Minimal models for monomial algebras

Abstract

Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_\infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of J\"ollenbeck, Welker and Sk\"oldberg. We then show how this result can be used to obtain models for algebras with a chosen Gr\"obner basis, and briefly outline how to compute some classical homological invariants with it.