Lemma of the day

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 gF. See Tag 080G.

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


A new website for the stacks 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.


Lemma of the day

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.


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 XA which is flat over A and has proper support over A. We want to write down some pseudo-coherent complex L on XA such that for every surjection of S-algebras A’ —> A with square zero kernel I the ext groups

ExtiXA(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) = Tor1OS(OX, A) ⊗ F. The ingredient I was missing is a canonical map

c : LXA/A —> Tor1OS(OX, 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 XA 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 —> LXA/A ⊗ F[1]

and compose it with the map above to get F —> Tor1OS(OX, A) ⊗ F[3]. The cone on this map is the desired complex L.


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

Lemma of the day

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 ⊗LR A under the map D(B ⊗R A) —> D(A). See Tag 08J2.

Lemma of the day

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.