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arXiv · 2603.01424

Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules

Souvik Dey·Jian Liu·Xue-Song Lu
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Abstract

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring $S$ admitting a canonical module $\omega$, and assume that $R$ is maximal Cohen--Macaulay over $S$. We prove that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\mathrm Ext$-orthogonal class of the thick subcategory generated by $R$ and ${\mathrm Hom}_S(R,\omega)$. As an application, finitely generated Gorenstein projective $R$-modules form the left half of a hereditary cotorsion pair. In the case of Cohen--Macaulay local rings, this yields an affirmative answer to a question of R. Takahashi. We further characterize when $R$ is left weakly Gorenstein. Finally, we prove that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\mathrm Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.

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Souvik Dey, Jian Liu, Xue-Song Lu. 2026-03-02. Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules. https://arxiv.org/abs/2603.01424

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