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 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
\[ P_ p(S, f , \epsilon ) = \sum \nolimits _{s \in S} \epsilon (s) f(s)^ p \]
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 $2$.
$\square$
Comments (0)