Communications in Algebra
ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20
Category of G-Groups and its Spectral Category
María José Arroyo Paniagua & Alberto Facchini
To cite this article: María José Arroyo Paniagua & Alberto Facchini (2017) Category of G-Groups and its Spectral Category, Communications in Algebra, 45:4, 1696-1710, DOI: 10.1080/00927872.2016.1222409
To link to this article: http://dx.doi.org/10.1080/00927872.2016.1222409
Accepted author version posted online: 07 Oct 2016. Published online: 07 Oct 2016.
Submit your article to this journal
Article views: 12
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lagb20
Download by: [UNAM Ciudad Universitaria] Date: 29 November 2016, At: 17:29 COMMUNICATIONS IN ALGEBRA® 2017, VOL. 45, NO. 4, 1696–1710 http://dx.doi.org/10.1080/00927872.2016.1222409
Category of G-Groups and its Spectral Category María José Arroyo Paniaguaa and Alberto Facchinib aDepartamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., México; bDipartimento di Matematica, Università di Padova, Padova, Italy
ABSTRACT ARTICLE HISTORY Let G be a group. We analyse some aspects of the category G-Grp of G-groups. Received 15 April 2016 In particular, we show that a construction similar to the construction of the Revised 22 July 2016 spectral category, due to Gabriel and Oberst, and its dual, due to the second Communicated by T. Albu. author, is possible for the category G-Grp. KEYWORDS Group actions; Semi-abelian categories; Spectral categories 2000 MATHEMATICS SUBJECT CLASSIFICATION 20J15
1. Introduction The original motivation of this paper was to give an adequate categorical setting to the techniques employed in [4]. In that paper, Andrea Lucchini and the second author obtained results concerning the direct product decompositions of a group G as a direct product of indecomposable normal subgroups employing techniques essentially based on the notion of normal homomorphism [12, p. 81]. Here, if N and M are normal subgroups of G, a homomorphism f : N → G/M is normal if it commutes with all inner automorphisms of G. In particular, a construction reminiscent of the construction of the spectral category [5] was applied to the set of all normal endomorphisms of the group G. In order to place these techniques into a proper setting, we study groups, where now is the group G, and G acts on its normal subgroups N and its quotient groups G/M via inner automorphisms. To this end, we consider the category G Grp of all pairs (H, ϕ), where H is any group and ϕ : G → Aut(H) is a group homomorphism. In other words, G Grp is the category whose objects are the so called G groups. The notion of G group is a classical one, and sometimes G is called the operator group acting on H ([1, De nition 5.5] and [13, De nition 8.1]). In this context, normal subgroups N of G and quotient groups G/M become objects of G Grp, and normal homomorphisms become morphisms in G Grp. Thus we focus on the study of the category G Grp, which is a semi abelian category in the sense of [9]; cf. [1] and [6]. We determine the free G groups and show that the injective objects in the category G Grp are only the trivial ones, like in the case of the category Grp of groups. There are two more categories in strict relation with the category G Grp. One is the category of all G semidirect products (i.e., the category of pointed objects over the group G [1, Section 5]) and the other is the category of all le kG module Hopf algebras. We adapt the Gabriel Oberst construction of spectral category and its ′ dual to the category G Grp, getting two categories Spec(G Grp) and CG. This setting works particularly well on the full subcategory CG of G Grp consisting of all objects (H, ϕ) of G Grp for which the image of the group homomorphism ϕ : G → Aut(H) contains the group Inn(H) of all inner automorphisms C C′ of H. Furthermore, there is a local canonical functor of G to the product category Spec(G Grp) × G (Theorem 6.7). Also, we obtain that the endomorphism monoid of an object H of the category Spec(G Grp) is a regular monoid whenever H is a G group in CG.
