Abelian hearts of twin cotorsion pairs
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Archiv der Mathematik
Abelian hearts of twin cotorsion pairs Yu Liu
Abstract. The heart of a cotorsion pair, which is a generalization of the heart of a t-structure, has been proven to be abelian on triangulated, exact, and extriangulated categories. On the other hand, the heart of a twin cotorsion pair is not always abelian (it is only semi-abelian). In this article, we study a special kind of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories. We will give a sufficient-necessary condition when such hearts become abelian. We will also see that such hearts are hereditary. Mathematics Subject Classification. 18E10, 16G99. Keywords. Heart, Twin cotorsion pair, Fully rigid subcategory, d-Cluster tilting subcategory.
1. Introduction. Koenig and Zhu [4] showed that any ideal quotient of a triangulated category modulo a cluster tilting subcategory is an abelian category and this abelian quotient category is the module category of a 1-Gorenstein algebra. It generalized the work of Keller and Reiten [3] for 2-Calabi–Yau triangulated categories. Nakaoka [8] introduced the notion of cotorsion pairs in triangulated categories and showed that from any cotorsion pair, one can construct an abelian category, which agrees with Koenig and Zhu’s abelian quotient category when the cotorsion pair comes from a cluster tilting subcategory. Similar results were proven on exact categories by Deoment and Liu [2,5]. After an extriangulated category, which is a generalization of both triangulated and exact categories, was introduced by Nakaoka and Palu [9], Liu and Nakaoka developed a generalized theory for the heart of the (twin) cotorsion pairs on extriangulated category. Yu Liu was supported by the Fundamental Research Funds for the Central Universities (Grants No. 2682019CX51) and the National Natural Science Foundation of China (Grants No. 11901479).
Y. Liu
Arch. Math.
We recall the definition of (twin) cotorsion pair and its heart on exact categories [5]. Throughout this article, let B be a Krull–Schmidt, Hom-finite exact category over a field k with enough projectives and enough injectives. Any subcategory we discuss in this article will be full and closed under isomorphisms. Definition 1.1. Let U and V be two subcategories of B which are closed under direct summands. We call (U, V) a cotorsion pair if it satisfies the following conditions: (a) Ext1B (U, V) = 0. (b) For any object B ∈ B, there exist two short exact sequences VB UB B,
B V B UB
satisfying UB , U B ∈ U and VB , V B ∈ V. A pair of cotorsion pairs (S, T ) and (U, V) is called a twin cotorsion pair if S ⊆ U. Definition 1.2. Let (S, T ), (U, V) be a twin cotorsion pair. Let H be the subcategory of B consisting of objects B which admit the following short exact sequences: VB WB B,
B W B SB
where W B , WB ∈ U ∩T , VB ∈ V, and S B ∈ S. The quotient category H/(U ∩T ) is called the heart of (S, T ), (U, V). The heart of a cotorsion pair is the heart of the special twin cotorsion pair (U, V), (U, V). The heart of a cotorsion pair is abe
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