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
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
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
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
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
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
is a polynomial. We conclude by a variant of Algebra, Lemma 10.58.5 for functions defined for all integers (details omitted). $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)