# Excision in algebraic K-theory revisited

## TL;DR

This paper provides a new, direct proof of Suslin's theorem on excision in algebraic K-theory for Tor-unital rings, extending to descent results for ring spectra and clarifying the role of Tor-unitality.

## Contribution

It offers a novel proof of Suslin's excision theorem and generalizes it to descent for ring spectra using an exact sequence of perfect modules.

## Key findings

- New proof of Suslin's theorem on excision in algebraic K-theory.
- Extension of excision to descent in ring spectra.
- Clarification of the role of Tor-unitality in K-theory.

## Abstract

By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic K-theory. We give a new and direct proof of Suslin's result based on an exact sequence of categories of perfect modules. In fact, we prove a more general descent result for a pullback square of ring spectra and any localizing invariant. Besides Suslin's result, this also contains Nisnevich descent of algebraic K-theory for affine schemes as a special case. Moreover, the role of the Tor-unitality condition becomes very transparent.