Codimension and connectedness of degeneracy loci over local rings

We deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules \(f:M\rightarrow N\) over a local noetherian ring \((A,\mathfrak {m})\). We show for instance that the expected determinantal bounds on the dimension of the t-th degeneracy locus of f hold if \(f\in \mathfrak {m}\mathrm{Hom}(M,N)\), and that this degeneracy locus is connected in the expected dimension provided \({\hat{A}}\) is a domain.