The cohomological Hall algebra of a surface and factorization cohomology

Abstract

For a smooth quasi-projective surface $S$ over $\mathbb{C}$ 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 case $S={\mathbb{A}}^2$, 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.