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Let $(A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a local homomorphism with $A$ a regular local ring. Assume further that this ring map is finite. How can we prove that $\operatorname{depth}B = \operatorname{depth}_A B$? What about if we replace $B$ with a finitely generated $B$-module $M$?

I believe I can prove this in a roundabout way using Auslander-Buschbaum by first proving that $A \to B$ is flat, but this is probably not optimal. As this is at the edge of my knowledge on commutative algebra, I am hoping someone can help me out with an elegant proof/ what are the valid results surrounding something like that.

For those wondering why I am asking this, well I was trying to do exercise 3.9.3 of Hartshorne.

user26857
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    Just a note: You can only apply Auslander-Buchsbaum if $B$ is module-finite over $A$ (which is a very special situation) – zcn Jan 20 '14 at 16:16

2 Answers2

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Exercise 1.2.26 from Bruns and Herzog, Cohen-Macaulay Rings:

Let $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ be a local homomorphism of noetherian local rings, and $M$ an $S$-module which is finitely generated as an $R$-module. Then $\operatorname{depth}_RM=\operatorname{depth}_SM$.

user26857
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In addition to the answer by user26857, if you assume $B$ is a finite $A$-module, one can use local cohomology. Since $\sqrt {\mathfrak mS} = \mathfrak n$, $$H^i_{\mathfrak m} (B) = H^i_{\mathfrak mB} (B) = H^i_{\mathfrak n} (B),$$ for all $i$. The equality between the first and the third modules give you the depth equality.

user26857
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Youngsu
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  • Dear Youngsu, thanks for your answer. I do wonder now if there's a way of expressing depth in terms of local cohomology. Do you mind giving me a reference? –  Jan 22 '14 at 05:50
  • @Benja: The first non vanishing $i$ is the depth. There are many references for this for instance, local cohomology, 24 hours of local cohomology, and more. You can google for books or lecture note easily. I don't know how the Algebraic geometry book by Hartshorne defines depth but there is an exercise which links sheaf cohomology and local cohomology. The above mentioned case is when $Z = V(m)$. I believe that this exercises shows why depth at least $2$ is useful. – Youngsu Jan 22 '14 at 06:47