Spectral sequences for the cohomology rings of a smash product

Negron, Cris

doi:10.1016/j.jalgebra.2015.02.021

Cited by 13 publications

(22 citation statements)

References 21 publications

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“…In Section 2, we produce a free resolution of the smash product algebra B#H; see Construction 2.5 and Theorem 2.10. This resolution is adapted from Guccione and Guccione [12]; Negron independently constructed a similar resolution [25]. Our resolution is used in the proof of Theorem 0.4, which is given in Section All of the examples of B#H above have nontrivial PBW deformations.…”

Section: Introductionmentioning

confidence: 99%

Poincaré–Birkhoff–Witt deformations of smash product algebras from Hopf actions on Koszul algebras

Walton

Witherspoon

2014

Algebra Number Theory

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Section: Introductionmentioning

confidence: 99%

Poincaré–Birkhoff–Witt deformations of smash product algebras from Hopf actions on Koszul algebras

Walton

Witherspoon

2014

Algebra Number Theory

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“…We see also that the cohomology is a graded object in Y D E E . One can check easily that the right E-action implicit in this Yetter-Drinfeld structure is exactly the right action considered in [32,25]. The coaction is induced by the coaction on B = A * E. Namely, by coinvariance of the elements in A, we will have for any f ∈ C • (A, B), ξ ∈ E * , and…”

Section: Identifications With Standard Hochschild Cohomology For Smasmentioning

confidence: 90%

Braided Hochschild cohomology and Hopf actions

Negron

2019

J. Noncommut. Geom.

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We show that the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second application we deduce some structural results for the usual Hochschild cohomology of a smash product, and discuss specific implications for finite group actions on smooth affine schemes.

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“…Section 4. ∆ 0 was introduced in [19, Definition 3.1] and applied to smash products whether to investigate Calabi-Yau duality (see [26,30]) or Hochschild cohomology (see [29]).…”

Section: Equivariant Modulesmentioning

confidence: 99%

Smash products of Calabi–Yau algebras by Hopf algebras

Meur

2019

J. Noncommut. Geom.

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Let H be a Hopf algebra and A be an H-module algebra. This article investigates when the smash product A♯H is (skew) Calabi-Yau, has Van den Bergh duality or is Artin-Schelter regular or Gorenstein. In particular, if A and H are skew Calabi-Yau, then so is A♯H and its Nakayama automorphism is expressed using the ones of A and H. This is based on a description of the inverse dualising complex of A♯H when A is a homologically smooth dg algebra and H is homologically smooth and with invertible antipode. This description is also used to explain the compatibility of standard constructions of Calabi-Yau dg algebras with taking smash products.Note that it is proved in [15, Theorem 2.3] that, when H is noetherian, H is Calabi-Yau if and only if S 2 is an inner automorphism and H is Artin-Schelter regular with trivial left homological integral.This article is based on the description of RHom Λ e (Λ, Λ e ). When S is invertible, there exists a dg A-bimodule D A which is H S 2 -equivariant in the sense of [30] (see Section 4) and such that D A ≃ RHom A e (A, A e ) in D(A e ). A suitable extension of D A is then isomorphic to RHom Λ e (Λ, Λ e ) in the following sense. See 5.5.1 for a general statement. See also [12,14,15,16,26,30,36,37] for previous results describing RHom Λ e (Λ, Λ e ) when A is a connected graded algebra and H is finitedimensional, semisimple or cocommutative.Proposition 2 (5.5.2). Let H be a Hopf algebra with Van den Bergh duality in dimension d. Let A be an H-module dg algebra. Assume that A is homologically smooth. Then, Λ is homologically smooth and RHom Λ e (Λ,The description of RHom Λ e (Λ, Λ e ) can be used to describe the deformed Calabi-Yau completions of Λ. Recall that HH n−2 (A) ≃ H 0 Hom A e (D A [n − 1], A[1]) when D A is cofibrant over A e , which is possible to assume. The following result was proved in [25] when H is the (semisimple) group algebra of a finite group.

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Spectral sequences for the cohomology rings of a smash product

Cited by 13 publications

References 21 publications

Poincaré–Birkhoff–Witt deformations of smash product algebras from Hopf actions on Koszul algebras

Poincaré–Birkhoff–Witt deformations of smash product algebras from Hopf actions on Koszul algebras

Braided Hochschild cohomology and Hopf actions

Smash products of Calabi–Yau algebras by Hopf algebras

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