Periodicity of d-cluster-tilted algebras

It is well-known that any maximal Cohen-Macaulay module over a hypersurface has a periodic free resolution of period 2. Auslander, Reiten (1996) [4] and Buchweitz (1998) [8] have used this periodicity to explain the existence of periodic projective resolutions over certain finite-dimensional algebras which arise as stable endomorphism rings of Cohen-Macaulay modules. These algebras are in fact periodic, meaning that they have periodic projective resolutions as bimodules and thus periodic Hochschild cohomology as well. The goal, of this article is to generalize this construction of periodic algebras to the context of Iyama's higher AR-theory. Let C be a maximal (d - 1)-orthogonal subcategory of an exact Frobenius category B, and start by studying the projective resolutions of finitely presented functors on the stable category (C) under bar, over both (C) under bar and C. Under the assumption that (C) under bar is fixed by ohm(d), we show that ohm(d) induces the (2 +d)th syzygy on mod-(C) under bar. If C has finite type, i.e., if C = add(T) for a d-cluster tilting object T. then we show that the stable endomorphism ring of T has a quasi-periodic resolution over its enveloping algebra. Moreover, this resolution will be periodic if some power of ohm(d) is isomorphic to the identity on (C) under bar. It follows, in particular, that 2-C.Y.-tilted algebras arising as stable endomorphism rings of Cohen-Macaulay modules over curve singularities, as in the work of Burban, Iyama, Keller and Reiten (2008) [9], have periodic bimodule resolutions of period 4. (c) 2012 Elsevier Inc. All rights reserved.