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I would like to know some applications of Anick's resolution in non-commutative algebras.

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Applications to what? What are you looking for? See these remarks in the site FAQ: mathoverflow.net/faq "MathOverflow is not an encyclopedia... MathOverflow is not the appropriate place to ask somebody to write an expository article for you." –  Yemon Choi Mar 31 '13 at 5:13
I think there should be a more focused question that you can ask... –  Yemon Choi Mar 31 '13 at 5:14
mathoverflow.net/questions/81415/… here is nice application given by Vladimir Dotsenko –  Alexander Chervov Apr 1 '13 at 10:27

2 Answers 2

The first paragraph of David Anick's paper, "On the Homology of Associative Algebras" (http://www.jstor.org/stable/2000383):

Let $k$ be a field and let $G$ be an associative augmented $k$-algebra. For many purposes one wishes to have a projective resolution of $k$ as a $G$-module. The bar resolution is always easy to define, but it is often too large to use in practice. At the other extreme, minimal resolutions may exist, but they are often hard to write down in a way that is amenable to calculations. The main theorem of this paper presents a compromise resolution. Though rarely minimal, it is small enough to offer some bounds but explicit enough to facilitate calculations. As it relies heavily upon combinatorial constructions, it is best suited for analyzing otherwise tricky algebras given via generators and relations.

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A question about the main object constructed in a paper which can be usefully answered by the first paragraph in that same paper is surely not a great question :-) –  Mariano Suárez-Alvarez Mar 31 '13 at 5:35
Hence my upvote of Yemon's comment on the question itself. I know it's bad form to answer a question which is (pretty clearly) not appropriate for MO, I just really liked this paragraph. :P –  Noah S Mar 31 '13 at 6:25
I have to say I really don't like "what are applications of X" questions, unless they come with specific qualifiers and signs of having thought about the question properly. As it happens, I think I once saw some paper using Anick-type resolutions to do explicit calculations for group cohomology, but I really don't feel like doing the work to hunt down the reference when the OP's question shows little sign of any work. –  Yemon Choi Mar 31 '13 at 8:36
@Yemon, I agree; see my previous comment. –  Noah S Mar 31 '13 at 18:21

In view of Yemon's reference to group cohomology, I would like to mention Graham Ellis' work on "Homological Algebra Programming". The key point is that he constructs free resolutions inductively together with a contracting homotopy: it is the latter that gives the computational aspect.

There is an explanation of some of this in Section 9.3 of the book Nonabelian algebraic topology, in terms of constructing a "home for a contracting homotopy", as against the more traditional "killing kernels", a method which is notably non algorithmic.

The spirit of this derives from Homological Perturbation Theory, in which also the homotopies are crucial.

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This is precisely how I always explain constructing resolutions to students. Nice to see it written down! –  Mariano Suárez-Alvarez Apr 1 '13 at 10:08
@Ronnie: thanks, it was exactly Ellis's work that I had (vaguely) in mind, having seen him give a talk on this at some conference –  Yemon Choi Apr 1 '13 at 19:02
@Mariano: Actually I think the first description appeared in my paper with Razak Salleh (with A. RAZAK SALLEH), `Free crossed crossed resolutions of groups and presentations of modules of identities among relations', LMS J. Comp. and Math. 2 (1999) 28-61, but using (the quite natural concepts of!) groupoids and crossed complexes, which as Graham felt probably currently puts off lots of people. The problem had bothered me since writing with Johannes Huebschmann our exposition on "Identities among relations" published in 1982. –  Ronnie Brown Apr 3 '13 at 14:19

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