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Let $X$ be a smooth algebraic variety over a field $\mathbb{k}$ of characteristic $0$. The analogue of the HKR Theorem here is this:
(Swan, Yekutieli). There is a canonical isomorphism
Here $\mathcal{T}_X$ is the tangent sheaf of $X$, and $X$ is embedded diagonally in $X^2$.
This is a consequence of the following result. Let $\mathcal{C}_{cd, X}$ be the sheaf of continuous Hochschild cochains of $X$. It is a bounded below complex of quasi-coherent $\mathcal{O}_X$-modules.
(Yekutieli).
There is a canonical isomorphism
in the derived category of $\mathcal{O}_X$-modules.
There is a canonical quasi-isomorphism of complexes of sheaves
Therefore there is a canonical isomorphism
in the derived category of $\mathcal{O}_X$-modules.
The relation to deformation quantization is this: $\mathcal{C}_{cd, X}$ is a shift by $1$ of the sheaf of $\mathcal{D}_{poly, X}$ of polydifferential operators (viewed only as a complex of quasi-coherent $\mathcal{O}_X$-modules). Similarly, $\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$ is the shift by $1$ of the sheaf $\mathcal{T}_{poly, X}$ of polyvector fields. Thus item 2 in the theorem above says that there is a canonical $\mathcal{O}_X$-linear quasi-isomorphism
Trying to replicate the global formality theorem of Kontsevich, one would like to upgrade this to an $\mathrm{L}_{\infty}$ quasi-isomorphism. However, it seems that in general this cannot be done directly, but only after a suitable resolution.
Here is our result. (See also Van den Bergh.) Any quasi-coherent sheaf $\mathcal{M}$ on $X$ admits a canonical flasque resolution called the mixed resolution:
This βmixesβ the jet resolution with the Cech resolution (corresponding to an affine open covering of $X$ that we supress). In particular we get quasi-isomorphisms of sheaves of DG algebras
and
(Yekutieli). There is an $\mathrm{L}_{\infty}$ quasi-isomorphism
whose $1$-st order term commutes with the HKR quasi-isomorphism above. It is independent on choices up to homotopy.
A Poisson deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of Poisson $\mathbb{k}[[\hbar]]$-algebras on $X$, with an isomorphism $\mathbb{k} \otimes_{\mathbb{k}[[\hbar]]} \mathcal{A} \cong \mathcal{O}_X$ called an augmentation. Likewise an associative deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of associative unital (but noncommutative) $\mathbb{k}[[\hbar]]$-algebras on $X$, with an augmentation to $\mathcal{O}_X$.
Theorem 3 implies:
(Yekutieli). Assume that the cohomology groups $\operatorname{H}^{1}(X, \mathcal{O}_X)$ and $\operatorname{H}^{2}(X, \mathcal{O}_X)$ vanish. Then there is a canonical bijection
A proof of this theorem when $X$ is affine is here. For the full statement see this paper.
For twisted (or stacky) deformations there is a corresponding (but much more difficult to state and prove). See the paper and the survey.
hello $x_2^3$ | *hello* $x_2^3$ |
hello $x_2^3$ | *hello* $x_2^3$ |
hello $x_2^3$ | *hello* $x_2^3$ |
hello $x_2^3$ | *hello* $x_2^3$ |
type theory | category theory | |
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syntax | semantics | |
natural deduction | universal construction | |
dependent sum type | dependent sum |
type theory | category theory | |
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syntax | semantics | |
natural deduction | universal construction | |
dependent sum type | dependent sum |
?? Grothendieck inequality $<>$ ***