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Serre dimension and Euler class groups of overrings of polynomial rings. (English) Zbl 1387.13024

Let \(R\) be a commutative Noetherian ring of dimension \(d\), \(B=R[X_1,\ldots,X_m,Y_1^{\pm1},\ldots,Y_n^{\pm1}]\) a Laurent polynomial ring over \(R\), and \(A=B[Y,f^{-1}]\) for some \(f\in R[Y]\). In the present paper, the authors prove that if \(f\) is a monic polynomial, then the Serre dimension of \(A\) is \(\leq d\). Let \(P\) be a projective \(R\)-module. Recall that an element \(p\in P\) is said to be unimodular if there exists \(\phi\in\text{Hom}_R(P,R)\) such that \(\phi(p)=1\). Furthermore, the Serre dimension of \(R\) is said to be \(\leq d\) if every projective \(R\)-module of rank \(\geq t+1\) has a unimodular element.
Also, the authors prove that the \(p\)th Euler class group \(E^{p}(A)\) of \(A\) is trivial for \(p\geq\{d+1, \dim A-p+3\}\). See [S. M. Bhatwadekar and R. Sridharan, Compos. Math. 122, No. 2, 183–222 (2000; Zbl 0999.13007)] for the definition of Euler class group.

MSC:

13B25 Polynomials over commutative rings
13C10 Projective and free modules and ideals in commutative rings

Citations:

Zbl 0999.13007
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