# The Gauss–Manin connection on the periodic cyclic homology

@article{Petrov2017TheGC, title={The Gauss–Manin connection on the periodic cyclic homology}, author={Alexander Petrov and Dmitry Vaintrob and V. Vologodsky}, journal={Selecta Mathematica}, year={2017}, volume={24}, pages={531-561} }

AbstractLet R be the algebra of functions on a smooth affine irreducible curve S over a field k and let $${A_{\bullet }}$$A∙ be a smooth and proper DG algebra over R. The relative periodic cyclic homology $$HP_* ({A_{\bullet }})$$HP∗(A∙) of $${A_{\bullet }}$$A∙ over R is equipped with the Hodge filtration $${\mathcal F}^{\cdot }$$F· and the Gauss–Manin connection $$\nabla $$∇ (Getzler, in: Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992… Expand

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#### References

SHOWING 1-10 OF 34 REFERENCES

CARTAN HOMOTOPY FORMULAS AND THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY

- 2002

such that m|ν=0 is the product on A. We will define a connection on the periodic cyclic bar complex of Aν for which the differential is covariant constant, thus inducing a connection on the periodic… Expand

On topological cyclic homology

- Mathematics
- 2017

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace… Expand

Bokstein homomorphism as a universal object

- Mathematics
- 2015

We give a simple construction of the correspondence between square-zero extensions $R'$ of a ring $R$ by an $R$-bimodule $M$ and second MacLane cohomology classes of $R$ with coefficients in $M$ (the… Expand

A künneth formula for the cyclic cohomology of ℤ/2-graded algebras

- Mathematics
- 1986

We define here Hochschild and cyclic (co)homology groups for Z/2-graded algebras. A definition of cyclic cohomology of such algebras over the complex numbers has already be given by Kastler [13-1 who… Expand

Cohomology of finite group schemes over a field

- Mathematics
- 1997

A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or… Expand

Geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case

- Mathematics
- 2006

Let X be a smooth projective curve over an algebraically closed field k of characteristic p>0. In this paper we explore the relation between algebraic D-modules on the moduli space $Bun_n$ of vector… Expand

Dyer-Lashof operations on Tate cohomology of finite groups

- Mathematics
- 2010

Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain… Expand

Co-periodic cyclic homology

- Mathematics
- 2015

Following an old suggestion of M. Kontsevich, and inspired by recent work of A. Beilinson and B. Bhatt, we introduce a new version of periodic cyclic homology for DG agebras and DG categories. We… Expand

Cyclic homology with coefficients

- Mathematics
- 2007

We propose a category which can serve as the category of coefficients for the cyclic homology HC ∗(A) of an associative algebra A over a field k. The construction is categorical in nature, and… Expand

Hochschild cohomology and Atiyah classes

- Mathematics
- 2010

Abstract In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the… Expand