Serre's Inequality On Height
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In algebra, specifically in the theory of
commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
s, Serre's inequality on height states: given a (Noetherian)
regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maxi ...
''A'' and a pair of
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
s \mathfrak, \mathfrak in it, for each prime ideal \mathfrak r that is a
minimal prime ideal In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal prime ideals. De ...
over the sum \mathfrak p + \mathfrak q, the following inequality on
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an e ...
s 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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums ...
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 In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' that satisfies any and all of the following equivalent conditions: # '' ...
. 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 In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
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^(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 {{commutative-algebra-stub