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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 18th 2012

started syzygy.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeSep 18th 2012

That looks good. I suggest that the other ’non-linear’ entries are called homotopy syzygies, homological syzygies and then probably higher homoty syzygies, If that is ok with you, then there is a clear distinction and we can avoid unnecessary work later on.

• CommentRowNumber3.
• CommentAuthorTim_Porter
• CommentTimeSep 18th 2012
• (edited Sep 18th 2012)

I added a reference to Eisenbud’s book as well at syzygy. There is also the topic of syzygies for polynomial equations and in Grobner base theory. A useful reference for this is Cox, Little, O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, see here.

I do not see where that could be usefully put at present, so will just give the link here.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeSep 21st 2012

Thanks, now I understand what a syzygy is!

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeSep 21st 2012

Ah! but the fun starts with homotopical syzygies. … (I forget how to do an evil grin with a smiley.. drat!)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeSep 21st 2012

Do we mean the same by “homotopical syzygies”?

To me, the non-abelian version of a (cofibrant) resolution by syzygies and higher syzygies is a computad.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeSep 21st 2012
• (edited Sep 21st 2012)

It will essentially be equivalent to a computad. Loday used the term for an explicit cellular representative of an element in a homotopy group constructed along the way from a presentation of a group to the construction of a K(G,1). There is a neat paper by Kapranov and Saito that links higher syzygies for the usual presentation of the Steinberg group to Stasheff polytopes. There are also links with explicit resolutions of groups due to Deligne (and some others that I could look up). The main point seems to be finding explicit generators that have to be killed. These often have geometric significance. (Kapranov and Saito did not give any meta reason why the higher Stasheff polytopes might be occuring. They also had a link with some Morse function phenomenon.) The construction of a $K(G,1)$ for G the Steinberg group is linked to one of the standard definitions of the algebraic K-theory of rings.

The lowest dimensional homotopical syzygies are the identities amongst the relations. This allows some very nice illustrative examples to be given as well as being of interest to combinatorial group theorists.