The Stacks project

Proof. We prove this for the total Chern class; the other two cases follow by the same arguments using Lemma 42.45.2 instead of Lemma 42.40.3.

As in Remark 42.38.10 in order to define the total chern class $c(\mathcal{E}^\bullet )$ we decompose $X$ into open and closed subschemes

such that the rank $\mathcal{E}^ n$ is constant on $X_ i$ for all $n$ and $i$. (Since these ranks are locally constant functions on $X$ we can do this.) Since $\mathcal{E}^\bullet $ is locally bounded, we see that only a finite number of the sheaves $\mathcal{E}^ n|_{X_ i}$ are nonzero for a fixed $i$. Hence we can define

as above. By Lemma 42.35.4 we have $A^ p(X) = \prod _ i A^ p(X_ i)$. Hence for each $p \in \mathbf{Z}$ we have a unique element $c_ p(\mathcal{E}^\bullet ) \in A^ p(X)$ restricting to $c_ p(\mathcal{E}^\bullet |_{X_ i})$ on $X_ i$ for all $i$.

Suppose we have a second locally bounded complex $\mathcal{F}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules representing $E$. Let $g : Y \to X$ be a morphism locally of finite type with $Y$ integral. By Lemma 42.35.3 it suffices to show that with $c(g^*\mathcal{E}^\bullet ) \cap [Y]$ is the same as $c(g^*\mathcal{F}^\bullet ) \cap [Y]$ and it even suffices to prove this after replacing $Y$ by an integral scheme proper and birational over $Y$. Then first we conclude that $g^*\mathcal{E}^\bullet $ and $g^*\mathcal{F}^\bullet $ are bounded complexes of finite locally free $\mathcal{O}_ Y$-modules of constant rank. Next, by More on Flatness, Lemma 38.40.3 we may assume that $H^ i(Lg^*E)$ is perfect of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. This reduces us to the case discussed in the next paragraph.

Assume $X$ is integral, $\mathcal{E}^\bullet $ and $\mathcal{F}^\bullet $ are bounded complexes of finite locally free modules of constant rank, and $H^ i(E)$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. We have to show that $c(\mathcal{E}^\bullet ) \cap [X]$ is the same as $c(\mathcal{F}^\bullet ) \cap [X]$. Denote $d_\mathcal {E}^ i : \mathcal{E}^ i \to \mathcal{E}^{i + 1}$ and $d_\mathcal {F}^ i : \mathcal{F}^ i \to \mathcal{F}^{i + 1}$ the differentials of our complexes. By More on Flatness, Remark 38.40.4 we know that $\mathop{\mathrm{Im}}(d_\mathcal {E}^ i)$, $\mathop{\mathrm{Ker}}(d_\mathcal {E}^ i)$, $\mathop{\mathrm{Im}}(d_\mathcal {F}^ i)$, and $\mathop{\mathrm{Ker}}(d_\mathcal {F}^ i)$ are finite locally free $\mathcal{O}_ X$-modules for all $i$. By additivity (Lemma 42.40.3) we see that

and similarly for $\mathcal{F}^\bullet $. Since we have the short exact sequences

we reduce to the problem stated and solved in the next paragraph.

Assume $X$ is integral and we have two short exact sequences

with $\mathcal{E}$, $\mathcal{E}'$, $\mathcal{F}$, $\mathcal{F}'$ finite locally free. Problem: show that $c(\mathcal{E})c(\mathcal{E}')^{-1} \cap [X] = c(\mathcal{F})c(\mathcal{F}')^{-1} \cap [X]$. To do this, consider the short exact sequence

defining $\mathcal{G}$. Since $\mathcal{Q}$ has tor dimension $\leq 1$ we see that $\mathcal{G}$ is finite locally free. A diagram chase shows that the kernel of the surjection $\mathcal{G} \to \mathcal{F}$ maps isomorphically to $\mathcal{E}'$ in $\mathcal{E}$ and the kernel of the surjection $\mathcal{G} \to \mathcal{E}$ maps isomorphically to $\mathcal{F}'$ in $\mathcal{F}$. (Working affine locally this follows from or is equivalent to Schanuel's lemma, see Algebra, Lemma 10.109.1.) We conclude that

as desired. $\square$

Proof. Fix $p \in \mathbf{Z}$. We will prove the lemma for the chern class $c_ p(E) \in A^ p(X)$ and omit the arguments for the other cases.

