So this is a follow up on the post about Burch’s theorem. Namely, I’ve just learned in the last month or so that the next case of this is in Eisenbud + Buchsbaum Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. It says that the resolution of a codimension 3 Gorenstein singularity R/I with R regular has a free resolution of the form
0 —> R —> R^n —f—> R^n —> R
where f is an alternating matrix and the other arrows are given by Pfaffians of f.
Moreover, if R/J is an almost complete intersection of grade 3, then R/J is linked to a Gorenstein R/I as above and a similar type of resolution can be obtained (results of Brown, kustin, etc).
OK, this is cool, very cool.
It seems completely clear that similarly to Burch’s theorem this implies that such a singularity is unobstructed, just as in the codimension 2 Cohen-Macaulay case. To be precise, as a simple consequence of the paper we obtain:
If R = k[[x, y, z]] and R —> S is an Artinian quotient ring such that either (1) S is Gorenstein, or (2) the kernel of R —> S is generated by at most 4 elements, then the miniversal deformation space of S is a power series ring over k.
Right…?
What I’d like is a reference to articles (with page and line numbers) stating exactly the above for (1) and (2). A generic reference to unobstructedness of determinantal singularities doesn’t count. I’ve googled and binged, but no luck so far. Can you help?
Or maybe this is just one of the innumerable results in our field that are so clearly true that you cannot formulate it in a paper as your paper will be immediately rejected?
I believe that Artin says that Burch’s theorem should also be attributed to Schaps (I believe they both discovered this theorem independently).
Ha, you previously commented on the history of Burch’s theorem here.
Yes.
You asked for a reference. Page 501, line 4 from the bottom of the following.
MR0743730 (85k:13015) Reviewed
Kustin, Andrew R.(1-SC); Miller, Matthew(1-TN)
Deformation and linkage of Gorenstein algebras.
Trans. Amer. Math. Soc. 284 (1984), no. 2, 501–534.
13D10 (13H10 14B07 14M05)
They, in turn, cite the following article of Juergen Herzog.
MR0579384 (81m:13012) Reviewed
Herzog, Jürgen
Deformationen von Cohen-Macaulay Algebren. (German)
J. Reine Angew. Math. 318 (1980), 83–105.
13D10 (13H10 14B07)
Unfortunately that article is behind a paywall, so I cannot give a page number and line number.
Jason, yes, I found that paper also, but i considered it not to be a reference of “exactly” the result. Because the discussion is more in the vein of “it is folklore that these types of singularities are unobstructed” and there is no precise statement of a theorem. Finally, in any case page 501 line -4 only covers case (1).
I also found and looked at the paper by Herzog. It seems to me that that paper talks about reduced CM algebras and it doesn’t prove (1) or (2) at all. Its goal is to prove that T^2 is *zero* for some algebras which is not true in the setting of the blog post. In fact it talks mostly about “strongly unobstructed” algebras that satisfy an even stronger condition. So the reference in Kustin-Miller to it is a bit weird.
So I was really hoping for an actual reference with an actual statement together with a short proof explaining how it follows from Eisenbud-Buchsbaum. The Kustin-Miller paper suggests there is another way to prove (1) and (2), namely by first proving such a ring is in the linkage class of a ci and then deducing it, but that is not what I want (although a precise statement and proof via that method would at least give me a reference).