Graded Algebra

Abstract

Since our approach to structures over valued fields relies in a fundamental way on filtrations and associated graded structures, our arguments often require information that is specific to graded modules. We collect in this chapter the basic definitions and results on graded algebras and modules that will be of constant use in subsequent chapters. In §2.1 we define graded rings, modules, and the gradings on their homomorphism groups and tensor products. For finite-dimensional semisimple graded algebras A we prove in §2.2 graded analogues to the classical Wedderburn Theorems. When A is graded simple we also prove graded versions of the Double Centralizer Theorem and Skolem–Noether Theorem. For A graded simple, its degree-0 component A ₀ is semisimple, though often not simple. In §2.3 we relate the grade set Γ _A to the structure of A ₀ via the map $\theta_{\mathsf {A}}\colon \Gamma^{\times}_{\mathsf {A}}\to \operatorname {\mathcal {G}}(Z(A_{0})/(Z(\mathsf {A}))_{0})$ induced by inner automorphisms of homogeneous units. We also describe inertial graded algebras A, which are completely determined by A ₀.

Appendices

Exercises

Exercise 2.1

Let D be a graded division ring.

• (i) Let γ, δ∈Γ. Show that the shifted right graded D-vector spaces D(γ) and D(δ) are isomorphic as graded D-modules if and only if γ≡δ (mod Γ _D ).

• (ii) Let V be a right graded D-vector space and W a left graded D-vector space. Establish a canonical isomorphism of graded Z(D)-vector spaces V(γ)⊗ _D W(δ)≅ _g (V⊗ _D W)(γ+δ).

Exercise 2.2

Let D be a graded division ring and let δ ₁,…,δ _n ,ε ₁,…,ε _n ∈Γ. Show that M _n (D)(δ ₁,…,δ _n )≅ _g M _n (D)(ε ₁,…,ε _n ) if and only if there is a permutation σ of {1,…,n} and a γ∈Γ such that δ _i −ε _σ(i)∈γ+Γ _D for all i=1, …, n.

Exercise 2.3

Let F be a field, let t be a commuting indeterminate over F, and let A=F[t]{x,y}, the free algebra in noncommuting indeterminates x  and  y over the polynomial ring F[t]. Thus, A has a base as a free F[t]-module consisting of all words of finite length in x and y. Clearly Z(A)=F[t]. Let I be the two-sided ideal of A generated by tx, and let B=A/I, which is generated as an F-algebra by the images $\overline{t}, \overline{x}, \overline{y}$ of t,x,y. Show that $Z(B) = F[\, \overline{t}\,] \cong F[t]$, and $\overline{x} \ne 0$, but $\overline{t}\, \overline{x} = 0$. Thus, q(Z(B))≅F(t), a rational function field over F, but B does not embed in its ring of central quotients q(B)=B⊗ _Z(B) q(Z(B))≅B⊗ _F[t] F(t). In fact, $q(B) \cong F(\overline{t})[\overline{y}]$, a commutative polynomial ring in $\overline{y}$ over the field F(t). (Hint: There is a $\mathbb {Z}\times \mathbb {Z}$-grading on A given by degree in t and total degree in x and y. Since I  is a homogeneous ideal of A with respect to this grading, there is an induced grading on B. From this one can determine an F-vector space base of B.)

Exercise 2.4

This exercise gives the graded version of a standard identity for symbol algebras, cf.  Draxl [63, Lemma 7, p. 81]. Let n=n ₁ n ₂ for some relatively prime integers n ₁, n ₂≥2, and suppose F is a graded field such that F ₀ contains a primitive n-th root of unity ω. If $m_{1}, m_{2}, n_{1},n_{2}\in \mathbb {Z}$ satisfy m ₁ n ₁+m ₂ n ₂=1, show that for all homogeneous elements a, b∈F ^×

$$\big(a,b/\mathsf {F}\big)_{\omega,n} \,\cong_g\, \big(a^{m_2},b/\mathsf {F}\big)_{\omega^{n_2},n_1} \otimes_\mathsf {F}\big(a^{m_1},b/\mathsf {F}\big)_{\omega^{n_1},n_2}. $$

Exercise 2.5

Use the same notation as in the preceding exercise, and let ω ₁, ω ₂∈F ₀ be primitive roots of unity of order n ₁ and n ₂ respectively. Show that for all homogeneous elements a ₁, b ₁, a ₂, b ₂∈F ^×

$$\big(a_1,b_1/\mathsf {F}\big)_{\omega_1,n_1} \otimes_\mathsf {F}\big(a_2,b_2/\mathsf {F}\big)_{\omega_2,n_2} \,\cong_g\, \big(a_1^{n_2}a_2^{n_1}, b_1^{n_2}b_2^{n_1}/\mathsf {F}\big)_{\omega_1\omega_2,n}. $$

Exercise 2.6

Let F be a graded field, let A be a central simple graded F-algebra, and let B be an arbitrary graded F-algebra. Show that there is a bijection between two-sided homogeneous ideals of A⊗ _F B and two-sided homogeneous ideals of B, which maps J⊆A⊗B to J∩(1⊗B)⊆B and K⊆B to A⊗ _F K⊆A⊗ _F B. (This result yields an alternative proof of Prop. 2.32.)

