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

Without the assumption on regularity, the inequality can fail; see scheme-theoretic 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 $$\mathfrak r$$, we assume $$(A, \mathfrak r)$$ is a local ring. Then the inequality is equivalent to the following inequality: for finite $$A$$-modules $$M, N$$ such that $$M \otimes_A N$$ 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. $$\square$$