§ 3. Graded algebras

The graduations considered in this paragraph will have as set of degrees a commutative monoid written additively whose identity element is denoted by 0.

1. Graded algebras

Definition 1

Let $\Delta$ be a commutative monoid, $A$ a graded ring of type $\Delta$ * (II, § 11, no. 2), $(A_\lambda)_{\lambda \in \Delta}$ its graduation and E an A-algebra. A graduation $(E_\lambda)_{\lambda \in \Delta}$ *of type $\Delta$ on the additive group E is said to be compatible with the A-algebra structure on E if it is compatible both with the A-module and with the ring structure on E, in other words, if, for all $\lambda, \mu$ in $\Delta$ ,

(1)

A_\lambda E_\mu \subset E_{\lambda + \mu}

(2)

E_\lambda E_\mu \subset E_{\lambda + \mu}.

The A-algebra E, with this graduation, is then called a graded algebra of type $\Delta$ over the graded ring A.

When the graduation on A is trivial (that is (II, § 11, no. 1) $A_0 = A, A_\lambda = \{0\}$ for $\lambda \neq 0$ ), condition (1) means that the $E_\lambda$ are sub-A-modules of E. This leads to the definition of the notion of graded algebra of type $\Delta$ over a non-graded commutative ring $A$ : $A$ is given the trivial graduation of type $\Delta$ and the above definition is applied.

When we consider graded $A$ -algebras $E$ with a unit element $e$ , it will always be understood that $e$ is of degree 0 (cf. Exercise 1).

It follows that if an invertible element $x \in E$ is homogeneous and of degree $p$ , its inverse $x^{-1}$ is homogeneous and of degree $-p$ : it suffices to decompose $x^{-1}$ as a sum of homogeneous elements in the relations $x^{-1}x = xx^{-1} = e$ .

Let $E$ and $E'$ be two graded algebras of type $\Delta$ over a graded ring $A$ of type $\Delta$ . An $A$ -algebra homomorphism $u : E \to E'$ is called a graded algebra homomorphism if $u(E_\lambda) \subset E'_\lambda$ for all $\lambda \in \Delta$ (where $(E_\lambda)$ and $(E'_\lambda)$ denote the respective graduations of $E$ and $E'$ ); where $E$ and $E'$ are associative and unital and $u$ is unital, this condition means that $u$ is a graded ring homomorphism (II, § 11, no. 2).

Let $E$ be a graded $A$ -algebra of type $N$ . $E$ is identified with a graded $A$ -algebra of type $\mathbf{Z}$ by writing $E_n = \{0\}$ for $n < 0$ .

Remark

Definition 1 can also be interpreted by saying that $E$ is a graded $A$ -module and that the $A$ -linear mapping

m : E \otimes_A E \to E

defining the multiplication on $E$ (§ 1, no. 3), is homogeneous of degree 0 when $E \otimes_A E$ is given its graduation of type $\Delta$ (II, § 11, no. 5).

To define a graded $A$ -algebra structure of type $\Delta$ on the graded ring $A$ , with $E$ as underlying graded $A$ -module, therefore amounts to defining for each ordered pair $(\lambda, \mu)$ of elements of $\Delta$ a $\mathbf{Z}$ -bilinear mapping

m_{\lambda \mu} : E_\lambda \times E_\mu \to E_{\lambda + \mu}

such that for every triple of indices $(\lambda, \mu, \nu)$ and for $\alpha \in A_\lambda,\ x \in E_\mu,\ y \in E_\nu$ ,

\alpha . m_{\mu \nu}(x, y) = m_{\lambda + \mu, \nu}(\alpha x, y) = m_{\mu, \lambda + \nu}(x, \alpha y).

Example

(1) Let $B$ be a graded ring of type $\Delta$ ; if $B$ is given its canonical $\mathbf{Z}$ -algebra structure (§ 1, no. 1, Example 2), $B$ is a graded $A$ -algebra ( $\mathbf{Z}$ being given the trivial graduation).

(2) Let $A$ be a graded commutative ring of type $\Delta$ and $M$ a graded $A$ -module of type $\Delta$ . Suppose that all the elements of the monoid $\Delta$ are cancellable, which allows (II, § 11, no. 6) us to define on $\mathrm{Homgr}_A(M, M) = \mathrm{Endgr}_A(M)$ a graded $A$ -module structure of type $\Delta$ ; as this graduation is compatible with the ring structure on $\mathrm{Endgr}_A(M)$ (II, § 11, no. 6), it defines a unital graded $A$ -algebra structure on the $A$ -algebra $\mathrm{Endgr}_A(M)$ .

(3) Algebra of a magma. Let $S$ be a magma and $\phi : S \to \Delta$ a homomorphism. For all $\lambda \in \Delta$ , we write $S_\lambda = \phi^{-1}(\lambda)$ ; then $S_\lambda S_\mu \subset S_{\lambda + \mu}$ . Let $A$ be a graded commutative ring of type $\Delta$ and $(A_\lambda)_{\lambda \in \Delta}$ its graduation; we shall define a graded $A$ -algebra structure on the algebra $E = A^{(S)}$ of the magma $S$ (§ 2, no. 6). To this end, let $E_{\lambda}$ denote the additive subgroup of $E$ generated by the elements of the form $\alpha.s$ such that $\alpha \in A_{\mu}, s \in S_{\nu}$ and $\mu + \nu = \lambda$ . As the $S_{\lambda}$ are pairwise disjoint, $E$ is the direct sum of the $A_{\mu}S_{\nu}$ and hence also the direct sum of the $E_{\lambda}$ and it is immediate that the $E_{\lambda}$ satisfy conditions (1) and (2) and therefore define on $E$ the desired graded $A$ -algebra structure. If $S$ admits an identity element $e$ , it may also be supposed that $\phi(e) = 0$ . A particular case is the one where the graduation of the ring $A$ is trivial; then $E_{\lambda}$ is the sub- $A$ -module of $E$ generated by $S_{\lambda}$ . More particularly, if we take $S = \mathbf{N}^{(I)}, \Delta = \mathbf{N}$ and $\phi$ the mapping such that $\phi((n_i)) = \sum_{i \in I} n_i$ , the ring $A$ having the trivial graduation, a graduation is thus obtained on the polynomial algebra $A[X_i]_{i \in I}$ , for which the degree of a homogeneous polynomial $\neq 0$ is the total degree defined in § 2, no. 9 (cf. § 6, no. 6).

