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

Codimension and connectedness of degeneracy loci over local rings