CONTACT Alberto Facchini [email protected] Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy. © 2017 Taylor & Francis COMMUNICATIONS IN ALGEBRA® 1697
The reader will notice the analogy between the category G Grp of G groups and the category R Mod of all le modules over a xed ring R. The category R Mod has as objects the pairs (M, λ), where M is an additive abelian group and λ: R → End(M) is a ring morphism. In the category G Grp the morphisms f : (H, ϕ) → (H′, ϕ′) are the group homomorphisms f : H → H′ such that f (gh) = gf (h) for every g ∈ G, h ∈ H, exactly as in the category R Mod. We write mappings on the le (we stress this, because in group theory mappings are o en written on the right, sometimes as exponents. Moreover, writing mappings on the le has as a consequence that le actions correspond to homomorphisms, and right actions correspond to antihomomorphisms. We will come back to this point in Remark 2.3). Let G be a group. As we have already said above, a (le ) G group is a pair (H, ϕ), where H is a group and ϕ : G → Aut(H) is a group homomorphism. Since we write mappings on the le , a G group can be equivalently de ned as a group H endowed with a mapping : G × H → H, (g, h) → gh, called le scalar multiplication, such that (a) g(hh′) = (gh)(gh′) (b) (gg′)h = g(g′h) (c) 1Gh = h for every g, g′ ∈ G and every h, h′ ∈ H. As far as terminology is concerned, it is convenient to be very precise. The notion of G group is a particular case of the notion of group as treated in [12, p. 28] (for any set , an group is a pair (H, ϕ), where H is a group and ϕ : → End(H) is a mapping). At the same time, this notion of group is a particular case of the more general notion of a group with multiple operators studied in [7]. Also, recall that a (le ) G set is a pair (X, σ), where X is a set and σ : G → SX is a group homomorphism into the symmetric group SX on X. Equivalently, a G set is a set X with a mapping : G × X → X, ′ ′ ′ (g, x) → gx, such that (gg )x = g(g x) and 1Gx = x for every g, g ∈ G and x ∈ X.
2. G-groups and their morphisms The rst part of this section is devoted to establishing the elementary terminology about G groups. A G group morphism f : (H, ϕ) → (H′, ϕ′) is a group homomorphism f : H → H′ such that f (gh) = gf (h) ′ ′ for every g ∈ G, h ∈ H. We will denote by HomG(H, H ) the set of all G group morphisms of H into H . Thus G groups form a category, which we will denote by G Grp. Notice that the category Grp of groups and the category 1 Grp, where 1 is the trivial group (with one element), are isomorphic categories. This is the analog of the fact that the category Ab of abelian groups and the category Z−Mod of modules over the ring Z of integers are isomorphic categories, because 1 and Z are the initial objects in the category of groups and the category of rings, respectively. It is clearly possible to generalize our study of the category G Grp to a more general setting, though this will not be our primary aim in this paper. The generalization is the following. Let C be a category and M a monoid. We can obviously de ne the category CM of all M objects in C as the category of all pairs (C, ϕ), where C is an object in C and ϕ : M → End(C) is a monoid homomorphism. If we view M as a category with one object, CM is the category of all functors M → C. When M is a group, every homomorphism M → End(C) factors uniquely though Aut(C), and so we can write ϕ : M → Aut(C) instead of ϕ : M → End(C). When C is the category Grp of groups and G is a group, CG turns out to be the category G Grp of G groups. Now for any category C and any monoid M, there is a forgetful functor U : CM → C, which associates to every object (C, ϕ) of CM (the category of M objects in C) the object C of C. Suppose now that C has M indexed coproducts. This is the case, for instance, for M = G and C = Grp, where, for any family of groups Hg indexed by the elements g of G, the M indexed coproduct is the free product of the groups Hg. The forgetful functor U : CM → C has a le adjoint F : C → CM. Here F is de ned on objects by F(C) = (M C, ϕ), where M C is the M indexed coproduct of |M| copies of C, and ϕ : M → End(M C) 1698 M. J. A. PANIAGUA AND A. FACCHINI is constructed as follows. For every m ∈ M, let ιm : C → M C be the coproduct injection. Thus, for every ′ ′ family of morphisms fm : C → C , where m ∈ M and C is an object of C, there exists a unique morphism ′ f : M C → C such that f ιm = fm for every m ∈ M. Now if m ∈ M is xed and we consider the family ′ of morphisms fm′ := ιmm′ : C → M C, m ∈ M, we get a unique morphism f := ϕ(m): M C → M C ′ such that ϕ(m)ιm′ = ιmm′ for every m ∈ M. Thus we get a mapping ϕ : M → End(M C) such that ′ ϕ(m)ιm′ = ιmm′ for every m, m ∈ M, and it is easily checked that ϕ is a monoid morphism. On the morphisms of C, F is de ned as follows. Let C, C′ be two objects in C and α : C → C′ be a morphism. Consider F(C) = (M C, ϕ) and F(C′) = (M C′, ϕ′). For each m ∈ M, let ′ ′ ′ ιm : C → M C and ιm : C → M C ′ ′ be the coproduct injections. Given the family of morphisms αm = ιmα : C → M C , F(α) is the ′ ′ unique morphism F(α): M C → M C such that F(α)ιm = αm = ιmα for every m ∈ M, so that ϕ′(m)F(α) = F(α)ϕ(m) for every m ∈ M. The forgetful functor U : CM → C has both a right adjoint and a le adjoint, provided that C has M indexed coproducts and products, which is the case for M a group and C the category of groups. For the reader’s convenience, we recall that, for a ring R, the forgetful functor U : R−Mod → Ab has as a right adjoint the functor HomZ(R, −): Ab → R−Mod and as a le adjoint the functor R ⊗Z −: Ab → R−Mod [10, Exercise 2, p. 89]. We now pass to study G groups from an algebraic point of view. We begin with an example, the −1 example of the regular G group (G, α). Here α : G → Aut(G), g → αg, where αg(x) = gxg for every g, x ∈ G. ′ ′ ′ Recall that a G subgroup H of a G group H, denoted H ≤G H, is a subgroup H of H such that gh′ ∈ H′ for every g ∈ G, h′ ∈ H′. For instance, the G subgroups of the regular G group G are the ′ ′ ′ normal subgroups of G. If f : (H, ϕ) → (H , ϕ ) is a G group morphism, A ≤G H implies f (A) ≤G H ′ ′ −1 ′ and A ≤G H implies f (A ) ≤G H.
An equivalence relation ∼ on H is compatible with the G group structure on H if, for every g ∈ G and h1, h2, h3, h4 ∈ H, h1 ∼ h2 and h3 ∼ h4 imply h1h3 ∼ h2h4 and gh1 ∼ gh2. The following lemma is a special case of a well known property of groups with operators.
Lemma 2.1. Let H be a G group. Then there is a canonical one to one correspondence between the set EH of all equivalence relations ∼ on H compatible with the G group structure on H and the set NG(H) of all ′ ′ normal subgroups H of H with H ≤G H. If ∼ ∈ EH, the corresponding normal subgroup in NG(H) is the ′ equivalence class [1H]∼ of 1H modulo ∼. If H ∈ NG(H), the corresponding compatible equivalence on H −1 ′ is the equivalence relation ∼H′ de ned, for every h1, h2 ∈ H, by h1 ∼H′ h2 if h1 h2 ∈ H .
′ ′ If H ∈ NG(H), the G group structure on H induces a G group structure on the quotient group H/H in such a way that the canonical projection H → H/H′ turns out to be a G group morphism. The le action of G on H/H′ is de ned by g(hH′) = (gh)H′ for every g ∈ G and h ∈ H. For every G group ′ ′ morphism f : (H, ϕ) → (H , ϕ ), the kernel ker f belongs to NG(H). Let us recall a few other standard observations adapted to G groups. Trivial G groups, that is, the G groups with only one element, are the null objects in the category G Grp. The monomorphisms in G Grp are precisely the G group morphisms that are injective mappings (to see this, compose a non injective G group morphism f : H → H′ with the embedding and the zero mapping of ker f into H). The kernels in the category G Grp are the G group morphisms that are injective mappings and have a normal image. The isomorphisms are precisely the bijective G group morphisms. In the next theorem, we show that the epimorphisms in G Grp are exactly the surjective mappings, as in Grp [10, Exercise 5, p. 21].
