and
and
nonabelian homological algebra
A graded algebra is an associative algebra which is with a labelling on its elements by elements of some monoid or group, and such that the multiplication in the algebra is reflected in the multiplication in the labelling group.
Let $G$ be a group. (Often $G$ will be abelian, and, in fact, one usually takes by default $G = \mathbb{Z}$ the additive group of integers, in which case the actual group being used is omitted from the terminology and notation.)
A graded ring is a ring $R$ equipped with a decomposition of the underlying abelian group as a direct sum $R = \oplus_{g \in G} R_g$ such that the product takes $R_{g} \times R_{g'} \to R_{g g'}$.
Analogously there is the notion of graded $k$-associative algebra over any commutative ring $k$.
Specifically for $k$ a field a graded algebra is a monoid in graded vector spaces over $k$.
An $\mathbb{N}$-graded algebra is called connected if in degree-0 it is just the ground ring
A differential graded algebra is a graded algebra $A$ equipped with a derivation $d : A\to A$ of degree +1 (or -1, dependig on conventions) and such that $d \circ d = 0$. This is the same as a monoid in the category of chain complexes.
For $R$ a commutative ring write $Spec R \in Ring^{op}$ for the corresponding object in the opposite category. Write $\mathbb{G}_m$ for the multiplicative group underlying the affine line.
There is a natural isomorphism between
$\mathbb{Z}$-gradings on $R$;
$\mathbb{G}_m$-actions on $Spec R$.
The proof is spelled out at affine line in the section Properties.
Let $G$ be any (discrete) group and $k[G]$, its group algebra. This has a direct sum decomposition as a $k$-module,
where each $L_g$ is a one dimensional free $k$-module, for which it is convenient, here, to give a basis $\{\ell_g\}$. The graded algebra structure is obtained by extending the multiplication rule,
given on basis elements, by $k$-linearity.
The Lazard ring, carrying the universal (1-dimensional, commutative) formal group law is naturally an $\mathbb{N}$-graded ring.