The spectrum of prime ideals in tensor triangulated categories

The Spectrum of Prime Ideals in Tensor Triangulated Categories

We define the spectrum of a tensor triangulated category K as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects of K. This construction is functorial with respect to all tensor triangulated functors. Several elementary properties of schemes hold for such spaces, e.g. the ...

Triangulated categories and the Ziegler spectrum

Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta

We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal of morphisms with certain properties, and that if has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study v...

Objects in Triangulated Categories

We introduce the Calabi-Yau (CY) objects in a Hom-finite Krull-Schmidt triangulated k-category, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and Auslander-Reiten triangles is provided. Finally we classify all the CY modules of selfinjective Nakayama algebras, determining this way the self-injectiv...

Localizations in Triangulated Categories and Model Categories

Recall that for a triangulated category T , a Bousfield localization is an exact functor L : T → T which is coaugmented (there is a natural transformation Id → L; sometimes L is referred to as a pointed endofunctor) and idempotent (there is a natural isomorphism Lη = ηL : L → LL). The kernel ker(L) is the collection of objects X such that LX = 0. If T is closed under coproducts, it’s a localizi...