Skew group rings

10 Pages·2016·0.276 MB·English

by  Edward Poon

SKEW GROUP RINGS
EDWARD POON - SUMMER 2016
Contents
1. Introduction 1
2. Skew group rings 2
3. Categorical relationships from group rings 5
4. Categorical relationships from skew group rings 7
References 10
1. Introduction
Skew group rings are a natural generalization of group rings, where one does not require that
the ground ring to commute with the group elements. This construction is analogous to that of a
semidirect product of groups (see Example 2.6).
Besidesgeneralizinggrouprings,skewgroupringsalsoappearinmanyareasofmathematics. For
instance, the category of representations of certain wreath product algebras (see Example 2.8) are
related to the geometry of Hilbert schemes [Wan02]. Similarly, they have also appeared via twisted
group algebras in relation to the geometry of flag varieties [HMLSZ14, KK86]. More recently, they
have also appeared in the field of categorification, where one develops a graphical categorification
of the Heisenberg algebra that depends on a Frobenius algebra [RS15]. The resulting graphical
category then acts naturally on modules over wreath product algebras.
There are many questions regarding group rings and skew group rings which are difficult to
answer. One such problem is giving a precise description of the center of an arbitrary group ring
or skew group ring. The goal of this document is not to answer these questions, but to provide
some examples of skew group rings and to explore some properties of skew group rings using the
language of categories. In particular, we will explore some interesting examples of adjunctions and
equivalences of categories that are induced by group rings and skew group rings.
Prerequisites. This document was written as an Undergraduate Honours Project at the Univer-
sity of Ottawa. For the most part, it should be accessible to students with a basic understanding
of algebra. However, some knowledge of category theory will be assumed for the latter parts of the
document.
Acknowledgements. The author would like to thank Alistair Savage for his guidance and helpful
comments throughout the project.
1
2 EDWARDPOON-SUMMER2016
2. Skew group rings
Definition 2.1 (Skew group ring). Let R be a ring, G a finite group and ϕ: G → Aut(R) a group
homomorphism. The skew group ring of G over R induced by ϕ is the ring of formal sums
(cid:40) (cid:41)
(cid:88)
R(cid:111) G = a g: a ∈ R
ϕ g g
g∈g
where the addition operation is component-wise and multiplication is given by ag·bh = aϕ(g)(b)gh
and then extending linearly.
Remark 2.2. Toreducethenumberofparentheses,wewillsometimeswriteg(b)orϕ (b)todenote
g
ϕ(g)(b).
Proposition 2.3. The skew group ring R(cid:111) G is a ring.
ϕ
Proof. By construction, the underlying abelian group is the free R-module generated by G. Thus
it remains to verify the necessary axioms for multiplication as well the compatability with the
addition structure.
Indeed, for a,b,c ∈ R and g,h,k ∈ G, we have that
(ag·bh)·ck = (aϕ (b)gh)·ck = aϕ (b)ϕ (c)ghk = aϕ (bϕ (c))agk = ag·(bϕ (c)hk) = ag·(bh·ck),
g g gh g h h
where the third equality follows from the fact that ϕ is a group homomorphism. That is, the
multiplication is associative. Moreover, we have that
ag·1 1 = aϕ (1 )g1 = ag = 1 ϕ (a)1 g = 1 1 ·ag.
R G g R G R 1G G R G
Thus, R(cid:111) G contains a multiplicative identity.
ϕ
We will proceed to prove that multiplication is distributive over addition. For all a,b,c ∈ R and
g,h,k ∈ G, we have the following set of equalities:
ag·(bh+ck) = a(ϕ (b)gh+ϕ (c)gk) = aϕ (b)gh+ϕ (c)gk = ag·bh+ag·ck,
g g g g
(bh+ck)·ag = (bϕ (a)h+cϕ (a)k)g = bϕ (a)hg+cϕ (a)kg = bh·ag+ck·ag.
h k h k
The proof then follows by extending linearly. (cid:3)
Let A be a ring and B a subring of A. Recall that A is a Frobenius extension of B if A is finitely
generatedandprojectiveasarightB-moduleandthereexistsahomomorphismof(B,B)-bimodules
tr: A → B such that
B B B B
• if tr(aA) = 0 for some a ∈ A, then a = 0,
• for every ϕ ∈ HomR(A ,B ), there exists an a ∈ A such that ϕ = tr◦(cid:96)a.