Theorem 2.2. Let G be a group and (H, ϕ), (H′, ϕ′) be G groups. A G group morphism f : H′ → H is an epimorphism in the category G Grp if and only if it is a surjective mapping. COMMUNICATIONS IN ALGEBRA® 1699
Proof. Clearly, any surjective G group morphism is an epimorphism in G Grp, so that it su ces to prove the converse. We go back to the general setting of the category CM described at the beginning of this section. We have see that the forgetful functor U : G Grp → Grp has a right adjoint. Thus U is a le adjoint functor, and le adjoint functors preserve colimits [10, Section V.5,p. 119] and, in particular, preserve coproducts and epimorphisms. Thus U : G Grp → Grp preserves epimorphisms. Thus if f is an epimorphism in G Grp, U(f ) is an epimorphism in Grp, hence a surjective mapping.
′ ′ ′ ′ If) f : (H, ϕ → (H , ϕ ) is a G group morphism, the normal closure of f (H) in H belongs to NG(H ). Thus f has a category theoretic cokernel given by the quotient group of H′ modulo the normal closure of f (H) in H′. It is well known that the category theoretic product in the category Grp of groups is just the direct product of groups. Now if (H1, ϕ1), (H2, ϕ2) are G groups, then the group morphisms ϕi : G → Aut(Hi), i = 1, 2, de ne a group morphism ϕ1 × ϕ2 : G → Aut(H1) × Aut(H2) ⊆ Aut(H1 × H2). The G group (H1 × H2, ϕ1 × ϕ2) turns out to be the direct product of (H1, ϕ1) and (H2, ϕ2) in the category G Grp. The action of g on H1 × H2 is de ned, equivalently, by g(h1, h2) = (gh1, gh2). Recall that as far as coproducts are concerned, the category theoretic coproduct of two groups H1, H2 in Grp is their free product H1 ∗ H2. Let εi : Hi → H1 ∗ H2, i = 1, 2, denotes the canonical embeddings. ′ Now let (H1, ϕ1) and (H2, ϕ2) be G groups. Their coproduct in the category G Grp is (H1 ∗ H2, ϕ ). ′ ′ Here, for every g ∈ G, ϕ (g) is the unique group endomorphism of H1 ∗ H2 such that ϕ (g)εi = εiϕi(g) for both i = 1 and i = 2. Equivalently, the action of G on H1 ∗ H2 can be described as follows. Each element [h] ∈ H1 ∗ H2 can be written uniquely in its normal form [h] = [h1][h2] [hr], (r ≥ 0), where
1 = hi ∈ Hλi , λi ∈ {1, 2} and λi = λi+1; if r = 0, [h] = 1 [12, p. 170]. Then g[h] = g([h1][h2] [hr]) = [gh1][gh2] [ghr]. It is easy to prove that the category G Grp has equalizers, hence it is nitely complete. It is also
a semi abelian category in the sense of [9]; cf. [1, De nition 1.3] and Gran [6]. Every semi abelian category C contains an abelian full subcategory Ab(C) whose objects are all the abelian objects of C. Here, an object C of a semi abelian category C is abelian when the identity of C commutes with itself, or, equivalently, when C can be provided with the structure of an abelian group, that is, there exists a morphism m: C × C → C in C such that m ◦ (idC, 0) = m ◦ (0, idC) = idC. In the particular case where C = G Grp and the object C is a G group H, the mapping m: H × H → H such that m ◦ (idC, 0) = m ◦ (0, idC) = idC is necessarily the multiplication on H [1, Proof of Example 6.2]. This is a group morphism if and only if the group H is abelian. When H is a G group which is an abelian group, then the group morphism multiplication m: H × H → H is a G group morphism, because m(g(h1, h2)) = m(gh1, gh2) = (gh1)(gh2) = g(h1h2) = gm(h1, h2). It follows that the abelian objects in the category G Grp are the objects (H, ϕ) of G Grp with H an abelian group. We call them abelian G groups. The full subcategory Ab(G Grp) of G Grp whose objects are all abelian G groups is clearly equivalent to the category ZG−Mod of all le modules over the group ring ZG, and ZG−Mod is an abelian category. It should be noted that the category of G sets is a Boolean topos (which does not satisfy the Axiom of Choice), and the category of G groups is the category of groups of this topos. We are grateful to Professor Janelidze for this remark and several other enlightening comments on this paper. His suggestions greatly improved the quality of the exposition. He also noticed that some results of this paper can be generalized from the category of G groups to a wide class of semi abelian and more general categories.
