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

in $A^*(X)$.

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

42.39.1.1

\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 (42.37.1.1) 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 (42.39.1.1) 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*}

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)