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?

7 thoughts on “Depth of the zero module

  1. I am not sure what should be the depth, but I know that the codepth must be negative infinity. The codepth is dimension minus depth. Auslander proves that codepth is upper semicontinuous (at least in the finitely presented setting), but only with the convention that the codepth of the zero module is negative infinity, cf. EGA IV.6.11.

      • Wait! I’m confused as in EGA, O Definition 16.4.9 they explicitly define coprof(zero module) = 0! So are you sure you need to define codepth to be -∞ for the zero module? I don’t think so!

        • You are correct. Since the codepth is always nonnegative, it suffices to make the codepth of the zero module any nonpositive number. Auslander’s theorem works with codepth(0) = 0.

  2. I agree with Huneke and Wiegand that the depth of the zero module over a local ring should be defined to be plus infinity. This agrees with Bourbaki and EGA. It’s based on an earlier convention in Bourbaki and EGA, that elements x_1,…,x_n in a ring R form an M-regular sequence (for an R-module) if multiplication by x_i on M/(x_1,…,x_{i-1}) is injective for each i. In other words, we don’t require that M/(x_1,…,x_n) is nonzero (as the Stacks Project does). The argument is that the key properties of regular sequences, like the exactness of the Koszul complex, work fine without assuming M/(x_1,…,x_n) to be nonzero. So I would argue for changing the Stacks Project’s convention on this point. (I admit it’s not the most important thing in the world.)

    The references I found are:
    N. Bourbaki. Algèbre. Chapitre 10. Algèbre Homologique. Springer (2006). X.9.6.
    A. Grothendieck. EGA IV, Part 1. Publ. Math. IHÉS 20 (1964).
    N. Bourbaki. Algèbre Commutative. Chapitre 10. Springer (2007). Th. X.4.2.

    • If EGA specifies that the depth of the zero module should be $+\infty$, and if they also specify that the codepth should be $0$, does that mean that they consider the zero module to have dimension $+\infty$?

  3. You know I’ve decided I really don’t like this thing where you can have an infinite length regular sequence on a module over a local Noetherian ring. The length of a regular sequence should *always* be less than or equal to the dimension of the ring. So I am not going to use the convention suggested by your comment and the references therein.

    But if more people write in to second your suggestion, then I will reconsider.

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