There exists a local ring R with a unique prime ideal and a nonzero ideal I ⊂ R which is a flat R-module. See Section Tag 05FZ.
Slogan: Zero dimensional ring with flat ideal.
There exists a local ring R with a unique prime ideal and a nonzero ideal I ⊂ R which is a flat R-module. See Section Tag 05FZ.
Slogan: Zero dimensional ring with flat ideal.
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!
Let F be a predeformation category which has a versal formal object. Then
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.
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
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
Then I is generated by a regular sequence and J/I is generated by a regular sequence. See Lemma Tag 09PW.
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:
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.