B B B
Proposition 2.4. Let R be a ring and G be a finite group equipped with a group homomorphism
ϕ: G → Aut(R). The skew group ring R(cid:111) G is a Frobenius extension of R.
ϕ
Proof. By construction, R(cid:111) G is free as a R-module. Now consider the map
ϕ
(cid:40)
a if g = 1 ,
tr: R(cid:111) G → R, ag (cid:55)→ G
ϕ
0 if g (cid:54)= 1 .
G
We claim that tr is a trace map. Let r ∈ R and (cid:80) a g,(cid:80) b g ∈ R(cid:111) G. Then
g∈G g g∈G g ϕ
   
(cid:88) (cid:88) (cid:88)
tra agg+ bgg = tr (aag +bg)g
g∈G g∈G g∈G
SKEW GROUP RINGS 3
(cid:88)
= δ (aa +b )
g,1G g g
g∈G
   
(cid:88) (cid:88)
= atr agg+tr bgg.
g∈G g∈G
That is, tr is a homomorphism of left R-modules. It remains to show that tr is nondegenerate.
Suppose that a g +···+a g is an element in R(cid:111) G such that tr((cid:80)n a g (R(cid:111) G)) = 0. Then
1 1 n n ϕ i=1 i i ϕ
for each g , we have that
i
0 = tr((a g +···+a g )g−1) = tr(a g g−1+···+a 1 +···+a g ) = a .
1 1 n n i 1 1 i i G n n i
(cid:3)
Example 2.5 (Group rings). Let R be a ring and G be a finite group and let ϕ be the trivial
group homomorphism
ϕ: G → Aut(R), g (cid:55)→ id for all g ∈ G.
R
Then R(cid:111) G is the usual group ring R[G].
ϕ
Example 2.6 (Group rings arising from the semidirect product of groups). Let R be a ring and G
be a finite group with a subgroup H and a normal subgroup N. Suppose that G is the semidirect
product of N and H, denoted as G = N (cid:111)H. There is a group homomorphism
ϕ: H → Aut(R[N]), h (cid:55)→ (an (cid:55)→ ahnh−1) a ∈ R,n ∈ N.
We claim that the function
U: R[N](cid:111) H → R[N (cid:111)H], (an)h (cid:55)→ anh a ∈ R,n ∈ N,h ∈ H.
ϕ
yields an isomorphism of rings. Let (an)h and (bm)k ∈ R[N](cid:111) H. The function U is clearly a
ϕ
homomorphism of groups with respect to addition since
U((an)h+(bm)k) = (a)nh+(bm)k = U((an)h)+U((bm)k) and U((0 1 )1 ) = 0 1 .
R G G R G
Moreover, observe that
U((an)h·(bm)k) = U((abnhmh−1)hk) = abnhmh−1hk = abnhmk = U((an)h)U((bm)k)
and
U((1 1 )1 ) = 1 1 1 = 1 1 .
R G G R G G R G
That is, U is a homomorphism of monoids with respect to the multiplication. Lastly, it is clear
that ψ is both injective and surjective, whence ψ is an isomorphism of rings.
Example 2.7 (Skew Laurent polynomial rings). Let R be a ring. Recall that the ring of Laurent
polynomials with coefficients in R is given by
R[x,x−1] = {a x−m+···+a x−1+a +a x+···+a xn: n,m ∈ N,a ∈ R}
−m −1 0 1 1 i
where the addition is defined component wise and the multiplication is given by a xm ·b xn =
m n
a b xm+n.
m n
Now suppose that there is a homomorphism of groups ϕ: Z → Aut(R), say n (cid:55)→ (a (cid:55)→ na). The
skew Laurent polynomial ring induced by ϕ is the ring R[x,x−1,ϕ] whose underlying abelian group
is R[x,x−1] but the multiplication is defined to be a xm ·b xn = a ϕ(m)(b)xm+n. As the name
m n m
suggests, this is an example of a skew group ring.
