# An elementary proof of a criterion for $k$ sufficiently large that $M_k = \Gamma(\mathbb{P}^n, \widetilde{M}(k))$

It is well known that coherent sheaves on $\mathbb{P}^n$ are equivalent, as a category, to finitely generated graded modules over the polynomial ring, provided that in the latter category, morphisms need only be defined on sufficiently large degrees. However, actually using this reason about sheaves can be problematic. For instance, given a homogeneous ideal $I$ of the polynomial ring $S$, if we let $X \subset \mathbb{P}^n$ be the associated closed subscheme, it can be difficult to determine the global sections of $\mathscr{O}_X(n)$ from $S/I$.

Such computations can be made more feasible by the following theorem, which can be extracted from Bayer and Stillman's 1987 paper, "A criterion for detecting $m$-regularity":

Theorem: Let $S = \Bbbk[x_0, \dotsc, x_n]$, where $\Bbbk$ is an infinite field. Let $M$ be a finitely generated graded $S$-module, with associated coherent sheaf $\mathscr{F}$ on $\mathbb{P}^n_{\Bbbk}$. Suppose that

1. $M$ admits a finite presentation with generators in degrees $\leq m$ and relations in degrees $\leq m+1$, and
2. There exist $h_1, \dotsc, h_j \in S_1$ for some $j \geq 0$ such that $(h_1, \dotsc, h_j)M_{m} = M_{m+1}$ and for all $i = 1, \dotsc, j$, the map $$\left(\frac{M}{(h_1, \dotsc, h_{i-1})M}\right)_{m+1} \stackrel{h_i}{\longrightarrow} \left(\frac{M}{(h_1, \dotsc, h_{i-1})M}\right)_{m+2}$$ is an injection.

Then it follows that the natural map $M_k \to \Gamma(\mathbb{P}^n_{\Bbbk}, \mathscr{F}(k))$ is an isomorphism for all $k \geq m+1$.

Bayer and Stillman's conclusion is actually a lot stronger than this (they show that the hypotheses 1. and 2. give necessary and sufficient conditions for $M$ to be $m$-regular), but it seems to me that the statement here requires less background to understand. One could present this statement to someone who had just read, say, section II.5 of Hartshorne ("Sheaves of Modules"), and they would understand its significance and probably find it useful. Bayer and Stillman's proof involves understanding not only one, but two equivalent definitions for Castelnuovo-Mumford regularity (one in terms of graded free resolutions, the other in terms of local cohomology).

Question: Is there a more elementary proof of just the theorem above?

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