# 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:[1][2]

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 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

- William Fulton. (1998),
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.,**2**(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323 - P. Serre,
*Local algebra*, Springer Monographs in Mathematics