Singular loci of algebras over ramified discrete valuation rings

Abstract

When is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated ‐algebra as the closed subset of an ideal generated by appropriate minors of the so‐called Jacobian matrix. Recently, Hochster‐Jeffries and Saito have extended this result for algebras over any unramified discrete valuation ring of mixed characteristic via the use of ‐derivations. Motivated by these results, in this paper, we state and prove an analogous Jacobian criterion for algebras over ramified discrete valuation rings of mixed characteristic.