The Equivariant Generating Hypothesis

On Freyd’s Generating Hypothesis

We revisit Freyd’s generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that I, the Brown-Comenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofiber of a self map of a wedge of spheres. We also show that a consequence of the generating hypothesi...

Freyd’s Generating Hypothesis with Almost Split Sequences

Freyd’s generating hypothesis for the stable module category of a nontrivial finite group G is the statement that a map between finitely generated kGmodules that belongs to the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. In this paper we show that Freyd’s generating hypothesis fails for kG when the Sylow p-subgroup of G has ord...

From falsification to generating an alternative hypothesis

Hypothesis-generating research and predictive medicine.

Genomics has profoundly changed biology by scaling data acquisition, which has provided researchers with the opportunity to interrogate biology in novel and creative ways. No longer constrained by low-throughput assays, researchers have developed hypothesis-generating approaches to understand the molecular basis of nature-both normal and pathological. The paradigm of hypothesis-generating resea...

The Generating Hypothesis in the Derived Category of a Ring

We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author [Loc07]. We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived catego...