**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 $S$.