# How to adapt exactly Brown's collapse from monoids to algebras?

In The geometry of rewriting systemsBrown describes a method to reduce the bar resolution of a monoid. Roughly:

• Given a simplicial set $$X$$ equipped with a scheme of collapse (a partition of the geometric realization $$lvert X rvert$$ in essential, redundant and reducible cells, which satisfy certain properties), it is possible to reduce $$lvert X rvert$$ in a smaller complex CW with one cell for each essential cell of $$lvert X rvert$$.
• A monoid $$M$$ presented with a complete rewrite system (the set of relations $$R$$ ends Church-Rosser) induces a schema of collapse on the simplicial set $$BM$$.
• the $$n$$-the cells of $$lvert BM rvert$$ are in correspondence with the generators of $$B_n$$ in the standard bar resolution of $$M$$. In particular, everyone $$B_n$$ can be collapsed in the same way that $$lvert BM rvert$$ is.
• Yes $$M$$ has a good set of normal forms of finite type ($$M$$ has a finished presentation $$(S, R)$$ and $$R$$ ends Church-Rosser), and then the classification space $$lvert BM rvert$$ can be reduced to a finite CW complex
• Under the assumptions on the last point, we can also reduce the resolution of the bar of $$M$$ in one where all $$B_n$$ are finished generated. In particular, $$M$$ is of type $$(FL) _ { infty}$$.

The states of Brown (that's me pointing out):

The method used in this section works, without essential change, if the ring $$mathbb {Z}[M]$$ is replaced by an increased arbitrariness $$k$$-algebra $$A$$ who comes equipped with a presentation satisfying the conditions of the Bergman Diamond Lemma (The diamond lemma for ring theory, Theorem 1.2). Right here $$k$$ can be any commutative ring. We start with the standard bar resolution $$C$$ of $$k$$ more than $$A$$, and we get a quotient resolution $$D$$with a generator for each "essential" generator of the bar resolution. In particular, we retrieve the theorem from Anick (On the homology of associative algebras, Theorem 1.4).

Allow me to state here the conditions of the diamond lemma:

Theorem: Let $$S$$ to be a system of reduction for a free associative algebra $$k langle X rangle$$ (a subset of $$langle X rangle$$ times k langle X rangle), and $$leq$$ a partial order of semi-group on $$langle X rangle$$, compatible with $$S$$and having a decreasing string condition. Then…

I do not understand two aspects of this adaptation:

• The conditions of the diamond lemma are stated for 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 $$k langle X rangle$$ satisfy the conditions, but what about the ideal $$I$$? I guess this is related to the reduction system, but how exactly?
• What would be the essential generators of standardized bar resolution? In the monoid context, they come from the essential cells of the monoid's classification space. In algebra mode, we no longer have this tool (unless we generalize the classification of spaces for internal monoids with respect to monoidal categories, which, in my opinion, is not the case).
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