Support Varieties for Selfinjective Algebras

Abstract

Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true.

Varieties for Modules of Quantum Elementary Abelian Groups

“…We use the definition of (Hochschild) support varieties for modules of finite dimensional algebras given in [29] and developed further in [14]. Let…”

“…In case m = 1, identify Λ with k[t]/(t ℓ ). There is a periodic Λ e -free resolution of Λ: We must verify that Λ and H satisfy the properties required by the theory of support varieties defined via Hochschild cohomology in [14]: As Λ is local, it is an indecomposable algebra. The cohomology algebra H = H * (A, k) is a polynomial ring in m variables, each of degree 2 (4.0.4), so it is a commutative, noetherian, graded subalgebra of HH * (Λ).…”

‘Ext-Finite Modules for Weakly Symmetric Algebras With Radical Cube Zero’

“…If a selfinjective algebra A has a non-projective ext finite module there is no support variety theory for A-modules via Hochschild cohomology. This follows from Corollary 2.3 in [10], it shows that the finite generation conditions (Fg1, 2) in [10] (and (Fg) of [16]) must fail. That is, existence of ext-finite non-projective modules gives information about action of the Hochschild cohomology on ext algebras of modules.…”

Maximal 𝑛-orthogonal modules for selfinjective algebras

“…Such an algebra, if it exists, would have very unusual homological properties. For example, if for such algebra the Nakayama automorphism ν has finite order, then the finite generation properties Fg1 and Fg2 in [12] must fail, since those imply that Ω-periodicity is the same as having complexity one. 5.5.…”