This paper deals with the notion of projective scheme in the framework of noncommutative \(\mathbb{N}\)-graded algebras. The paper does not give a direct definition of the projective spectrum as a ringed space. The bright solution is not to define the projective spectrum itself, but rather, the category of coherent sheaves on it, a category that contains information of many geometrical properties of the projective spectrum, like cohomology. This is carried out by taking Serre’s theorem as the definition of the projective spectrum. In the commutative case, if \(A = k[\xi_1, \ldots, \xi_r]\) and \(X = \text{Proj} (A)\), the functor \(\Gamma_* : \text{qcoh} X \to \text{QGR}A\) described as \(\Gamma_* ({\mathcal F}) = \bigoplus^\infty_{d = - \infty} (X, {\mathcal F} (d))\) is an equivalence of categories between the category \(\text{qcoh} X\) of quasi-coherent sheaves on \(X\) and the quotient category \(\text{QGR} A = \text{GR} A/\)TORS, where GRA is the category of graded \(A\)-modules and TORS be subcategory of graded torsion modules (that is, such that for every \(x \in M\), one has \(x \cdot A_{\geq s} = 0\) for \(s\) big enough). In this categorical equivalence, the structure ring of \(X\) corresponds to the module \(A_A\) (which is \(A\) itself considered as an \(A\)-module) and the operation of twisting with \({\mathcal O} (1)\) corresponds to the automorphism \(s\) of \(\text{qgr} A\) defined by the shift of degrees. If \(\text{gr} A\) is the subcategory of \(\text{GR}A\) corresponding to the graded modules equivalent to finite graded modules modulo torsion, the category \(\text{coh} X\) of coherent sheaves on \(X\) is equivalent to \(\text{qgr} A = \text{gr} A/ \text{tors}\).
Keeping this equivalence in mind, the authors define the notion of general projective spectrum of a noncommutative graded algebra \(A\), as the triple \(\text{Proj} A = (\text{QGR} A,A_A,s)\). The algebra \(A\) is recovered from this triple since one has \(A = \bigoplus^\infty_{d = - \infty} \operatorname{Hom} (A_A, s^d (A))\). Similarly, the noetherian projective scheme is defined as the triple \(\text{proj} A = (\text{qgr} A,A_A,s)\). This corresponds in the commutative case to define \(X = \text{proj} A\) as the category \(\text{coh} X\).
Once this definition of \(\text{proj} A\) has been given, a natural question is to characterize those triples \(X = ({\mathcal C}, {\mathcal A}, s)\), consisting of a \(k\)-linear category \({\mathcal C}\), and object \({\mathcal A}\), and an autoequivalence \(s\) of \({\mathcal C}\), which are equivalent to \(\text{proj} A\), a characterization that can be considered as Serre’s theorem for noncommutative rings. To do that, the authors define, for every object \({\mathcal M}\) of \({\mathcal C}\) an object \(\Gamma ({\mathcal M}) = \bigoplus^\infty_{d = - \infty} \operatorname{Hom} ({\mathcal A}, s^d ({\mathcal M}))\). Then \(\Gamma ({\mathcal A})\) is a graded algebra and \(\Gamma ({\mathcal M})\) is a graded right module over \({\mathcal A}\). The authors succeed in proving the noncommutative correspondent to Serre’s theorem:
If \(A = \Gamma ({\mathcal A})_{\geq 0}\), then \(({\mathcal C}, {\mathcal A}, s) \simeq \text{proj} A\) provided that \({\mathcal A}\) satisfies certain noetherian conditions, \(\operatorname{Hom} ({\mathcal A}, {\mathcal M})\) is finite over \(k\) for every \({\mathcal M}\) and \(s\) is ample in a sense that is precised in the paper. Conversely, a right noetherian \(k\)-algebra \(A\) that satisfies a certain technical condition, can be recovered up to torsion from \(\text{proj} A\) by \(\Gamma\). The cohomology over \(\text{proj} A\) (to be thought of as the cohomology of coherent sheaves on the projective spectrum) can be defined as \(\text{proj} A = (\text{qgr} A, A_A, s)\). Then, there exist minimal injective resolutions, thus allowing the definition of Ext’s and of cohomology groups. A generalization of Serre’s finiteness of the cohomology to this case is proved, by means of a careful analysis of the involved injective resolution. Some properties of the cohomological dimension of \(\text{proj} A\) are studied, together with bounds in certain cases.

Reviewer: D.Hernández Ruipérez (Salamanca)