The cohomological Hall algebra of a surface and factorization cohomology

Source 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.