# Differential-algebra(2)

From Encyclopedia of Mathematics

(Redirected from Differential graded algebra)

An algebra $A$ over a field (ring) $K$ which is a differential ring and such that, moreover, any derivation $\partial$ commutes with multiplications by elements of $K$, i.e. $\partial(\alpha x) = \alpha \partial(x)$, where $\alpha \in K$, $x \in A$.

#### Comments

Cf. also Derivation in a ring.

A **differential graded algebra** (or DGA) over a ring $K$ is a graded algebra $A$ equipped with a graded $K$-module homomorphism $\partial : A \rightarrow A$ of degree $-1$ such that $\partial^2 = 0$ and such that $\partial$ is a derivation in the graded sense, i.e. such that $\partial(uv) = \partial(u) v + (-1)^{\deg u} u \partial(v)$. They are of importance in (co)homology theory.

#### References

[a1] | S. MacLane, "Homology" , Springer (1963) |

**How to Cite This Entry:**

Differential graded algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Differential_graded_algebra&oldid=39578