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I would like to experiment with André-Quillen (co)homology. Especially for singular rings. A key problem is that the construction of a simplicial resolution of a ring seems to require a rather big effort [even in dimension 1]. In fact, I have not found any technique in the literature to construct "small" resolutions [except for maybe a method in Iyengar's notes for local complete intersections; but this does not help me in the cases I am interested)

Now my greatest hope would be that some computer algebra system constructs simplicial resolutions automatically?

Or, at least maybe someone knows a good reference where the explicit construction of manageable resolutions is discussed in detail?

Finally, so far on my search I found numerous papers on equipping free module resolutions with a dg algebra structure (à la Tate, Gulliksen). I must admit, it's not clear to me whether such resolutions (modulo the simplicial<->dg dictionary) can be used as simplicial (well, dg) resolutions for the purposes of André-Quillen (co)homology?

Weibel's book on homological algebra mentions an alternative way to compute AQ-cohomology (via sth called Barr's Theorem and summands of Hochschild homology). Of course, if some computer algebra system can automatically compute for example such summands of Hochschild homology, that would also solve my problem.

Thank you very much for your help. Any sort of answer will be appreciated.

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In characteristic zero the theorem of Barr that you mention identifies AQ cohomology with Harrison cohomology, which should in principle be computable from the Hochschild as it consists of those classes represented by cocycles satisfying a certain generalized symmetry condition. – Yemon Choi May 12 '12 at 17:39

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