In *The Geometry of Rewriting Systems*, Brown describes a method to collapse the bar resolution of a monoid. Roughly:

- Given a simplicial set $ X$ equipped with a collapsing scheme (a partition of the geometric realization $ \lvert X \rvert$ into essential, redundant and collapsible cells, which satisfy some properties) it is possible to collapse $ \lvert X \rvert$ into a smaller CW-complex with a cell for each essential cell of $ \lvert X \rvert$ .
- A monoid $ M$ presented with a complete rewriting system (the set of relations $ R$ is terminating Church-Rosser) induces a collapsing scheme on the simplicial set $ BM$ .
- The $ n$ -cells of $ \lvert BM \rvert$ are in correspondence with the generators of $ B_n$ in the normalized bar resolution of $ M$ . In particular, each $ B_n$ can be collapsed in a similar way that $ \lvert BM \rvert$ is.
- If $ M$ has a good set of normal forms of finite type ($ M$ has a finite presentation $ (S,R)$ and $ R$ is terminating Church-Rosser), then the classifying space $ \lvert BM \rvert$ can be collapsed into a finite CW-complex
- Under the assumptions on the last bullet, we can also collapse the bar resolution of $ M$ into one where all $ B_n$ are finitely generated. In particular, $ M$ is of type $ (FL)_{\infty}$ .

Brown states (emphasis mine):

The Method used in this section works, with no essential change, if the ring $ \mathbb{Z}[M]$ is replaced by an arbitrary augmented $ k$ -algebra $ A$ which comes equipped with a **presentation satisfying the conditions of Bergman’s diamond lemma** (*The diamond lemma for ring theory*, Theorem 1.2). Here $ k$ can be any commutative ring. One starts with the normalized bar resolution $ C$ of $ k$ over $ A$ , and one obtains a quotient resolution $ D$ , with **one generator for each “essential” generator of the bar resolution**. In particular, we recover Anick’s Theorem (*On the homology of associative algebras*, Theorem 1.4).

Let me state here the conditions of the diamond lemma:

**Theorem**: Let $ S$ be a reduction system for a free associative algebra $ k\langle X \rangle$ (a subset of $ \langle X \rangle$ \times k\langle X \rangle), and $ \leq$ a semigroup partial ordering on $ \langle X \rangle$ , compatible with $ S$ , and having descending chain condition. Then…

I can’t understand two aspects of this adaptation:

- The conditions of the diamond lemma are stated for the free associative algebra $ k\langle X \rangle$ . What does it mean for the presentation of $ A$ to satisfy these conditions? If we assume $ A \cong k\langle X \rangle/I$ we can interpret Brown’s sentence as $ k\langle X \rangle$ satisfying the conditions, but what about the ideal $ I$ ? I assume it is related somehow to the reduction system, but how exactly?
- What would be the essential generators of the normalized bar resolution? In the monoid setting, they originate from the essential cells of the classifying space of the monoid. In the algebra setting, we don’t have that tool anymore (unless we generalize classifying spaces for internal monoids over monoidal categories, which I don’t think is the case).