Questions tagged [differential-graded-algebras]

A differential graded algebra (or simply DG-algebra) $A$ is a graded algebra equipped with a map $ d\colon A\to A$ which has either degree $1$ (cochain complex convention) or degree $ -1$ (chain complex convention) that satisfies two conditions:

• $d\circ d=0$.
  This says that $d$ gives $A$ the structure of a chain complex or cochain complex (accordingly as the differential reduces or raises degree).
• $ d(a\cdot b)=(da)\cdot b+(-1)^{\deg(a)}a\cdot (db) $, where $\deg$ is the degree of homogeneous elements.
  This says that the differential $d$ respects the graded Leibniz rule.

A more succinct (but esoteric) way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes. A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential $d$.

A differential graded augmented algebra (also called a DGA-algebra, an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground ring.

The homology $H_{*}(A)=\ker(d)/\operatorname {im} (d)$ of a DG-algebra $ (A,d) $ is a graded algebra. The homology of a DGA-algebra is an augmented algebra.

Source: Wikipedia