Example 2.8. Let K be a field, A a K-algebra and let S denote the symmetric group on n
n
elements. Moreover, let G be a subgroup of Sn. The algebra An = A⊗n ⊗K K[G] is a skew group
ring where an element σ ∈ G acts on A⊗n via a permutation of the factors. When G = S , then
n
the algebra A is called a wreath product algebra.
n
4 EDWARDPOON-SUMMER2016
Example 2.9. Let R be a commutative ring and let G = (cid:104)g(cid:105) be a cyclic group of order n. Then
we have a group homomorphism
ϕ: G → Aut(Rn) g (cid:55)→ ((a ,a ,··· ,a ) (cid:55)→ (a ,a ,a ,···a )).
1 2 n n 1 2 n−1
We claim that Rn(cid:111) G ∼= Mat (R).
ϕ n
Let diag(a ,a ,··· ,a ) denote the matrix whose diagonal is given by (a ,a ,··· ,a ) and with
1 2 n 1 2 n
zeroeselsewhere. Furthermore,letσ denotethepermutation(12 ··· n)andI thematrixobtained
σ
by permuting the row vectors of the identity matrix by σ. That is, the ith row of I is given by the
σ
σ(i)th row of the identity matrix.
Now consider the map ψ: Rn(cid:111) G → Mat (R) given by
ϕ n
(a ,a ,··· ,a )g(cid:96) (cid:55)→ diag(a ,a ,··· ,a )I .
1 2 n 1 2 n σ(cid:96)
and then extending linearly.
We will proceed to verify that ψ is a homomorphism of rings. By construction, the map ψ
preserves the addition. Furthermore, ψ also preserves the multiplicative and additive identities
since
ψ((0 ,0 ,···0 )1 ) = diag(0,0,··· ,0)·I = diag(0 ,0 ,···0 ),
R R R G σ0 R R R
ψ((1 ,1 ,···1 )1 ) = diag(1 ,1 ,···1 )·I = diag(1 ,1 ,··· ,1 ).
R R R G R R R σ0 R R R
o Before proving that ψ preserves the multiplication, first observe that for all (a ,a ,··· ,a )g(cid:96) and
1 2 n
(b ,b ,··· ,b )gk ∈ R(cid:111) G, we have that
1 2 n ϕ
(a ,a ,··· ,a )g(cid:96)·(b ,b ,···b )gk = (a ,a ,··· ,a )g(cid:96)−1·((1,1,··· ,1)g·(b ,b ,··· ,b )gk)
1 2 n 1 2 n 1 2 n 1 2 n
= (a ,a ,··· ,a )g(cid:96)−1·(b ,b ,··· ,b )gk+1
1 2 n n 1 n−1
= (a ,a ,··· ,a )g(cid:96)−1·(b ,b ,··· ,b )gk+1
1 2 n σ−1(1) σ−1(2) σ−1(n)
= ··· = (a b ,a b ,··· ,a b )gk+(cid:96).
1 σ−(cid:96)(1) 2 σ−(cid:96)(2) n σ−1(n)
Furthermore, for any matrix A ∈ Mat (R), the matrix I ·A is the matrix obtained by permutating
n σ
the row vectors by σ. In particular, if we take A = I , we have that I ·I = I . It follows by an
σ σ σ σ2
induction argument that I ·I = I for all (cid:96),k ∈ Z. Using the two facts above, we obtain the
σ(cid:96) σk σ(cid:96)+k
equalities:
ψ((a ,a ,··· ,a )g(cid:96)·(b ,b ,··· ,b )gk) = ψ((a (b ,a b ,··· ,a b )g(cid:96)+k)
1 2 n 1 2 n 1 σ−(cid:96)(1) 2 σ−(cid:96)(2) n σ−(cid:96)(n)
= diag(a b ,a b ,··· ,a b )I
1 σ−(cid:96)(1) 2 σ−(cid:96)(2) n σ−(cid:96)(n) σl+k
= diag(a ,a ,··· ,a )I ·diag(b ,b ,···b )I
1 2 n σ(cid:96) σ(cid:96)(1) σ(cid:96)(2) σ(cid:96)(n) σk
= ψ((a ,a ,··· ,a )g(cid:96))·ψ((b ,b ,··· ,b )gk).
