# Cyclic homology of categories of matrix factorizations

@article{Efimov2012CyclicHO, title={Cyclic homology of categories of matrix factorizations}, author={Alexander I. Efimov}, journal={arXiv: Algebraic Geometry}, year={2012} }

In this paper, we will show that for a smooth quasi-projective variety over $\C,$ and a regular function $W:X\to \C,$ the periodic cyclic homology of the DG category of matrix factorizations $MF(X,W)$ is identified (unde Riemann-Hilbert correspondence) with vanishing cohomology $H^{\bullet}(X^{an},\phi_W\C_X),$ with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of $(\Omega_X^{\bullet},dW\wedge).$
One can show that the image of the…

## 27 Citations

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## References

SHOWING 1-10 OF 27 REFERENCES

Étale descent for hochschild and cyclic homology

- Mathematics
- 1991

AbstractIfB is an étale extension of ak-algebraA, we prove for Hochschild homology thatHH*(B)≅HH*(A)⊗AB. For Galois descent with groupG there is a similar result for cyclic homology:HC*≅HC*(B)G if…

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

- Mathematics
- 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…

Non-commutative Hodge structures: Towards matching categorical and geometric examples

- Mathematics
- 2011

The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas…

Triangulated categories of singularities and D-branes in Landau-Ginzburg models

- Mathematics
- 2003

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a…

Hochschild (co)homology of the second kind I

- Mathematics
- 2012

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG-categories. An…

Global matrix factorizations

- Mathematics
- 2011

We study matrix factorization and curved module categories for Landau-Ginzburg models (X,W) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these…

THE HODGE FILTRATION AND CYCLIC HOMOLOGY Charles Weibel

- Mathematics
- 2008

We relate the “Hodge filtration” of the cohomology of a complex algebraic variety X to the “Hodge decomposition” of its cyclic homology. If X is smooth and projective, HC (i) n (X) is the quotient of…

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

- Mathematics
- 2012

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of…