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 [math]\displaystyle{ \mathfrak{p}, \mathfrak{q} }[/math] in it, for each prime ideal [math]\displaystyle{ \mathfrak r }[/math] that is a minimal prime ideal over the sum [math]\displaystyle{ \mathfrak p + \mathfrak q }[/math], the following inequality on heights holds:[1][2]
- [math]\displaystyle{ \operatorname{ht}(\mathfrak r) \le \operatorname{ht}(\mathfrak p) + \operatorname{ht}(\mathfrak q). }[/math]
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.[3]
By replacing [math]\displaystyle{ A }[/math] by the localization at [math]\displaystyle{ \mathfrak r }[/math], we assume [math]\displaystyle{ (A, \mathfrak r) }[/math] is a local ring. Then the inequality is equivalent to the following inequality: for finite [math]\displaystyle{ A }[/math]-modules [math]\displaystyle{ M, N }[/math] such that [math]\displaystyle{ M \otimes_A N }[/math] has finite length,
- [math]\displaystyle{ \dim_A M + \dim_A N \le \dim A }[/math]
where [math]\displaystyle{ \dim_A M = \dim(A/\operatorname{Ann}_A(M)) }[/math] = the dimension of the support of [math]\displaystyle{ M }[/math] and similar for [math]\displaystyle{ \dim_A N }[/math]. To show the above inequality, we can assume [math]\displaystyle{ A }[/math] is complete. Then by Cohen's structure theorem, we can write [math]\displaystyle{ A = A_1/a_1 A_1 }[/math] where [math]\displaystyle{ A_1 }[/math] is a formal power series ring over a complete discrete valuation ring and [math]\displaystyle{ a_1 }[/math] is a nonzero element in [math]\displaystyle{ A_1 }[/math]. Now, an argument with the Tor spectral sequence shows that [math]\displaystyle{ \chi^{A_1}(M, N) = 0 }[/math]. Then one of Serre's conjectures says [math]\displaystyle{ \dim_{A_1} M + \dim_{A_1} N \lt \dim A_1 }[/math], which in turn gives the asserted inequality. [math]\displaystyle{ \square }[/math]
References
- ↑ Serre 2000, Ch. V, § B.6, Theorem 3.
- ↑ Fulton 1998, § 20.4.
- ↑ Serre 2000, Ch. V, § B. 6.
- Fulton, William (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
- Serre, Jean-Pierre (2000) (in de). Local Algebra. Springer Monographs in Mathematics. doi:10.1007/978-3-662-04203-8. ISBN 978-3-662-04203-8. OCLC 864077388.
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