
TL;DR
This paper explores nilpotence detection in the homotopy of _n-algebras, revealing limitations of classical criteria and constructing examples with non-nilpotent torsion at various chromatic heights.
Contribution
It demonstrates that classical nilpotence detection via the Hurewicz homomorphism does not always hold for _n-algebras and provides explicit counterexamples at different heights.
Findings
Constructed _{2n-1}-algebras with non-nilpotent p^n-torsion
Showed the bound is sharp at height 1 using Bousfield-Kuhn functor and Rezk's logarithm
Discussed the situation at height 2 and limitations of classical detection methods
Abstract
Nilpotence in the homotopy of -ring spectra is detected by the classical -Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of -ring spectra. For all odd primes and all chromatic heights , we use the Cohen-Moore-Neisendorfer theorem to construct examples of -local, -algebras with non-nilpotent -torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound is sharp at height , and remark on the situation at height .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Advanced Topology and Set Theory
