Support varieties and Hochschild cohomology rings

Abstract

We define a support variety for a finitely generated module over an artin algebra Λ over a commutative artinian ring k, with Λ flat as a module over k, in terms of the maximal ideal spectrum of the algebra HH*(Λ) of Λ. This is modelled on what is done in modular representation theory, and the varieties defined in this way are shown to have many of the same properties as those for group rings. In fact the notions of a variety in our sense and those for principal and non‐principal blocks are related by a finite surjective map of varieties. For a finite‐dimensional self‐injective algebra over a field, the variety is shown to be an invariant of the stable component of the Auslander–Reiten quiver. Moreover, we give information on nilpotent elements in HH*(Λ), give a thorough discussion of the ring HH*(Λ) on a class of Nakayama algebras, give a brief discussion of a possible notion of complexity, and make a comparison with support varieties for complete intersections. 2000 Mathematics Subject Classification 16E40, 16G10, 16P10, 16P20 (primary), 13D03, 16G70, 20J06 (secondary).

Quillen stratification for Hochschild cohomology of blocks

“…In Linckelmann's paper [16], he also deals more generally with the support varieties of modules over a block B, defined via the block cohomology LH * (B). An analogous theory defined via the Hochschild cohomology HH * (B) was developed in the unpublished work of Siegel [21], some of which has now been done in greater generality by Snashall and Solberg [24] The analogous result for ordinary cohomology is the well-known Alperin-Evens Theorem ( [3]). …”

“…There are a number of recent papers that use Hochschild cohomology to study various types of finite-dimensional algebras and their modules, for example see [10,13,24]. Theories of support varieties for modules have been built from Hochschild cohomology [21,24] and from Linckelmann's block cohomology [16].…”

Orbit algebras that are invariant under stable equivalences of Morita type

“…They were studied by several authors (see [3,7,8,10,11,19,20]). A general reason to study these algebras is the local analysis of the endofunctor F .…”

On the vanishing of cohomology in triangulated categories

“…Details concerning the following can be found in [30] and [31]. Let k be a field and Λ a finite dimensional k-algebra, and denote the enveloping algebra Λ ⊗ k Λ op of Λ by Λ e .…”