nLab graded algebra



Differential-graded objects

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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 GG be a group. (Often GG will be abelian, and, in fact, one usually takes by default G=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 RR equipped with a decomposition of the underlying abelian group as a direct sum R= gGR gR = \oplus_{g \in G} R_g such that the product takes R g×R gR ggR_{g} \times R_{g'} \to R_{g g'}.

Analogously there is the notion of graded kk-associative algebra over any commutative ring kk.

Specifically for kk a field a graded algebra is a monoid in graded vector spaces over kk.

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 AA equipped with a derivation d:AAd : A\to A of degree +1 (or -1, depending on conventions) and such that dd=0d \circ d = 0. This is the same as a monoid in the category of chain complexes.

A \mathbb{N}-graded algebra is called strongly \mathbb{N}-graded (in Ardizzoni & Menini (2007), Def. 3.2) if for every n,p0n,p \ge 0, the multiplication A nA pA n+pA_{n} \otimes A_{p} \rightarrow A_{n+p} is an epimorphism.



For RR a commutative ring write SpecRRing opSpec R \in Ring^{op} for the corresponding object in the opposite category. Write 𝔾 m\mathbb{G}_m for the multiplicative group underlying the affine line.

There is a natural isomorphism between

  • \mathbb{Z}-gradings on RR;

  • 𝔾 m\mathbb{G}_m-actions on SpecRSpec R.

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


Group ring

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

k[G]= gGL gk[G] = \bigoplus_{g\in G}L_g

where each L gL_g is a one dimensional free kk-module, for which it is convenient, here, to give a basis { g}\{\ell_g\}. The graded algebra structure is obtained by extending the multiplication rule,

g 1 g 2= g 1g 2,\ell_{g_1}\cdot \ell_{g_2} = \ell_{g_1g_2},

given on basis elements, by kk-linearity.

Lazard ring

The Lazard ring, carrying the universal (1-dimensional, commutative) formal group law is naturally an \mathbb{N}-graded ring.


Textbook account:

For Hopf algebras:

  • Ken Brown, Paul Gilmartin, James J. Zhang, Connected (graded) Hopf algebras (arXiv:1601.06687)

The notion of strongly \mathbb{N}-graded algebra is defined in:

  • Alessandro Ardizzoni, Claudia Menini, Associated graded algebras and coalgebras (arXiv:0704.2106)

Last revised on November 23, 2022 at 17:22:36. See the history of this page for a list of all contributions to it.