1 2 n 1 2 n
Thus, ψ is a ring homomorphism. It remains to show that ψ is a bijection. Indeed, any matrix in
Mat (R) can be obtained via
n
 
a a ··· a
1,1 1,2 1,n
a2,1 a2,2 ··· a2,n (cid:32)(cid:88)n (cid:33)
 ... ... ... ...  = ψ (a1,σ−(cid:96)(1),a2,σ−(cid:96)(2),··· ,an,σ−(cid:96)(n))g(cid:96) .
(cid:96)=1
a a ··· a
n,1 n,2 n,n
Thus ψ is surjective. Lastly, we have that
(cid:32) n (cid:33)
(cid:88)
0 = ψ a gi ⇐⇒ a = (0,0,··· ,0) for all g ∈ G.
Matn(R) g g
i=1
SKEW GROUP RINGS 5
So ψ is injective.
3. Categorical relationships from group rings
Let R be a ring and let R-Mod denote the category of R-modules whose objects are R-modules
and whose morphism are R-module homomorphisms. We will also denote Grp and R-Alg as the
category of groups and the category of unital associative R-algebras respectively, with the usual
morphisms. The ring R induces the functor
f R[f]
R[−]: Grp → R-Alg, G (cid:55)→ R[G] and (G −→ K) (cid:55)→ (R[G] −−→ R[K]),
where R[G] is the usual group ring and R[f] is the ring homomorphism defined to be
R[f](ag) = af(g) for a ∈ R,g ∈ G,
and then extending linearly.
Proposition 3.1. The functor R[−] is right adjoint to the functor
−×: R-Alg → Grp A (cid:55)→ A× and f: A → B (cid:55)→ f| : A× → B×.
A×
That is, the functor −× sends a ring to its group of units and restricts the domain and codomain
of a ring homomorphism to the corresponding group of units.
Proof. Let G and H be groups and f: G → H a group homomorphism. The map R[f]: R[G] →
R[H] is clearly a ring homomorphism since for all a,b ∈ R and g,h ∈ G, we have
R[f](ag+bh) = af(g)+bf(h) = aR[f](g)+bR[f](h),
R[f](1 1 ) = 1 f(1 ) = 1 1 ,
R G R G R H
R[f](ag·bh) = R[f](abgh) = abf(gh) = abf(g)f(h) = af(g)·bf(h) = R[f](ag)·R[f](bh),
R[f](0 1 ) = 0 f(1 ) = 0 1 .
R G R G R H
We claim that unit of the adjunction is given by the map η: id ⇒ −×◦R[−], where it is defined
Grp
on an object G ∈ ObGrp by
η : G → R[G]×, g (cid:55)→ 1 g for g ∈ G.
G R
We proceed by verifying that η is a natural transformation. Let G ∈ ObGrp. The map η is a
G
group homomorphism because for all g,h ∈ G,
η (gh) = 1 gh = 1 g·1 h = η (g)·η (h) and η (1 ) = 1 1 .
G R R R G G G G R G
Naturality of η then follows from the following commutative diagrams:
f (cid:47)(cid:47) (cid:31) (cid:47)(cid:47)
G H g f(g)
(cid:95) (cid:95)
ηG ηH
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
R[G]× (cid:47)(cid:47) R[H]× 1 g (cid:31) (cid:47)(cid:47) 1 f(g)
R R
R[f]|R[G]×
Moreover, we claim that the counit of the adjunction is given by the map (cid:15): R[−]◦−× ⇒ id ,
R-Alg
where for A ∈ ObR-Alg, the map is defined by
(cid:15) : R[A×] → A, cu (cid:55)→ c·u c ∈ R,u ∈ A×.
