Serre's inequality on height
In algebra, specifically in the theory of commutative rings, Serre's inequality on height states: given a 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:
Without the assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection.Sketch of Proof
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.
By replacing by the localization at, we assume is a local ring. Then the inequality is equivalent to the following inequality: for finite -modules such that has finite length,
where = the dimension of the support of and similar for. To show the above inequality, we can assume is complete. Then by Cohen's structure theorem, we can write where 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.
