graded algebra


Differential-graded objects

Homological algebra

homological algebra


nonabelian homological algebra


Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




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.


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 internal to graded vector spaces.

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



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.

Revised on February 4, 2016 13:27:28 by Urs Schreiber (