42.46 Chern classes and the derived category

In this section we define the total Chern class of an object of the derived category which may be represented globally by a finite complex of finite locally free modules.

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let

$\mathcal{E}^ a \to \mathcal{E}^{a + 1} \to \ldots \to \mathcal{E}^ b$

be a bounded complex of finite locally free $\mathcal{O}_ X$-modules of constant rank. Then we define the total Chern class of the complex by the formula

$c(\mathcal{E}^\bullet ) = \prod \nolimits _{n = a, \ldots , b} c(\mathcal{E}^ n)^{(-1)^ n} \in \prod \nolimits _{p \geq 0} A^ p(X)$

Here the inverse is the formal inverse, so

$(1 + c_1 + c_2 + c_3 + \ldots )^{-1} = 1 - c_1 + c_1^2 - c_2 - c_1^3 + 2c_1 c_2 - c_3 + \ldots$

We will denote $c_ p(\mathcal{E}^\bullet ) \in A^ p(X)$ the degree $p$ part of $c(\mathcal{E}^\bullet )$. We similarly define the Chern character of the complex by the formula

$ch(\mathcal{E}^\bullet ) = \sum \nolimits _{n = a, \ldots , b} (-1)^ n ch(\mathcal{E}^ n) \in \prod \nolimits _{p \geq 0} (A^ p(X) \otimes \mathbf{Q})$

We will denote $ch_ p(\mathcal{E}^\bullet ) \in A^ p(X) \otimes \mathbf{Q}$ the degree $p$ part of $ch(\mathcal{E}^\bullet )$. Finally, for $P_ p \in \mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$ as in Example 42.43.6 we define

$P_ p(\mathcal{E}^\bullet ) = \sum \nolimits _{n = a, \ldots , b} (-1)^ n P_ p(\mathcal{E}^ n)$

in $A^ p(X)$. Then we have $ch_ p(\mathcal{E}^\bullet ) = (1/p!)P_ p(\mathcal{E}^\bullet )$ as usual. The next lemma shows that these constructions only depends on the image of the complex in the derived category.

Lemma 42.46.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be an object such that there exists a locally bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_ X$-modules representing $E$. Then a slight generalization of the above constructions

$c(\mathcal{E}^\bullet ) \in \prod \nolimits _{p \geq 0} A^ p(X),\quad ch(\mathcal{E}^\bullet ) \in \prod \nolimits _{p \geq 0} A^ p(X) \otimes \mathbf{Q},\quad P_ p(\mathcal{E}^\bullet ) \in A^ p(X)$

are independent of the choice of the complex $\mathcal{E}^\bullet$.

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

$X = \coprod \nolimits _{i \in I} X_ i$

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

$c(\mathcal{E}^\bullet |_{X_ i}) = \prod \nolimits _ n c(\mathcal{E}^ n|_{X_ i})^{(-1)^ n} \in \prod \nolimits _{p \geq 0} A^ p(X_ i)$

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

$c(\mathcal{E}^\bullet ) = \prod \nolimits _ i c(\mathop{\mathrm{Ker}}(d_\mathcal {E}^ i))^{(-1)^ i} c(\mathop{\mathrm{Im}}(d_\mathcal {E}^ i))^{(-1)^ i}$

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

$0 \to \mathop{\mathrm{Im}}(d_\mathcal {E}^ i) \to \mathop{\mathrm{Ker}}(d_\mathcal {E}^ i) \to H^ i(E) \to 0 \quad \text{and}\quad 0 \to \mathop{\mathrm{Im}}(d_\mathcal {F}^ i) \to \mathop{\mathrm{Ker}}(d_\mathcal {F}^ i) \to H^ i(E) \to 0$

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

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

$0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{Q} \to 0 \quad \text{and}\quad 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{Q} \to 0$

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

$0 \to \mathcal{G} \to \mathcal{E} \oplus \mathcal{F} \to \mathcal{Q} \to 0$

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

$c(\mathcal{E})c(\mathcal{F}') = c(\mathcal{G}) = c(\mathcal{F})c(\mathcal{E}')$

as desired. $\square$

Lemma 42.46.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Assume there exists an envelope $f : Y \to X$ (Definition 42.22.1) 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. Then there exists unique bivariant classes $c(E) \in A^*(X)$, $ch(E) \in A^*(X) \otimes \mathbf{Q}$, and $P_ p(E) \in A^ p(X)$, independent of the choice of $f : Y \to X$ and $\mathcal{E}^\bullet$, such that the restriction of these classes to $Y$ are equal to $c(\mathcal{E}^\bullet ) \in A^*(Y)$, $ch(\mathcal{E}^\bullet ) \in A^*(Y) \otimes \mathbf{Q}$, and $P_ p(\mathcal{E}^\bullet ) \in A^ p(Y)$.

