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On the geometric fixed-points of real topological cyclic homology

Emanuele Dotto, Kristian Moi, Irakli Patchkoria

Abstract

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to carry out computations in the fundamental examples of spherical group-rings, perfect $\mathbb{F}_p$-algebras, and $2$-torsion free rings with perfect modulo $2$ reduction. Our calculations agree with the normal L-theory spectrum in the cases where the latter is known, as conjectured by Nikolaus.

On the geometric fixed-points of real topological cyclic homology

Abstract

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to carry out computations in the fundamental examples of spherical group-rings, perfect -algebras, and -torsion free rings with perfect modulo reduction. Our calculations agree with the normal L-theory spectrum in the cases where the latter is known, as conjectured by Nikolaus.

Paper Structure

This paper contains 21 sections, 39 theorems, 270 equations.

Table of Contents

  1. Preliminaries
  2. Equivariant spectra
  3. Ring spectra with anti-involution and real topological Hochschild homology
  4. Real cyclotomic spectra and real topological cyclic homology
  5. The Geometric fixed-points of TCR
  6. The odd primary case
  7. The prime 2
  8. The geometric fixed-points of TRR
  9. The geometric fixed-points of TFR and TCR
  10. TCR of spherical monoid-rings
  11. TCR of spherical monoid-rings and assembly
  12. TCR of spherical group-rings for some discrete groups
  13. TCR of perfect fields
  14. TCR of perfect fields of odd characteristic
  15. TCR of perfect fields of characteristic 2
  16. ...and 6 more sections

Key Result

Theorem 1

Let $A$ be a ring spectrum with anti-involution, and suppose that the underlying spectrum and the $\mathop{\mathrm{\mathbb{Z}}}\nolimits/2$-fixed-points of $A$ are bounded below. Then for every odd prime $p$ there is a natural equivalence of spectra For the prime $2$, there is a natural equivalence with the homotopy equaliser where $f$ forgets the fixed-points and $r$ maps to the $C_2$-geometric

Theorems & Definitions (90)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Definition 1.4
  • ...and 80 more