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:

$\operatorname {ht} ({\mathfrak {r}})\leq \operatorname {ht} ({\mathfrak {p}})+\operatorname {ht} ({\mathfrak {q}}).$

Without the assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection.

Sketch of Proof

(Serre, Ch. V, § B. 6.) 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,

$\dim _{A}M+\dim _{A}N\leq \dim A$

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

• William Fulton. (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
• P. Serre, Local algebra, Springer Monographs in Mathematics