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A Quillen model structure of local homotopy equivalences

Devarshi Mukherjee, Guillermo Cortiñas

Abstract

In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring $V$, the homotopy category of this model structure is the derived category of the quasi-abelian category $\overleftarrow{\mathsf{Ind}(\mathsf{Ban}_F)}$. This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree $V$-algebras and $\mathbb{F}$-algebras. When the base field is $\mathbb{C}$, the homotopy category is the target of local and analytic cyclic homology for pro-bornological $\mathbb{C}$-algebras, which includes the subcategory of pro-$C^*$-algebras.

A Quillen model structure of local homotopy equivalences

Abstract

In this note, we construct a closed model structure on the category of -graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field of a complete discrete valuation ring , the homotopy category of this model structure is the derived category of the quasi-abelian category . This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree -algebras and -algebras. When the base field is , the homotopy category is the target of local and analytic cyclic homology for pro-bornological -algebras, which includes the subcategory of pro--algebras.
Paper Structure (7 sections, 23 theorems, 30 equations)

This paper contains 7 sections, 23 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. An exact structure on pro-objects in E
  3. From exact to model categories
  4. Bivariant local and analytic cyclic homology
  5. Local cyclic homology for nonarchimedean Banach algebras
  6. Local cyclic homology for pro-$C^*$-algebras
  7. Chern character

Key Result

Theorem 1

Let $\mathcal{C}$ be an exact category with enough projectives. Then the category $\overleftarrow{\mathsf{Kom}(\mathsf{Ind}((\mathcal{C})))}$ carries an injective model structure where the weak equivalences are the local weak equivalences. Thus for the associated homotopy category we have This applies, in particular, when $\mathcal{C}$ is the category of Banach spaces over $\mathbb R$, $\mathbb C

Theorems & Definitions (48)

  • Theorem 1
  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • remark 1: Indisation of an exact category
  • Definition 3
  • lemma 1
  • ...and 38 more