A
andthenextendinglinearly. Byconstruction, (cid:15) isahomomorphismofabeliangroupswithrespect
A
to the addition. In fact, the map (cid:15) is a homomomorphism of R-algebras since for all c,d ∈ R and
A
u,v ∈ A×, we have that
(cid:15) (cu·dv) = (cid:15) (cduv) = cd·uv = (c·u)·(d·v) = (cid:15) (cu)·(cid:15) (dv),
A A A A
6 EDWARDPOON-SUMMER2016
(cid:15) (c(du)) = (cid:15) (cdu) = cd·u = c(d·u) = c(cid:15) (du).
A A A
Naturality of (cid:15) follows from the following commutative diagrams:
R[A×] R[f|A×] (cid:47)(cid:47) R[B×] a(cid:95)u (cid:31) (cid:47)(cid:47) af(cid:95)(u)
(cid:15)A (cid:15)B
(cid:15)(cid:15) (cid:47)(cid:47) (cid:15)(cid:15) (cid:15)(cid:15) (cid:31) (cid:47)(cid:47) (cid:15)(cid:15)
A B a·u f(a·u) = a·f(u)
f
Lastly, the zig-zag equations are satisfied by the commutative diagrams below for arbitrary G ∈
ObGrp and A ∈ ObR-Alg:
R[G] R[ηG(cid:47)(cid:47)]R[R[G]×] ag(cid:7)(cid:31) (cid:47)(cid:47) a(1(cid:95)Rg) A× ηA×(cid:47)(cid:47) R[A×]× u(cid:9)(cid:31) (cid:47)(cid:47) 1R(cid:95)u
idR[G] (cid:37)(cid:37) (cid:15)(cid:15) (cid:15)R[G] (cid:35)(cid:35) (cid:15)(cid:15) idA× (cid:36)(cid:36) (cid:15)(cid:15) (cid:15)A|R[A×]× (cid:36)(cid:36) (cid:15)(cid:15)
R[G] ag A× 1 ·u = u
R
(cid:3)
Recall that any group H gives rise to the groupoid B(H) whose object is a formal object ∗
and Mor (∗,∗) = H. Let Fun(B(H),R-Mod) denote the functor category whose objects are
B(H)
functors from B(H) to R-Mod and whose morphisms are natural transformations.
Proposition3.2. ThereisanequivalenceofcategoriesbetweenR[H]-ModandFun(B(H),R-Mod).
Proof. Recall that if M is an R[H]-module, there exists a ring homomorphism
ϕ: R[H] → End (M), ah (cid:55)→ (m (cid:55)→ ahm), a ∈ R,h ∈ H,m ∈ M.
R
By restricting the domain of ϕ to H, we have a group homomorphism from H to Aut (M).
R
Now consider the functor F: R[H]-Mod → Fun(B(H),R-Mod) defined as follows:
(cid:40)
F(M)(∗) = M, ∗ ∈ ObB(H),
M (cid:55)→ (F(M): B(H) → R-Mod),
F(M)(h) = ϕ(h), h ∈ H,
(f: M → N) (cid:55)→ (F(f): F(M) → F(N)), F(f)(m) = f(m), m ∈ M.
The map F(M) is truly a functor since for all h,k ∈ H, we have that
F(M)(h◦k) = ϕ(h·k) = ϕ(h)◦ϕ(k) = F(M)(h)◦F(M)(k) and F(M)(1 ) = ϕ(1 ) = id .
