Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact triangulated categories. In this paper, we show that the idempotent completion an extriangulated category admits natural structure. As application, prove recollement is still recollement.

Recollements of abelian, resp. triangulated, categories are exact sequences of abelian, resp. triangulated, categories where the inclusion functor as well as the quotient functor have left and right adjoints. They appear quite naturally in various settings and are omnipresent in representation theory. Recollements which all categories involved are module categories (abelian case) or derived cat...

The Frobenius–Perron theory of an endofunctor a category was introduced in recent years (Chen et al. Algebra Number Theory 13(9):2005–2055, 2019; Chen for projective schemes. Preprint. arXiv:1907.02221 , 2019). We apply this to monoidal (or tensor) triangulated structures quiver representations.

We find stability conditions ([D2], [Br]) on some derived categories of differential graded modules over a graded algebra studied in [RZ], [KS]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derive...

We propose a general method to construct new triangulated categories, relative stable as additive quotients of given one. This construction enhances results Beligiannis, particularly in the tensor-triangular setting. prove birationality result showing that original category and its quotient are equivalent on some open piece their spectrum.

In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with natural, universal structure category. This construction unifies Serre quotient abelian categories and Verdier triangulated categories. Indeed give such for bit wider class morphisms, so it covers several other localizations appeared in literat...

Let k be a field and let D be a k-linear algebraic triangulated category with split idempotents. Let Σ be the suspension functor of D and let s be a 2-spherical object of D, that is, the morphism space D(s,Σs) is k for i = 0 and i = 2 and vanishes otherwise. Assume that s classically generates D, that is, each object of D can be built from s using (de)suspensions, direct sums, direct summands, ...

We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension zero, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of t...

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...

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 ...