---
tag: GQVAXW5X09
kind: concept
name: Differential graded algebra
msc: ["16E45", 18G35]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_graded_algebra
same_as:
  wikidata: Q2916283
  wikipedia: Differential graded algebra
---

## Summary

A differential graded algebra, or dg algebra, is a [[graded algebra|8KB4SQGC0C]] with a differential $d$ of degree one.
The differential squares to zero and obeys a signed Leibniz rule.
Forgetting the product it is a [[chain complex|WQ82C0M3QB]], and its cohomology is again a graded algebra.
The de Rham complex of a smooth manifold, the singular cochains of a space with the cup product, and Koszul complexes are standard examples.
Dg algebras are the basic objects of homological algebra with products and of rational homotopy theory.

## Explanations

### L1 human

Differential forms on a manifold can be multiplied by the wedge product and differentiated by $d$.
Applying $d$ twice always gives zero, and $d$ of a product follows a product rule with a sign.
A differential graded algebra is any algebra with these two features.
A small example is polynomials $f(x)$ in degree $0$ together with the forms $g(x)\,dx$ in degree $1$, with $d f = f'(x)\, dx$.