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On the K-theory of algebraic Cuntz-Pimsner rings

Thibaut Lescure

Abstract

We establish a long exact sequence for the homotopy K-theory groups of the algebraic Cuntz-Pimsner rings introduced by Carlsen and Ortega [CO11] by adapting Pimsner's original proof [Pim97] to Cuntz's formalism.

On the K-theory of algebraic Cuntz-Pimsner rings

Abstract

We establish a long exact sequence for the homotopy K-theory groups of the algebraic Cuntz-Pimsner rings introduced by Carlsen and Ortega [CO11] by adapting Pimsner's original proof [Pim97] to Cuntz's formalism.

Paper Structure

This paper contains 5 sections, 15 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Burgstaller's functional modules and $M$-stable functors
  4. Algebraic Cuntz-Pimsner rings
  5. Proofs of main results

Key Result

Theorem A

Let $R$ be a ring with local units. Let $\mathcal{X}$ be an $R$-correspondence. Let $\mathcal{T}_{\mathcal{X}}$ be the corresponding Toeplitz ring. Every homotopy invariant, split-exact and $M$-stable functor $E$ sends the inclusion $R \rightarrow \mathcal{T}_{\mathcal{X}}$ to an isomorphism $E(R) \

Theorems & Definitions (38)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • ...and 28 more