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. $\square$

Definition 42.46.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object.

1. We say the Chern classes of $E$ are defined1 if there exists an envelope $f : Y \to X$ such that $Lf^*E$ is isomorphic in $D(\mathcal{O}_ Y)$ to a locally bounded complex of finite locally free $\mathcal{O}_ Y$-modules.

2. If the Chern classes of $E$ are defined, then we define

$c(E) \in \prod \nolimits _{p \geq 0} A^ p(X),\quad ch(E) \in \prod \nolimits _{p \geq 0} A^ p(X) \otimes \mathbf{Q},\quad P_ p(E) \in A^ p(X)$

by an application of Lemma 42.46.2.

This definition applies in many but not all situations envisioned in this chapter, see Lemma 42.46.4. Perhaps an elementary construction of these bivariant classes for general $E/X/(S,\delta )$ as in the definition exists; we don't know.

Lemma 42.46.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. If one of the following conditions hold, then the Chern classes of $E$ are defined:

1. there exists an envelope $f : Y \to X$ such that $Lf^*E$ is isomorphic in $D(\mathcal{O}_ Y)$ to a locally bounded complex of finite locally free $\mathcal{O}_ Y$-modules,

2. $E$ can be represented by a bounded complex of finite locally free $\mathcal{O}_ X$-modules,

3. the irreducible components of $X$ are quasi-compact,

4. $X$ is quasi-compact,

5. there exists a morphism $X \to X'$ of schemes locally of finite type over $S$ such that $E$ is the pullback of a perfect object $E'$ on $X'$ whose chern classes are defined, or

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 37.77.3. 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. $\square$

Lemma 42.46.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Assume the Chern classes of $E$ are defined. For $g : W \to X$ locally of finite type with $W$ integral, there exists a commutative diagram

$\xymatrix{ W' \ar[rd]_{g'} \ar[rr]_ b & & W \ar[ld]^ g \\ & X }$

with $W'$ integral and $b : W' \to W$ proper birational such that $L(g')^*E$ is represented by a bounded complex $\mathcal{E}^\bullet$ of locally free $\mathcal{O}_{W'}$-modules of constant rank and we have $res(c_ p(E)) = c_ p(\mathcal{E}^\bullet )$ in $A^ p(W')$.

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. $\square$

Lemma 42.46.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect. If the Chern classes of $E$ are defined then

1. $c_ p(E)$ is in the center of the algebra $A^*(X)$, and

2. if $g : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_ p(E) = c_ p(Lg^*E) \circ c$.

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

$c \cap c_ p(\mathcal{E}^\bullet ) \cap [X] = c_ p(\mathcal{E}^\bullet ) \cap c \cap [X]$

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 $\mathcal{E}^ n$. $\square$

Lemma 42.46.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let

$E_1 \to E_2 \to E_3 \to E_1[1]$

be a distinguished triangle of perfect objects in $D(\mathcal{O}_ X)$. If one of the following conditions holds

1. there exists an envelope $f : Y \to X$ such that $Lf^*E_1 \to Lf^*E_2$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ Y$-modules,

2. $E_1 \to E_2$ can be represented be a map of locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules,

3. the irreducible components of $X$ are quasi-compact,

4. $X$ is quasi-compact, or

then the Chern classes of $E_1$, $E_2$, $E_3$ are defined and we have $c(E_2) = c(E_1) c(E_3)$, $ch(E_2) = ch(E_1) + ch(E_3)$, and $P_ p(E_2) = P_ p(E_1) + P_ p(E_3)$.

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 $\mathcal{E}_1^\bullet$ $\mathcal{E}_2^\bullet$ 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. $\square$

Remark 42.46.8. The Chern classes of a perfect complex, when defined, satisfy a kind of splitting principle. Namely, suppose that $(S, \delta ), X, E$ are as in Definition 42.46.3 such that the Chern classes of $E$ are defined. Say we want to prove a relation between the bivariant classes $c_ p(E)$, $P_ p(E)$, and $ch_ p(E)$. To do this, we may choose an envelope $f : Y \to X$ and a locally bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_ X$-modules representing $E$. By the uniqueness in Lemma 42.46.2 it suffices to prove the desired relation between the bivariant classes $c_ p(\mathcal{E}^\bullet )$, $P_ p(\mathcal{E}^\bullet )$, and $ch_ p(\mathcal{E}^\bullet )$. Thus we may replace $X$ by a connected component of $Y$ and assume that $E$ is represented by a bounded complex $\mathcal{E}^\bullet$ of finite locally free modules of fixed rank. Using the splitting principle (Lemma 42.43.1) we may assume each $\mathcal{E}^ i$ has a filtration whose successive quotients $\mathcal{L}_{i, j}$ are invertible modules. Settting $x_{i, j} = c_1(\mathcal{L}_{i, j})$ we see that

