# Four approaches to cohomology theories with reality

## TL;DR

This paper compares four methods for computing RO(Q)-graded coefficients in basic Q-equivariant cohomology theories, highlighting the Tate square's advantages and providing a unified translation between different approaches.

## Contribution

It demonstrates the effectiveness of the Tate square over slice spectral sequences and unifies various computational approaches in Q-equivariant cohomology.

## Key findings

- Tate square outperforms slice spectral sequences in simple cases
- Unified framework for translating between different computational approaches
- Explicit calculations of RO(Q)-graded coefficients for basic theories

## Abstract

We give an account of well known calculations of the RO(Q)-graded coefficient rings of some of the most basic Q-equivariant cohomology theories, where Q is a group of order 2. One purpose is to advertise the effectiveness of the Tate square, showing it has advantages over the slice spectral sequences in algebraically simple cases. A second purpose is to give a single account showing how to translate between the languages of different approaches. [v2 corrects some typos and adds some thanks and references, v3 corrects a few more typos].