The <i>n</i>
-order of algebraic triangulated categories

Schwede, Stefan

doi:10.1112/jtopol/jtt014

Cited by 9 publications

(12 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Examples include the model categories of spectra [33], K (2) -local spectra [30], and modules over Eilenberg-Mac Lane ring spectra (see [36]). Moreover, the paper [34] shows that the homotopy category of spectra, also known as the stable homotopy category, is not algebraic.…”

Section: Introductionmentioning

confidence: 99%

The derived category of complex periodic K-theory localized at an odd prime

Patchkoria

2017

Advances in Mathematics

View full text Add to dashboard Cite

We prove that for an odd prime p, the derived category D(KU (p) ) of the p-local complex periodic K-theory spectrum KU (p) is triangulated equivalent to the derived category of its homotopy ring π * KU (p) . This implies that if p is an odd prime, the triangulated category D(KU (p) ) is algebraic.The derived category of complex periodic K-theory localized at an odd primeThe category D(R) has the important extra structure of a triangulated category. The triangulated structure comes from the mapping cone sequences (distinguished triangles)and in particular encodes how the objects of D(R) are built out of R using suspensions, desuspensions and cell attachments. Similarly, the derived category D(π * R) together with the algebraic mapping cone sequences is a triangulated category (see Subsection 2.2 for the details). The question arises whether the equivalence D(R) ∼ D(π * R) preserves the triangulated structures. More specifically, we are interested whether the equivalence D(KU ) ∼ D(π * KU ) is compatible with the distinguished triangles. In order to be able to formulate these questions more precisely, we recall some terminology:Definition. Let T 1 and T 2 be triangulated categories. A triangulated equivalence between T 1 and T 2 is an equivalence of categoriesHo(M) is called distinguished (or exact ) if it isomorphic to an elementary distinguished triangle in Ho(M). Example 2.2.1. For any ring k, the category Ch(k) of unbounded chain complexes of k-modules with the projective model structure is a stable model category [19, 2.3.11]. The weak equivalences and fibrations in this model structure are quasi-isomorphisms (i.e., homology isomorphisms) and degreewise epimorphisms, respectively.

show abstract

Section: Introductionmentioning

confidence: 99%

The derived category of complex periodic K-theory localized at an odd prime

Patchkoria

2017

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…An account of the classical argument using Steenrod operations can be found in [29,Proposition 4]. It is more subtle to show that the p-local stable homotopy category is not algebraic when p is an odd prime, and this is the main result of the companion paper; see [30,Theorem 3.1].…”

Section: Topological Triangulated Categoriesmentioning

confidence: 99%

“…Our main result is Theorem 5.3, saying that the p-order of a topological triangulated category is at least p − 1. In the companion paper [30], we prove some complementary results: the porder of every algebraic triangulated category is infinite [30,Theorem 2.2] and the p-order of the p-local stable homotopy category is at most p − 1 [30, Theorem 3.1]. Hence the p-order of the p-local stable homotopy category is exactly p − 1, and the p-local stable homotopy category is not algebraic for any prime p.…”

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

The p -order of topological triangulated categories

Schwede

2013

Journal of Topology

Self Cite

View full text Add to dashboard Cite

The p‐order of a triangulated category is an invariant that measures ‘how strongly’ p annihilates objects of the form Y/p. In this paper, we show that the p‐order of a topological triangulated category is at least p−1; here we call a triangulated category topological if it admits a model as a stable cofibration category. Our main new tools are enrichments of cofibration categories by Δ‐sets; in particular, we generalize the theory of ‘framings’ (or ‘cosimplicial resolutions’) from model categories to cofibration categories.

show abstract

“…Moreover, they proved that this particular triangulated category is neither algebraic nor topological. For a discussion on algebraic and topological triangulated categories, see, for example, [14][15][16].…”

Section: Introductionmentioning

confidence: 99%