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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