[Paper Review] Simple-minded reductions of triangulated categories

Haibo Jin|arXiv (Cornell University)|Jul 11, 2019

Algebraic structures and combinatorial models20 references7 citations

TL;DR

This paper introduces a new reduction process for triangulated categories using pre-simple-minded collections (pre-SMCs), constructing a new triangulated category where SMCs bijectively correspond to those in the original category containing the pre-SMC. It establishes an analogue of Buchweitz's theorem for singularity categories and shows that SMS reduction is the shadow of this SMC reduction, paralleling the silting–Calabi-Yau reduction relationship.

ABSTRACT

We will introduce a new reduction process of triangulated category, which is analogue to the silting reduction and Calabi-Yau reduction. For a triangulated category $\cal T$ with a pre-simple-minded collection (=pre-SMC) $\cal R$, we construct a new triangulated category $\cal U$ such that the SMCs in $\cal U$ bijectively correspond to those in $\cal T$ containing $\cal R$. Secondly, we give an analogue of Buchweitz's theorem for the singularity category $\cal T_{ m sg}$ of a SMC quadruple $(\cal T,\cal T^{ m p},\mathbb S, \cal S)$: the category $\cal T_{ m sg}$ can be realized as the stable category of an extriangulated subcategory $\cal F$ of $\cal T$. Finally, we show the SMS (simple-minded system) reduction due to Coelho Simoes and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi-Yau reduction is the shadow of silting reduction due to Iyama and Yang.

Motivation & Objective

• To develop a new reduction mechanism in triangulated categories based on pre-SMCs, analogous to silting and Calabi-Yau reductions.
• To establish a categorical realization of the singularity category as a stable category of an extriangulated subcategory, extending Buchweitz's theorem to SMC quadruples.
• To clarify the relationship between SMS reduction (by Coelho Simões and Pauksztello) and the proposed SMC reduction, showing the former as a shadow of the latter.

Proposed method

• Define a new triangulated category 𝒰 from a triangulated category 𝒯 and a pre-SMC 𝒓, such that SMCs in 𝒰 correspond bijectively to SMCs in 𝒯 containing 𝒓.
• Construct an extriangulated subcategory 𝒇 of 𝒯 such that the singularity category 𝒯ₛg is equivalent to the stable category of 𝒇, generalizing Buchweitz’s theorem.
• Use the structure of SMC quadruples (𝒯, 𝒯ᵖ, 𝒔, 𝒔) to define the singularity category and relate it to the extriangulated subcategory.
• Demonstrate that the SMS reduction of Coelho Simões and Pauksztello arises as a shadow of the proposed SMC reduction, mirroring the silting–Calabi-Yau reduction duality.
• Leverage the categorical duality between reduction processes to unify perspectives on triangulated category reductions.
• Apply the theory to show that the correspondence of SMCs under reduction is both natural and fully faithful in the derived category context.

Experimental results

Research questions

• RQ1 How can a new reduction process for triangulated categories be defined using pre-SMCs, and what properties does it satisfy?
• RQ2 Can Buchweitz’s theorem on singularity categories be extended to SMC quadruples using an extriangulated subcategory?
• RQ3 What is the relationship between the SMS reduction of Coelho Simões and Pauksztello and the proposed SMC reduction?
• RQ4 Is the SMC reduction a categorical lift of the SMS reduction, analogous to how Calabi-Yau reduction lifts silting reduction?
• RQ5 How does the new reduction process preserve or reflect the structure of SMCs in the original category?

Key findings

• The construction of the new triangulated category 𝒰 ensures a bijective correspondence between SMCs in 𝒰 and those in 𝒯 that contain the pre-SMC 𝒓.
• The singularity category 𝒯ₛg of an SMC quadruple is realized as the stable category of an extriangulated subcategory 𝒇 of 𝒯, extending Buchweitz’s theorem.
• The SMS reduction of Coelho Simões and Pauksztello is shown to be the shadow of the proposed SMC reduction, establishing a categorical duality.
• The SMC reduction process generalizes the silting and Calabi-Yau reduction frameworks, providing a unifying perspective on reduction in triangulated categories.
• The reduction mechanism preserves essential categorical structures, including the correspondence of SMCs and the stability of singularity categories.
• The duality between SMC reduction and SMS reduction mirrors the known duality between silting reduction and Calabi-Yau reduction, confirming a deeper structural analogy.