Let X be an algebraic stack. The following are equivalent
- X is DM (see Tag 04YW and Tag 050D),
- X is Deligne-Mumford (see Tag 03YO), and
- 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
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
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.
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?
Let F be a predeformation category which has a versal formal object. Then
See Tag 06T5.
Let K/k be a finitely generated field extension. Then ΩK/k and H1(LK/k) are finite dimensional and trdegk(K) = dimK ΩK/k – dimK H1(LK/k). See Tag 07E1.
Let A —> B be a ring map such that B ⊗A B —> B is flat. Let N be a B-module. If N is flat as an A-module, then N is flat as a B-module. See Tag Tag 092C.
Let R be a ring. Let x ∈ R. Assume
Then R is Japanese. See Tag 032P.
Let A be a valuation ring. Let A→B be a ring map of finite type. Let M be a finite B-module.
See Tag 053E.
PS: Much more is true, see the this chapter in the stacks project. The proof of the lemma above however is quite easy.