_1, \ldots , \mathcal{L}_{d + 1}$ on $X$ we have $c_1(\mathcal{L}_1) \circ \ldots \circ c_1(\mathcal{L}_{d + 1}) = 0$ in $A^{d + 1}(X)$.
Proof. We prove this by induction on $d$. The base case $d = 0$ is true because in this case $X$ is a finite set of closed points and hence every invertible module is trivial. Assume $d > 0$. By Divisors, Lemma 31.15.12 we can write $\mathcal{L}_{d + 1} \cong \mathcal{O}_ X(D) \otimes \mathcal{O}_ X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D' \subset X$. Then $c_1(\mathcal{L}_{d + 1})$ is the difference of $c_1(\mathcal{O}_ X(D))$ and $c_1(\mathcal{O}_ X(D'))$ and hence we may assume $\mathcal{L}_{d + 1} = \mathcal{O}_ X(D)$ for some effective Cartier divisor.
Denote $i : D \to X$ the inclusion morphism and denote $i^* \in A^1(D \to X)$ the bivariant class given by the gysin hommomorphism as in Lemma 42.33.3. We have $i_* \circ i^* = c_1(\mathcal{L}_{d + 1})$ in $A^1(X)$ by Lemma 42.29.4 (and Lemma 42.33.4 to make sense of the left hand side). Since $c_1(\mathcal{L}_ i)$ commutes with both $i_*$ and $i^*$ (by definition of bivariant classes) we conclude that
\[ c_1(\mathcal{L}_1) \circ \ldots \circ c_1(\mathcal{L}_{d + 1}) = i_* \circ c_1(\mathcal{L}_1) \circ \ldots \circ c_1(\mathcal{L}_ d) \circ i^* = i_* \circ c_1(\mathcal{L}_1|_ D) \circ \ldots \circ c_1(\mathcal{L}_ d|_ D) \circ i^* \]
Thus we conclude by induction on $d$. Namely, we have $\dim (D) < d$ as none of the generic points of $X$ are in $D$. $\square$
Remark 42.34.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $Z \to X$ be a closed immersion of schemes locally of finite type over $S$ and let $p \geq 0$. In this setting we define
\[ A^{(p)}(Z \to X) = \prod \nolimits _{i \leq p - 1} A^ i(X) \times \prod \nolimits _{i \geq p} A^ i(Z \to X). \]
Then $A^{(p)}(Z \to X)$ canonically comes equipped with the structure of a graded algebra. In fact, more generally there is a multiplication
\[ A^{(p)}(Z \to X) \times A^{(q)}(Z \to X) \longrightarrow A^{(\max (p, q))}(Z \to X) \]
In order to define these we define maps
\begin{align*} A^ i(Z \to X) \times A^ j(X) & \to A^{i + j}(Z \to X) \\ A^ i(X) \times A^ j(Z \to X) & \to A^{i + j}(Z \to X) \\ A^ i(Z \to X) \times A^ j(Z \to X) & \to A^{i + j}(Z \to X) \end{align*}
For the first we use composition of bivariant classes. For the second we use restriction $A^ i(X) \to A^ i(Z)$ (Remark 42.33.5) and composition $A^ i(Z) \times A^ j(Z \to X) \to A^{i + j}(Z \to X)$. For the third, we send $(c, c')$ to $res(c) \circ c'$ where $res : A^ i(Z \to X) \to A^ i(Z)$ is the restriction map (see Remark 42.33.5). We omit the verification that these multiplications are associative in a suitable sense.
Remark 42.34.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $Z \to X$ be a closed immersion of schemes locally of finite type over $S$. Denote $res : A^ p(Z \to X) \to A^ p(Z)$ the restriction map of Remark 42.33.5. For $c \in A^ p(Z \to X)$ we have $res(c) \cap \alpha = c \cap i_*\alpha $ for $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(Z)$. Namely $res(c) \cap \alpha = c \cap \alpha $ and compatibility of $c$ with proper pushforward gives $(Z \to Z)_*(c \cap \alpha ) = c \cap (Z \to X)_*\alpha $.
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