Flat modules and Gr\"obner bases over truncated discrete valuation rings

TL;DR

This paper studies Gr"obner bases over polynomial rings with coefficients in truncated discrete valuation rings, providing criteria for flatness of modules and exploring their properties in both graded and non-graded contexts.

Contribution

It introduces new properties of Gr"obner bases over truncated discrete valuation rings and offers a flatness criterion for finitely generated modules.

Findings

Characterization of Gr"obner bases over truncated discrete valuation rings

Flatness criterion for finitely generated modules over polynomial rings

Extension of results to non-graded modules

Abstract

We present basic properties of Gr\"obner bases of submodules of a free module of finite rank over a polynomial ring $R$ with coefficients in a graded truncated discrete valuations ring $A$. As an application, we give a criterion for a finitely generated $R$-module to be flat over $A$. Its non-graded version is also given.