In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}_ i$ be a finite collection of finite locally free $\mathcal{O}_ X$-modules. By Lemma 82.28.4 we see that the Chern classes
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
are zero for all morphisms $f : Y \to X$ of good algebraic spaces over $B$. By Lemma 82.26.9 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
is zero in $\mathop{\mathrm{CH}}\nolimits _{\dim (Y) - 5}(Y)$.
A specific example is the relation
proved in Lemma 82.18.2. More generally, here is what happens when we tensor an arbitrary locally free sheaf by an invertible sheaf.
Lemma 82.29.1. In Situation 82.2.1 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 in $A^*(X)$.
82.29.1.1
\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}
Proof. The proof is identical to the proof of Chow Homology, Lemma 42.39.1 replacing the lemmas used there by Lemmas 82.26.9 and 82.28.1. $\square$
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