Differential graded algebra

The tensor algebra is a DG-algebra with differential similar to that of the Koszul complex. For a vector space ${\displaystyle V}$ over a field ${\displaystyle K}$ there is a graded vector space ${\displaystyle T(V)}$ defined as

${\displaystyle T(V)=\bigoplus _{i\geq 0}T^{i}(V)=\bigoplus _{i\geq 0}V^{\otimes i}}$

where ${\displaystyle V^{\otimes 0}=K}$.

If ${\displaystyle e_{1},\ldots ,e_{n}}$ is a basis for ${\displaystyle V}$ there is a differential ${\displaystyle d}$ on the tensor algebra defined component-wise

${\displaystyle d:T^{k}(V)\to T^{k-1}(V)}$

sending basis elements to

${\displaystyle d(e_{i_{1}}\otimes \cdots \otimes e_{i_{k}})=\sum _{1\leq j\leq k}e_{i_{1}}\otimes \cdots \otimes d(e_{i_{j}})\otimes \cdots \otimes e_{i_{k}}}$

In particular we have ${\displaystyle d(e_{i})=(-1)^{i}}$ and so

${\displaystyle d(e_{i_{1}}\otimes \cdots \otimes e_{i_{k}})=\sum _{1\leq j\leq k}(-1)^{i_{j}}e_{i_{1}}\otimes \cdots \otimes e_{i_{j-1}}\otimes e_{i_{j+1}}\otimes \cdots \otimes e_{i_{k}}}$

One of the foundational examples of a differential graded algebra, widely used in commutative algebra and algebraic geometry, is the Koszul complex. This is because of its wide array of applications, including constructing flat resolutions of complete intersections, and from a derived perspective, they give the derived algebra representing a derived critical locus.

Differential forms on a manifold, together with the exterior derivation and the exterior product form a DG-algebra. These have wide applications, including in derived deformation theory.[2] See also de Rham cohomology.

• The singular cohomology of a topological space with coefficients in ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence ${\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to \mathbb {Z} /p^{2}\mathbb {Z} \to \mathbb {Z} /p\mathbb {Z} \to 0}$, and the product is given by the cup product. This differential graded algebra was used to help compute the cohomology of Eilenberg–MacLane spaces in the Cartan seminar.[3][4]