Algebrization of some complete modules

TL;DR

This paper investigates when complete modules over a Noetherian local ring can be derived from modules over the ring itself, introducing strongly algebraic modules and providing new criteria and applications in algebraic geometry and homological algebra.

Contribution

It introduces the concept of strongly algebraic modules, unifies existing results, and offers new homological criteria and examples for algebrization of modules.

Findings

Characterization of strongly algebraic modules

New criteria for algebrization involving homological properties

Applications to Grothendieck group calculations and vector bundles

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is algebraic. We introduce and investigate the class of \emph{strongly algebraic} modules, those complete modules all of whose direct summands are algebraic. Our approach unifies and extends several known results of Levy--Odenthal, Weston, Peskine--Szpiro, Puthenpurakal, and several others, and provides new examples and homological criteria for algebrization. Applications include a computation of the Grothendieck group $G_0(R)$ in dimension one and new algebrization results for generalized Cohen--Macaulay modules and vector bundles along with a connection to local cohomology modules.