Hearts of twin cotorsion pairs on extriangulated categories

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.

Abelian Categories Arising from Cluster Tilting Subcategories

“…We first recall the following proposition ( [LN,Proposition 1.20]), which (also the dual of it) will be used many times in the article.…”

“…We can also define cluster tilting subcategory on extriangulated categories. Liu and Nakaoka [LN,Theorem 3.2] showed that any quotient…”

Localizations of the Hearts of Cotorsion Pairs

“…End B (C ) op ) is equivalent to the heart of (C, C ⊥ 1 ) (resp. (C ⊥ 1 , M )) (see [12,Proposition 4.15]). Moreover, if B has a Serre functor S, then we also have cotorsion pairs (…”

“…We denote the subcategory of all the objects B such that U B , V B ∈ U ∩ V in the conflations in (b) by H. We call the ideal quotient H/(U ∩ V) the heart of (U , V). It is an abelian category by [12,Theorem 3 When B has enough projectives and injectives, a rigid subcategory C which is contravariantly finite and contains P induces a cotorsion pair (C, C ⊥ 1 ) (see Lemma 2.7 for details); the functor category (see [2]) mod(C/P) is equivalent to the heart of (C, C ⊥ 1 ). Let D ⊂ C, when we consider the mutation of C: C = CoCone(D, C) ∩ D ⊥ 1 (see [9,Definition 2.5]) that induces a cotorsion pair (C , C ⊥ 1 ) where C is also rigid, we can investigate the relation of two functor categories mod(C/P) and mod(C /P) by studying the hearts.…”

“…Let D ⊂ C, when we consider the mutation of C: C = CoCone(D, C) ∩ D ⊥ 1 (see [9,Definition 2.5]) that induces a cotorsion pair (C , C ⊥ 1 ) where C is also rigid, we can investigate the relation of two functor categories mod(C/P) and mod(C /P) by studying the hearts. We know if C ⊥ 1 = C ⊥ 1 , the hearts are equivalent [12,Proposition 3.12]. But here it is obviously not the case, since C…”