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In algebra, specifically in the theory of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s, Serre's inequality on height states: given a (Noetherian) regular ring ''A'' and a pair of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s \mathfrak, \mathfrak 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(\mathfrak r) \le \operatorname(\mathfrak p) + \operatorname(\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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
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_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 Tor, TOR or ToR may refer to: Places * Tor, Pallars, a village in Spain * Tor, former name of Sloviansk, Ukraine, a city * Mount Tor, Tasmania, Australia, an extinct volcano * Tor Bay, Devon, England * Tor River, Western New Guinea, Indonesia ...
shows that \chi^(M, N) = 0. Then one of Serre's conjectures says \dim_ M + \dim_ N < \dim A_1, which in turn gives the asserted inequality. \square


References

* * Commutative algebra {{algebra-stub