Graded algebra

A graded algebra over a field $k$ is a $k$-algebra $A={⨁}_{n\ge 0}{A}_n$ whose multiplication satisfies $A_i{A}_j\subseteq A_{i+j}$.

For an augmented graded algebra $A$, its positive augmentation ideal $A^{+}$ has indecomposable quotient $Q(A)=A^{+}/(A^{+}{)}^2$. For a Sullivan minimal model $(\Lambda V,d)$, this quotient is $V$ with zero differential because every $dv$ is decomposable. The map $H^{+}(\Lambda V,d)\to V$ induced by projection is dual to the rational Hurewicz homomorphism under the finite-type model identification.

The tensor product of graded algebras uses the displayed multiplication, incorporating the sign for interchanging homogeneous factors. This is the algebra structure in the cohomological Künneth theorem, and explains the signs of mixed intersection products on product manifolds.

The bracket of homogeneous elements in a parity-graded associative algebra. It is a commutator if at least one element is even and an anticommutator if both are odd. This is the natural bracket for the action of an odd BRST charge on quantum operators.

A graded algebra equipped with a degree-one differential satisfying $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$ and $d^2=0$. Its cohomology inherits a graded algebra structure. In rational homotopy, a Sullivan minimal model is a free graded-commutative differential graded algebra with decomposable generator differentials.

A commutative differential graded algebra has multiplication $ab=(-1{)}^{|a||b|}ba$ and a degree-one differential with $d^2=0$ and $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$. Over $\mathbb{Q}$, its free algebra on graded generators is polynomial on even-degree generators and exterior on odd-degree generators. Rational rational polynomial differential forms on a space supply such an algebra, modelled up to quasi-isomorphism by a Sullivan minimal model.

For an even-degree generator, take a free graded $R$-module with basis $a_k$, $k\ge 0$, and multiplication $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$, with $a_0=1$ and $|a_k|=k|a|$. Associativity follows from the multinomial identity. Over $\mathbb{Z}$, $a_1^k=k!a_k$, so the algebra need not be polynomial on $a_1$.

An odd graded derivation satisfies $Q(ab)=(Qa)b+(-1{)}^{|a|}a(Qb)$ for homogeneous $a$. Its square is an even derivation; consequently checking $Q^2=0$ on algebra generators establishes nilpotence everywhere.

A graded algebra is a graded commutative algebra if $uv=(-1{)}^{pq}vu$ for homogeneous elements of a graded algebra of degrees $p,q$. In characteristic not equal to two, every odd-degree element then squares to zero. The exterior algebra and Hochschild cohomology with the Hochschild cup product are examples.

For a graded vector space $V$, $\mathrm{Sym}(V)$ is the quotient of the tensor algebra by $uv-(-1{)}^{|u||v|}vu$. Over $\mathbb{Q}$ it is polynomial on even-degree basis elements and exterior on odd-degree ones. With each generator primitive, it is a graded cocommutative Hopf algebra. Its primitive subspace is exactly $V$: applying the diagonal to a polynomial of symmetric length greater than one produces mixed terms, or equivalently the polynomial's formal derivatives cannot all vanish in characteristic zero. This is the associated graded algebra in the graded Poincaré–Birkhoff–Witt theorem.

An element of a single graded component $A_n$ of a graded algebra is homogeneous of degree $n$. The degree sign rules for an exterior product or a graded Lie bracket apply first to homogeneous elements, then extend by linearity.

A Gerstenhaber algebra has an associative product making it a graded commutative algebra and a degree-minus-one graded Lie bracket. In the left convention, $[u,vw]=[u,v]w+(-1{)}^{(|u|-1)|v|}v[u,w]$. The shifted degrees $|u|-1$ determine the signs of antisymmetry and the Jacobi identity. Hochschild cohomology and the exterior algebra of polynomial derivations are examples.

On a graded algebra, a left derivation of degree $r$ satisfies the displayed rule for homogeneous $a$. A right derivation of degree $r$ satisfies $R(ab)=aR(b)+(-1{)}^{r|b|}R(a)b$. For odd $r$, these reduce to the familiar parity sign rules. The left Gerstenhaber bracket makes $[u,-]$ a left derivation of degree $|u|-1$ on Hochschild cohomology; this rule holds for the induced cohomology operations, not necessarily for their cochain representatives.

A standard graded algebra over a field $k$ has $A_0=k$ and is generated as a $k$-algebra by finitely many elements of degree one. Its Hilbert function agrees eventually with a polynomial.