Khovanov-Rozansky homology via Cohen-Macaulay approximations and Soergel bimodules

TL;DR

This thesis simplifies the construction of Khovanov-Rozansky link homology by connecting matrix factorizations with Cohen-Macaulay approximations of Soergel bimodules, enabling more efficient computations.

Contribution

It introduces a novel approach linking Cohen-Macaulay approximations to Soergel bimodules in Khovanov-Rozansky homology, simplifying calculations and revealing new algebraic insights.

Findings

Matrix factorizations equal Cohen-Macaulay approximations of Soergel bimodules

Cohen-Macaulay approximation commutes with tensor products without cycles

Braid closure relates to Hochschild cohomology, with trivial approximations for certain bimodules

Abstract

This is the author's diploma thesis. We describe a simplification in the construction of Khovanov-Rozansky's categorification of quantum sl(n) link homology using the theory of maximal Cohen-Macaulay modules over hypersurface singularities and the combinatorics of Soergel bimodules. More precisely, we show that the matrix factorizations associated to basic MOY-graphs equal Cohen-Macaulay approximations of certain Soergel bimodules, and prove that taking Cohen-Macaulay approximation commutes with tensor products as long as the MOY-graph under consideration does not possess oriented cycles. It follows that the matrix factorization associated to a MOY-braid equals the Cohen-Macaulay approximation of the Soergel bimodule corresponding to the endofunctor on BGG-category O associated to the braid by Mazorchuk and Stroppel. This reduces certain computations in the category of matrix factorizations to known combinatorics of the Hecke-algebra. Finally, we describe braid closure as some kind of Hochschild cohomology and prove that the indecomposable Soergel bimodules corresponding to Young tableaux with more than n rows have trivial Cohen-Macaulay approximation, in analogy to the fact that the corresponding projective functors on category O vanish on restriction to parabolics with at most n parts.