📝 Original Info
- Title: Formality of the homotopy calculus algebra of Hochschild (co)chains
- ArXiv ID: 0807.5117
- Date: 2008-08-01
- Authors: Researchers from original ArXiv paper
📝 Abstract
The Kontsevich-Soibelman solution of the cyclic version of Deligne's conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^*(A), C_*(A)) ``Hochschild cochains + Hochschild chains'' of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^*(O_X), C_*(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author.
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Deep Dive into Formality of the homotopy calculus algebra of Hochschild (co)chains.
The Kontsevich-Soibelman solution of the cyclic version of Deligne’s conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^(A), C_(A)) ``Hochschild cochains + Hochschild chains’’ of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^(O_X), C_(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author.
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arXiv:0807.5117v1 [math.KT] 31 Jul 2008
Formality of the homotopy calculus algebra of
Hochschild (co)chains
Vasiliy Dolgushev, Dmitry Tamarkin, and Boris Tsygan
To Mikhail Olshanetsky on the occasion of his 70th birthday.
Abstract
The Kontsevich-Soibelman solution of the cyclic version of Deligne’s conjecture and
the formality of the operad of little discs on a cylinder provide us with a natural homo-
topy calculus structure on the pair (C•(A), C•(A)) “Hochschild cochains + Hochschild
chains” of an associative algebra A. We show that for an arbitrary smooth algebraic
variety X over a field K of characteristic zero the sheaf (C•(OX), C•(OX)) of homotopy
calculi is formal. This result was announced in paper [29] by the second and the third
author.
Contents
1
Introduction
2
2
Preliminaries
3
2.1
(Co)operads and (co)algebras
. . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Hochschild (co)chain complexes . . . . . . . . . . . . . . . . . . . . . . . . .
9
3
The operads Ho (calc) , Ho (e2) , and Ho (Lie+
δ )
11
3.1
Description of the operads Ho (calc) and Ho (e2)
. . . . . . . . . . . . . . .
11
3.2
Description of the operad Ho (Lie+
δ ) . . . . . . . . . . . . . . . . . . . . . . .
15
4
The Kontsevich-Soibelman operad and the operad of little discs on a cylin-
der
19
4.1
The Kontsevich-Soibelman operad KS
. . . . . . . . . . . . . . . . . . . . .
19
4.2
The operad of little discs on a cylinder . . . . . . . . . . . . . . . . . . . . .
23
4.3
Required results from [23]
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.4
A useful property of the operad KS . . . . . . . . . . . . . . . . . . . . . . .
34
5
The homotopy calculus on the pair (C•
norm(A), Cnorm
•
(A)) .
37
6
Formality theorem
46
6.1
Enveloping algebra of a Gerstenhaber algebra
. . . . . . . . . . . . . . . . .
46
6.2
Sheaves of Hochschild (co)chains on an algebraic variety
. . . . . . . . . . .
48
6.3
Morita equivalence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
6.4
Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
1
7
Applications and generalizations
62
1
Introduction
The standard Cartan calculus on polyvector fields and exterior forms can be naturally ex-
tended to the Hochschild cohomology HH•(A, A) and the Hochschild homology HH•(A, A)
of an arbitrary associative algebra A [11], [24]. This calculus is induced by simple opera-
tions on Hochschild (co)chains, and the identities of this algebraic structure hold for these
operations up to homotopy.
The Kontsevich-Soibelman proof of the cyclic version of Deligne’s conjecture [23] and
the formality of the operad of little discs on a cylinder1 imply that this nice collection
of the operations on the pair (C•
norm(A), Cnorm
•
(A)) “(normalized) Hochschild cochains +
(normalized) Hochschild chains” can be extended to an ∞- or homotopy calculus structure.
This homotopy calculus structure on the pair (C•
norm(A), Cnorm
•
(A)) is a natural general-
ization of the homotopy Gerstenhaber algebra structure on the cochains C•
norm(A) . In paper
[13] we proved the formality of this homotopy Gerstenhaber algebra on C•
norm(A) for an ar-
bitrary regular commutative algebra A over a field K of characteristic zero. In this paper we
extend this result to the homotopy calculus algebra on the pair (C•
norm(A), Cnorm
•
(A)) .
As well as in [13] we also consider the situation when the algebra A is replaced by the
structure sheaf OX of a smooth algebraic variety X over the field K . More precisely, we con-
sider the homotopy calculus algebra on the pair (C•
norm(OX), Cnorm
•
(OX)) where C•
norm(OX)
and Cnorm
•
(OX) is, respectively, the sheaf of (normalized) Hochschild cochains and the sheaf
of (normalized) Hochschild chains of OX . In this paper we show that the sheaf of homotopy
calculi (C•
norm(OX), Cnorm
•
(OX)) is formal.
If A is an associative algebra (with unit), the pair (C•
norm(A), Cnorm
•
(A)) is also equipped
with an algebraic structure defined by a degree −1 Lie bracket on C•
norm(A) , a degree −1 Lie
module structure on Cnorm
•
(A) over C•
norm(A) , and Connes’ operator on Cnorm
•
(A) which is
compatible with the Lie module structure. In the paper we refer to such algebra structures
as ΛLie+
δ -algebra. (See Definition 4.)
In paper [31] the third author conjectured that if A is a regular commutative algebra
then this ΛLie+
δ -algebra structure on (C•
norm(A), Cnorm
•
(A)) is formal. This conjecture was
proved in [33] (at least in the case R ⊂K) by Willwacher who used the constructions of B.
Shoikhet [25] and the first author [12].
In general Ho(ΛLie+
δ )-part of the homotopy calculus structure on (C•
norm(A), Cnorm
•
(A))
derived from [23] may not coincide with the ΛLie+
δ -algebra on the pair (C•
norm(A), Cnorm
•
(A)) .
However, we show that this homotopy calculus algebra on (C•
norm(A), Cnorm
•
(A)) is quasi-
isomorphic to another homotopy calculus algebra on (C•
norm(A), Cnorm
•
(A)) whose Ho(ΛLie+
δ )-
part is the ordinary ΛLie+
δ -a
…(Full text truncated)…
Reference
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