H H M
Furthermore, for f ∈ Mor (M,N), the map F(f) is a natural transformation because the
R[H]-Mod
following diagrams commute for all h ∈ Mor (∗,∗):
B(H)
F(M)(h)(cid:47)(cid:47) (cid:31) (cid:47)(cid:47)
M = F(M)(∗) F(M)(∗) = M m(cid:95) h(cid:95)m
F(f) F(f)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:31) (cid:47)(cid:47)
N = F(N)(∗) F(N)(∗) = N f(m) f(hm) = hf(m)
F(N)(h)
To prove the proposed equivalence of categories, we will use the characterization that a functor
yields an equivalence of categories if and only if it is essentially surjective on objects and fully
faithful on morphisms. Let G ∈ ObFun(B(H),R-Mod). The R-module G(∗) can be viewed as a
R[H]-module where the action of R[H] on G(∗) is given by
ah·m = aG(h)(m) a ∈ R,h ∈ H,m ∈ G(∗),
SKEW GROUP RINGS 7
and then extending linearly. It follows that G = F(G(∗)), that is, F is essentially surjective on
objects. Now let f,g ∈ Mor (M,N) and suppose that F(f) = F(g). Then for all m ∈ M,
R[H]-Mod
F(f)(m) = F(g)(m) ⇐⇒ f(m) = g(m) ⇐⇒ f = g.
Thus, F is faithful on morphisms. Now let G,G(cid:48) ∈ ObFun(B(H),R-Mod) and α: G ⇒ G(cid:48). As
before, we can view G(∗) and G(cid:48)(∗) as R[H]-modules and α as a R[H]-module homomorphism
∗
since
α (ah·m) = α (aG(h)(m)) = aG(cid:48)(h)(α (m)) = ah·α (m), a ∈ R,h ∈ H,m ∈ G(∗),
∗ ∗ ∗ ∗
where the second equality follows from the naturality of α. That is, F is full. (cid:3)
4. Categorical relationships from skew group rings
Let Hom(Grp,Aut(R)) denote the category whose objects consist of group homomorphisms
α: G → Aut(R). A morphism between two objects α: G → Aut(R) and β: H → Aut(R) is a
group homomorphism f: G → H such that α = β ◦f. That is, the following diagram commutes:
α (cid:47)(cid:47)
G Aut(R)
(cid:56)(cid:56)
f
(cid:15)(cid:15) β
H
We define the composition of morphisms in the category to be the usual composition of group
homomorphisms. It is an easy exercise to show that the composition of two morphisms will satisfy
the commutative diagram above and that the identity morphisms are precisely the identity group
homomorphisms. Lastly, composition of morphisms is associatve since the composition of group
homomorphisms is associative. We will sometimes refer to objects of Hom(Grp,Aut(R)) as pairs
(G,α), where G is a group and α: G → Aut(R) is a homomorphism of groups.
Proposition 4.1. Let Hom (Grp,Aut(R)) denote the full subcategory of Hom(Grp,Aut(R))
triv
where the objects of Hom (Grp,Aut(R)) are trivial group homomorphisms with codomain Aut(R).
triv
There is an equivalence of categories between Hom (Grp,Aut(R)) and Grp.
triv
Proof. ConsiderthefunctorF: Hom (Grp,Aut(R))whichsendsα: G → Aut(R)toGandleaves
triv
f: G → H unaltered. Since every group G is equipped with a group homomorphism to Aut(R)
whose image is {id }, it is clear that F is essentially surjective on objects.
R
It is also immediately obvious that F is faithful on morphisms. Moreover, every group homo-
morphism f: G → H satisfies the following commutative diagram:
(cid:47)(cid:47)
G Aut(R)
(cid:59)(cid:59)
f
(cid:15)(cid:15)
H
where the arrows without labels are the trivial group homomorphisms. Thus, F is full. (cid:3)
The following proposition illustrates how the construction of skew group rings depends on the
input data of a group homomorphism with codomain Aut(R).
Proposition 4.2. The map R(cid:111)− that sends a group homomorphism α: G → Aut(R) to the skew
group ring R(cid:111) G and a group homomorphism f: G → H to the map
α
(R(cid:111)f)(ag) = af(g), a ∈ R,g ∈ G
and then extending linearly, is a functor from Hom(Grp,Aut(R)) → R-Alg.
