Proposition 52.25.4 (0F27)—The Stacks project

Proposition 52.25.4. Let $(\mathcal{F}, \mathcal{F}_0, \alpha )$ be a coherent triple. Let $(\mathcal{L}, \mathcal{L}_0, \lambda )$ be an invertible coherent triple. Then the function

$\mathbf{Z} \longrightarrow \mathbf{Z},\quad n \longmapsto \chi ((\mathcal{F}, \mathcal{F}_0, \alpha ) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda )^{\otimes n})$

is a polynomial of degree $\leq \dim (\text{Supp}(\mathcal{F}))$ .

Proof. We will prove this by induction on the dimension of the support of $\mathcal{F}$ .

The base case is when $\mathcal{F} = 0$ . Then either $\mathcal{F}_0$ is zero or its support is $\{ \mathfrak m\}$ . In this case we have

$(\mathcal{F}, \mathcal{F}_0, \alpha ) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda )^{\otimes n} = (0, \mathcal{F}_0 \otimes \mathcal{L}_0^{\otimes n}, 0) \cong (0, \mathcal{F}_0, 0)$

Thus the function of the lemma is constant with value equal to the length of $\mathcal{F}_0$ .

Induction step. Assume the support of $\mathcal{F}$ is nonempty. Let $\mathcal{G}_0 \subset \mathcal{F}_0$ denote the submodule of sections supported on $\{ \mathfrak m\}$ . Then we get a short exact sequence

$0 \to (0, \mathcal{G}_0, 0) \to (\mathcal{F}, \mathcal{F}_0, \alpha ) \to (\mathcal{F}, \mathcal{F}_0/\mathcal{G}_0, \alpha ) \to 0$

This sequence remains exact if we tensor by the invertible coherent triple $(\mathcal{L}, \mathcal{L}_0, \lambda )$ , see discussion above. Thus by additivity of $\chi$ (Lemma 52.25.3) and the base case explained above, it suffices to prove the induction step for $(\mathcal{F}, \mathcal{F}_0/\mathcal{G}_0, \alpha )$ . In this way we see that we may assume $\mathfrak m$ is not an associated point of $\mathcal{F}_0$ .

Let $T = \text{Ass}(\mathcal{F}) \cup \text{Ass}(\mathcal{F}/f\mathcal{F})$ . Since $U$ is quasi-affine, we can find $s \in \Gamma (U, \mathcal{L})$ which does not vanish at any $u \in T$ , see Properties, Lemma 28.29.7. After multiplying $s$ by a suitable element of $\mathfrak m$ we may assume $\lambda (s \bmod f) = s_0|_{U_0}$ for some $s_0 \in \Gamma (X_0, \mathcal{L}_0)$ ; details omitted. We obtain a morphism

$(s, s_0) : (\mathcal{O}_ U, \mathcal{O}_{X_0}, 1) \longrightarrow (\mathcal{L}, \mathcal{L}_0, \lambda )$

in the category of coherent triples. Let $\mathcal{G} = \mathop{\mathrm{Coker}}(s : \mathcal{F} \to \mathcal{F} \otimes \mathcal{L})$ and $\mathcal{G}_0 = \mathop{\mathrm{Coker}}(s_0 : \mathcal{F}_0 \to \mathcal{F}_0 \otimes \mathcal{L}_0)$ . Observe that $s_0 : \mathcal{F}_0 \to \mathcal{F}_0 \otimes \mathcal{L}_0$ is injective as it is injective on $U_0$ by our choice of $s$ and as $\mathfrak m$ isn't an associated point of $\mathcal{F}_0$ . It follows that there exists an isomorphism $\beta : \mathcal{G}/f\mathcal{G} \to \mathcal{G}_0|_{U_0}$ such that we obtain a short exact sequence

$0 \to (\mathcal{F}, \mathcal{F}_0, \alpha ) \to (\mathcal{F}, \mathcal{F}_0, \alpha ) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda ) \to (\mathcal{G}, \mathcal{G}_0, \beta ) \to 0$

By induction on the dimension of the support we know the proposition holds for the coherent triple $(\mathcal{G}, \mathcal{G}_0, \beta )$ . Using the additivity of Lemma 52.25.3 we see that

$n \longmapsto \chi ((\mathcal{F}, \mathcal{F}_0, \alpha ) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda )^{\otimes n + 1}) - \chi ((\mathcal{F}, \mathcal{F}_0, \alpha ) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda )^{\otimes n})$

is a polynomial. We conclude by a variant of Algebra, Lemma 10.58.5 for functions defined for all integers (details omitted). $\square$

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