Benjamin Briggs: Syzygies of the cotangent complex

Abstract

The cotangent complex is an important but difficult to understand object associated to a map of commutative rings (or schemes). It is connected with some easier to compute invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies of the cotangent complex. In general, these syzygies are known as the higher cotangent modules. Quillen conjectured that, for maps of finite flat dimension, the cotangent complex can only be bounded for locally complete intersection homomorphisms. This was proven by Avramov in 1999, and I will explain how to get a new proof by paying attention to the cotangent modules, as well as how to simultaneously prove a conjecture of Vasconcelos on the conormal module. The arguments rely on an object known as the homotopy Lie algebra, whose use was largely pioneered in Stockholm (starting with Jan-Erik Roos and his community). This is all joint work with Srikanth Iyengar.