# Basics(?) about quasi-coherent modules on projective schemes

EDIT. (05-04-12) I have revised and improved the questions.

Let $A$ be a commutative $\mathbb{N}$-graded $R$-algebra, which is finitely generated by $A_1$ as an $A_0$-algebra. You may also assume that $A_1$ is of finite type over $R$ and that $R$ is noetherian, if this is necessary. Let us denote by $\mathrm{grMod}(A)$ the category of graded $A$-modules and by $\mathrm{Proj}(A)$ the Proj scheme associated to $A$. It is well-known that the functor $\mathrm{grMod}(A) \to \mathrm{Qcoh}(\mathrm{Proj}(A)), M \mapsto \widetilde{M}$ is left-adjoint to $\mathrm{Qcoh}(\mathrm{Proj}(A)) \to \mathrm{grMod}(A), F \mapsto \Gamma_*(F) := \oplus_n \Gamma(F(n))$ and that the counit $c : \widetilde{\Gamma_*(F)} \to F$ is an isomorphism for all $F$; see EGA II.

By general nonsense, it follows that $\mathrm{Qcoh}(\mathrm{Proj}(A))$ is equivalent to some full subcategory $G$ of $\mathrm{grMod}(A)$, consisting of those graded $A$-modules $M$ such that the unit $M \to \Gamma_*(\tilde{M})$ is an isomorphism.

Question 1. Is there any explicit description of these graded modules, which only refers to the abstract tensor category of graded modules? Does this category $G$ appear in the literature?

On the other hand, we also get by general nonsense (Gabriel-Zisman, p. 7) that $\mathrm{Qcoh}(\mathrm{Proj}(A))$ is equivalent to the localization of $\mathrm{grMod}(A)$ with respect to those morphisms $\phi : M \to N$ of graded modules which induce an isomorphism $\tilde{\phi} : \widetilde{M} \to \widetilde{N}$ of quasi-coherent modules. It is easy to see that this happens iff for every $f \in A_1$ the induced homomorphism of $A_{(f)}$-modules $M_{(f)} \to N_{(f)}$ is an isomorphism. If I am not mistaken, this even implies that $M_f \to N_f$ is an isomorphism. Obviously these morphisms constitute a right multiplicative system $\Sigma$ in $\mathrm{grMod}(A)$. It is even closed under orthogonality consequences in the sense of the paper "A Logic of Orthogonality" by Adamek, Hebert, Sousa.

Question 2. How can we describe $\Sigma$ in terms of the abstract tensor category $\mathrm{grMod}(A)$?

For example, it is easy to see that $\Sigma$ contains $\mathcal{T}$, which consists of TN-isomorphisms in the language of EGA II (there is some integer $a$ so that in degrees $\geq a$ we have an isomorphism). The kind of definition of $\mathcal{T}$ is what I'm looking for; it doesn't refer to elements, but can be formulated abstractly. However, $\mathcal{T}$ is not closed under infinite direct sums. So consider the colimit closure $\mathcal{T}_c$, it still satisfies $\mathcal{T}_c \subseteq \Sigma$.

Do we have $\mathcal{T}_c = \Sigma$?

Let me make even more precise what I mean by an abstract description of $\Sigma$: Let $C$ be a nice cocomplete symmetric monoidal category (say Grothendieck and presentable), $\mathcal{A}$ an $\mathbb{N}$-graded commutative algebra object in $C$, then $\mathrm{grMod}(\mathcal{A})$ is a nice cocomplete symmetric monoidal category. Now I wonder how to define the class $\Sigma$ of morphisms in this abstract context; of course in such a way that for $C=\mathrm{Mod}(R)$ one recovers the class defined above.

I would also appreciate a lot any information about one of these two representations of $\mathrm{Qcoh}(\mathrm{Proj}(A))$ in the literature. So far I've only found the description of $\mathrm{Coh}(\mathrm{Proj}(A))$ as a quotient category of $\mathrm{grMod}_f(A)$, where $A$ is assumed to be noetherian. Another remark: I am also interested in the "global case" when $A$ is a graded $\mathcal{O}_S$-algebra for some (noetherian) scheme $S$.

