The Bass-Murthy question: Serre dimension of Laurent polynomial extensions
The Serre dimension of a commutative ring R is the smallest t such that every projective R-module is a direct sum of a free module and a module of rank \(\leq t.\) This paper is an important contribution to the study of the relation between the Krull dimension of R and the Serre dimension of Laurent and polynomial extensions of R. The main result is that the Serre dimension is independent of the number and type of variables: Theorem: Let R be a noetherian commutative ring of Krull dimension d, and let \(A=R[X_ 1,...,X_ n,Y_ 1^{\epsilon_ 1},...,Y_ m^{\epsilon_ m}]\) where \(\epsilon_ i=\pm 1\). Then every projective A-module of rank \(\geq d+1\) has a unimodular element. The algebraic technique employed is to obtain a 2-cover of spec(A), on each piece of which the projective module has a unimodular element, and then to patch these unimodular elements by a split automorphism.
scientific article; zbMATH DE number 3975062
scientific article