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.

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

Let X be an algebraic space. Let x ∈ |X|. If X is separated, locally Noetherian, and the dimension of the local ring of X at x is ≤1 (Definition Tag 04NA), then there exists an open subspace of X containing x which is a scheme. See Lemma Tag 0ADD.

Slogan: Separated algebraic spaces are schemes in codimension 1.

Let (C, O) be a ringed site. Given K,L,M in D(O) there is a canonical morphism RHom(L,M) ⊗ RHom(K,L) ⟶ RHom(K,M) in D(O). See Lemma Tag 0A98.

Slogan: Composition on RSheafHom.

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!

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.

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.

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

A regular local ring is a UFD. See Lemma Tag 0AG0.

Let (A,I) be a henselian pair. Set X = Spec(A) and Z = Spec(A/I). For any torsion abelian sheaf F on X_{e´tale} we have H^q_{e´tale}(X, F) = H^q_{e´tale}(Z, F|Z). See Theorem Tag 09ZI.

Slogan: Affine analogue of the proper base change theorem (due to Gabber; can also be found in a paper by Huber)