A flat base change of a perfect morphism is perfect. See Tag 06C0.

A simplicial ring A_• is just a simplicial object in the category of rings. What is a simplicial module over A_•? Well it is a simplicial object in the category of systems (A, M, +, *, +, *) where A is a ring and M is an A-module (so the + and * are multiplication and addition on A and M respectively) such that forgetful functor to the category of rings gives back A_•.

Of course this is annoying. Better: A simplicial ring A_• is a sheaf on Δ (the category of finite ordered sets endowed with the chaotic topology). Then a simplicial module over A_• is just a sheaf of modules.

You can extend this to simplicial sheaves of rings over a site C. Namely, consider the category C x Δ together with the projection C x Δ —> C. This is a fibred category hence we get a topology on C x Δ inherited from C. Then a simplicial sheaf of rings A_• is just a sheaf of rings on C x Δ and we define a simplicial module over A_• as a sheaf of modules on C x Δ over this sheaf of rings. There is a derived category D(A_*) and a derived lower shriek functor

Lπ_! : D(A_•) ———-> D(C)

as discussed in Tag 08RV. Moreover, a map A_• —> B_• of simplicial rings on C gives rise to a morphism of ringed topoi, and hence a derived base change functor

D(A_•) ———-> D(B_•)

as well as a restriction functor the other way.

Why am I pointing this out? The reason is to use it for the following. If A —> B is a map of sheaves of rings and M is a B-module, then a priori the Atiyah class “is” the extension of principal parts

0 —> Ω_P•/A ⊗ M —> E —> M —> 0

over the polynomial simplicial resolution P_• of B over A. To get it in D(B) Illusie uses the base change along the map P_• —> B. I was worried that we’d have to introduce lots of new stuff in the Stacks project to even define this, but all the nuts and bolts are already there. Cool!

PS: Warning! The category D(A_•) is not the same as the category D_•(A_•) defined in Illusie.

Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let b : S′ —> S be the blowing up of Z in S. Let g : X —> Y be an affine morphism of schemes over S. Let F be a quasi-coherent sheaf on X. Let g′ : X ×_S S′ —> Y ×_S S′ be the base change of g. Let F′ be the strict transform of F relative to b. Then g′_∗F′ is the strict transform of g_∗F. See Tag 080G.

This tag has one of the densest initial trees in the project:

Huge thanks to Pieter Belmans who did an enormous amount of work coding the stacks project website.

Please visit the new version of the website and play around. A major new feature is the dynamic creation of graphs vizualing the logical connections between results in the stacks project, for each and every result. Here is an example

Edit: To view a graph, browse the project online, choose a chapter, choose a section, choose a lemma, and then click on one of the three types of graphs.

If you have a comment, suggestion, etc then please come back here. If you find a bug in the operation of the website, then either leave a comment here, or email the maintainer, or create an issue on the github repository.

Enjoy!

Let C be a site. Let O’ —> O be a surjection of sheaves of rings whose kernel I is an ideal of square zero. Let F’ be an O’-module and set F = F’/I F’. The following are equivalent

1. F’ is a flat O’-module, and
2. F is a flat O-module and I ⊗_O F —> F’ is injective.

See Tag 08M4.

There exists a countable ring R and a projective module M which is a direct sum of countably many locally free rank 1 modules such that M is not locally free. See Tag 05WL.

Ok, so I’ve finally found (what I think will be) a “classical” solution to getting a deformation theory for the stack of coherent sheaves in the non-flat setting. I quickly recall the setting.

The problem: Suppose you have a finite type morphism X —> S of Noetherian algebraic spaces. Let A be a finite type S-algebra. Let F be a coherent sheaf on the base change X_A which is flat over A and has proper support over A. We want to write down some pseudo-coherent complex L on X_A such that for every surjection of S-algebras A’ —> A with square zero kernel I the ext groups

Extⁱ_XA(L, F ⊗_A I), i = 0, 1, 2

give infinitesimal automorphisms, infinitesimal defos, and obstructions.

Derived solution: If you know derived algebraic geometry, then you know how to solve this problem. I tried to sketch the approach in this remark and now I can answer the question formulated at the end of that remark as follows.

Namely, the question is to construct a complex L such that H⁰(L) = F and H^-2(L) = Tor₁^O_S(O_X, A) ⊗ F. The ingredient I was missing is a canonical map

c : L_XA/A —> Tor₁^O_S(O_X, A)[2]

You get this map quite easily from the Lichtenbaum-Schlessinger description of the cotangent complex (again, in terms of derived schemes, this follows as X_A is cut out in the derived base change by an ideal which starts with the Tor₁ sheaf sitting in cohomological degree -1, but remember that the point here is to NOT use derived methods). OK, now use the Atiyah class

F —> L_XA/A ⊗ F[1]

and compose it with the map above to get F —> Tor₁^O_S(O_X, A) ⊗ F[3]. The cone on this map is the desired complex L.

Yay!

PS: Of course, to actually prove that L “works” may be somewhat painful.

Let R —> A and R —> B be ring maps. In general there does not exist a functor T : D(B) —> D(B ⊗_R A) of triangulated categories such that a B-module M gives an object T(M) of D(B ⊗_R A) which maps to M ⊗^L_R A under the map D(B ⊗_R A) —> D(A). See Tag 08J2.

Let X be an algebraic stack. The following are equivalent

1. X is DM (see Tag 04YW and Tag 050D),
2. X is Deligne-Mumford (see Tag 03YO), and
3. there exists a scheme W and a surjective étale morphism W→X.

See Tag 06N3.

Let P be a property of morphisms of algebraic spaces. Assume

1. P is smooth local on the source,
2. P is smooth local on the target, and
3. P is stable under postcomposing with smooth morphisms: if f : X —> Y has P and Y —> Z is a smooth morphism then X —> Z has P.

Then P is smooth local on the source-and-target. See Tag 06FB.