On the Bass Stable Rank of Stein Algebras

Abstract

We compute the Bass stable rank of the ring Γ(X, OX ) of global sections of the structure sheaf OX on a finite-dimensional Stein space (X, OX ) and then apply this result to the problem of the factorization of invertible holomorphic matrices on X.

Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

“…Certain K-theory arguments guarantee that the number of unipotent matrices needed for factorizing an element in SL n (O(Ω)) is a non-increasing function of n (see [7]). So, as done in [3], combining the above property and results from [11], we obtain the following estimates:…”

“…Certain K-theory arguments guarantee that the number of unipotent matrices needed for factorizing an element in SL n (O(Ω)) is a non-increasing function of n (see [7]). So, as done in [3], combining the above property and results from [11], we obtain the following estimates: E(1, n) ≤ N (1, n) = 4 for all n, E(2, n) ≤ N (2, n) ≤ 5 for all n, and for each k, there exists n(k) such that E(k, n) ≤ N (k, n) ≤ 6 for all n ≥ n(k).…”

Factorization of Holomorphic Matrices and Kazhdan's property (T)

“…Our first result is concerned with the classical K-theoretic question about the K 1 groups of the ring R = O(X) and bounded generation of the corresponding elementary subgroups with concrete bounds. Since we assume that the Stein space X has finite dimension n (defined as the complex dimension of the manifold X \ Sing(X), called the smooth part of X), these rings are interesting rings since they have finite Bass stable rank bsr(O(X)) = 1 2 dim X + 1 as established by Alexander Brudnyi in [Bru19].…”