Support varieties for selfinjective algebras

Abstract

Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg 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.

Cited by in corpus (19)

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• Multiplicative structures for Koszul algebras
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• On support varieties for Lie superalgebras and finite supergroup schemes
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• Finite generation for Hochschild cohomology of Gorenstein monomial algebras
• Homological epimorphisms, recollements and Hochschild cohomology - with a conjecture by Snashall-Solberg in view
• (Co)homology of quantum complete intersections
• Vanishing of self-extensions over symmetric algebras
• Bounds on cohomological support varieties
• Ext-finite modules for weakly symmetric algebras with radical cube zero
• Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions
• Representations of monomial algebras have poly-exponential complexities
• On the representation dimension of skew group algebras, wreath products and blocks of Hecke algebras
• Reduction techniques of singular equivalences
• Examples of support varieties for Hopf algebras with noncommutative tensor products
• On l-parabolic Hecke algebras of symmetric groups
• Hochschild cohomology, finiteness conditions and a generalisation of $d$-Koszul algebras
• A combinatorial characterisation of d-Koszul and (D,A)-stacked monomial algebras that satisfy (Fg)