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:

$\mathrm{ht}(\mathfrak{𝔯})\le \mathrm{ht}(\mathfrak{𝔭})+\mathrm{ht}(\mathfrak{𝔮}).$

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 $\mathfrak{𝔯}$, 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 ${\dim }_{A}M=\dim (A/{\mathrm{Ann}}_A(M))$ = 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. $◻$

References

• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Serre, Jean-Pierre (2000). Local Algebra. Springer Monographs in Mathematics (in Deutsch). doi:10.1007/978-3-662-04203-8. ISBN 978-3-662-04203-8. OCLC 864077388.