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

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.

• Your whole premise is wrong: there are nonzero graded modules $M$ which have associated sheaf zero. Consider the $k[x_0,x_1]$-module given by $k$ in degree $0$ and $0$ elsewhere: it's associated sheaf on $\Bbb P^1$ is just the zero sheaf. – KReiser Jun 2 at 23:01
• True, but I what I meant (apologies for not being clear) is that I'm interested in elements which are not of degree zero. We can test cohomologically if there exist non-trivial elements of degree strictly greater then zero the way I describe above. I'll edit the question accordingly. – curious math guy Jun 2 at 23:24
• That's still wrong - just shift the previous module around (for instance, place it in degree $n$ instead of degree $0$). – KReiser Jun 2 at 23:27
• Fair, but I belive my premise and thus my question is still valid if we assume that $R_{>0}$ is not acting trivially on $M$, i.e. is not contained in the annihilator of $M$ as then the module can not be concentrated in one single degree, and in fact the if $M$ is non-trival in one degree, then it is non-trival in in all high degrees (which we can detect cohomologically). – curious math guy Jun 3 at 0:31

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.