[Paper Review] Proper classes and Gorensteinness in extriangulated categories

Jiangsheng Hu, Dongdong Zhang+1 more|arXiv (Cornell University)|Jun 26, 2019

Algebraic structures and combinatorial models17 references4 citations

TL;DR

This paper introduces the concept of a proper class of $\mathbb{E}$-triangles in extriangulated categories, constructing a new extriangulated structure that generalizes both exact and triangulated categories. It defines $\xi$-Gorenstein projective objects and establishes an admissible model structure via Hovey twin cotorsion pairs, generalizing results from triangulated and module categories.

ABSTRACT

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let $\mathcal{C}$ be an extriangulated category and $ξ$ a proper class in $\mathcal{C}$. We prove that $\mathcal{C}$ admits a new extriangulated structure. This construction gives extriangulated categories which are neither exact categories nor triangulated categories. Moreover, we introduce and study $ξ$-Gorenstein projective objects in $\mathcal{C}$ and demonstrate that $ξ$-Gorenstein projective objects share some basic properties with Gorenstein projective objects in module categories or in triangulated categories. In particular, we refine a result of Asadollahi and Salarian [Gorenstein objects in triangulated categories, J. Algebra 281(2004), 264-286]. As an application, the $ξ$-$\mathcal{G}$projective model structures on extriangulated categories are obtained.

Motivation & Objective

• To define and study proper classes of $\mathbb{E}$-triangles in extriangulated categories, generalizing exact and triangulated structures.
• To introduce $\xi$-Gorenstein projective objects in extriangulated categories and investigate their homological properties.
• To establish an admissible model structure on extriangulated categories using Hovey twin cotorsion pairs.
• To generalize known results on Gorenstein projective objects from triangulated and module categories to the broader setting of extriangulated categories.

Proposed method

• Define a proper class $\xi$ of $\mathbb{E}$-triangles in an extriangulated category $\mathcal{C}$, ensuring closure under isomorphisms and satisfying exactness axioms.
• Construct a new extriangulated structure $(\mathcal{C}, \mathbb{E}_\xi, \mathfrak{s}_\xi)$ by restricting the original extriangulated structure to $\xi$, proving it satisfies the axioms of an extriangulated category.
• Define $\xi$-Gorenstein projective objects as objects admitting a $\xi$-projective resolution with $\xi$-projective terms in the kernel of the differential.
• Establish that $\xi$-Gorenstein projective objects satisfy key homological properties, such as vanishing of $\mathbb{E}_\xi$-extensions beyond a certain degree.
• Prove that the pair $((\mathcal{P}(\xi), \mathcal{C}), (\mathcal{GP}(\xi), \mathcal{P}^{\leq n}(\xi)))$ forms a Hovey twin cotorsion pair, leading to an admissible model structure.
• Use the equivalence between finite $\xi$-Gorenstein projective dimension and the existence of such a model structure to derive the main theorem.

Experimental results

Research questions

• RQ1 Can a proper class of $\mathbb{E}$-triangles be defined in an extriangulated category such that the resulting structure remains extriangulated?
• RQ2 What are the homological properties of $\xi$-Gorenstein projective objects in extriangulated categories, and how do they relate to classical Gorenstein projective objects?
• RQ3 Under what conditions does an admissible model structure exist on an extriangulated category equipped with a proper class $\xi$?
• RQ4 How does the $\xi$-Gorenstein projective dimension relate to the existence of a Hovey twin cotorsion pair and an admissible model structure?
• RQ5 To what extent do results on Gorenstein projective objects in triangulated and module categories extend to the general setting of extriangulated categories?

Key findings

• The category $\mathcal{C}$ equipped with a proper class $\xi$ of $\mathbb{E}$-triangles admits a new extriangulated structure $(\mathcal{C}, \mathbb{E}_\xi, \mathfrak{s}_\xi)$, which is neither necessarily exact nor triangulated.
• The class of $\xi$-Gorenstein projective objects $\mathcal{GP}(\xi)$ is well-defined and shares key homological properties with Gorenstein projective objects in module and triangulated categories.
• The pair $((\mathcal{P}(\xi), \mathcal{C}), (\mathcal{GP}(\xi), \mathcal{P}^{\leq n}(\xi)))$ forms a Hovey twin cotorsion pair if and only if the supremum of $\xi$-Gorenstein projective dimension over all objects in $\mathcal{C}$ is at most $n$.
• An admissible model structure exists on $(\mathcal{C}, \mathbb{E}_\xi, \mathfrak{s}_\xi)$ precisely when $\sup\{\xi\text{-} \mathcal{G}{\rm pd}A \mid A \in \mathcal{C}\} \leq n$.
• The result refines a theorem of Asadollahi and Salarian on Gorenstein objects in triangulated categories by generalizing it to extriangulated categories.
• The construction provides a new class of extriangulated categories that are neither exact nor triangulated, expanding the scope of relative homological algebra.