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42.39 Polynomial relations among Chern classes

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}_ i$ be a finite collection of finite locally free sheaves on $X$. By Lemma 42.38.9 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 algebra $A^*(X)$. 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 $f : Y \to X$ which are locally of finite type. By Lemma 42.35.3 this is equivalent to the requirement that given any morphism $f : Y \to X$ where $Y$ is an integral scheme locally of finite type over $S$ 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 42.25.2. More generally, here is what happens when we tensor an arbitrary locally free sheaf by an invertible sheaf.

Lemma 42.39.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. 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
\begin{equation} \label{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. This should hold for any triple $(X, \mathcal{E}, \mathcal{L})$. In particular it should hold when $X$ is integral and by Lemma 42.35.3 it is enough to prove it holds when capping with $[X]$ for such $X$. Thus assume that $X$ is integral. Let $(\pi : P \to X, \mathcal{O}_ P(1))$, resp. $(\pi ' : P' \to X, \mathcal{O}_{P'}(1))$ be the projective space bundle associated to $\mathcal{E}$, resp. $\mathcal{E} \otimes \mathcal{L}$. Consider the canonical morphism

\[ \xymatrix{ P \ar[rd]_\pi \ar[rr]_ g & & P' \ar[ld]^{\pi '} \\ & X & } \]

see Constructions, Lemma 27.20.1. It has the property that $g^*\mathcal{O}_{P'}(1) = \mathcal{O}_ P(1) \otimes \pi ^* {\mathcal L}$. This means that we have

\[ \sum \nolimits _{i = 0}^ r (-1)^ i (\xi + x)^ i \cap \pi ^*(c_{r - i}(\mathcal{E} \otimes \mathcal{L}) \cap [X]) = 0 \]

in $\mathop{\mathrm{CH}}\nolimits _*(P)$, where $\xi $ represents $c_1(\mathcal{O}_ P(1))$ and $x$ represents $c_1(\pi ^*\mathcal{L})$. By simple algebra this is equivalent to

\[ \sum \nolimits _{i = 0}^ r (-1)^ i \xi ^ i \left( \sum \nolimits _{j = i}^ r (-1)^{j - i} \binom {j}{i} x^{j - i} \cap \pi ^*(c_{r - j}(\mathcal{E} \otimes \mathcal{L}) \cap [X]) \right) = 0 \]

Comparing with Equation ( it follows from this that

\[ c_{r - i}(\mathcal{E}) \cap [X] = \sum \nolimits _{j = i}^ r \binom {j}{i} (-c_1(\mathcal{L}))^{j - i} \cap c_{r - j}(\mathcal{E} \otimes \mathcal{L}) \cap [X] \]

Reworking this (getting rid of minus signs, and renumbering) we get the desired relation. $\square$

Some example cases of ( are

\begin{align*} c_1(\mathcal{E} \otimes \mathcal{L}) & = c_1(\mathcal{E}) + r c_1(\mathcal{L}) \\ c_2(\mathcal{E} \otimes \mathcal{L}) & = c_2(\mathcal{E}) + (r - 1) c_1(\mathcal{E}) c_1(\mathcal{L}) + \binom {r}{2} c_1(\mathcal{L})^2 \\ c_3(\mathcal{E} \otimes \mathcal{L}) & = c_3(\mathcal{E}) + (r - 2) c_2(\mathcal{E})c_1(\mathcal{L}) + \binom {r - 1}{2} c_1(\mathcal{E})c_1(\mathcal{L})^2 + \binom {r}{3} c_1(\mathcal{L})^3 \end{align*}

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