Remark 2.3. Instead of considering le G groups, as we do in this paper, we could have de ned and studied right G groups. Because of our original choice of writing mappings on the le , a right G group must be de ned as a pair (H, ϕ′), where H is a group and ϕ′ : G → Aut(H) is a group antihomomorphism. Equivalently, a right G group is a group H endowed with a mapping : H×G → H, (h, g) → hg, called right scalar multiplication, such that (a) becomes (hh′)g = (hg)(h′g), etc. We can thus construct the category of Grp G of all right G groups. Let Gop be the opposite group of the group 1700 M. J. A. PANIAGUA AND A. FACCHINI
G. From a le G group, we can also construct a right Gop group, and so we see that the category G Grp is isomorphic to the category Grp Gop and Grp G is isomorphic to Gop Grp. It is interesting to note that the categories G Grp and Grp G are always isomorphic. More precisely, let (H, ϕ) be a le G group. Then ϕ′ : G → Aut(H), de ned by ϕ′(g) = ϕ(g−1) for every g ∈ G, is a group antihomomorphism, so that (H, ϕ′) turns out to be a right G group. This assignment gives a category isomorphism G Grp → Grp G, in which the regular le G group GG corresponds to the regular right G group GG. Thus the situation is completely di erent from that of le and right modules over a ring R. Namely, for a ring R, denoting by Rop its opposite ring, the category R−Mod is isomorphic to the category Mod−Rop, the category Mod−R is isomorphic to the category Rop−Mod, and the category R−Mod is isomorphic to the category Mod−R when R is a commutative ring. In general, the categories R−Mod and Mod−R are not equivalent.
3. Free G-groups and injective objects From the point of view of Universal Algebra, G groups are algebras −1 H, , , 1H, (ϕg)g∈G , −1 where is a binary operation, is unary, 1H is nullary, and each ϕg is unary, such that the following −1 hold: H, , , 1H is a group, ′ ′ ′ ′ ϕg(hh ) = (ϕg(h))(ϕg(h )), ϕgg (h) = ϕg(ϕg (h)) and ϕ1G (h) = h for g, g′ ∈ G, h, h′ ∈ H.
Proposition 3.1. Let G be a group and X be a set. The free G group over the set X is the free group FG×X over the cartesian product G × X on which the group G acts via g(g′, x) = (gg′, x) for every g, g′ ∈ G and
every x ∈ X.
Proof. We go back to the general setting of the category CM described at the beginning of Section 2, where now M is the group G and C is the category Set of sets, so that CM turns out to be the category G Set of all G sets. As we have seen there, the forgetful functor U : G Set → Set has a le adjoint functor F : Set → G Set. The functor F is de ned on objects by F(X) = (G×X, ϕ), where the cartesian product G × X is the G indexed coproduct of |G| copies of the set X and ϕ : G → Aut(G × X) is de ned by ϕ(g)(g′, x) = (gg′, x). The forgetful functor U′ : G Grp → G Set also has a le adjoint F′ : G Set → G Grp. Given any ′ ′ ′ object (X, ϕ) of G Set, F is de ned on objects by F (X, ϕ) = (FX, ϕ ), where FX is the free group with free ′ ′ set X of generators and ϕ : G → Aut(FX) is the group morphism for which ϕ (g) is the automorphism ′ of FX that extends the bijection ϕ(g): X → X on the set X of free generators of FX. Thus F F : Set → G ′ ′ Grp, X → (FG×X, ϕ ), is the le adjoint of the forgetful functor UU : G Grp → Set. This proves the proposition.
