The cohomological Hall algebra of a surface and factorization cohomology
Mikhail Kapranov
Tokyo University, Kashiwa, Chiba, JapanÉric Vasserot
Université Paris Cité and Institut Universitaire de France, Paris, France
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Abstract
For a smooth quasi-projective surface over we consider the Borel–Moore homology of the stack of coherent sheaves on 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 , the Hecke algebra of -dimensional sheaves. For the case , we show that is an enveloping algebra and identify it, as a vector space, with the symmetric algebra of an explicit graded vector space. For a general , we find the graded dimension of , using the techniques of factorization cohomology.
Cite this article
Mikhail Kapranov, Éric Vasserot, The cohomological Hall algebra of a surface and factorization cohomology. J. Eur. Math. Soc. 25 (2023), no. 11, pp. 4221–4289
DOI 10.4171/JEMS/1264