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

Theorem of the day

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)

Lemma of the day

Let k be a field. Let G be a separated group algebraic space locally of finite type over k. There does not exist a nonconstant morphism f : P^1_k → G over Spec(k). See Lemma Tag 0AEN.

Slogan: no (complete) rational curves on groups.

Lemma of the day

Let X be a quasi-compact and quasi-separated scheme. Let U, V be quasi-compact disjoint open subschemes of X. Then there exist a (U ∪ V)-admissible blowup b : X′ → X such that X′ is a disjoint union of open subschemes X′ = X′1 ⨿ X′2 with b^{−1}(U) ⊂ X′1 and b^{−1}(V) ⊂ X′2. See Lemma Tag 080P.

Slogan: separate irreducible components by blowing up.

Lemma of the day

Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let C be the full subcategory of (Sch/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y. Then the blowing up b : X′→X of Z in X is a final object of C. See Lemma Tag 0806.

Slogan: Universal property of blowing up

Lemma of the day

Let S be a scheme. Let {Xi → X}i ∈ I be an fppf covering of algebraic spaces over S. Assume I is countable (we can allow larger index sets if we bound the size of the algebraic spaces or if we don’t worry about set theoretic issues). Then any descent datum for algebraic spaces relative to {Xi → X}i ∈ I is effective. See Lemma Tag 0ADV.

Slogan: fppf descent data for algebraic spaces are effective.