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A general Mayer-Vietoris sequence in algebraic $K$-theory

Yakun Zhang

Abstract

This paper investigates the Mayer-Vietoris sequence for the Milnor square. While such sequences often involve elusive intermediate terms, we provide an explicit characterization of the key group $X$ in a new, more general variant of the sequence. By identifying $X$ as a categorical pullback, we provide a full, constructive proof of the modified Mayer-Vietoris sequence. Furthermore, we show that $X$ fits into a structural exact sequence involving the relative $K$-groups $K_{*}(A, B, I)$. Finally, we provide a homotopy-theoretic description of $X$ as the homotopy group of a suitable fiber, clarifying its structure, kernel , and image.

A general Mayer-Vietoris sequence in algebraic $K$-theory

Abstract

This paper investigates the Mayer-Vietoris sequence for the Milnor square. While such sequences often involve elusive intermediate terms, we provide an explicit characterization of the key group in a new, more general variant of the sequence. By identifying as a categorical pullback, we provide a full, constructive proof of the modified Mayer-Vietoris sequence. Furthermore, we show that fits into a structural exact sequence involving the relative -groups . Finally, we provide a homotopy-theoretic description of as the homotopy group of a suitable fiber, clarifying its structure, kernel , and image.
Paper Structure (2 sections, 3 theorems, 5 equations)

This paper contains 2 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Generalized Mayer-Vietoris sequence

Key Result

Proposition 2.1

Let $R$ be a commutative ring and $A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D$ be an exact sequence of $R$-modules.

Theorems & Definitions (4)

  • Proposition 2.1: Stability of Exactness
  • Proposition 2.2: Lifting of Exactness
  • Theorem 2.3
  • proof