What is the correct convention for the depth of the zero module over a local ring?<\/p>\n
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<\/a>) and the zero module has no regular sequence whatsoever (Tag 00LI<\/a>).<\/p>\n In this erratum<\/a> 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.<\/p>\n Hmm, I’m not so sure.<\/p>\n 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.<\/p>\n 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.<\/p>\n But what do you<\/strong> think?<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading \n
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