Schematic Algebras and the Auslander-Gorenstein Property

Noncommutative algebraic geometry studies a certain quotient category Rqgr of the category of graded R-modules which for commutative R is equivalent to the category of quasi-coherent sheaves by a famous theorem of Serre. For a large class of graded algebras, the so-called schematic algebras, we are able to construct a kind of scheme such that the coherent sheaves on it are equivalent to R-qgr. We give a brief survey on the results so-far on schematic algebras and include some new results on cohomological properties of Auslander-Gorenstein algebras which might be useful in determining the strength of the schematic property. Résumé En géométrie algébrique noncommutative on étudie un certain quotient R-qgr de la catégorie des R-modules gradués, qui pour R commutatif, est équivalente à la catégorie des faisceaux quasi-cohérents par un théorème bien connu de Serre. Pour une grande classe d’algèbres, les algèbres schématiques, nous pouvons construire une sorte de schéma sur lequel les faisceaux cohérents forment une catégorie équivalente à R-qgr. Nous rappelons les résultats connus sur les algèbres schématiques et donnons quelques résultats nouveaux sur les propriétés cohomologiques des algèbres de Auslander-Gorenstein. 1 Preliminaries and introduction Let k be an algebraically closed field of characteristic zero and consider a k-algebra R of the following kind : R is connected (i.e. R is positively graded and R0 = k), Noetherian and generated in degree 1. The category of graded R-modules will be denoted by R-gr and the two-sided ideal ⊕i≥1Ri by R+. A graded R-module M is said to be torsion if each of its elements is annihilated by some power of R+ : ∀m ∈ M ∃n ∈ : (R+)m = 0. The set {(R+) : n ∈ } is an idempotent filter (cf. [11]). Hence we have a localization functor Qκ+ available. Moreover, the AMS 1980 Mathematics Subject Classification (1985 Revision): 14A15, 14A22, 16E40, 16W50 ∗Research assistant of the N.F.W.O. (Belgium) — Department of Mathematics, University of Antwerp, U.I.A., Universiteitsplein 1, B-2610 Wilrijk Société Mathématique de France