Compact Corigid Objects in Triangulated Categories and Co-t-structures

In the work of Hoshino, Kato and Miyachi, the authors look at t-structures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on T whose heart es equivalent to Mod(End(C)^op). Rigid objects in a triangulated category can be thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, S, of a triangulated category, T, induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End(S)^op), and hence an abelian subcategory of T.Comment: 21 pages, reorganised paper with added material and examples of t-structures and co-t-structure

Homological Epimorphisms of Differential Graded Algebras

Let R and S be differential graded algebras. In this paper we give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories M\otimes - :D(S)--> D(R). In particular, this characterisation generalises the theory of Geigle and Lenzing's homological epimorphisms of rings. Furthermore, there is an application of the main result to Dwyer and Greenlees's Morita theory.Comment: 14 page

Torsion pairs in a triangulated category generated by a spherical object

We extend Ng's characterisation of torsion pairs in the 2-Calabi-Yau triangulated category generated by a 2-spherical object to the characterisation of torsion pairs in the w-Calabi-Yau triangulated category, $T_w$, generated by a w-spherical object for any integer w. Inspired by the combinatorics of $T_w$ for w < 0, we also characterise the torsion pairs in a certain w-Calabi-Yau orbit category of the bounded derived category of the path algebra of Dynkin type A.Comment: v2: 36 pages, 11 figures, added Section 4 which deals with extensions whose outer terms are decomposable, minor changes in presentation, accepted in J. Algebr

Averaging t-structures and extension closure of aisles

We ask when a finite set of t-structures in a triangulated category can be `averaged' into one t-structure or, equivalently, when the extension closure of a finite set of aisles is again an aisle. There is a straightforward, positive answer for a finite set of compactly generated t-structures in a big triangulated category. For piecewise tame hereditary categories, we give a criterion for when averaging is possible, and an algorithm that computes truncation triangles in this case. A finite group action on a triangulated category gives a natural way of producing a finite set of t-structures out of a given one. If averaging is possible, there is an induced t-structure on the equivariant triangulated category.Comment: 26 pages, 11 figures. v2: fixed minor mistakes, improved presentation. Comments still welcome

The co-stability manifold of a triangulated category

Stability conditions on triangulated categories were introduced by Bridgeland as a 'continuous' generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold which has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is the compact derived category of the dual numbers over an algebraically closed field. This is one of the motivations in this paper for introducing co-stability conditions as a 'continuous' generalisation of co-t-structures. Our main result is that the set of nice co-stability conditions on a triangulated category is a manifold. In particular, we show that the co-stability manifold of the compact derived category of the dual numbers is the complex numbers.Comment: 14 page