---
tag: JWTTXSGD0E
kind: concept
name: Steenrod algebra
msc: [55S10]
sources:
  - url: https://en.wikipedia.org/wiki/Steenrod_algebra
same_as:
  wikidata: Q7605884
  wikipedia: Steenrod algebra
---

## Summary

The mod $2$ Steenrod algebra $\mathcal{A}_2$ is the graded algebra over $\Z/2\Z$ generated by the [[Steenrod squares|VR7RHP41M0]], subject to the [[Adem relations|GS8SCGYPX7]].
It is the algebra of all stable [[cohomology operations|K8T15QHN3W]] on mod $2$ cohomology.
So the mod $2$ cohomology of every space is a module over it.
For each odd prime $p$ there is an analogous algebra built from the reduced powers and the Bockstein.
The Steenrod algebra is a Hopf algebra, and it is the input of the [[Adams spectral sequence|DD4D0EKX74]], which computes stable homotopy groups.

## Explanations

### L1 human

Steenrod squares can be composed, and compositions of them are again cohomology operations.
The Steenrod algebra collects all such compositions and their sums.
The squares do not commute freely: certain compositions can be rewritten in terms of others, and the Adem relations list the rules.
For example, applying $\operatorname{Sq}^1$ twice always gives zero.