Noncommutative proj and coherent algebras (Q1774277)

The main result of the paper is a generalization of the theorem of \textit{M. Artin} and \textit{J. J. Zhang} [Adv. Math. 109, 228--287 (1994; Zbl 0833.14002)] characterizing certain classes of abelian categories that can be viewed as noncommutative analogues of categories of coherent sheaves on projective schemes. The main idea of this approach to noncommutative geometry is to associate to a noncommutative graded algebra the quotient category \(\text{QGRA}(A)\) of of the category of graded \(A\)-modules by the subcategory of torsion modules. This generalizes a theorem of Serre who proves that in the commutative case the category of quasicoherent sheaves on \(\text{Proj}(A)\) is equivalent to \(\text{QGRA} (A)\). Then Artin and Zhang [loc. cit.] gave a criterion (the AZ-theorem) for a locally Noetherian abelian category \(\mathcal C\) to be equivalent to \(\text{QGRA} (A)\) for some \(A\), namely that the category has the essential properties of coherent sheaves on a projective scheme \(X\) (i.e. has an ample sequence). The main goal of the present paper is to prove that if one removes the assumption that the category is Noetherian in the AZ-theorem, then the corresponding graded algebra (constructed from the ample sequence) is still coherent and the abelian category in question is equivalent to the quotient of the category of coherent modules by the subcategory of finite-dimensional modules. The article proves partial results to develop techniques for checking whether a given graded algebra is coherent. This gives a connection between the coherency of a graded algebra and its Veronese subalgebras. Notice that the paper extends the class of graded algebras to the wider class consisting of \(\mathbb{Z}\)-algebras. This makes some connections to the non-obstructed cases of O.A. Laudals noncommutative geometry (which is not referred to). The paper is nicely written, and also explicit examples are given.