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
in it, for each prime ideal
that is a
minimal prime ideal over the sum
, the following inequality on
heights holds:
:
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
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
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
. Then one of Serre's conjectures says
, which in turn gives the asserted inequality.
References
*
*
Commutative algebra
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