5 research outputs found
A chromatic vanishing result for TR
In this note, we establish a vanishing result for telescopically localized
. More precisely, we prove that -local
vanishes on connective -acyclic -rings for every and deduce consequences for connective Morava K-theory and the
Thom spectra . The proof relies on the relationship between
and the spectrum of curves on K-theory together with fact that algebraic
K-theory preserves infinite products of additive -categories which was
recently established by C\'{o}rdova Fedeli.Comment: 8 pages. Comments very welcome
Polygonic spectra and TR with coefficients
We introduce the notion of a polygonic spectrum which is designed to
axiomatize the structure on topological Hochschild homology
of an -ring with coefficients in an -bimodule . For
every polygonic spectrum , we define a spectrum as the
mapping spectrum from the polygonic version of the sphere spectrum
to . In particular if applied to this gives a
conceptual definition of .
Every cyclotomic spectrum gives rise to a polygonic spectrum and we prove
that TR agrees with the classical definition of TR in this case. We construct
Frobenius and Verschiebung maps on by exhibiting
as the -fixedpoints of a quasifinitely genuine
-spectrum. The notion of quasifinitely genuine -spectra
is a new notion that we introduce and discuss inspired by a similar notion over
introduced by Kaledin. Besides the usual coherences for genuine
spectra, this notion additionally encodes that admits certain
infinite sums of Verschiebung maps.Comment: 61 pages, comments are welcom