Singular loci of algebras over ramified discrete valuation rings

When $k$ is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated $k$-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 $p$-derivations. Motivated by these results, in this paper, we state and prove an analogous Jacobian criterion for algebras over a ramified discrete valuation ring of mixed characteristic.