On the stable rank of localizations of affine algebras (Q2762282)

For a commutative ring \(A\), let \(\text{sr}(A)\) denote the stable rank of \(A\), \(\dim A\) the Krull dimension of \(A\), and \(\dim\text{Max}(A)\) the dimension of the maximal spectrum of \(A\). A natural problem is to study the relationship between \(\text{sr}(A)\), \(\dim A\), and \(\dim\text{Max}(A)\). It was shown by \textit{H. Bass} [``Algebraic \(K\)-theory'', Benjamin, New York (1968; Zbl 0174.30302)] that \(\text{sr}(A)\leq \dim\text{Max}(A)+1\). When \(A\) is an affine algebra over a sufficiently large (e.g., uncountable) field, \textit{A. A. Suslin} [Math. USSR Izv. 15, 589-623 (1980; Zbl 0452.13007)] proved that \(\text{sr}(A)= \dim\text{Max}(A)+1\).NEWLINENEWLINENEWLINEThis paper deals with the case where \(A\) is a localization of an integral finitely generated algebra over an uncountable \(k\). The authors describe all triples of integers \((n,m,d)\) for which there exists a ring \(A\) of the above-mentioned type with \(\text{sr}(A)=n\), \(\dim A =m\), and \(\dim\text{Max}(A)=d\). Evidently, in this case, \(d\leq m\) and \(1\leq n\leq d+1\). The authors of this paper prove that these two conditions on \(n\), \(m\), \(d\) are also sufficient. Also, it is shown that if \(A\) is a local domain of geometric origin with \(\dim A\geq 3\), then there is \(0\not=\pi\in A\) such that \(\text{sr}(A_{\pi})\geq 2\).