# The cohomological Hall algebra of a surface and factorization cohomology

## TL;DR

This paper constructs a cohomological Hall algebra for a surface using derived moduli stacks and explores its structure, including Hecke operators and the algebra of 0-dimensional sheaves, with explicit results for the affine plane.

## Contribution

It introduces a new algebraic structure on the Borel-Moore homology of coherent sheaves on surfaces, incorporating derived geometry and factorization cohomology techniques.

## Key findings

- R(S) is an enveloping algebra for S=A^2.
- The graded dimension of R(S) is computed for general surfaces.
- The algebra structure involves virtual pullbacks and derived moduli stack data.

## Abstract

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we study R(S), the Hecke algebra of 0-dimensional sheaves. For the flat case S=A^2, we show that R(S) is an enveloping algebra and identify it, as a vector space, with the symmetric algebra of an explicit graded vector space. For a general S we find the graded dimension of R(S), using the techniques of factorization cohomology.