On the cogeneration of t-structures

Abstract.

Let C be a set of objects in a triangulated compactly generated category $\textsf{T}.$ We denote by ${\cal U}_C\,(_C{\cal U})$ the smallest suspended subcategory closed under coproducts which contains C (the smallest cosuspended subcategory closed under products which contains C). We prove that if C is a set of compacts objects then $({^{\perp}}_{T_{C}I}{\cal U}, \, _{T_{C}I}{\cal U}[1]),$ is a t-structure in $\textsf{T},$ where T _C I is the dual of C with respect to an injective cogenerator I in the category Mod C. Moreover, we show that: C is a tilting set if and only if ${\cal U}_C^{\perp}[1] = _{T_{C}I}\cal U.$ And, this is equivalent to T _C I is a cotilting object in $\textsf{T}.$