We now take $S$ to be the free monoid $Mo(B)$ of a set $B$ (I, § 7, no. 2) and $\phi$ the homomorphism $Mo(B) \to \mathbf{N}$ which associates with each word its length. Thus a graded $A$ -algebra structure is obtained on the free associative algebra of the set $B$ (§ 2, no. 7; cf. § 5, no. 5).

2. Graded subalgebras, graded ideals of a graded algebra

Let $E$ be a graded algebra of type $\Delta$ over a graded ring $A$ of type $\Delta$ . If $F$ is a sub- $A$ -algebra of $E$ which is a graded sub- $A$ -module, then the graduation $(F_{\lambda})$ on $F$ is compatible with its $A$ -algebra structure, since $F_{\lambda} = F \cap E_{\lambda}$ ; in this case $F$ is called a graded subalgebra of $E$ and the canonical injection $F \to E$ is a graded algebra homomorphism.

Similarly, if $a$ is a left (resp. right) ideal of $E$ which is a graded sub- $A$ -module, then $E_{\lambda}a_{\mu} \subset a_{\lambda+\mu}$ (resp. $a_{\lambda}E_{\mu} \subset a_{\lambda+\mu}$ ), since $a_{\lambda} = a \cap E_{\lambda}$ ; then $a$ is called a graded ideal of the algebra $E$ . If $b$ is a graded two-sided ideal of $E$ the quotient graduation on the module $E/b$ is compatible with the algebra structure on $E/b$ and the canonical homomorphism $E \to E/b$ is a graded algebra homomorphism.

If $u : E \to E'$ is a graded algebra homomorphism, $\operatorname{Im}(u)$ is a graded subalgebra of $E'$ , $\operatorname{Ker}(u)$ is a graded two-sided ideal of $E$ and the bijection $E/\operatorname{Ker}(u) \to \operatorname{Im}(u)$ canonically associated with $u$ is a graded algebra isomorphism.

Proposition 1

Let $A$ be a graded commutative ring of type $\Delta$ , $E$ a graded $A$ -algebra of type $\Delta$ and $S$ a set of homogeneous elements of $E$ . Then the sub- $A$ -algebra (resp. left ideal, right ideal, two-sided ideal) generated by $S$ is a graded subalgebra (resp. graded ideal).*.

The subalgebra of $E$ generated by $S$ is the sub- $A$ -module generated by the finite products of elements of $S$ , which are homogeneous; similarly, the left (resp. right) ideal generated by $S$ is the sub- $A$ -module generated by the elements of the form $u_1(u_2(\ldots(u_n s))\ldots)$ (resp. $(\ldots((s u_n) u_{n-1})\ldots) u_2$ ) $u_1$ ) where s ∈ S and the u_j ∈ E are homogeneous (n arbitrary) and these products are homogeneous, whence in this case the conclusion by virtue of II, § 11, no. 3, Proposition 2); finally the two-sided ideal generated by S is the union of the sequence (\mathfrak{I}_n)_{n \geq 1}, where \mathfrak{I}_1 is the left ideal generated by S and \mathfrak{I}_{2n} (resp. \mathfrak{I}_{2n+1}) the right (resp. left) ideal generated by \mathfrak{I}_{2n-1} (resp. \mathfrak{I}_{2n}), which completes the proof.

3. Direct limits of graded algebras

Let (A_\alpha, \phi_{\beta \alpha}) be a directed direct system of graded commutative rings of type \Delta (II, § 11, no. 3, Remark 3) and for each \alpha let E_\alpha be a graded A_\alpha-algebra of type \Delta; for \alpha \leq \beta let f_{\beta \alpha}: E_\alpha \to E_\beta be an A_\alpha-homomorphism of graded algebras and suppose that f_{\gamma \alpha} = f_{\gamma \beta} \circ f_{\beta \alpha} for \alpha \leq \beta \leq \gamma; then we shall call (E_\alpha, f_{\beta \alpha}) a directed direct system of graded algebras of type \Delta over the directed direct system (A_\alpha, \phi_{\beta \alpha}) of graded commutative rings of type \Delta. Then we know (II, § 11, no. 3) that E = \lim \to E_\alpha has canonically a graded module structure of type \Delta over the graded ring A = \lim \to A_\alpha and a multiplication such that E^\lambda E^\mu \subset E^{\lambda + \mu} (where (E^\lambda) denotes the graduation on E); then this multiplication and the graded A-module structure on E define on E a graded A-algebra structure of type \Delta; E, with this structure, is called the direct limit of the direct system (E_\alpha, f_{\beta \alpha}) of graded algebras. The canonical homomorphisms E_\alpha \to E are taken A_\alpha-homomorphisms of graded algebras. Moreover, if F is a graded A-algebra of type \Delta and (u_\alpha) a direct system of A_\alpha-homomorphisms u_\alpha: E_\alpha \to F, u = \lim \to u_\alpha is an A-homomorphism of graded algebras.

Exercises

See the exercises for § 3.