Braided Hochschild cohomology and Hopf actions

Negron, Cris

doi:10.4171/jncg/308

Cited by 3 publications

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“…Remark 7.3. Theorem 7.2 can also be deduced from the fact that the cohomology HH • (X ) is a braided commutative algebra in the category of Yetter-Drinfeld modules for G. See [42].…”

Section: 2mentioning

confidence: 99%

The Hochschild cohomology ring of a global quotient orbifold

Negron

Schedler

2020

Advances in Mathematics

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We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T poly X on X, and is generated as a T poly X -algebra by the sum of the determinants det(N X g ) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne-Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruan's cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature. G .(1.3)In the unpublished manuscript [2], a version of the theorem above appears in the affine setting (in terms of the obvious explicit formula for cup product, rather than (i)-(iv); see Section 7.5). We note that the vector space identification (1.3) is due to Ginzburg-Kaledin in the affine case, and Arinkin-Cȃldȃraru-Hablicsek in general [3].The subalgebra of (iii) is denoted SA(X ) := ⊕ g∈G det(N X g ) throughout. We see from (ii), or more precisely Theorem 7.2 below, that this subalgebra provides all of the novel features of the Hochschild cohomology of the orbifold X , as compared with that of a smooth variety. Furthermore, as the points of the fixed spaces X g account for the points of X which admit automorphisms, this algebra also represents a direct contribution of the stacky points of X to the Hochschild cohomology.An explicit description of the multiplicative structure on the subalgebra SA(X ) is given in Theorem 7.5, and the multiplicative structure on the entire cohomology HH • (X ) is subsequently obtained from Theorem 7.2. The algebra identification (1.3) appears in Corollary 7.8.

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“…Remark 7.3. Theorem 7.2 can also be deduced from the fact that the cohomology HH • (X ) is a braided commutative algebra in the category of Yetter-Drinfeld modules for G. See [42].…”

Section: 2mentioning

confidence: 99%