nLab
graded algebra

Idea

A graded algebra is an associative algebra 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.

Definition

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 $R_g\times R_{g\prime }\to R_{gg\prime }$.

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 internal to graded vector spaces.

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.

Properties

Proposition

For $R$ a commutative ring write $\mathrm{Spec}R\in {\mathrm{Ring}}^{\mathrm{op}}$ for the corresponding object in the opposite category. Write ${𝔾}_m$ for the multiplicative group underlying the affine line.

There is a natural isomorphism between

• $\mathbb{Z}$-gradings on $R$;

• ${𝔾}_m$-actions on $\mathrm{Spec}R$.

The proof is spelled out at affine line in the section Properties.

Example with $G$ not necessarily abelian.

Let $G$ be any (discrete) group and $k[G]$, its group algebra. This has a direct sum decomposition as a $k$-module,

$$k[G]=\underset{g\in G}{⨁}{L}_g$$k[G] = \bigoplus_{g\in G}L_g

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,

$${\ell }_{g_1}\cdot {\ell }_{g_2}={\ell }_{g_1{g}_2},$$\ell_{g_1}\cdot \ell_{g_2} = \ell_{g_1g_2},

given on basis elements, by $k$-linearity.