Remark 3.2. We will use the “standard additive notation” for mappings between multiplicative groups. That is, if H and H′ are multiplicative groups and α, β : H → H′ are any two mappings, we denote by α + β : H → H′ the mapping de ned by (α + β)(h) = α(h)β(h) for every h ∈ H. If α and β are group morphisms, the mapping α + β turns out to be a group morphism if and only if the two morphisms α and β commute in the category of groups [1, Example 6.2], that is, if and only if α(h1) commutes with β(h2) for all h1, h2 ∈ H. The identity endomorphisms and the trivial homomorphisms will be denoted by 1 and 0, respectively. Now let G be a group and H a G group. The set HH of all mappings H → H, endowed with the addition + de ned above and the composition of mappings ◦ as multiplication, is a right near ring (that COMMUNICATIONS IN ALGEBRA® 1701 is, a group with respect to +, a monoid with respect to ◦, right distributivity holds, but le distributivity does not in general.) It is tempting to say that HH is a “right near G ring”, in the sense, that it is a right near ring, which is also a G group (viewing HH as a product of copies of the G group H, so that the action of G on HH is de ned by (gf )(h) = g(f (h)) for every g ∈ G, f ∈ HH, h ∈ H), and the two structures are compatible, in the sense that g(f ◦ f ′) = (gf ) ◦ f ′ for g ∈ G and f , f ′ ∈ HH. H The subset EndG(H) := HomG(H, H) ⊆ H is a partial ring with identity, in the following sense. Set S = { (α, β) ∈ EndG(H) × EndG(H) | α(h1) commutes with β(h2) for all h1, h2 ∈ H }, so that the H addition + is a mapping +: S → EndG(H). Then EndG(H) is a multiplicative submonoid of H , and End(G) also has the partially de ned operation +. Also, the equality 0+α = α+0 = α always holds, and the identities α +β = β +α, (α +β)+γ = α +(β +γ ), α(β +γ ) = αβ +αγ , (α +β)γ = αγ +βγ hold, for α, β, γ ∈ End(G), whenever both members of the equalities are de ned. Notice that both distributivity laws hold in EndG(H), when the operations are de ned. Finally, EndG(H) is a ring if and only if H is abelian.
It is interesting to notice that the automorphisms in G Grp of the regular object G are exactly the central automorphisms of G, that is, the automorphisms of G that belong to the kernel of the canonical mapping Aut(G) → Aut(G/ζ(G)) [4, Proposition 4.4(d)]. Here ζ(G) denotes the center of G. Thus the automorphisms of G in G Grp are exactly the automorphisms of the group G of the form 1 + ϕ for some endomorphism ϕ of G with ϕ(G) ⊆ ζ(G) [4, Lemma 4.3]. It is easy to see that if ϕ is a group endomorphism of G with ϕ(G) ⊆ ζ(G), then 1+ϕ is an automorphism of G if and only if the restriction of 1 + ϕ to ζ(G) is an automorphism of the abelian group ζ(G). In Section 5, we will present a construction very similar to the construction of the spectral category of a Grothendieck category due to Gabriel and Oberst [5]. In the case of a Grothendieck category C, the spectral category Spec(C) re ects the behaviour of injective objects in the category, because two objects A, B of C are isomorphic in Spec(C) if and only if their injective envelopes E(A), E(B) are isomorphic C in . The situation for the category G Grp is necessarily completely di erent, because in this category the unique injective objects are the trivial G groups with one element, as we will see in Theorem 3.3. This is exactly what happens in the category Grp, see e.g., Eilenberg and Moore [2, p. 21–22]. Recall that an object E of a category C is injective if for any monomorphism f : A → B in C and any morphism ℓ: A → E, there exists a morphism m: B → E such that mf = ℓ.
Theorem 3.3. If H is a G group that is an injective object in the category G Grp, then H = 1.
