I'm trying to understand the following. Any reference to this fact, or hints, would be very much appreciated! I am reading the section on Castelnuevo-Mumford Regularity in Positivity I, and the following appears in the setup of 1.8.B.
Let $\mathbb{P} = \mathbb{P}(V)$ be some projective space, with $\mathcal{F}$ a coherent sheaf on $\mathbb{P}$. Then, we denote $F = \bigoplus_{k \in \mathbb{Z}} H^0(\mathbb{P}, \mathcal{F}(k))$ the associated graded module. It is then claimed that if $F_k = 0$ for $k << 0$, then $F$ is a finitely generated $S$-module.
To explain this, the author writes that the finite generation of $F$ is equivalent to the assumption that none of the associated primes of $\mathcal{F}$ have zero dimensional support.
Why does this equivalence hold, and how can one use this to show that $F$ is finitely generated?
Again, I would appreciate any reference or explanation of this.
Thanks!
Edit: The stacks project has this fact (that $F$ is finitely generated) as part of Lemma 01YS. This still doesn't really explain the fact about associated primes though.
Edit2: So I was able to get one direction. I was having trouble with the other direction, but I suspect one can use a similar argument, along with the fact that twisting by a line bundle preserves associated points.
Anyway, if $F$ is finitely generated, it follows easily that $F_k = 0$ for $k << 0$. I claim that in this case $\mathcal{F}$ has no associated points $x$ of dimension 0.
Indeed, suppose $\mathcal{F}$ has an associated point $x$ of dimension $0$. Then, we can find a local section $s$ of $\mathcal{F}$ so that the zero locus of $s$ is exactly a closed point $x \in \mathbb{P}$. Twisting by large $d$, and replacing $\mathcal{F}$ with $\mathcal{F}(d)$, we may assume that $s \in H^0(\mathbb{P}, \mathcal{F})$. We will conclude by showing that this implies $H^0(\mathbb{P}, \mathcal{F}(d)) \neq 0$ for all $d \in \mathbb{Z}$.
Indeed if $\mathfrak{m}_x \subset \mathcal{O}_{\mathbb{P}}$ is the maximal ideal of $x$, then since $\mathfrak{m}_x$ kills $s$, the nonzero map $s: \mathcal{O}_{\mathbb{P}} \to \mathcal{F}$ descends to a nonzero map $\mathcal{O}_{\mathbb{P}}/\mathfrak{m}_x = k_x \to \mathcal{F}$. Since twisting by line bundles leaves skyscraper sheaves unchanged, it follows that for all $d \in \mathbb{Z}$, there is a nonzero map $k_x \to \mathcal{F}(d)$, which implies that $\mathcal{F}(d)$ has a nonzero global section.
