Hebrew University topology and geometry seminar

June 15, 2016
Special time: 1400-1535
Ross building, Seminar Room 70A

Ezra Getzler

Northwestern University

The derived Maurer-Cartan locus

Abstract: Derived algebraic geometry is a nonlinear analogue of homological algebra, in which one keeps track of syzygies among the relations among the defining equations of a variety, and higher analogues. It has important applications to intersection theory and enumerative geometry.
Quillen showed that cosimplicial schemes form a good model for derived geometry: in characteristic zero, the theory of cosimplicial schemes is equivalent to that of differential graded schemes, while in positive characteristic, the use of cosimplicial schemes is essential, since differential graded schemes do not behave well in that case. In this talk, however, I focus on the case of characteristic zero.
The most important construction in derived geometry is the derived Maurer-Cartan locus of a differential graded Lie algebra, which is a derived scheme having the Maurer-Cartan locus as its classical locus. This construction is in fact closely related to the Chevalley-Eilenberg complex of cochains for differential graded Lie algebras. In this talk, I present a natural cosimplicial scheme realizing the derived Maurer-Cartan locus, whose definition has the advantage of not requiring the theory of tensor products for its construction. For this reason, the construction may be expected to have applications in analytic geometry and other generalizations of algebraic geometry.
The talk is based on the following preprint: http://front.math.ucdavis.edu/1508.03007