Given an augmented graded associative $K$ -algebra $A$, we can construct a free resolution of $K$ given by $K_A$ modules which gives nice combinatorial informations about the homology classes of the algebra $A$. This type of resolution is first constructed by David Anick which was given in his paper
"On the Homology of Associative Algebras", Transactions of the American Mathematical Society, vol. 296, no. 2, (Aug 1986), pp 641-659, AMS.

One can also find a nice description of Anick's resolution in the book $\textit{Combinatorial and Asymptotic Methods in Algebra}$, EMS, Algebra VI by V.A. Ufnarovskij.

I would like to know what is the geometric significance of defining such resolution? It may be true that for a general associative $K$ algebra $A$, it is difficult to tell about some geometric significance but for some specific algebras over a given field $K$, it maybe possible. If anyone knows some example with geometric interpretation explained properly please let me know.

## 1 Answer

You might look at Ken Brown's https://www.math.cornell.edu/~kbrown/scan/1992.0000.0137.pdf where he gives a geometric proof of the resolution for groups given by a complete rewriting system and the reinterprets it algebraically for monoids. The same sort of thing can be done for the two-sided bar construction for algebras with a Grobner-Shirshov basis to get a geometric view on it all.

Nowadays, people call what Ken Brown did discrete Morse theory and what he did at the chain level discrete Morse theory for chain complexes and I think you can find proofs of the Anick resolution using this language as well, but it is basically the same as what Ken Brown does.