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A cyclotomic spectrum is an -equivariant spectrum with fixed points for all the finite cyclic groups inside the circle group, and equipped with -equivariant identifications of the -fixed points with the full object.
The topological Hochschild homology-spectra are naturally cyclotomic spectra, and this is where the concept originates: by the discussion at Hochschild cohomology is the E-infinity ring of functions on the free loop space of , and cyclotomic structure reflects the structure of free loop spaces: loops that repeat with period are equivalent to plain loops.
Cyclotomic structure is the origin of the cyclotomic trace map from topological Hochschild homology to topological cyclic homology.
Throughout, for a prime number write for the cyclic group of order , regarded as a subgroup of the circle group.
A definition says that a cyclotomic spectrum is an circle group-genuine equivariant spectrum (modeled on orthogonal spectra) equipped with equivalences to its geometric fixed point spectra for all the cyclic subgroups .
A more abstract definition was given in Nikolaus-Scholze 17:1
A cyclotomic spectrum is
a spectrum
a circle group ∞-action on , i.e. an (∞,1)-functor which takes the unique point of to ;
for each prime number a homomorphism of spectra with such circle group action
to the Tate spectrum (the homotopy cofiber of the norm map),
where the circle action on the Tate spectrum comes from the canonical identification .
(These morphisms are called the Frobenius morphisms of the cyclotomic structure, due to this def., this example).
(Nikolaus-Scholze 17, def. 1.3, def. II.1.1).
For a spectrum with stable homotopy groups bounded below, then def. is equivaent to the traditional:
There is an (∞,1)-functor
from traditional (“genuine”) cyclotomic spectra bounded below to bounded below cyclotomic spectra in the sense of def. , and this is an equivalence of (∞,1)-categories.
(Nikolaus-Scholze 17, prop. II.3.4, theorem II.6.9).
(topological Hochschild homology)
For every A-∞ ring , the topological Hochschild homology spectrum naturally carries the structure of a cyclotomic spectrum (def. ).
(Nikolaus-Scholze 17, section II.2, def. III.2.3)
(trivial cyclotomic spectra)
Every spectrum becomes a cyclotomic spectrum in the sense of def. by equipping it
with the trivial circle group ∞-action
for each prime with the composite morphism
(the first being the -unit into the homotopy fixed points, the second the defining morphism into the Tate spectrum )
the -equivariant structure on these morphisms given under the adjunction between trivial action and homotopy fixed points by the adjunct morphisms
as the composite
This construction constitutes a left adjoint (infinity,1)-functor to taking topological cyclic homology
(Nikolaus-Scholze 17, example II.1.2 (ii) and middle of p. 126)
(cyclotomic sphere spectrum)
The sphere spectrum regarded as a cyclotomic spectrum via example is called the cyclotomic sphere spectrum.
As such it is equivalently its topological Hochschild homology according to example :
(Nikolaus-Scholze 17, example II.1.2 (ii))
The tensor unit in the symmetric monoidal (infinity,1)-category of cyclotomic spectra is the cyclotomic sphere spectrum from example (Blumberg-Mandell 13, example 4.9)
Andrew Blumberg, Michael Mandell, The homotopy theory of cyclotomic spectra, Geom. Topol. 19 (2015) 3105-3147 [arXiv:1303.1694, doi:10.2140/gt.2015.19.3105]
Clark Barwick, Saul Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin (arXiv:1602.02163)
Clark Barwick, Saul Glasman, Noncommutative syntomic realization (pdf)
Cary Malkiewich: A visual introduction to cyclic sets and cyclotomic spectra, talk at Young Topologists Meeting Lausanne, Switzerland (2015) [pdf, pdf]
Thomas Nikolaus, Peter Scholze, On topological cyclic homology, Acta Math. 221 2 (2018) 203-409 [arXiv:1707.01799, doi:10.4310/ACTA.2018.v221.n2.a1]
Last revised on September 26, 2024 at 11:25:04. See the history of this page for a list of all contributions to it.