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Search: MSC category 16E40 ( (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) )

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1. CJM 2014 (vol 66 pp. 902)

Levandovskyy, Viktor; Shepler, Anne V.
 Corrigendum to Example in "Quantum Drinfeld Hecke Algebras" The last example of the article contains an error which we correct. We also indicate some indices in Theorem 11.1 that were accidently transposed. Keywords:quantum/skew polynomial rings, noncommutative Groebner basesCategories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40

2. CJM 2013 (vol 66 pp. 874)

Levandovskyy, Viktor; Shepler, Anne V.
 Quantum Drinfeld Hecke Algebras We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a PoincarÃ©-Birkhoff-Witt property using the theory of noncommutative GrÃ¶bner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the PBW conditions. Keywords:skew polynomial rings, noncommutative GrÃ¶bner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomologyCategories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40

3. CJM 2002 (vol 54 pp. 1319)

Yekutieli, Amnon
 The Continuous Hochschild Cochain Complex of a Scheme Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$ of topological $\mathcal{O}_X$-modules, called the complete Hochschild chain complex of $X$. To any $\mathcal{O}_X$-module $\mathcal{M}$---not necessarily quasi-coherent---we assign the complex $\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ of continuous Hochschild cochains with values in $\mathcal{M}$. Our first main result is that when $X$ is smooth over $\mathbb{K}$ there is a functorial isomorphism $$\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R \mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M})$$ in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, where $X^2 := X \times_{\mathbb{K}} X$. The second main result is that if $X$ is smooth of relative dimension $n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps $\pi \colon \widehat{\mathcal{C}}^{-q} (X) \to \Omega^q_{X/ \mathbb{K}}$ induce a quasi-isomorphism $$\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/ \mathbb{K}} [q], \mathcal{M} \Bigr) \to \mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).$$ When $\mathcal{M} = \mathcal{O}_X$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology $$\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong \bigoplus_q \H^{i-q} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X} \mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M} \Bigr),$$ where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf. Keywords:Hochschild cohomology, schemes, derived categoriesCategories:16E40, 14F10, 18G10, 13H10
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