# Govorov--Lazard and finite deconstructibility for Gorenstein and restricted homological dimensions

## TL;DR

This paper investigates the properties of modules with bounded restricted homological dimensions over Cohen--Macaulay and Gorenstein rings, establishing finite deconstructibility and Govorov-Lazard properties, with implications for module theory.

## Contribution

It proves finite deconstructibility and Govorov-Lazard properties for modules with restricted homological dimensions over Cohen--Macaulay and Gorenstein rings, extending existing results.

## Key findings

- Modules of restricted projective dimension are finitely deconstructible.
- Modules of restricted flat dimension satisfy Govorov-Lazard property.
- Results apply to Gorenstein projective and flat dimensions over Gorenstein rings.

## Abstract

Over Cohen--Macaulay rings admitting a pointwise dualizing module, we show that the class of modules of restricted projective dimension bounded by any integer is finitely deconstructible and that the class of modules of restricted flat dimension bounded by any integer satisfies the Govorov-Lazard property. Along the way, we prove the corresponding result for Gorenstein projective and flat dimensions over (locally) Gorenstein rings. Outside of Cohen--Macaulay rings, we consider analogous properties for restricted projective dimension zero and restricted flat dimension zero and establish them for commutative noetherian rings of finite Krull dimension. This has consequences for the corresponding classes of finitely generated modules being preenveloping in certain cases and provides generalizations of Holm's results on structure of balanced big Cohen--Macaulay modules in various directions.