66 research outputs found

    Vertex and source determine the block variety of an indecomposable module

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    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

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    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

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    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

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    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 SymondsĘĽ proof of BensonĘĽs 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
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