In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geometric space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure.

In geometry, the de Rham algebra of differential forms on a manifold has the structure of a differential graded algebra, and it encodes the de Rham cohomology of the manifold. In algebraic topology, the singular cochains of a topological space form a DGA encoding the singular cohomology. Moreover, American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces.

Let ${\displaystyle A_{\bullet }=\bigoplus \nolimits _{i\in \mathbb {Z} }A_{i}}$ be a ${\displaystyle \mathbb {Z} }$-graded algebra, with product ${\displaystyle \cdot }$, equipped with a map ${\displaystyle d\colon A_{\bullet }\to A_{\bullet }}$ of degree ${\displaystyle -1}$ (homologically graded) or degree ${\displaystyle +1}$ (cohomologically graded). We say that ${\displaystyle (A_{\bullet },d,\cdot )}$ is a differential graded algebra if ${\displaystyle d}$ is a differential, giving ${\displaystyle A_{\bullet }}$ the structure of a chain complex or cochain complex (depending on the degree), and satisfies a graded Leibniz rule. In what follows, we will denote the "degree" of a homogeneous element ${\displaystyle a\in A_{i}}$ by ${\displaystyle |a|=i}$. Explicitly, the map ${\displaystyle d}$ satisfies the conditions

Often one omits the differential and multiplication and simply writes ${\displaystyle A_{\bullet }}$ or ${\displaystyle A}$ to refer to the DGA ${\displaystyle (A_{\bullet },d,\cdot )}$.

A linear map ${\displaystyle f:A_{\bullet }\to B_{\bullet }}$ between graded vector spaces is said to be of degree n if ${\displaystyle f(A_{i})\subseteq B_{i+n}}$ for all ${\displaystyle i}$. When considering (co)chain complexes, we restrict our attention to chain maps, that is, maps of degree 0 that commute with the differentials ${\displaystyle f\circ d_{A}=d_{B}\circ f}$. The morphisms in the category of DGAs are chain maps that are also algebra homomorphisms.

One can also define DGAs more abstractly using category theory. There is a category of chain complexes over a ring ${\displaystyle R}$, often denoted ${\displaystyle \operatorname {Ch} _{R}}$, whose objects are chain complexes and whose morphisms are chain maps. We define the tensor product of chain complexes ${\displaystyle (V,d_{V})}$ and ${\displaystyle (W,d_{W})}$ by

with differential

This operation makes ${\displaystyle \operatorname {Ch} _{R}}$ into a symmetric monoidal category. Then, we can equivalently define a differential graded algebra as a monoid object in ${\displaystyle \operatorname {Ch} _{R}}$. Heuristically, it is an object in ${\displaystyle \operatorname {Ch} _{R}}$ with an associative and unital multiplication.