# A few examples of $R-$good and $R-$bad classifying spaces

## TL;DR

This paper explores the properties of classifying spaces related to a commutative ring R, highlighting the limitations of their 'good' and 'bad' classifications beyond prime-based completions.

## Contribution

It provides examples illustrating the constraints on classifying spaces being simultaneously 'good' and 'bad' for arbitrary rings R.

## Key findings

- Spaces cannot be arbitrarily good or bad for a given ring R.
- The contrast with Bousfield-Kan completion at a prime is emphasized.
- Examples demonstrate the specific behavior of classifying spaces in this context.

## Abstract

For a commutative ring $R$, in contrast to the completion in the sense of Bousfield and Kan at just a prime integer, there cannot exist spaces which are good and bad in an arbitrary way.