Discreteness of silting objects and t-structures in triangulated categories

Abstract

We introduce the notion of ST‐pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite‐dimensional algebra. For an ST‐pair (C,D), we construct an injective order‐preserving map from silting objects in sans-serifC to bounded t‐structures on sans-serifD and show that the map is bijective if and only if sans-serifC is silting‐discrete if and only if sans-serifD is t‐discrete. Based on the work of Qiu and Woolf, the above result is applied to show that if sans-serifC is silting‐discrete then the stability space of sans-serifD is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi–Yau triangulated categories associated to Dynkin quivers.

Silting Objects

“…As a consequence, it is shown that the stability manifold of a derived‐discrete algebra of finite global dimension is always contractible. More recently, this result has been extended to the class of all silting‐discrete algebras in , and independently, in . Further results relating stability conditions with silting theory can be found in .…”

“…A few words on the last condition are in order. In [, Proposition 3.27], it is shown that for any basic silting complex with endomorphism ring , there is a bijection between the subset of consisting of the objects such that and the functorially finite torsion classes in . Keeping in mind that the latter are in bijection with isomorphism classes of basic support ‐tilting ‐modules (Theorem ), one obtains the equivalence of conditions (4) and (5).…”

Spherical objects and stability conditions on 2-Calabi--Yau quiver categories

“…In the main text, Theorem 1.2 is Corollary 5.5. Theorem 1.2 has been proved in [5,6] for type A, and in [1] for types A, D, and E.…”

“…In the main text, Theorem 1.1 is Corollary 5.2. A proof of this theorem in type A appears in [5,6] and may also follow for all ADE types from the ideas of [1].…”

Silting and Tilting for Weakly Symmetric Algebras

“…The silting-discreteness property also has particularly nice implications on the Bridgeland stability manifold associated to D b (mod A) [2,15] -a topological invariant related to…”