Let X be a quasi-compact and quasi-separated algebraic space such that for every quasi-coherent O_X-module F we have H¹(X, F) = 0. Then X is an affine scheme. See Tag 07V6.

There exist a zero dimensional local ring with a nonzero flat ideal. See Tag 05FZ.

Let X be a quasi-separated algebraic space. Let E be an object of D_QCoh(O_X). Let a ≤ b. The following are equivalent

1. E has tor amplitude in [a,b], and
2. for all F in QCoh(O_X) we have Hⁱ(E ⊗^L F)=0 for i not in [a,b].

See Tag 08IL.

Let A —> B be a local homomorphism of Noetherian local rings. Assume A —> B is formally smooth in the m_B-adic topology. Then A —> B is flat. See Tag 07NP.

PS: Of course much more is true, see Tag 07NQ.

Let A be a ring. Let I ⊂ J ⊂ A be ideals. If M is J-adically complete and I is finitely generated, then M is I-adically complete. See Tag 090T.

Let X be a scheme. Let a : X —> Spec(k₁) and b : X —> Spec(k₂) be morphisms from X to spectra of fields. Assume a,b are locally of finite type, and X is reduced, and connected. Then we have k′₁ = k′₂, where k′_i ⊂ Γ(X,O_X) is the integral closure of k_i in Γ(X,O_X). See Tag 04MK.

Let X —> Y —> Z be morphism of schemes. Let P be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that X —> Z has P and that {X —> Y} can be refined by an fppf covering of Y. Then Y —> Z is P. See Tag 06NB.

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?

Let (A_n) be an inverse system of abelian groups. The following are equivalent

1. (A_n) is zero as a pro-object,
2. lim A_n = 0 and R¹lim A_n = 0 and the same holds for ⨁ _{i ∈ N} (A_n).

See Tag 091C.

Let A be a Noetherian ring and I ⊂ A an ideal. For every n let M_n be a flat A/I^n-module. Let M_{n + 1} —> M_n be a surjective A-module map. Then the inverse limit M =lim M_n is a flat A-module (see Tag 0912).