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The dimension and components of symmetric algebras. (English) Zbl 0584.13010

Let M be a finitely generated R-module, with R a Noetherian ring. The authors first give the following formula for the dimension of the symmetric algebra of \(M: \dim S(M)=Max\{\dim R/p+\mu (M_ p)\}\) over all \(p\in Spec R\). Here, \(\mu (M_ p)\) is the minimal number of generators of \(M_ p\). A consequence is that dim S(M)\(\geq \mu (M)\). The authors then give a sufficient condition for a prime in R to be the contraction of a minimal prime in S(M). They present an example (due to G. Valla) showing that S(M) can have arbitrarily large numbers of minimal primes with widely varying dimensions, even if R is a polynomial ring over a field and M is a prime ideal of R.
Reviewer: S.McAdam

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13A15 Ideals and multiplicative ideal theory in commutative rings
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References:

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