Let $g : T \to X$ be a morphism locally of finite type such that there exists a locally bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ T$-modules representing $Lg^*E$ in $D(\mathcal{O}_ T)$. The bivariant class $c_ p(\mathcal{E}^\bullet ) \in A^ p(T)$ is independent of the choice of $\mathcal{E}^\bullet $ by Lemma 42.46.1. Let us write $c_ p(Lg^*E) \in A^ p(T)$ for this class. For any further morphism $h : T' \to T$ which is locally of finite type, setting $g' = g \circ h$ we see that $L(g')^*E = L(g \circ h)^*E = Lh^*Lg^*E$ is represented by $h^*\mathcal{E}^\bullet $ in $D(\mathcal{O}_{T'})$. We conclude that $c_ p(L(g')^*E)$ makes sense and is equal to the restriction (Remark 42.33.5) of $c_ p(Lg^*E)$ to $T'$ (strictly speaking this requires an application of Lemma 42.38.7).

Let $f : Y \to X$ and $\mathcal{E}^\bullet $ be as in the statement of the lemma. We obtain a bivariant class $c_ p(E) \in A^ p(X)$ from an application of Lemma 42.35.6 to $f : Y \to X$ and the class $c' = c_ p(Lf^*E)$ we constructed in the previous paragraph. The assumption in the lemma is satisfied because by the discussion in the previous paragraph we have $res_1(c') = c_ p(Lg^*E) = res_2(c')$ where $g = f \circ p = f \circ q : Y \times _ X Y \to X$.

Finally, suppose that $f' : Y' \to X$ is a second envelope such that $L(f')^*E$ is represented by a bounded complex of finite locally free $\mathcal{O}_{Y'}$-modules. Then it follows that the restrictions of $c_ p(Lf^*E)$ and $c_ p(L(f')^*E)$ to $Y \times _ X Y'$ are equal. Since $Y \times _ X Y' \to X$ is an envelope (Lemmas 42.22.3 and 42.22.2), we see that our two candidates for $c_ p(E)$ agree by the unicity in Lemma 42.35.6. $\square$

Proof. Condition (1) is just Definition 42.46.3 part (1). Condition (2) implies (1).

As in (3) assume the irreducible components $X_ i$ of $X$ are quasi-compact. We view $X_ i$ as a reduced integral closed subscheme over $X$. The morphism $\coprod X_ i \to X$ is an envelope. For each $i$ there exists an envelope $X'_ i \to X_ i$ such that $X'_ i$ has an ample family of invertible modules, see More on Morphisms, Proposition 37.80.3. Observe that $f : Y = \coprod X'_ i \to X$ is an envelope; small detail omitted. By Derived Categories of Schemes, Lemma 36.36.7 each $X'_ i$ has the resolution property. Thus the perfect object $L(f|_{X'_ i})^*E$ of $D(\mathcal{O}_{X'_ i})$ can be represented by a bounded complex of finite locally free $\mathcal{O}_{X'_ i}$-modules, see Derived Categories of Schemes, Lemma 36.37.2. This proves (3) implies (1).

Part (4) implies (3).

Let $g : X \to X'$ and $E'$ be as in part (5). Then there exists an envelope $f' : Y' \to X'$ such that $L(f')^*E'$ is represented by a locally bounded complex $(\mathcal{E}')^\bullet $ of $\mathcal{O}_{Y'}$-modules. Then the base change $f : Y \to X$ is an envelope by Lemma 42.22.3. Moreover, the pulllback $\mathcal{E}^\bullet = g^*(\mathcal{E}')^\bullet $ represents $Lf^*E$ and we see that the chern classes of $E$ are defined. $\square$

Proof. Choose an envelope $f : Y \to X$ such that $Lf^*E$ is isomorphic in $D(\mathcal{O}_ Y)$ to a locally bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ Y$-modules. The base change $Y \times _ X W \to W$ of $f$ is an envelope by Lemma 42.22.3. Choose a point $\xi \in Y \times _ X W$ mapping to the generic point of $W$ with the same residue field. Consider the integral closed subscheme $W' \subset Y \times _ X W$ with generic point $\xi $. The restriction of the projection $Y \times _ X W \to W$ to $W'$ is a proper birational morphism $b : W' \to W$. Set $g' = g \circ b$. Finally, consider the pullback $(W' \to Y)^*\mathcal{E}^\bullet $. This is a locally bounded complex of finite locally free modules on $W'$. Since $W'$ is integral it follows that it is bounded and that the terms have constant rank. Finally, by construction $(W' \to Y)^*\mathcal{E}^\bullet $ represents $L(g')^*E$ and by construction its $p$th chern class gives the restriction of $c_ p(E)$ by $W' \to X$. This finishes the proof. $\square$

Proof. Part (1) follows immediately from part (2). Let $g : X' \to X$ and $c \in A^*(X' \to X)$ be as in (2). To show that $c \circ c_ p(E) - c_ p(Lg^*E) \circ c = 0$ we use the criterion of Lemma 42.35.3. Thus we may assume that $X$ is integral and by Lemma 42.46.5 we may even assume that $E$ is represented by a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules of constant rank. Then we have to show that

in $\mathop{\mathrm{CH}}\nolimits _*(X')$. This is immediate from Lemma 42.38.9 and the construction of $c_ p(\mathcal{E}^\bullet )$ as a polynomial in the chern classes of the locally free modules $\mathcal{E}^ n$. $\square$

