On a question of Cid-Ruiz: positive results and counterexamples

In this paper we give positive criteria and counterexamples to a question of Cid-Ruiz on whether intrinsic polar multiplicities detect reductions of modules. Over an excellent regular local surface with infinite residue field, we compute the entire polar sequence of every finite-colength submodule of a free module, showing that its only variable entry is the order of the determinant ideal. This computation yields classes for which equality of polar multiplicities characterizes reduction, while also producing counterexamples in every positive rank, including integrally closed modules with identical intrinsic Hilbert functions and arbitrarily large relative Buchsbaum--Rim multiplicity. In arbitrary dimension, we prove that, under equidimensionality and a sharp dimension bound on the kernel of the induced map of associated graded Rees algebras, equality of the sums of the polar multiplicities characterizes reduction.