Exercise 2.7

Show that if A and B are finite-dimensional graded algebras over a graded field F, then A×B is inertial over F if and only if A and B are each inertial over F.

Exercise 2.8

Let A and B be (finite-dimensional) central graded division algebras over a graded field F. Assume that Γ _A ∩Γ _B =Γ _F .

• (i) Prove that $(\mathsf {A}\otimes_{\mathsf {F}}\mathsf {B})_{0} = \mathsf {A}_{0} \otimes_{\mathsf {F}_{0}}\mathsf {B}_{0}$, $\Gamma_{\mathsf {A}\otimes_{\mathsf {F}}\mathsf {B}} = \Gamma^{\times}_{\mathsf {A}\otimes_{\mathsf {F}}\mathsf {B}} = \Gamma_{\mathsf {A}}+\Gamma_{\mathsf {B}}$ and that $\theta_{\mathsf {A}\otimes_{\mathsf {F}}\mathsf {B}}\colon \Gamma_{\mathsf {A}}+ \Gamma_{\mathsf {B}}\to \operatorname {\mathit{Aut}}\big(Z(\mathsf {A})_{0} \otimes_{\mathsf {F}_{0}}Z(\mathsf {B})_{0}\,\big/ \mathsf {F}_{0}\big)$ is given by γ+δ↦θ _A (γ)⊗θ _B (δ), for all γ∈Γ _A , δ∈Γ _B .

Let S be a separable closure of F ₀. Since Z(A)₀ (resp.  Z(B)₀) is Galois over  F ₀, we may identify it with its unique F ₀-isomorphic copy in S. Then the compositum Z(A)₀⋅Z(B)₀ and the intersection Z(A)₀∩Z(B)₀ are well-defined subfields of S. Moreover, as Z(A)₀ is abelian Galois over F ₀, the identification of $\operatorname {\mathcal {G}}(Z(\mathsf {A})_{0}/\mathsf {F}_{0})$ with the Galois group of the image of Z(A)₀ in S is independent of the choice of F ₀-homomorphism Z(A)₀↪S. Likewise for  Z(B)₀. See the remarks preceding Prop. 2.45.

• (ii) Let D be the graded division algebra associated to A⊗ _F B. Prove that Z(D ₀)≅Z(A)₀⋅Z(B)₀ and that D ₀ is the division algebra associated to

  $$\mathsf {A}_0 \otimes _{Z(\mathsf {A})_0} \big(Z(\mathsf {A})_0 \cdot Z(\mathsf {B})_0\big) \otimes_{Z(\mathsf {B})_0} \mathsf {B}_0. $$

Let Z=Z(A)₀∩Z(B)₀⊆S. Recall (see Pierce [178, Lemma b, p. 256]) that since Z is Galois over F ₀, we have

$$Z\otimes_{\mathsf {F}_0} Z \, \cong\, \operatorname*{ \mathchoice{\textstyle\prod}{\prod}{\prod}{\prod}}_{\sigma \in \operatorname {\mathcal {G}}(Z/\mathsf {F}_0)} e_\sigma Z, $$

where the primitive idempotents $\{e_{\sigma}\mid \sigma \in \operatorname {\mathcal {G}}(Z/\mathsf {F}_{0})\}$ of $Z\otimes_{\mathsf {F}_{0}}Z$ are characterized by the condition that e _σ (c⊗1)=e _σ (1⊗σ(c)) for all c∈Z.

• (iii) Prove that Γ _D ={γ+δ∣γ∈Γ _A , δ∈Γ _B , and θ _A (γ)| _Z =θ _B (δ)| _Z }. Hence $\lvert \Gamma_{\mathsf {A}}+\Gamma_{\mathsf {B}}\,{\mspace{1mu}:\mspace{1mu}}\,\Gamma_{\mathsf {D}}\rvert = [Z{\mspace{1mu}:\mspace{1mu}}\mathsf {F}_{0}]$.

Notes

Graded rings and modules are a classical topic, which is well-documented in the literature; see for example Bourbaki [31, § II.11, § III.3]. The idea to consider a graded ring in which nonzero homogeneous elements are invertible as a “graded field” can be traced back to Năstăsescu [169] in the special case where the grade group is $\mathbb {Z}$. Elaborating on this idea, it is natural to develop for semisimple graded algebras the analogue of the Wedderburn theory of semisimple algebras; this was done by Năstăsescu–Van Oystaeyen [170, § II.9] (for $\Gamma=\mathbb {Z}$); see also Boulagouaz [25] and Hwang–Wadsworth [103, §1]. The graded version of the Skolem–Noether Theorem (Th. 2.37) is due to Hwang–Wadsworth [103, Prop. 1.6]. The description of the zero-component of a simple graded algebra in §2.3 comes from Tignol–Wadsworth [246, §2].

Exercise 2.8 is a graded version of Morandi–Wadsworth [163, Cor. 3.12], which is for valued division algebras.