The pointwise-local-global principle for solutions of generic linear equations (Q1058568)

The author studies the problem of finding conditions on a matrix A over a ring R which ensure that: (*) the matrix equation \(Au=f\) is solvable over R if it is solvable over \(\kappa\) (p) for each \(p\in Spec R (\kappa (p)\) denotes the residue field of the localization \(R_ p)\). In the case that R is the ring of all real-valued \(C^{\infty}\) functions on a real n- dimensional ball B, sufficient conditions for (*) to hold have been found by \textit{J. N. Mather} [Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 185-193 (1973; Zbl 0272.26008)]. Using similar methods, the present paper proves a version of Mather's theorem for local rings, and as consequences, versions for polynomial rings and rings of holomorphic functions. It is shown (Theorem 5.1) that a sufficient condition for (*) to hold in the case of a regular local ring R is that the entries of A form part of a minimal generating set for the maximal ideal of R.