Serre's inequality on height

In algebra, specifically in the theory of commutative rings, Serre's inequality on height states: given a (Noetherian) regular ring A and a pair of prime ideals

in it, for each prime ideal

that is a minimal prime ideal over the sum

, the following inequality on heights holds:[1][2] Without the assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection.

Serre gives the following proof of the inequality, based on the validity of Serre's multiplicity conjectures for formal power series ring over a complete discrete valuation ring.

is a local ring.

Then the inequality is equivalent to the following inequality: for finite

has finite length, where

dim

= the dimension of the support of

To show the above inequality, we can assume

Then by Cohen's structure theorem, we can write

is a formal power series ring over a complete discrete valuation ring and

is a nonzero element in

Now, an argument with the Tor spectral sequence shows that

Then one of Serre's conjectures says

, which in turn gives the asserted inequality.

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