Vertex and source determine the block variety of an indecomposable module

AbstractThe block variety VG,b(M) of a finitely generated indecomposable module M over the block algebra of a p-block b of a finite group G, introduced in (J. Algebra 215 (1999) 460), can be computed in terms of a vertex and a source of M. We use this to show that VG,b(M) is connected, and that every closed homogeneous subvariety of the affine variety VG,b defined by block cohomology H*(G,b) (cf. Algebras Rep. Theory 2 (1999) 107) is the variety of a module over the block algebra. This is analogous to the corresponding statements on Carlson's cohomology varieties in (Invent. Math. 77 (1984) 291)

Quillen stratification for block varieties

AbstractThe classical results on stratifications for cohomology varieties of finite groups and their modules due to Quillen (Ann. Math. 94 (1971) 549–572; 573–602) and Avrunin–Scott (Invent. Math. 66 (1982) 277–286) carry over to the varieties associated with finitely-generated modules over p-blocks of finite groups, introduced in Linckelmann (J. Algebra 215 (1999) 460–480)

Tate Duality and Transfer for Symmetric Algebras Over Complete Discrete Valuation Rings

We show that dualising transfer maps in Hochschild cohomology of symmetric algebras over complete discrete valuations rings commutes with Tate duality. This is analogous to a similar result for Tate cohomology of symmetric algebras over fields. We interpret both results in the broader context of Calabi-Yau triangulated categories

On the Hilbert series of Hochschild cohomology of block algebras

We show that the degrees and relations of the Hochschild cohomology of a p-block algebra of a finite group over an algebraically closed field of prime characteristic p are bounded in terms of the defect groups of the block and that for a fixed defect d, there are only finitely many Hilbert series of Hochschild cohomology algebras of blocks of defect d. The main ingredients are Symondsʼ proof of Bensonʼs regularity conjecture and the fact that the Hochschild cohomology of a block is finitely generated as a module over block cohomology, which is an invariant of the fusion system of the block on a defect group

Quasi-hereditary twisted category algebras

We give a sufficient criterion for when a twisted finite category algebra over a field is quasi-hereditary, in terms of the partially ordered set of L-classes in the morphism set of the category. We show that this is a common generalisation of a long list of results in the context of EI-categories, regular monoids, Brauer algebras, Temperley–Lieb algebras, partition algebras

The indecomposability of a certain bimodule given by the Brauer construction

Broué’s abelian defect conjecture [3, 6.2] predicts for a p-block of a finite group G with an abelian defect group P a derived equivalence between the block algebra and its Brauer correspondent. By a result of Rickard [11], such a derived equivalence would in particular imply a stable equivalence induced by tensoring with a suitable bimodule - and it appears that these stable equivalences in turn tend to be obtained by “gluing” together Morita equivalences at the local levels of the considered blocks; see e.g. [4, 6.3], [8, 3.1], [12, 4.1], and [13, 5.6, A.4.1]. This note provides a technical indecomposability result which is intended to verify in suitable circumstances the hypotheses that are necessary to apply gluing results as mentioned above. This is used in [7] to show that Broué’s abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group