Cohomological criterion for non-triviality of negative part of graded module

Question

Let $R$ be a graded ring and $M$ a graded module. Then for sufficently large $n$, we have $$H^0(\operatorname{Proj}(R), \widetilde{M}(n))\cong M_n.$$ Hence if I want to show that $M_{>0}$ is non-trivial, I can use cohomology. However, this fails for the negative part $M_{<0}$. Even if $H^0(\operatorname{Proj}(R),\widetilde{M}(-n))\cong M_{-n}$, then most of my tools to deal with cohomology are useless, as the usally deal with sheaves of the form $\mathcal{F}(n)$ for $n\gg0$. Of course, if I could relate $H^0(\operatorname{Proj}(R),\widetilde{M}(-n))$ with $H^0(\operatorname{Proj}(R),\widetilde{M}^{\vee}(n))$, then I could deal with it. But I'm unaware of such a relation. So I guess I'm here for guidance or any sort of hint.

Answer 1

This is maybe not quite an answer to your question, but let me take the time to let you know about some things to file under "$\operatorname{Proj}$ exhibits interesting behavior with quasicoherent sheaves not found with $\operatorname{Spec}$". See here for some more examples.

Let us explain an important feature of the functor $\widetilde{-}:R\text{-mod}\to \operatorname{QCoh}(\operatorname{Proj} R)$ which takes graded modules $M$ on a graded ring $R$ to their associated sheaf on $\operatorname{Proj R}$. If $M$ is a graded module which is nonzero in only finitely many degrees, then $\widetilde{M}$ is the zero sheaf on $\operatorname{Proj} R$.

Proof: by definition, the sections of $\widetilde{M}$ on $D(f)$ for $f$ homogeneous of positive degree are given by $M_{(f)}$, the degree-zero elements of $M_{f}$. If $s$ is an element of $M_{(f)}$, then $s=\frac{f^ns}{f^n}$ is also such an element for $n$. But by picking $n$ large enough, we get that $f^ns$ lies in a degree where $M$ is zero, so $s=0$ and in fact $M_{(f)}=0$. So our sheaf is the zero sheaf.

This means that there's no invariant of quasi-coherent sheaves on $\operatorname{Proj} R$ which can tell apart the sheaves associated to two graded modules which differ by some module supported in finitely many degrees - we already destroyed that information just by applying the associated sheaf functor.

In particular, if $R$ has no elements of negative degree (a very common assumption) then unless $M$ is nonzero in infinitely many negative degrees, you can't tell it apart from a module which is zero in negative degrees. As a consequence, $M$ has to be infinitely generated as an $R$-module with generators in arbitrarily large negative degrees (and you'll need to say something about the annihilators of these generators not being the whole of $R_+$, etc). None of this is so bad, but you need to know what's up.