Proof. Recall that any functor, which has a le adjoint that preserves monomorphisms, preserves injective objects [8, Theorem 3.8.12∗]. Thus let us go back to the general setting of the category CM described at the beginning of Section 2 , where M is the group G and C is the category Grp, so that the forgetful functor U : G Grp → Grp has F : Grp → G Grp as a le adjoint. The functor F maps any group morphism α : C → C′ to the G group morphism F(α): G C → G C′ such that ′ ′ F(α)ιg = ιgα : C → G C for every g ∈ G. In particular, if α : C → C′ is a monomorphism in Grp, that is, an injective morphism, then F(α): G C → G C′ is also clearly an injective mapping, hence a monomorphism in G Grp. Thus F preserves monomorphisms, so that U preserves injective objects. Therefore if H is an injective object in the category G Grp, then U(H) is an injective object in the category Grp, so U(H) = 1, hence H = 1.
4. Other categories related to G-Grp 4.1. The category G-Sdp of all G-semidirect products Recall that a group P is a semidirect product of its normal subgroup H and its subgroup G, written P = H ⋊ G, if and only if there exists a homomorphism P → G with kernel H which is the identity 1702 M. J. A. PANIAGUA AND A. FACCHINI on G. Hence we can introduce a category G Sdp of all G semidirect products de ned as follows. Fix a group G. The objects of G Sdp are the triples (P, α, β), where P is a group and α : G → P, β : P → G are morphisms such that the composite mapping βα is the identity automorphism idG of the group G. The morphisms from (P, α, β) to (P′, α′, β′) in G Sdp are all group morphisms a: P → P′ which make the diagram
α β G −→ P −→ G ↓ a α′ β′ G −→ P′ −→ G commute. Here, the vertical arrows on the right and on the le are the identity morphism idG of G. Thus the diagram commutes if and only if aα = α′ and β′a = β. The category G Sdp is also called the category of pointed objects over the G group G [1, p. 28]. Let (H, ϕ) be a G group. Let P be the semidirect product H ⋊ G, that is, the cartesian product H × G with the operation de ned by (h1, g1)(h2, g2) = (h1ϕ(g1)(h2), g1g2). Let α : G → P and let β : P → G be de ned by α(g) = (1, g) and β(h, g) = g. Then α and β are group morphisms and βα = idG, so that (P, α, β) is an object of G Sdp. ′ ′ If (H, ϕ) and (H , ϕ ) are G groups, (P, α, β) and (P′, α′, β′) are the corresponding objects of G Sdp e e and f : (H, ϕ) → (H′, ϕ′) is a G group morphism, de ne f : P = H ⋊ G → P′ = H′ ⋊ G by f (h, g) = e (f (h), g) for every (h, g) ∈ P = H ⋊ G. Then f is a morphism in the category G Sdp. The assignments e (H, ϕ) → (P, α, β) and f → f de ne a functor F : G Grp → G Sdp. Conversely, let L: G Sdp → G Grp be the functor that assigns to any object (P, α, β) of G Sdp the pair (ker β, ϕ), where ϕ : G → Aut(ker β) is de ned by ϕ(g)(k) = kα(g) = α(g)kα(g)−1 for every g ∈ G, k ∈ ker β. If a: (P, α, β) → (P′, α′, β′) is a morphism in G Sdp, L assigns to a the morphism L(a): ker β → ker β′ de ned by L(a)(k) = a(k) for every k ∈ ker β. Then L is a quasi inverse to F, so that the categories G Grp and G Sdp are equivalent [1, Proposition 5.7]. In particular, the category
G Sdp is semi abelian. We will now examine the correspondence, induced by the functor L: G Sdp → G Grp, between subobjects of H in G Grp and subobjects of P := H ⋊ G in G Sdp.
