# Depth of the zero module

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 ExtiR(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 pdR(M). Then we have depth(R) = pdR(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?

# Lemma of the day

Let F be a predeformation category which has a versal formal object. Then

1. F has a minimal versal formal object,
2. minimal versal objects are unique up to isomorphism, and
3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

See Tag 06T5.

# Cartier Equality

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.

# Lemma of the day

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.

# Lemma of the day

Let R be a ring. Let x ∈ R. Assume

1. R is a normal Noetherian domain,
2. R/xR is a Japanese domain,
3. R = lim R/xnR is complete with respect to x.

Then R is Japanese. See Tag 032P.

# Lemma of the day

Let A be a valuation ring. Let A→B be a ring map of finite type. Let M be a finite B-module.

1. If B is flat over A, then B is a finitely presented A-algebra.
2. If M is flat as an A-module, then M is finitely presented as a 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.

# Lemma of the day

Let A be a Grothendieck abelian category. Then

1. D(A) has both direct sums and products,
2. direct sums are obtained by taking termwise direct sums of any complexes,
3. products are obtained by taking termwise products of K-injective complexes.

See Tag 07D9.

# Lemma of the day

Let A —> B be a ring map. Assume

1. A ⊂ B is an extension of domains,
2. A is Noetherian,
3. A —> B is of finite type, and
4. the extension f.f.(A) ⊂ f.f.(B) is finite.

Let p ⊂ A be a prime such that dim(Ap) = 1. Then there are at most finitely many primes of B lying over p. See Tag 02MA.

# Lemma of the day

Let R —> S be a ring map. Let p ⊂ R be a prime. Assume that

1. there exists a unique prime q ⊂ S lying over p, and
2. either
1. going up holds for R —> S, or
2. going down holds for R —> S and there is at most one prime of S above every prime of R.

Then Sp=Sq. See Tag 00EA.

# Proposition of the day

Let X be a quasi-compact and separated algebraic space. Let U be an affine scheme, and let f : U —> X be a surjective étale morphism. Let d be an upper bound for the size of the fibres of |U| —> |X|. Then for any quasi-coherent OX-module F we have Hq(X,F)=0 for q ≥ d. See Tag 072B.

Note: This is interesting even when X is a scheme.