$c(E) = \prod \nolimits _{i\text{ even}} (1 + x_{i, j}) \prod \nolimits _{i\text{ odd}} (1 + x_{i, j})^{-1}$

and

$P_ p(E) = \sum \nolimits _{i\text{ even}} (x_{i, j})^ p - \sum \nolimits _{i\text{ odd}} (x_{i, j})^ p$

Formally taking the logarithm for the expression for $c(E)$ above we find that

$\log (c(E)) = \sum (-1)^{p - 1}\frac{P_ p(E)}{p}$

Looking at the construction of the polynomials $P_ p$ in Example 42.43.6 it follows that $P_ p(E)$ is the exact same expression in the Chern classes of $E$ as in the case of vector bundles, in other words, we have

\begin{align*} P_1(E) & = c_1(E), \\ P_2(E) & = c_1(E)^2 - 2c_2(E), \\ P_3(E) & = c_1(E)^3 - 3c_1(E)c_2(E) + 3c_3(E), \\ P_4(E) & = c_1(E)^4 - 4c_1(E)^2c_2(E) + 4c_1(E)c_3(E) + 2c_2(E)^2 - 4c_4(E), \end{align*}

and so on. On the other hand, the bivariant class $P_0(E) = r(E) = ch_0(E)$ cannot be recovered from the Chern class $c(E)$ of $E$; the chern class doesn't know about the rank of the complex.

Lemma 42.46.9. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object whose Chern classes are defined. Then $c_ i(E^\vee ) = (-1)^ i c_ i(E)$, $P_ i(E^\vee ) = (-1)^ iP_ i(E)$, and $ch_ i(E^\vee ) = (-1)^ ich_ i(E)$ in $A^ i(X)$.

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 $E$. $\square$

Lemma 42.46.10. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$ whose Chern classes are defined. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

$c_ i(E \otimes \mathcal{L}) = \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c_{i - j}(E) c_1(\mathcal{L})^ j$

provided $E$ has constant rank $r \in \mathbf{Z}$.

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

$\frac{\prod _{a = 1, \ldots , n} (1 + x_ a)}{\prod _{n = 1, \ldots , m} (1 + y_ b)} = 1 + c_1 + c_2 + c_3 + \ldots$

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

$\frac{\prod _{a = 1, \ldots , n} (1 + x_ a + t)}{\prod _{b = 1, \ldots , m} (1 + y_ b + t)} = \sum \nolimits _{i \geq 0} \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c_{i - j} t^ j$

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

Lemma 42.46.11. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $E$ and $F$ be perfect objects of $D(\mathcal{O}_ X)$ whose Chern classes are defined. Then we have

$c_1(E \otimes _{\mathcal{O}_ X}^\mathbf {L} F) = r(E) c_1(\mathcal{F}) + r(F) c_1(\mathcal{E})$

and for $c_2(E \otimes _{\mathcal{O}_ X}^\mathbf {L} F)$ we have the expression

$r(E) c_2(F) + r(F) c_2(E) + {r(E) \choose 2} c_1(F)^2 + (r(E)r(F) - 1) c_1(F)c_1(E) + {r(F) \choose 2} c_1(E)^2$

and so on for higher Chern classes in $A^*(X)$. Similarly, we have $ch(E \otimes _{\mathcal{O}_ X}^\mathbf {L} F) = ch(E) ch(F)$ in $A^*(X) \otimes \mathbf{Q}$. More precisely, we have

$P_ p(E \otimes _{\mathcal{O}_ X}^\mathbf {L} F) = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(E) P_{p_2}(F)$

in $A^ p(X)$.

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 compuation 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

$P_ p(S, f , \epsilon ) = \sum \nolimits _{s \in S} \epsilon (s) f(s)^ p$

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

$P_ p(S \times S', f + f' , \epsilon \epsilon ') = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(S, f, \epsilon ) P_{p_2}(S', f', \epsilon ')$

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

$c_1(E \otimes F) = P_1(E \otimes F) = r(F)P_1(E) + r(E)P_1(F) = r(F)c_1(E) + r(E)c_1(F)$

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

\begin{align*} & 2c_2(E \otimes F) \\ & = c_1(E \otimes F)^2 - P_2(E \otimes F) \\ & = (r(F)c_1(E) + r(E)c_1(F))^2 - r(F)P_2(E) - P_1(E)P_1(F) - r(E)P_2(F) \\ & = (r(F)c_1(E) + r(E)c_1(F))^2 - r(F)(c_1(E)^2 - 2c_2(E)) - c_1(E)c_1(F) - \\ & \quad r(E)(c_1(F)^2 - 2c_2(F)) \end{align*}

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

[1] See Lemma 42.46.4 for some criteria.

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