Fixed all issues

Hello again. This just a quick post to let you know that I worked through your recent comments and updated the Stacks project website with the latest version. So get back there and find more mistakes. Thank you all!

Lemma of the day

Let F be a predeformation category which has a versal formal object. Then

  1. F has a minimal versal formal object,
  2. minimal versal objects are unique up to isomorphism, and
  3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

See Lemma Tag 06T5.

What is fun about this lemma is that it produces a minimal versal object (as defined in Definition Tag 06T4) from a versal one without assuming Schlessinger’s axioms. If Schlessinger’s axioms are satisfied and one is in the classical case (see Definition Tag 06GC), then a minimal versal formal object is a versal formal object defined over a ring with minimal tangent space. This is discussed in Section Tag 06IL.

Lemma of the day

Let h : X –> Y, g : Y –> B be morphisms of algebraic spaces with composition f : X –> B. Let b ∈ |B| and let Spec(k) → B be a morphism in the equivalence class of b. Assume

  1. X → B is a proper morphism,
  2. Y → B is separated and locally of finite type,
  3. one of the following is true:
    1. the image of |X_k| → |Y_k| is finite,
    2. the image of |f|^{−1}({b}) in |Y| is finite and B is decent.

Then there is an open subspace B′ ⊂ B containing b such that X_{B′} → Y_{B′} factors through a closed subspace Z ⊂ Y_{B′} finite over B′. See Lemma Tag 0AEJ.

Slogan: Collapsing a fibre of a proper family forces nearby ones to collapse too.

Lemma of the day

Let (A,m) be a Noetherian local ring. Let I ⊂ J ⊂ A be proper ideals. Assume

  1. A/J has finite tor dimension over A/I, and
  2. J is generated by a regular sequence.

Then I is generated by a regular sequence and J/I is generated by a regular sequence. See Lemma Tag 09PW.

Here is the graph of this lemma
09PW

Comments on the Stacks project

We’ve recently been having a few frequent commenters on the Stacks project which is great; take a look at the most recent comments. They’ve pointed out not only trivial typos and other idiocies, but also actual errors which we’ve repaired with their help. (One of the goals of the Stacks project is to fix errors as soon as possible in every case.) Huge thanks to all!

Anyway, I encourage you to waste a few hours doing the same. Here are some tips:

  • If you refer to a different tag say ABCD in a comment use \ref{ABCD}.
  • If you have a comment about a particular lemma then please leave the comment on the page corresponding to the lemma not the enclosing section.
  • Please make suggestions as explicit as possible. For example, if you think a proof is missing an argument, try to say exactly what part is missing (even just write the argument in the comment if you can) and where to put it.
  • Precise outside references for material are VERY WELCOME, including references to your own work. We now have a working system for dealing with this which is very easy and quick for me to use, so bring it on.
  • Funny comments are welcome too!

Closed comments on older posts

This is just to let you know that I have closed comments on posts which are older than 30 days. The spam comments on those older posts are just too annoying to deal with. If you want to comment on an older post, you can just email me.

Lemma of the day

Let (C,O) be a ringed site. Let (K_n)_{n ∈ N} be a system of perfect objects of D(O). Let K= hocolim K_n be the derived colimit (Definition Tag 090Z). For E in D(O) we have

RHom(K, E) = Rlim E ⊗ L_n

where L_n = RHom(K_n, O) is the inverse system of duals. See Lemma Tag 0A0A.

Slogan: Trivial duality for systems of perfect objects.

Lemma of the day

Let X be a quasi-compact and quasi-separated algebraic space. Let T⊂|X| be a closed subset such that |X| – T is quasi-compact. The category D_{QCoh,T}(O_X) of complexes with quasi-coherent cohomology sheaves supported on T is generated by a single perfect object. See Lemma Tag 0AEC

This result for schemes is in the paper “Dimensions of triangulated categories” by Raphaël Rouquier