SILTING OBJECTS, t-STRUCTURES AND DERIVED EQUIVALENCES CHRYSOSTOMOS PSAROUDAKIS Abstract. This note is an extended abstract of my talk given in the conference: “Maurice Auslander Distinguished Lectures and International Conference”, April 29 - May 4, 2015. It is based on [6] which is joint work with Jorge Vit´ oria. Contents 1. Motivation and Preliminaries 1 2. (Co)Silting Objects in Triangulated Categories 2 3. The Realisation Functor and Derived Equivalences 3 References 5 1. Motivation and Preliminaries Let R and S be two unital associative rings such that there is a triangle equivalence φ : D b (R) ≃ −→ D b (S) between the bounded derived categories of Mod-R and Mod-S. Then it is known that the object T := φ(R) is a tilting complex in D b (S) and that the ring R is the endomorphism ring End D b (S) (T ). Hence, via the equivalence φ we recover the module category of R via the module category of End D b (S) (T ). Another approach is to look where the standard t-structure generated by R is mapped via φ. For this reason, we recall from Beilinson-Bernstein-Deligne [4] the notion of a t-structure in triangulated categories. Definition 1.1. [4] Let T be a triangulated category. A t-structure in T is a pair (T ≤0 , T ≥0 ) of full subcategories such that, for T ≤n =Σ −n (T ≤0 ) and T ≥n =Σ −n (T ≥0 )(n ∈ Z), the following conditions are satisfied : (i) Hom T (T ≤0 , T ≥1 ) = 0, i.e. Hom T (X, Y ) = 0 for all X in T ≤0 and Y in T ≥1 . (ii) T ≤0 ⊆ T ≤1 and T ≥1 ⊆ T ≥0 . (iii) For every object D in T there exists a triangle X −→ D −→ Y −→ Σ(X) such that X lies in T ≤0 and Y lies in T ≥1 . In the above situation we denote by τ ≤0 (D) := X and τ ≥1 (D) := Y . Note that these objects are functorially determined. One of the main features of a t-structure is that it provides a full abelian subcategory (the heart) in T together with a cohomological functor. In particular, we have the next result. Theorem 1.2. [4] Let T be a triangulated category with a t-structure (T ≤0 , T ≥0 ). Then the heart H := T ≤0 ∩ T ≥0 is an abelian category and there is a cohomological functor H 0 : T −→ H given by H 0 = τ ≤0 τ ≥0 = τ ≥0 τ ≤0 . Note that the exact structure of H is induced from T and there is an isomorphism Ext 1 H (X, Y ) ∼ = Hom T (X, Y [1]) for all X, Y ∈ H. Now we return to the discussion we had in the beginning of this section. For a ring R we have the standard t-structure : D ≤0 R =  X ∈ D b (R) | H i (X)=0 ∀i> 0  and D ≥0 R =  X ∈ D b (R) | H i (X)=0 ∀i< 0  in D b (R), where H i is the usual cohomology of complexes. We leave to the reader to check that this is indeed a t-structure. Using that H i (X) = 0 if and only if Hom D b (R) (R, X[i]) = 0 we have D ≤0 R = R ⊥>0 :=  X ∈ D b (R) | Hom D b (R) (R, X[i]) = 0 ∀i> 0  The author is grateful to the organizers: Kiyoshi Igusa, Alex Martsinkovsky and Gordana Todorov, for the invitation to give this talk. 1