8 EDWARDPOON-SUMMER2016
Proof. Let(G,ϕ)and(K,ψ)beobjectsinHom(Grp,Aut(R))andletf ∈ Hom(Grp,Aut(R))((G,ϕ),(K,ψ)).
We claim that the map (R(cid:111)f) is a homomorphism of rings.
The map (R(cid:111)f) clearly preserves the multiplicative and additive identities since
(R(cid:111)f)(1 1 ) = 1 f(1 ) = 1 1 and (R(cid:111)f)(0 1 ) = 0 f(1 ) = 0 1 .
R G R G R K R G R G R K
Moreover, by construction, (R (cid:111) f) is linear. It remains to show that (R (cid:111) f) preserves the
multiplication. Let a,b ∈ R and g,h ∈ G. Then we have the following equalities:
(R(cid:111)f)(ag·bh) = (R(cid:111)f)(aϕ (b)gh)
g
= aϕ (b)f(gh)
g
= aϕ (b)f(g)f(h)
g
= aψ (b)f(g)f(h)
f(g)
= af(g)·bf(h)
= (R(cid:111)f)(ag)·(R(cid:111)f)(bh),
where the fourth equality follows from the property that ϕ = ψ◦f.
Furthermore, for all a ∈ R and g ∈ G, we have that
(R(cid:111)idG)(ag) = aidG(g) = idR(cid:111)idG(ag).
That is, (R(cid:111)f) preserves the identiy maps of Hom(Grp,Aut(R)).
We proceed to prove that the functor (R(cid:111)−) preserves the composition of morphisms. Suppose
we have the following data in Hom(Grp,Aut(R)):
G
α
f
(cid:15)(cid:15) (cid:35)(cid:35)
β (cid:47)(cid:47)
H Aut(R)
(cid:59)(cid:59)
γ
g
(cid:15)(cid:15)
K
Then for a ∈ A and x ∈ G, we have
(R(cid:111)(g◦f))(ax) = ag◦f(x) = (R(cid:111)g)(af(x)) = (R(cid:111)g)◦(R(cid:111)f)(ax). (cid:3)
Let G be a group. We can view the groupoid B(G) as a 2-category where the 2-morphisms
of B(G) are elements in the center of G. For the remainder of the document, we will be viewing
B(G) as a 2-category consisting of only one 2-morphism; 1 . To avoid confusion, we will denote
G
the identity element of G as 1 when we referring to it as a 1-morphism and as 1 when we are
G 1G
referring to it as a 2-morphism.
Definition4.3(2-representationofagroupinacategory). LetC bea2-category. A2-representation
of G on C is a pseudo 2-functor ρ from B(G) to C.
Example 4.4 (The trivial 2-representation). Let G be a group and C a small 2-category. The
trivial 2-representation of G on C is the functor 1: B(G) → Cat that sends ∗ to C and every g in
G to the identity functor.
Proposition 4.5. Let R be a ring, G a group and ϕ: G → Aut(R) a group homomorphism. Let
Cat denote the 2-category of small categories with the usual morphisms. The map ρ: B(G) → Cat
defined by
ρ(∗) = R-Mod and ρ(g): R-Mod → R-Mod,
SKEW GROUP RINGS 9
ρ(g)(M) = g−1M for M ∈ R-Mod,
ρ(g)(f) = f for f: M → M,
ρ(1 ) = id .
1G M M
is a 2-representation of G on Cat.
Proof. The map ρ is clearly a 1-functor since ρ(1 )(M) = 1GM = M for M ∈ R-Mod and for any
G
g and h ∈ G, we have that ρ(gh)(M) = (gh)−1M = h−1g−1M = ρ(g)◦ρ(h)(M). It follows trivially
that ρ preserves the vertical and horizontal composition of 2-morphisms. (cid:3)
Definition 4.6 (G-action, categorical representation of G). We say that a small category C has
a G-action or a categorical representation of G if there exists a 2-representation of G on Cat;
ρ: B(G) → C such that ρ(∗) = C. We will define the 2-category of categorical representations of G
2Rep(G) to be the category whose objects are categories equipped with G-actions.
