Periodicity of d-cluster-tilted algebras

TL;DR

This paper generalizes the concept of periodicity in algebras, extending it to d-cluster-tilted algebras within higher Auslander-Reiten theory, revealing conditions for periodic resolutions and their implications.

Contribution

It introduces a framework for periodic resolutions of stable endomorphism rings in higher AR-theory, expanding the understanding of periodicity in d-cluster-tilted algebras.

Findings

Stable endomorphism rings have quasi-periodic resolutions.

Periodic bimodule resolutions of period 4 for certain 2-Calabi-Yau tilted algebras.

Conditions for periodicity involve higher syzygy functors and stable categories.

Abstract

It is well-known that any maximal Cohen-Macaulay module over a hypersurface has a periodic free resolution of period 2. Auslander, Reiten and Buchweitz 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. We start by considering projective resolutions of functors on a maximal (d-1)-orthogonal subcategory C of an exact Frobenius category B. If C is fixed by the d-th syzygy functor of B, then we show that this d-th syzygy functor induces the (2+d)-th syzygy on the category of finitely presented functors on the stable category of C. 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 higher syzygy functor is isomorphic to the identity on the stable category of C. 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 have periodic bimodule resolutions of period 4.