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

TL;DR

This paper advances algebraic K-theory by explicitly characterizing the key group in a generalized Mayer-Vietoris sequence, providing constructive proofs and a homotopy-theoretic perspective.

Contribution

It offers a new, explicit description of the group X in the Mayer-Vietoris sequence for Milnor squares, including a categorical and homotopy-theoretic analysis.

Findings

Explicit characterization of the group X as a categorical pullback

Construction of a structural exact sequence involving relative K-groups

Homotopy-theoretic description of X as a homotopy group of a fiber

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.