---
tag: RDT4785M4D
kind: concept
name: Cohomology ring
msc: [55N45, 55N10]
sources:
  - url: https://en.wikipedia.org/wiki/Cohomology_ring
same_as:
  wikidata: Q5141399
  wikipedia: Cohomology ring
---

## Summary

The cohomology ring of a space $X$ with coefficients in a commutative ring $R$ is the direct sum $H^\bullet(X; R) = \bigoplus_{k \ge 0} H^k(X; R)$ of its [[singular cohomology|6N7BC70CX0]] groups.
Its multiplication is the [[cup product|JX47MS1R1H]].
It is a graded commutative ring and a homotopy invariant of $X$.
It separates spaces that have isomorphic homology groups, and it is the setting for characteristic classes and cohomology operations.

## Explanations

### L1 human

The cohomology groups of a space can be added, and with the cup product they can also be multiplied.
Putting all degrees together gives one ring, much like a ring of polynomials where each variable has a degree.
For the complex projective plane, the ring is generated by one class $x$ of degree $2$ with $x^3 = 0$, while $x^2$ is not zero.
The space made by joining a $2$-sphere and a $4$-sphere at a point has the same groups, but there the square of the degree $2$ class is zero.
So the two spaces are not homotopy equivalent.