Proof. Let $f : Y \to X$ be an envelope and let $\alpha ^\bullet : \mathcal{E}_1^\bullet \to \mathcal{E}_2^\bullet $ be a map of locally bounded complexes of finite locally free $\mathcal{O}_ Y$-modules representing $Lf^*E_1 \to Lf^*E_2$. Then the cone $C(\alpha )^\bullet $ represents $Lf^*E_3$. Since $C(\alpha )^ n = \mathcal{E}_2^ n \oplus \mathcal{E}_1^{n + 1}$ we see that $C(\alpha )^\bullet $ is a locally bounded complex of finite locally free $\mathcal{O}_ Y$-modules. We conclude that the Chern classes of $E_1$, $E_2$, $E_3$ are defined. Moreover, recall that $c_ p(E_1)$ is defined as the unique element of $A^ p(X)$ which restricts to $c_ p(\mathcal{E}_1^\bullet )$ in $A^ p(Y)$. Similarly for $E_2$ and $E_3$. Hence it suffices to prove $c(\mathcal{E}_2^\bullet ) = c(\mathcal{E}_1^\bullet ) c(C(\alpha )^\bullet )$ in $\prod _{p \geq 0} A^ p(Y)$. In turn, it suffices to prove this after restricting to a connected component of $Y$. Hence we may assume the complexes $\mathcal{E}_1^\bullet $ and $\mathcal{E}_2^\bullet $ are bounded complexes of finite locally free $\mathcal{O}_ Y$-modules of fixed rank. In this case the desired equality follows from the multiplicativity of Lemma 42.40.3. In the case of $ch$ or $P_ p$ we use Lemmas 42.45.2.

In the previous paragraph we have seen that the lemma holds if condition (1) is satisfied. Since (2) implies (1) this deals with the second case. Assume (3). Arguing exactly as in the proof of Lemma 42.46.4 we find an envelope $f : Y \to X$ such that $Y$ is a disjoint union $Y = \coprod Y_ i$ of quasi-compact (and quasi-separated) schemes each having the resolution property. Then we may represent the restriction of $Lf^*E_1 \to Lf^*E_2$ to $Y_ i$ by a map of bounded complexes of finite locally free modules, see Derived Categories of Schemes, Proposition 36.37.5. In this way we see that condition (3) implies condition (1). Of course condition (4) implies condition (3) and the proof is complete. $\square$

Proof. First proof: argue as in the proof of Lemma 42.46.6 to reduce to the case where $E$ is represented by a bounded complex of finite locally free modules of fixed rank and apply Lemma 42.43.3. Second proof: use the splitting principle discussed in Remark 42.46.8 and use that the chern roots of $E^\vee $ are the negatives of the chern roots of $E$. $\square$

Proof. In the case where $E$ is locally free of rank $r$ this is Lemma 42.39.1. The reader can deduce the lemma from this special case by a formal computation. An alternative is to use the splitting principle of Remark 42.46.8. In this case one ends up having to prove the following algebra fact: if we write formally

with $c_ i$ homogeneous of degree $i$ in $\mathbf{Z}[x_ i, y_ j]$ then we have

where $r = n - m$. We omit the details. $\square$

Proof. After choosing an envelope $f : Y \to X$ such that $Lf^*E$ and $Lf^*F$ can be represented by locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules this follows by a computation from the corresponding result for vector bundles in Lemmas 42.43.4 and 42.45.3. A better proof is probably to use the splitting principle as in Remark 42.46.8 and reduce the lemma to computations in polynomial rings which we describe in the next paragraph.

Let $A$ be a commutative ring (for us this will be the subring of the bivariant chow ring of $X$ generated by Chern classes). Let $S$ be a finite set together with maps $\epsilon : S \to \{ \pm 1\} $ and $f : S \to A$. Define

in $A$. Given a second triple $(S', \epsilon ', f')$ the equality that has to be shown for $P_ p$ is the equality

To see this is true, one reduces to the polynomial ring on variables $S \amalg S'$ and one shows that each term $f(s)^ if'(s')^ j$ occurs on the left and right hand side with the same coefficient. To verify the formulas for $c_1(E \otimes _{\mathcal{O}_ X}^\mathbf {L} F)$ and $c_2(E \otimes _{\mathcal{O}_ X}^\mathbf {L} F)$ we use the splitting principle to reduce to checking these formulae in a torsion free ring. Then we use the relationship between $P_ j(E)$ and $c_ i(E)$ proved in Remark 42.46.8. For example

the middle equation because $r(E) = P_0(E)$ by definition. Similarly, we have

which the reader can verify agrees with the formula in the statement of the lemma up to a factor of $2$. $\square$