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Under nice hypotheses (which I do not remember, but which might include the assumption that $A_0$ is a field), there is an exact sequence $0 \to H^0_{\mathfrak{m}}(M) \to M \to \Gamma_*(\widetilde{M}) \to H^1_{\mathfrak{m}}(M) \to 0$, where $H^i_{\mathfrak{m}}$ denote the local cohomology groups with respect to the irrelevant ideal. Thus, the condition that $H^1_m(M) = 0$ and $H^0_m(M) = 0$ may give what you want. On the other hand, perhaps I'm misremembering or misunderstanding the subtleties of your question. –  Charles Staats Apr 4 '12 at 21:51
This sounds reasonable, but it is formulated not within the category of graded modules. –  Martin Brandenburg Apr 5 '12 at 6:31
If, in the description given by Charles, you use the fact that, under the additional hypothesis that $A_0$ is noetherian, $\Gamma_*(\widetilde{M})$ is canonically isomorphic to the $\mathfrak{m}$-transform of $M$ (as a graded module), then it is formulated within the category of graded modules. –  Fred Rohrer Apr 5 '12 at 12:14
Where can I find the isomorphism $\Gamma_*(\tilde{M}) \cong \lim_n \mathrm{Hom}(\mathfrak{m}^n,M)$? –  Martin Brandenburg Apr 5 '12 at 15:15
@Martin: Brodmann-Sharp, Local Cohomology, Chapter 20. –  Fred Rohrer Apr 5 '12 at 17:32

Question 1: The category $C$ is often described as the quotient $$grMod(A)/TN(A)$$ where we denote by $TN(A)$ we denote The subcategory of $grMod(A)$ of graded modules $M$ such that $I^n m= 0$ for $n \gg 0$ for every $m \in M$, where $I := \oplus_{n > 0} A_n$, the irrelevant ideal. This subcategory constitutes the torsion part of a torsion theory so you can describe $Qco(Proj(A))$ using Gabriel's localization techniques.

Edit: For a description of quasi coherent modules as objects in a certain subcategory of $grMod(A)$, look at pages 198 and 199 in Stenström Rings and Modules, Springer, 1975. For the adaptation of these theory top the graded situation, see Van Oystaeyen, On graded rings and modules of quotients. Comm. Algebra 6 (1978), no. 18, 1923–1959. See also page 199 in Jeremías López, A.; López López, M. P.; Villanueva Nóvoa, E. Localization in symmetric closed Grothendieck categories. Third Week on Algebra and Algebraic Geometry (SAGA III) (Puerto de la Cruz, 1992). Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 197–221.

Question 2: Morphisms are usual morphisms making invertible those whose kernel and cokerenel belongs to $TN(A)$. It is therefore a localization situation. The functor $\Gamma_*\widetilde{(-)}$ is the quotient functor.

Bonus

1. The coherator functor is easily described as: $\widetilde{\Gamma_*(-)}$

2. You can find additional information on these topics on the mentioned paper by Jeremías, A.; López, M. P. and Villanueva, E.

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Thanks Leo for the answer. Meanwhile I have observed similar things. If $A$ generated by $A_1$ and not necessarily noetherian, $M$ is torsion iff for every element $m$ and every $f \in A_1$ there is some $n$ with $f^n m$ vanishes, right? Can you give some references for your claims? Of course I know Gabriels thesis and the work by Artin/Zhang. –  Martin Brandenburg Apr 4 '12 at 12:04
Besides I would like to understand $C$ as a full subcategory of graded modules, not as a quotient. Will these be just the torsionfree ones? –  Martin Brandenburg Apr 4 '12 at 12:05
@Martin Brandenburg: Not the torsion free but the closed ones, see EGA IV, definition (5.9.9) adapted to the graded situation. Remarque (5.9.12) gives you extra information. There are references in which the graded case is treated in detail, but I don't have them at hand now, I'll tell you later about them. –  Leo Alonso Apr 4 '12 at 22:24
Thanks. But this closure-condition is a trivial reformulation of what I have said about Question 2. I need a condition which only talks about graded modules. –  Martin Brandenburg Apr 5 '12 at 6:30
@Martin Brandenburg: I have edited the answer giving additional information and some precise references. –  Leo Alonso Apr 5 '12 at 11:36

This is an addition to the answer of Leo Alonso. On the level of derived categories there is a semiorthogonal decomposition $$D(grMod(A)) = \langle D_T(grMod(A)), D(Qcoh(Proj(A))) \rangle,$$ the first term being complexes with torsion cohomology modules. In particular, $D(Qcoh(Proj(A)))$ embeds into $D(grMod(A))$ as the orthogonal to $k[i]$ for all $i \in Z$.

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Thanks, this is interesting. –  Martin Brandenburg Apr 5 '12 at 15:15