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Analytic cyclic homology in positive characteristic

Ralf Meyer, Devarshi Mukherjee

Abstract

Let $V$ be a complete discrete valuation ring with residue field $\mathbb{F}$. We define a cyclic homology theory for algebras over $\mathbb{F}$, by lifting them to free algebras over $V$, which we enlarge to tube algebras and complete suitably. We show that this theory may be computed using any pro-dagger algebra lifting of an $\mathbb{F}$-algebra. We show that our theory is polynomially homotopy invariant, excisive, and matricially stable.

Analytic cyclic homology in positive characteristic

Abstract

Let be a complete discrete valuation ring with residue field . We define a cyclic homology theory for algebras over , by lifting them to free algebras over , which we enlarge to tube algebras and complete suitably. We show that this theory may be computed using any pro-dagger algebra lifting of an -algebra. We show that our theory is polynomially homotopy invariant, excisive, and matricially stable.

Paper Structure

This paper contains 9 sections, 36 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Bornologies and inductive systems
  3. The definition of analytic cyclic homology
  4. Bivariant analytic cyclic homology
  5. A variant of the definition
  6. Comparison of analytic cyclic homology theories
  7. Independence of the choice of lifting
  8. Liftings to pro-dagger algebras
  9. Homotopy invariance, matrix stability and excision

Key Result

Lemma 2.1

Let $M$ be a bornological $F$-vector space. There is a natural isomorphism $\varinjlim \mathop{\mathrm{diss}}\nolimits(M) \cong M$.

Theorems & Definitions (84)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 74 more