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Selecta Mathematica

, Volume 20, Issue 3, pp 885–928 | Cite as

Lefschetz type formulas for dg-categories

  • Alexander PolishchukEmail author
Article

Abstract

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230–256, 2012) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngô (Bull Math Sci 1(1):129–199, 2011). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity \({\varvec{w}}\). We apply this formula to get some restrictions on the Betti numbers of a \({\mathbb Z}/2\)-equivariant module over \(k[[x_1,\ldots ,x_n]]/({\varvec{w}})\) in the case when \({\varvec{w}}(-x)={\varvec{w}}(x)\).

Keywords

dg-category dg-functor Hochschild homology Matrix factorization 

Mathematics Subject Classification (1991)

Primary 16E40 Secondary 18E30 14A22 14B05 

Notes

Acknowledgments

I am grateful to Luchezar Avramov and David Eisenbud for helpful discussions.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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