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1. CJM Online first
Corrigendum to "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 bases Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 |
2. CJM Online first
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 cohomology Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 |
3. CJM 2002 (vol 54 pp. 1319)
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 categories Categories:16E40, 14F10, 18G10, 13H10 |