The cohomological Hall algebra of a surface and factorization cohomology

For a smooth quasi-projective surface S over 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 nonderived 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 = 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.