# 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}
}
• A. Efimov
• Published 12 December 2012
• Mathematics
• arXiv: Algebraic Geometry
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…
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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
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
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
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