Tilting theory via stable homotopy theory

Abstract We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie’s strongly (co)cartesian n-cubes. As applications we construct abstract Auslander–Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi–Yau property. This is used to offer a new perspective on May’s axioms for monoidal, triangulated categories.

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Cohomologically triangulated categories I

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is007011018jkt019 ◽

2008 ◽

Vol 1 (1) ◽

pp. 3-48 ◽

Cited By ~ 4

Author(s):

H.-J. Baues ◽

F. Muro

Keyword(s):

Cohomology Class ◽

Homotopy Theory ◽

Stable Homotopy ◽

Triangulated Categories ◽

Stable Homotopy Theory ◽

Additive Category

AbstractWe introduce cohomologically triangulated categories as triples (A,t,▽) given by an additive category A, an additive equivalence t:AA and a cohomology class ▽ in the translation cohomology H3(A,t). A stable homotopy theory C with A = HoC yields such a triple and the class of distinguished triangles in A is deduced from ▽.

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Tilting theory for trees via stable homotopy theory

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2015.11.009 ◽

2016 ◽

Vol 220 (6) ◽

pp. 2324-2363 ◽

Cited By ~ 2

Author(s):

Moritz Groth ◽

Jan Šťovíček

Keyword(s):

Homotopy Theory ◽

Stable Homotopy ◽

Stable Homotopy Theory ◽

Tilting Theory

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The Picard Group of Equivariant Stable Homotopy Theory

Advances in Mathematics ◽

10.1006/aima.2001.1997 ◽

2001 ◽

Vol 163 (1) ◽

pp. 17-33 ◽

Cited By ~ 7

Author(s):

H. Fausk ◽

L.G. Lewis ◽

J.P. May

Keyword(s):

Homotopy Theory ◽

Picard Group ◽

Stable Homotopy ◽

Stable Homotopy Theory ◽

Equivariant Stable Homotopy

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v 1 - and v 2 -Periodicity in Stable Homotopy Theory

American Journal of Mathematics ◽

10.2307/2374044 ◽

1981 ◽

Vol 103 (4) ◽

pp. 615 ◽

Cited By ~ 13

Author(s):

Donald M. Davis ◽

Mark Mahowald

Keyword(s):

Homotopy Theory ◽

Stable Homotopy ◽

Stable Homotopy Theory

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Spectra of derived module homomorphisms

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066603 ◽

1987 ◽

Vol 101 (2) ◽

pp. 249-257 ◽

Cited By ~ 6

Author(s):

Alan Robinson

Keyword(s):

Spectral Sequence ◽

Homotopy Theory ◽

Stable Homotopy ◽

Homotopy Groups ◽

Stable Homotopy Theory ◽

Ring Spectrum ◽

Homotopy Invariant ◽

New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

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