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.

- dim(M) ≥ depth(M), see Lemma Tag 00LK.
- M is Cohen-Macaulay if dim(M) = depth(M), see Definition Tag 00N3.
- depth(M) is equal to the smallest integer i such that Ext
^{i}_{R}(R/m, M) is nonzero, see Lemma Tag 00LW - Let 0 —> N′ —> N —> N′′→0 be a short exact sequence of finite R-modules. Then
- depth(N′′) ≥ min{depth(N), depth(N′) − 1}
- depth(N′) ≥ min{depth(N), depth(N′′) + 1}

- 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?