Homotopical and effective methods for associative algebras

Abstract

This thesis contains four main chapters based on four different papers. In the third chapter, we solve the problem of computing the minimal model of an arbitrary associative monomial algebra. Our methods are combinatorial and depend on a detailed analysis of a homotopy retract datum on the bar resolution of the trivial module obtained from the algebraic discrete Morse theory of Jöllenbeck Welker. In it, we also explain how the bimodule resolution of Bardzell can be obtained as a by-product of our resolution through a twisting cochain argument, and give some interesting examples of vanishing and non-vanishing patterns for higher multiplications of homotopy associative algebras.

The fourth chapter centers on a homotopically invariant effective computation of the Tamarkin Tsygan calculus of an associative algebra: we construct out of each cofibrant resolution of an associative algebra a pair of complexes and show they are ∞-quasi-isomorphic to the standard pair of Hochschild complexes defining the calculus structure on this algebra, and give computational examples at the end of the chapter. At the same time, we prove that the coloured operad defining Tamarkin Tsygan calculi is inhomogenous Koszul, which is essential for our proof to go through.

The fifth chapter is based on joint work with Vladimir Dotsenko and Vincent Gélinas. In it, we construct, through the use of Anick chains and perfect paths in a Gorenstein monomial algebra Λ, a monogenic polynomial subalgebra k[x] of the A∞-centre of the Yoneda algebra Ext_Λ(k;k) exhibiting this higher centre and the Yoneda algebra itself as module finite over k[x], ultimately showing that Λ satisfies the FG conditions of Snashall Solberg through the methods of Briggs and Gélinas.

The sixth and last chapter, based on joint work with Vladimir Dotsenko, presents a new way to interpret and prove the celebrated Diamond Lemma of Bergman from the viewpoint of homotopical algebra, while at the same time providing a general framework to state and prove the respective result for other algebraic structures. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman s condition of "resolvable ambiguities" becomes the first non-trivial component of the Maurer Cartan equation in the corresponding tangent complex.