A NOTE ON COMPACTLY GENERATED CO-t-STRUCTURES

Abstract. The idea of a co-t-structure is almost ‘dual ’ to that of a t-structure, but with some important differences. This note establishes co-t-structure analogues of Beligiannis and Reiten’s corresponding results on compactly generated t-structures. The notion of a co-t-structure on a triangulated category was introduced in-dependently by the author in [15] and Bondarko in [7]. In [7] they are referred to as weight structures; in this note we continue the terminology of [15]. In [7] they are introduced in the context of understanding Grothendieck’s weight filtration in Voevodsky’s triangulated category of motives; see also [6]. In [1] co-t-structures are important ingredients in the proofs of purity and decompo-sition theorems for ‘staggered sheaves’, and in [18] and [19] they are studied in the context of Chow motives and Artin-Tate motives, respectively. In the representation theoretic setting, co-t-structures have recently been studied in connection with the Auslander-Buchweitz context in [13]. In view of their recent proliferation into different branches of mathematics, it is useful to establish basic results regarding the structure and behaviour of co-t-structures, and, in particular, their relation with, similarities to, and differences from t-structures. In this sense, the present note should be viewed as an extension of [15] providing co-t-structure analogues of the corresponding results for t-structures in [4]. Throughout this note, T will be a triangulated category with set indexed coproducts and and Σ: T → T will denote its suspension functor. We direct the reader to [9] for an introduction to triangulated categories. We first recall some definitions. An object S of T is called rigid if HomT (S,ΣiS) = 0 for all i> 0 (see [11]); S is called compact if for any set indexed family of objects {Xi}i∈I of T one has the natural isomorphism HomT (S, i∈I Xi) ∼=∐ i∈I HomT (S,Xi). Recall also that a generating set for T is a set of objects G such that if HomT (G,X) = 0 for all G ∈ G then X = 0. In [10, Theorem 1.3], if one takes a compact rigid object S of T such that {ΣiS | i ∈ Z} is a generating set, then one obtains a canonical t-structure on