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 Iyamaʼs 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 , over both and C. Under the assumption that is fixed by Ωd, we show that Ωd induces the (2+d)th syzygy on mod-. 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 Ωd is isomorphic to the identity on . 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.