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.

Let R —> Λ be a ring map. Let I ⊂ R be an ideal. Assume that

1. I is nilpotent,
2. Λ/IΛ is a filtered colimit of smooth R/I-algebras, and
3. R —> Λ is flat.

Then Λ is a colimit of smooth R-algebras. See Tag 07CM.

Let k be a field. If X is smooth over Spec(k) then the set {x∈X closed such that k⊂κ(x) is finite separable} is dense in X. See Tag 056U.

What is the correct convention for the depth of the zero module over a local ring?

With our current conventions we have depth(0) = – ∞. This is because the depth of a module is the supremum of all the lengths of regular sequences (Tag 00LF) and the zero module has no regular sequence whatsoever (Tag 00LI).

In this erratum the authors say that the correct convention is to set the depth of the zero module equal to +∞. They say this is better than setting it equal to -1.

Hmm, I’m not so sure.

To help you think about the question I will list some results that use depth. Let M be a finite module over a Noetherian local ring R.

1. dim(M) ≥ depth(M), see Lemma Tag 00LK.
2. M is Cohen-Macaulay if dim(M) = depth(M), see Definition Tag 00N3.
3. depth(M) is equal to the smallest integer i such that Extⁱ_R(R/m, M) is nonzero, see Lemma Tag 00LW
4. Let 0 —> N′ —> N —> N′′→0 be a short exact sequence of finite R-modules. Then
   1. depth(N′′) ≥ min{depth(N), depth(N′) − 1}
   2. depth(N′) ≥ min{depth(N), depth(N′′) + 1}
5. Let M be a finite R-module which has finite projective dimension pd_R(M). Then we have depth(R) = pd_R(M) + depth(M). This is Auslander-Buchsbaum, see Tag 090V.

To me these examples suggest that -∞ isn’t a bad choice, especially if we define the Krull dimension of the empty topological space to be -∞ as well (again this makes sense as it is the supremum of an empty set of integers). And I just discovered that this is what Bourbaki does, so I’ll probably go with that.

But what do you think?