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On the map induced on Hochschild homology of matrix factorization categories by the inclusion of a divisor

Ville Nordstrom

Abstract

Given a smooth variety $X$ over $\mathbb{C}$, a smooth divisor $i:Y\hookrightarrow X$ and a global function $f$ on $X$ which vanishes on $Y$ and on its critical locus we compute the map induced on Hochschild homology by the pushforward functor $i_*:D^b(Y)\to D^{abs}(MF(X,f))$ in terms of the Hochschild-Kostant-Rosenberg isomorphisms.

On the map induced on Hochschild homology of matrix factorization categories by the inclusion of a divisor

Abstract

Given a smooth variety over , a smooth divisor and a global function on which vanishes on and on its critical locus we compute the map induced on Hochschild homology by the pushforward functor in terms of the Hochschild-Kostant-Rosenberg isomorphisms.
Paper Structure (22 sections, 24 theorems, 94 equations)

This paper contains 22 sections, 24 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Summary of paper
  3. Notation and conventions
  4. Acknowledgments
  5. Background on cdg categories, modules and Hochschild invariants
  6. Cdg categories and functors
  7. Two kinds of Hochschild homology
  8. Pseudo equivalences and projective modules
  9. Pseudo functors between dg categories
  10. Presheaves of cdg categories
  11. Hochschild homology of category of matrix factorizations
  12. Affine matrix factorizations
  13. Global matrix factorizations
  14. Hochschild homology of $\text{vect}_{\textit{\v{C}}}(X,f)$
  15. The case $f=0$
  16. ...and 7 more sections

Key Result

Theorem 1

There is a commutative diagram in $D_{\mathbb{Z}/2}(\mathbb{C})$\begin{tikzcd} HH_\bullet(Y)\arrow{d}{\sim}\arrow{rrr}{HH_\bullet(i_*)}&&&HH_\bullet(X,f)\arrow{d}{\sim}\\ \Cech(\mathscr{U},\Omega_Y^\bullet)\arrow{rr}{\bar{\wedge}Td(-Y)^{-1}}&&\Cech(\mathscr{U},\Omega_Y^\bullet)\arrow{r}{\delta}&

Theorems & Definitions (52)

  • Theorem 1
  • Example 2
  • Example 3
  • Example 4
  • Remark 5
  • Proposition 6: Pol-Pol
  • Proposition 7: Pol-Pol, section 2.6
  • Example 8
  • Definition 9
  • Proposition 10
  • ...and 42 more