The Stacks project

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.

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