The Stacks project

81.29 Polynomial relations among Chern classes

In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}_ i$ be a finite collection of finite locally free $\mathcal{O}_ X$-modules. By Lemma 81.28.4 we see that the Chern classes

\[ c_ j(\mathcal{E}_ i) \in A^*(X) \]

generate a commutative (and even central) $\mathbf{Z}$-subalgebra of the Chow cohomology $A^*(X)$. Thus we can say what it means for a polynomial in these Chern classes to be zero, or for two polynomials to be the same. As an example, saying that $c_1(\mathcal{E}_1)^5 + c_2(\mathcal{E}_2)c_3(\mathcal{E}_3) = 0$ means that the operations

\[ \mathop{\mathrm{CH}}\nolimits _ k(Y) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - 5}(Y), \quad \alpha \longmapsto c_1(\mathcal{E}_1)^5 \cap \alpha + c_2(\mathcal{E}_2) \cap c_3(\mathcal{E}_3) \cap \alpha \]

are zero for all morphisms $f : Y \to X$ of good algebraic spaces over $B$. By Lemma 81.26.9 this is equivalent to the requirement that given any morphism $f : Y \to X$ where $Y$ is an integral algebraic space locally of finite type over $X$ the cycle

\[ c_1(\mathcal{E}_1)^5 \cap [Y] + c_2(\mathcal{E}_2) \cap c_3(\mathcal{E}_3) \cap [Y] \]

is zero in $\mathop{\mathrm{CH}}\nolimits _{\dim (Y) - 5}(Y)$.

A specific example is the relation

\[ c_1(\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}) = c_1(\mathcal{L}) + c_1(\mathcal{N}) \]

proved in Lemma 81.18.2. More generally, here is what happens when we tensor an arbitrary locally free sheaf by an invertible sheaf.

Lemma 81.29.1. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then we have

81.29.1.1
\begin{equation} \label{spaces-chow-equation-twist} c_ i({\mathcal E} \otimes {\mathcal L}) = \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c_{i - j}({\mathcal E}) c_1({\mathcal L})^ j \end{equation}

in $A^*(X)$.

Proof. The proof is identical to the proof of Chow Homology, Lemma 42.39.1 replacing the lemmas used there by Lemmas 81.26.9 and 81.28.1. $\square$


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