Sheaf of sections supported at a point is coherent?

Question

Let $\mathcal{F}$ be a coherent analytic sheaf over some open subset of $U ⊆ \mathbb{C}^n$. I read in a book that if $p \in U$ then the subsheaf $\mathcal{G}$ defined by \begin{equation} \mathcal{G}(V) = \left\{s \in \mathcal{F} ~\middle|~ s|_{V ∖ \{p\}} = 0\right\} \end{equation} is a coherent subsheaf. How can you see this?

One idea: It is my understanding that "finite-dimensional skyscrapers" supported at a point are coherent. That is, you have a $\mathcal{O}_p$-module $M$ that is finite-dimensional as a $\mathbb{C}$-vector space and you consider the sheaf \begin{equation} V \mapsto \begin{cases} M & p \in V \\ 0 & p \not\in V \end{cases} \end{equation} So it would be sufficient to check that $\mathcal{G}_p$ is finite-dimensional. For this I think it would suffice to show that $\mathfrak{m}^k \mathcal{G}_p = 0$ for sufficiently large $k$.

Simonsays · Answer 1 · 2022-10-03T06:54:30.477

1

your understanding is right, skyscrapers of finite dimensional vector spaces are coherent! That is, if you define skyscraper sheaves to be coherent in the first place (see also here).

Generally you should think of a coherent skyscraper sheaf as a finitely generated $\mathcal{O}_{X,p}$-module $\mathbf{M}$ over a point $i: \{p\} \hookrightarrow X$ and you get the coherent sheaf of $\mathcal{O}_X$- modules on $X$ by considering $ i_*\mathbf{M}$ on $X$.
Check that $\mathcal{G}_q = 0$ if $q \neq p$. Indeed, $\mathcal{G}_q = \underset{\longrightarrow}\lim \mathcal{G}(V)$ where $V$ ranges over opens in $U$ containing $q$. Now if $q$ and $p$ are distinct then you can always find opens $V$ not containing $p$, hence $\mathcal{G}(V)\equiv 0 $ there after passing to the limit, so $\mathcal{G}_q = 0$ for $q\neq p$.

Now $\mathcal{G}_p$ is a $\mathbf{C}$-subspace of $H^0(\mathcal{F}_p)=Hom(\mathcal{O}_p,\mathcal{F}_p)\cong Hom(\mathcal{O}_X,i_*A)$ where $A$ is the $\mathcal{O}_p$-module $\mathcal{F}_p$ seen as a skyscraper sheaf over $p$ which is again coherent hence has finite dimensional global sections over a point.
The last equality uses that talking stalks and skyscrapers are adjoint.

Edit: Concerning the confusion regarding $\mathcal{F}_p$: $\mathcal{F_p}/({\mathcal{F}_p}m_p)$ is finite dimensional vector space and any basis lifts to a minimal generating set of $\mathcal{F}_p$ (that is Nakayamas lemma) which are precisely the elements of $H^0(\mathcal{F}_p)$.

edited Oct 03 '22 at 06:54

answered Oct 02 '22 at 13:48

Simonsays

1,823

Certainly you need the scyscraper to be a finite-dimensional $\mathbb{C}$-vector space, otherwise you contradict finiteness theorems for $H^0$. – Carlos Esparza Oct 02 '22 at 18:20
ok I don't understand your remark and your downvote: I said its finite dimensional? Please correct my if I'm wrong but finiteness for $H^0$ is a property of projective varieties? – Simonsays Oct 02 '22 at 19:16
how do you know that the $\mathcal{G}_p$ is finite−dimensional? – Carlos Esparza Oct 02 '22 at 19:18
also there is no projective variety in the question. (And the sheaves are analytic although that probably makes no difference?) – Carlos Esparza Oct 02 '22 at 19:19
it makes no difference over proper spaces. So affine spaces excluded. But I think you only need coherence over a point if I'm not mistaken – Simonsays Oct 02 '22 at 20:17
I edited the answer concerning the other question – Simonsays Oct 02 '22 at 20:20
Exactly, just the inclusion. – Simonsays Oct 02 '22 at 20:26
What is your last paragraph arguing? That $\mathcal{G}_p$ is finite-dimensional? (I don't see how that would follow, given that $\mathcal{F}_p$ doesn't have to be) – Carlos Esparza Oct 02 '22 at 20:26
$\mathcal{F}_p$ is finite! Its a coherent sheaf over a point (which is projective if you prefer to see it that way) – Simonsays Oct 02 '22 at 20:32
I'm confused. By $\mathcal{F}_p$ don't you mean the stalk of $\mathcal{F}$ at $p$? That would not be finite-dimensional. – Carlos Esparza Oct 02 '22 at 21:14
I edited the final bit! – Simonsays Oct 03 '22 at 06:54

Carlos Esparza · Answer 2 · 2022-10-03T05:44:10.110

It seems like this works for a proof:

As @Sisi points out (I think) we have the following
Lemma: If $M$ is a finitely generated (hence noetherian) $\mathcal{O}_p$-module and let $\mathcal{M}$ denote the skyscraper sheaf $i_* M$, where $i : \{p\} \to U$. Then $\mathcal{M}$ is a coherent $\mathcal{O}$-module if multiplication $\mathcal{O}(U) \times \mathcal{M}(U) \to \mathcal{M}(U)$ is defined by $f \cdot m = f_p m$ when $U \ni p$.
Proof: In a nbh $U$ of $p$ it is finitely generated by generators $m_1, \dots, m_n$ of $M$. Furthermore the kernel of the map $\mathcal{O}(U)^{\oplus n} \to \mathcal{M}(U)$ given by $(m_1, \dots, m_n)$ is a submodule of $M$ and hence finitely generated. $\square$

Now in this example $\mathcal{F}_p$ is a finitely generated $\mathcal{O}_p$-module (by coherence of $\mathcal{F}$) and the sheaf $\mathcal{G}$ is the skyscraper associated to $M = \left\{s \in \mathcal{F}_p \,\middle\vert\, s \text{ defined on $U \in p$ and } s|_{U ∖ p} = 0 \right\}⊆ \mathcal{F}_p$.

The idea of showing finite $\mathbb{C}$-dimension seems to be a red herring as is trying to use some kind of finiteness of global sections?

Sheaf of sections supported at a point is coherent?

2 Answers2