Let R = L
g∈GRg be a G-graded algebra over the ring O, and let X be an
R-module. A G-grading on X is a family of O-submodules (Xg)g∈G, such that
X =M
g∈G
Xg and ∀(g, h) ∈ G2, RgXh ⊆ Xgh.
If two R-modules X and Y admit G-gradings (Xg)g∈G and (Yg)g∈G, then a
morphism of G-graded R-modules u : X → Y is an algebra morphism such that u(Xg) ⊆ Yg for any g ∈ G. The following result is [Da-1980, Theorem 2.8].
Lemma 2.8. Let (A, ?) be a G-equivariant O-algebra, and A?G be the cor-
responding strongly G-graded O-algebra. For any A-module X, the induced A?G-module X0 = A?G ⊗ X admits the G-grading X0 =
L
g∈GX 0
g, where
Xg0 = g?A ⊗A X for g ∈ G. In particular, there is a natural isomorphism
X10 ' X. This construction defines an equivalence between the category AMod
of A-modules, and the category A?GGrMod of G-graded A?G-modules.
We now introduce another important notion. A G-invariant A-module is a pair (X, χ), where X is an A-module and χ is a family of isomorphisms χg :
g?X → X for g ∈ G, such that the following diagram commutes for any g, h in
G: (A4) g?h?X θ? g,h(X) // g?χh (gh)?X χgh g?X χg //X .
We may also say that the family χ = (χg)g∈G is a G-invariant structure on the
A-module X. Let (X, χ) and (Y, ψ) be two G-invariant A-modules. A morphism of A-modules u : X → Y is said to be G-invariant if u ◦ χg = ψg ◦ g?u for any
g ∈ G.
By calling the pair (X, χ) a G-invariant module, we follow [Ta-2001]. In the specific context of strongly graded algebras, [Da-1981] calls X a G-invariant A-module if there exists an isomorphism g?X ' X for each g ∈ G, and an
extendible A-module if there exists a consistent family of isomorphisms (χg)g∈G
2.3. GRADED MODULES AND INVARIANT MODULES following result is [Da-1980, Theorem 2.8, Theorem 2.12]. It is also very close to [Ta-2001, Theorem 3.1], in a different context.
Lemma 2.9. Let (A, ?) be a G-equivariant algebra and (X, χ) be a G-invariant A-module. The A-module X extends naturally to an A?G-module. This con-
struction defines an equivalence between the category (AMod)G of G-invariant
A-modules and G-invariant A-linear maps, and the category A?GMod of all
A?G-modules.
Proof. For the convenience of the reader, we explain the extension of X to an A?G-module. If (X, χ) is a G-invariant A-module, then we have the isomorphism
χg : g?X → X. We compose it with the isomorphism εg?(X) : g?A ⊗AX → g?X
of Morita’s theorem (cf. Section 2.1) to obtain a map g?A ⊗AX → X for any
g ∈ G. Since A?G =Lg∈Gg?A, these induce by linearity a map A?G⊗AX → X.
The naturality and compatibility conditions on the action ? and the invariant structure χ imply that the latter map makes X an A?G-module. The equivalence
of categories is proven in either of the above references.
Let (A, ?) be a G-equivariant algebra and (X, χ) be a G-invariant A-module. The G-invariant structure χ on the A-module X induces a structure of G-algebra on the endomorphism algebra EndA(X). Indeed, for any element g of the group
G and any endomorphism u of the A-modules X, we let γg(u) be the unique
endomorphism of the A-module X which makes the following diagram commu- tative; remember that χg in an isomorphism of A-modules:
g?X g?u // χg g?X χg X γg(u) //X
The functoriality of g? implies γg(v ◦ u) = γg(v) ◦ γg(u) for any two endomor-
phisms u, v of X. Thus the map γg is an automorphism of the O-algebra B.
Moreover, it follows from the commutativity of Diagram (A4) that γg◦ γh = γgh
for any two elements g, h in G. So the map g 7→ γg is a group morphism
G → AutAlg(EndA(X)) and makes EndA(X) a G-algebra.
Lemma 2.10. Let (A, ?) be a G-equivariant algebra and X be an A?G-module.
By Lemma 2.9, there is a natural G-invariant structure on the A-module X which makes EndA(X) a G-algebra. Moreover EndA(X)G= EndA?G(X).
Proof. By Lemma 2.9, an endomorphism of the A?G-module X is a G-invariant
endomorphism of the A-module X, i.e., precisely a fixed point of the action of the group G on the algebra EndA(X)G. This also appears in [Da-1982, Theorem
2.1].
With the notations of Lemma 2.10, let H 6 K be two subgroups of G. Then we can define a relative trace map TrKH : EndA?H(X) → EndA?K(X) by
setting TrKH(u) = P
g∈K/Hγg(u). Thus Higman’s criterion may be used to deal
with relative projectivity in the category of modules over a G-graded algebra, as appears in [Da-1982, Proposition 3.3].