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: Without the assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection. Sketch of Proof 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. 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. References Commutative algebra