Let $S:=k[X\_1,\ldots,X\_n]$ be a polynomial ring over a field $k$, with its natural grading, and let $\mathfrak{p}$ be a *homogeneous* prime ideal of $S$. Also, let $M:=\bigoplus\_{i} M\_i$ be a finitely generated graded $(S/\mathfrak{p})$-module. Write $\mathcal{F}\_P$ for the coherent sheaf on $\mathrm{Proj}\:\:\: S/\mathfrak{p}$ associated to $M$, and $\mathcal{F}\_A$ for the coherent sheaf on $\mathrm{Spec}\:\:\:S/\mathfrak{p}$ associated to $M$, where in the later case we forget about all gradings.


**Question:** Is there any relation between rank of $\mathcal{F}\_P$ at the generic point of $\mathrm{Proj}\:\:\: S/\mathfrak{p}$ and rank of $\mathcal{F}\_A$ at the generic point of $\mathrm{Spec}\:\:\: S/\mathfrak{p}$?



Generic rank of a coherent sheaf on a projective variety vs. generic rank on the cone
The ranks are the same. Since $M$ is a finitely generated $(S/\mathfrak{p})$-module you can write down a finite presentation of $M$ by twisted (in the sense of twisting the grading) copies of $S/\mathfrak{p}$:


$$ \oplus\_j (S/\mathfrak{p})(-b\_j) \stackrel{\Phi}{\longrightarrow} \oplus\_{i=1}^k (S/\mathfrak{p})(-a\_i) \longrightarrow M \longrightarrow 0.$$


Passing to $\operatorname{Proj} S/\mathfrak{p}$ this becomes a presentation of $\mathcal{F}\_{P}$ with the terms $(S/\mathfrak{p})(-a\_i)$ (respectively $-b\_j$) replaced by copies of the structure sheaf twisted by $-a\_i$ (respectively $-b\_j$). Thus the rank of $\mathcal{F}\_P$ at a generic point of $\operatorname{Proj} S/\mathfrak{p}$ is $k$ minus the rank of the map $\Phi$ at a generic point of the $\operatorname{Proj}$.


On the other hand, passing to $\operatorname{Spec} S/\mathfrak{p}$, the presentation above becomes a presentation of $\mathcal{F}\_A$ with the terms $(S/\mathfrak{p})(-a\_i)$ (respectively $-b\_j$) replaced by the structure sheaf. The rank of $\mathcal{F}\_P$ at a generic point of $\operatorname{Proj} S/\mathfrak{p}$ is again $k$ minus the rank of the map $\Phi$ at a generic point of the $\operatorname{Spec}$.


The scheme $\operatorname{Spec} S/\mathfrak{p}$ is the affine cone over $\operatorname{Proj} S/\mathfrak{p}$, and any point on the $\operatorname{Proj}$ gives rise to a line on the $\operatorname{Spec}$. The rank of $\Phi$ at any point of the $\operatorname{Proj}$ is equal to the rank of $\Phi$ at every point of the corresponding line in the $\operatorname{Spec}$ (except possibly the origin). Thus the rank of $\Phi$ at a generic point of the $\operatorname{Proj}$ is the same as the rank of $\Phi$ at a generic point of the $\operatorname{Spec}$, and therefore the ranks of $\mathcal{F}\_P$ and $\mathcal{F}\_A$ at the respective generic points are also the same.


