[Paper Review] The Gorenstein projective modules are precovering

Peter Jørgensen|ArXiv.org|Dec 12, 2003

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper proves that over a noetherian commutative ring with a dualizing complex, the Gorenstein projective modules form a precovering class in the module category. Using Bousfield localization in the homotopy category of complexes of projective modules, the author constructs a right adjoint to the inclusion of exact complexes with vanishing Hom to projectives, which descends to yield Gorenstein projective precovers, thereby resolving a long-standing gap in relative homological algebra for Gorenstein rings.

ABSTRACT

The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex.

Motivation & Objective

• To resolve the longstanding open problem of whether Gorenstein projective modules form a precovering class in the module category of a general class of rings.
• To extend the precovering property beyond Gorenstein rings, where it was previously known only in special cases.
• To establish a foundational result in Gorenstein homological algebra by enabling the use of Gorenstein projective resolutions in place of projective resolutions.
• To generalize the result from commutative noetherian rings with dualizing complexes to non-commutative algebras with dualizing complexes.
• To provide a homotopical method—based on Bousfield localization—for constructing precovers, offering a new perspective distinct from prior approaches.

Proposed method

• The proof proceeds by passing to the homotopy category K(Pro A) of complexes of projective A-modules.
• It defines the subcategory E(A) ⊆ K(Pro A) consisting of exact complexes E such that Hom(E, Q) is exact for all projective modules Q.
• It shows that E(A) is the kernel of a homological functor respecting small coproducts, enabling the application of Bousfield localization.
• Using Bousfield localization, it establishes that the inclusion E(A) → K(Pro A) has a right adjoint, implying E(A) is precovering in K(Pro A).
• It then shows that the Gorenstein projective modules, as kernels of complexes in E(A), inherit the precovering property via descent from E(A).
• The method is generalized to non-commutative algebras by replacing commutative rings with left- and right-noetherian k-algebras with dualizing complexes, preserving the core homotopical argument.

Experimental results

Research questions

• RQ1 Does the class of Gorenstein projective modules form a precovering class over a noetherian commutative ring with a dualizing complex?
• RQ2 Can the precovering property of Gorenstein projective modules be established in a broader class of rings beyond Gorenstein rings?
• RQ3 Is there a homotopical or categorical method—specifically using Bousfield localization—that can be used to prove the precovering property for Gorenstein projective modules?
• RQ4 Can the result be extended to non-commutative algebras with dualizing complexes using the same homotopical framework?
• RQ5 What is the role of the subcategory E(A) of exact complexes with vanishing Hom to projectives in constructing precovers?

Key findings

• The Gorenstein projective modules form a precovering class in the module category of any noetherian commutative ring A that admits a dualizing complex.
• The key technical innovation is the use of Bousfield localization to construct a right adjoint to the inclusion of the subcategory E(A) in K(Pro A), which is shown to be precovering.
• The precovering property for Gorenstein projective modules is established by descending the precovering structure from E(A) to the module category via the kernel construction.
• The result extends to non-commutative algebras: for any left- and right-noetherian k-algebra A with a dualizing complex, the Gorenstein projective A-modules form a precovering class.
• The method applies to large classes of non-commutative rings, including complete semi-local PI algebras and filtered algebras with noetherian graded rings, as shown in Corollary 4.5.
• The proof avoids traditional resolution techniques and instead uses homotopical algebra, marking a novel approach distinct from prior work on precovering classes.