The Continuous Hochschild Cochain Complex of a Scheme

By Amnon Yekutieli. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2001

110 citations recorded by the source corpus.

Abstract Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$ . There is a complex ${{\hat{C}}^{\cdot }}\left( X \right)$ 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 $Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,\mathcal{M} \right)$ 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 $$Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{C}^{\cdot }}\left( X \right),\,M \right)\,\cong \,\text{R}\,Hom_{{{\mathcal{O}}_{X}}^{2}}^{{}}\,\left( {{\mathcal{O}}_{X}},\,M \right)$$ in the derived category $\text{D}\left( \text{Mod}\,{{\mathcal{O}}_{{{X}^{2}}}} \right)$ , 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 $\text{ }\!\!\pi\!\!\text{ }\,\text{:}\,{{\hat{C}}^{-q}}\left( X \right)\,\to \,\Omega _{X/\mathbb{K}}^{q}$ induce a quasi-isomorphism $$Hom_{{{\mathcal{O}}_{X}}}^{{}}\,\left( \underset{q}{\mathop \oplus }\,\,\Omega _{X/\mathbb{K}}^{q}\,\left[ q \right],\,M \right)\,\,\to \,Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,M \right).$$ When $M\,=\,{{\mathcal{O}}_{X}}$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology $$\text{Ext}_{{{\mathcal{O}}_{{{X}^{2}}}}}^{i}\,\left( {{\mathcal{O}}_{X}}\,,\,M \right)\,\cong \,\underset{q}{\mathop \oplus }\,\,\,{{\text{H}}^{i-q}}\,\left( X,\,\left( \underset{{{\mathcal{O}}_{X}}}{\overset{q}{\mathop \Lambda }}\,\,{{T}_{X/\mathbb{K}}} \right)\,{{\otimes }_{{{\mathcal{O}}_{X}}}}\,M \right),$$ where ${{T}_{X/\mathbb{K}}}$ is the relative tangent sheaf.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/59053c7f39bb2a5d3ed609817344bbe8b2d75497

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Citations in the corpus

The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem (2006, background): “This construction has appeared in the literature; see [7, 14, 15].”
The relative Riemann-Roch theorem from Hochschild homology (2006, methodology): “Yekutieli [9] showed that applying the functor HomcontOX (−,OX) (OX is given the discrete topology) to Ĉ•(X) ∈ Ch−(OX − mod) yields D•poly(X) ∈ Ch+(OX − mod).”
Derived equivalences of K3 surfaces and orientation (2007, background): “The algebraic case was studied in [36], for the absolute case see also [28, 7].”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “ΦH K, associated to chclass(K) ∈ H•(X×Y;C). Proof. The HKR isomorphism in the aﬃne case is due to[HKR62]. In this generality, it is due to Swan [Swa96, Corollary 2.6] and Kontsevich [Kon03], see also [Yek02]. The preservation of the Chern character was stated in [Mar01] and proven as [Cal05, Theorem”
Deformation quantization in algebraic geometry (2003, background): “By Yekutieli [ Ye1 , Lemma 4.3] the C-linear map U1 : Tpoly(C) → G(C) is a quasi-isomorphism, and by Theorem 3.12 the map U1 : Tpoly(C) → D nor poly(C) is a quasi-isomorphism.”
Global matrix factorizations (2011, methodology): “ived category of the endomorphism dg algebra of the generator is enough to apply the derived Morita theory of [To¨e]. To reach our Hochschild cohomology result, we next take a detour into the work of [Yek] on the global Hochschild–Kostant–Rosenberg theorem, and we employ the calculations of [CalTu].”
Hochschild cohomology of abelian categories and ringed spaces (2004, methodology): “We recall that for a smooth scheme the Hochschild complex defined by Swan is quasi-isomorphic to the one defined by Kontsevich [23] in terms of differential operators (see [39, 42]).”
Hochschild cohomology and Atiyah classes (2007, methodology): “ch refers only to the cupproduct was proved by Kontsevich (see [5, Thm 5.1]). For the cupproduct one can use Swan’s deﬁnition of Hochschild cohomology [22] (1.3) HHi(X) = Exti X×X(OX,OX) as Yekutieli [28] shows that there is an isomorphism (1.4) Hi(X,Dpoly(OX)) →Exti X×X(OX,OX) which is compatible w”
When is the self-intersection of a subvariety a fibration? (2010, background): “Equivalently one has the HKR isomorphism [16], [19] ∆∗O∆ ∼= S(Ω 1 X[1]) which identifies W with the linear fibration TX[−1] over X, where TX is the tangent bundle of X. 0.5.”
Deformations and Fourier-Mukai transforms (2005, methodology): “The following theorem is due to Hochschild-Kostant-Rosenberg [8] , Kontsevich [11], Swan [17], and Yekutieli [19].”
What is geometry in quantum theory (2004, methodology): “At the same time, the formality theorem is also applied to quantization of several algebro-geometric structures such as algebraic varieties [50, 172].”
The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism (2003, methodology): “(This is nothing but a brief recounting of the results in [25], and the reader should consult [loc.”