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