Homology, Homotopy and Applications

Volume 23 (2021)

Number 1

Minimal models for monomial algebras

Pages: 341 – 366

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a18


Pedro Tamaroff (School of Mathematics, Trinity College Dublin, Ireland)


We give, for any monomial algebra $A$, an explicit description of its minimal model, which 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 Jöllenbeck, Welker and Sköldberg. We then show how this result can be used to obtain models for algebras with a chosen Gröbner basis, and briefly outline how to compute some classical homological invariants with it.


monomial algebra, minimal model, $A_\infty$-algebra, rewriting theory, higher structure

2010 Mathematics Subject Classification

16E05, 16E40, 16E45, 18G15

Copyright © 2020, Pedro Tamaroff. Permission to copy for private use granted.

Received 4 September 2020

Received revised 7 September 2020

Accepted 7 September 2020

Published 9 December 2020