Tilting theory via stable homotopy theory

Abstract

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. Contents

Abstract representation theory of Dynkin quivers of type A

“…We show that stable derivators of abstract representations of A n -quivers are equivalent to derivators of coherent Auslander-Reiten quivers, i.e., derivators of certain representations of mesh categories (Theorem 4.15). These latter derivators allow us to conveniently encode reflection functors (see also [33] and [32]), (partial) Coxeter functors, and Serre functors. Along the way we establish an abstract fractionally Calabi-Yau property and give an explanation of it in down-to-earth terms.…”

“…The fact that these epimorphisms are split is a consequence of the abstract fractionally Calabi-Yau property. To illustrate how explicit the spectral bimodules are, we also revisit an example of [33] and compute the spectral bimodule realizing a strong stable equivalence between the commutative square and D 4 -quivers.…”

“…First steps of this program are carried out in [27] and [28]. While [29] and [33, § §9-10] are formal studies of the interaction of stability and monoidal structure, the papers [33] and [32] mainly provide first examples of strong stable equivalences which is to say abstract tilting results. In this paper the techniques of those papers are combined in order to develop aspects of what we think of as abstract representation theory of A n -quivers.…”

“…This is mainly to fix some notation and to recall a few facts which we use constantly in later sections. Derivators were introduced independently by Heller [39], Grothendieck [34], and Franke [23]. They were recently studied further by Maltsiniotis [63], Cisinski [15] and others.…”

“…This seemingly abstract calculus turns out to capture many typical constructions showing up in different areas of algebra, geometry, and topology; see for example the recent [27][28][29]33,32,73,72].…”