Definition 4.7 (Category of G-equivariant objects). We define the category of G-equivariant ob-
jects in V, to be the category whose objects are pairs (X,((cid:15) : ρ(g)(X) → X) ) where (cid:15) is an
g g∈G g
isomorphism satisfiying the following two conditions:
(i) For g = 1 , we have that the following equality:
G
(cid:15) = φ : ρ(1 )(X) (cid:55)→ X
1G 1,X G
(ii) For any g and h in G, the following diagram commutes:
(cid:111)(cid:111) (cid:15)g
X(cid:79)(cid:79) ρ(g)(cid:79)(cid:79)(X)
(cid:15)gh ρ(g)((cid:15)h)
(cid:111)(cid:111) ∼=
ρ(gh)(X) ρ(g)(ρ(h)(X))
A morphism f between objects (X,((cid:15) : X → ρ(g)(X)) ) and (Y,(η : Y → ρ(g)(Y)) ) is a
g g∈G g g∈G
morphismf: X → Y inV thatintertwineswith(cid:15) andη . Thatis,thefollowingdiagramcommutes
g g
for each g ∈ G:
(cid:111)(cid:111) (cid:15)g
X ρ(g)(X)
f ρ(g)(f)
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:111)(cid:111) ηg
Y ρ(g)(Y)
Proposition 4.8. The category R-ModG is equivalent to the category (R(cid:111) G)-Mod.
ϕ
Proof. Consider the functor F: R-ModG → (R (cid:111) G)-Mod that is constant on morphisms but
ϕ
sends an object (M,((cid:15) : g−1M → M) ) to (cid:15)M, where (cid:15)M is the (R(cid:111) G)-module defined by
g g∈G ϕ
rg·m = r(cid:15) (m) and then extending linearly. The object (cid:15)M is truly a (R(cid:111) G)-module since we
g ϕ
have that
(rg·sh)·m = (rg(s)gh)·m = rg(s)(cid:15) (m) = rg(s)(cid:15) ((cid:15) (m)) = rg·s(cid:15) (m) = rg·(sh·m),
gh g h h
wherethethirdequalityfollowsfromDefinition4.7(ii). Theprooffortheremainingmoduleaxioms
is straightforward and is left as an exercise for the reader.
We claim that the functor F is essentially surjective. We first observe that an (R(cid:111) G)-module
ϕ
M can be regarded as a R-module by restricting the action to R. Moreover, every g ∈ G induces
an R-linear map
(cid:15) : g−1M → M, m (cid:55)→ g·m.
g
10 EDWARDPOON-SUMMER2016
The map (cid:15) is indeed linear because for all r ∈ R and m ∈ M, we have that
g
(cid:15) (r·m) = (cid:15) (g−1(r)m) = rg(m) = r(cid:15) (m).
g g g
Thus for M ∈ (R (cid:111) G)-Mod, F((M,((cid:15) : g−1M → M) )) = M. That is, F is essentially
ϕ g g∈G
surjective.
Finally, we may regard an R-module homomorphism
f: (M,((cid:15) : g−1M → M) ) → (N,(η : g−1N → N) )
g g∈G g g∈G
as an (R(cid:111) G)-module homomorphism from (cid:15)M to ηN. Indeed, for r ∈ R,g ∈ G and m ∈ M, we
ϕ
have that
f(rg·m) = f(r(cid:15) (m)) = rf((cid:15) (m)) = rη (f(m)) = rg·f(m),
g g g
where the third equality follows from the fact f intertwines with (cid:15) and η . It follows that F is
g g
fully faithful on morphisms, thus F induces an equivalence of categories between R-ModG and
(R× G)-Mod. (cid:3)
ϕ
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