[Paper Review] Hearts of twin Cotorsion pairs on extriangulated categories

Yu Liu, Hiroyuki Nakaoka|arXiv (Cornell University)|Feb 1, 2017

Algebraic structures and combinatorial models16 references6 citations

TL;DR

This paper establishes a unified framework for studying hearts of cotorsion pairs in extriangulated categories—generalizing exact and triangulated categories—proving the heart is abelian and constructing a cohomological functor to it. The key result shows that when the extriangulated category has enough projectives, the heart is equivalent to the category of coherent functors over the coheart modulo projectives, and $n$-cluster tilting subcategories induce families of cotorsion pairs with equivalent hearts.

ABSTRACT

In this article, we study the heart of a cotorsion pairs on an exact category and a triangulated category in a unified meathod, by means of the notion of an extriangulated category. We prove that the heart is abelian, and construct a cohomological functor to the heart. If the extriangulated category has enough projectives, this functor gives an equivalence between the heart and the category of coherent functors over the coheart modulo projectives. We also show how an n-cluster tilting subcategory of an extriangulated category gives rise to a family of cotorsion pairs with equivalent hearts.

Motivation & Objective

• To unify the study of hearts of cotorsion pairs across exact and triangulated categories using the framework of extriangulated categories.
• To prove that the heart of a cotorsion pair in an extriangulated category is abelian, generalizing known results in exact and triangulated settings.
• To construct a cohomological functor from the extriangulated category to the heart, providing a homological bridge.
• To establish an equivalence between the heart and the category of coherent functors over the coheart modulo projectives when the category has enough projectives.
• To show that $n$-cluster tilting subcategories induce a sequence of cotorsion pairs with equivalent hearts, linking higher homological algebra to tilting theory.

Proposed method

• Use the notion of extriangulated categories as a unifying framework for exact and triangulated categories.
• Define and analyze twin cotorsion pairs, proving their hearts are semi-abelian via adjoint properties and conflation structures.
• Specialize to single cotorsion pairs to prove the heart is abelian and the associated functor is cohomological.
• Introduce the coheart and kernel of a cotorsion pair, and establish conditions under which the heart has enough projectives.
• Prove that the heart is equivalent to the category of coherent functors over the coheart modulo projectives when the extriangulated category has enough projectives.
• Construct a sequence of cotorsion pairs from an $n$-cluster tilting subcategory, showing they share equivalent hearts via core and coheart structures.

Experimental results

Research questions

• RQ1 How can the theory of hearts of cotorsion pairs be unified across exact and triangulated categories?
• RQ2 Under what conditions is the heart of a cotorsion pair in an extriangulated category abelian?
• RQ3 When does the heart of a cotorsion pair have enough projectives, and how does this relate to the category of coherent functors over the coheart modulo projectives?
• RQ4 How do $n$-cluster tilting subcategories in an extriangulated category give rise to families of cotorsion pairs with equivalent hearts?
• RQ5 What is the role of the kernel and coheart in determining the structure and equivalence of hearts of cotorsion pairs?

Key findings

• The heart of a cotorsion pair in an extriangulated category is abelian, generalizing results from exact and triangulated categories.
• A cohomological functor from the extriangulated category to the heart is constructed, providing a homological link.
• When the extriangulated category has enough projectives, the heart is equivalent to the category of coherent functors over the coheart modulo projectives.
• The heart has enough projectives if and only if the kernel of the cotorsion pair satisfies a certain condition related to the coheart.
• An $n$-cluster tilting subcategory induces $n-1$ cotorsion pairs whose hearts are all equivalent, and each is equivalent to $\operatorname{mod}(\mathcal{M}/\mathcal{P})$.
• All induced cotorsion pairs from an $n$-cluster tilting subcategory share the same core $\Sigma^{\ell-1}\mathcal{M}$ and coheart $\mathcal{M}$, leading to equivalent hearts.