Using the localization property, we construct triangulated categories of motives over quasi-projective $T$-schemes for any coefficient where $T$ is a noetherian separated scheme, and prove Grothendieck six operations formalism. We also integral étale realization motives.

We use a K-theory recipe of Thomason to obtain classi cations of triangulated subcategories via re ning some standard thick subcategory theorems. We apply this recipe to the full subcategories of nite objects in the derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classi cation of the triangulated subcategories of perfect...

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 D ( k[X]/(X) ) , the compact deri...

For a Calabi-Yau triangulated category C of Calabi-Yau dimension d with a d−cluster tilting subcategory T , the decomposition of C is determined by the decomposition of T satisfying ”vanishing condition” of negative extension groups, namely, C = ⊕i∈ICi, where Ci, i ∈ I are triangulated subcategories, if and only if T = ⊕i∈ITi, where Ti, i ∈ I are subcategories with HomC(Ti[t],T j) = 0,∀1 ≤ t ≤ ...

Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4-manifolds usi...

We prove a structure theorem for triangulated Calabi-Yau categories: An algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable categor...