Serre's inequality on height explained

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

ak{p}, ak{q} in it, for each prime ideal ak r that is a minimal prime ideal over the sum ak p + ak q , the following inequality on heights holds:\operatorname{ht}( ak r) \le \operatorname{ht}( ak p) + \operatorname{ht}( ak q).

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

A by the localization at ak r , we assume (A, ak r) is a local ring. Then the inequality is equivalent to the following inequality: for finite A -modules M, N such that M ⊗ _{A} N has finite length,\dim _{A} M + \dim _{A} N \le \dim A where \dim _{A} M = \dim(A/\operatorname{Ann} _{A(M))} = the dimension of the support of M and similar for \dim _{A} N . To show the above inequality, we can assume A is complete. Then by Cohen's structure theorem, we can write A = A _{1/a} _{1} A _{1} where A _{1} is a formal power series ring over a complete discrete valuation ring and a _{1} is a nonzero element in A _{1} . Now, an argument with the Tor spectral sequence shows that \chi ^{A _{1}} (M, N) = 0 . Then one of Serre's conjectures says \dim _{A _{1}} M + \dim _{A _{1}} N < \dim A _{1} , which in turn gives the asserted inequality. \square

References

• • Book: Serre, Jean-Pierre . Local Algebra . 2000 . 978-3-662-04203-8 . Springer Monographs in Mathematics . 10.1007/978-3-662-04203-8 . 864077388 . de.