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

Zbl 0999.13007
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