Proposition 4.1. Let H be a G group. Then: (a) The G subgroups of H correspond to the subgroups of P containing α(G). (b) For any subgroup P′ of P containing α(G), we have that P′ = (P′ ∩ ker β) ⋊ G. (c) If a: (P, α, β) → (P′′, α′′, β′′) is a morphism in G Sdp,P′ is the kernel of a in G Sdp and N is the kernel of L(a): ker β → ker β′′, then P′ = N ⋊ G. (d) The G subgroups of H that are normal subgroups of H correspond to the subgroups P′ = N ⋊ G of P, where N ranges in the set of the normal subgroups of H.
4.2. The category of left A-module Hopf algebras In the theory of group representations, there is a standard technique of associating to every group representation ρ : G → GLn(k) (G a group, n ≥ 1 an integer, k a eld) a le module over the group algebra kG of dimension n as a vector space over k. This yields an equivalence between the category of all representations of a group G over a eld k and the full subcategory of kG−Mod whose objects are the le kG modules nite dimensional over k. In the case of G groups (H, ϕ), we have group morphisms ϕ : G → Aut(H), and thus we are in a situation very similar to that of a group representation (i.e., group morphism) ρ : G → GLn(k). Hence it is tempting to try to nd an analog for G groups of the technique mentioned in the previous paragraph COMMUNICATIONS IN ALGEBRA® 1703 for group representations of G. The analog for G groups seems to be the following, in the setting of Hopf algebras over a xed eld k. Cf. [11, p. 40].
De nition 4.2. Let (A, mA, uA, A, εA, SA) be a Hopf algebra. A Hopf algebra (M, mM, uM, M, εM, SM) is a le A module Hopf algebra if (a) M is a le A module, i.e., a le module over the algebra (A, mA, uA), via A⊗M → M, h⊗a → h a. (b) mM, uM, M and εM are le A module morphisms.
We understand that the term we use, “le A module Hopf algebra", is somewhat misleading, because it could be interpreted as a Hopf algebra in the category of le A modules (such a concept makes sense if A is a quasitriangular Hopf algebra). But we have not been able to nd a better name for our le A module Hopf algebras. Notice that a le A module Hopf algebra M is a Hopf algebra, a le A module such that M is a le A module algebra (i.e., mM and uM are A module morphisms) and a le A module coalgebra (i.e., M and εM are A module morphisms). Notice that from this it follows that SM is a le A module morphism. Indeed, it is easy to check that SMmM is a le inverse, and mM(I ⊗ SM) is a right inverse for mM in the algebra Hom(A ⊗ M, M) with the convolution product, so that mM(I ⊗ SM) = SMmM, i.e., SM is a a le A module morphism. We denote by A−ModH the category of all le A module Hopf algebras. The morphisms in this category are the mappings that preserve both the structure as a le A module and the structure as a Hopf algebra. Given any G group H, the group algebras kG and kH are Hopf algebras and kH turns out to be a le kG module Hopf algebra simply extending by k bilinearity the le scalar multiplication G × H → H to a le scalar multiplication kG ⊗ kH → kH. In this way, we get a faithful functor k from the category G Grp to kG−ModH, which is de ned on objects by H → kH, i.e., associates to each object H of G Grp the le kG module Hopf algebra kH. Thus the category G Grp turns out to be equivalent
to a subcategory of kG−ModH. Now, for every Hopf algebra A, the set G(A) := { g ∈ A | (g) = g ⊗ g and ε(g) = 1 } of group like elements of A is a group, and the elements of G(A) are k independent [11, p. 4]. If A is a Hopf algebra and M is a le A module Hopf algebra, le multiplication λg : M → M by a group like element g ∈ G(A) is a Hopf algebra automorphism of M, hence it induces a group automorphism λg|G(M) of G(M). The assignment g → λg|G(M) is a group homomorphism G(A) → Aut(G(M)), so that G(M) is a G(A) group. In this way, we get a functor G : A−ModH → G(A) Grp. Cf. [11, Example 4.1.6]. For a xed group G and the Hopf algebra kG, we have that G(kG) = G. The group algebra functor k : G Grp → kG−ModH is a le adjoint to G : kG−ModH → G Grp, hence there are natural one to one correspondences ∼ HomkG−ModH(kH, M) = HomG Grp(H, G(M)) for every G group H and le kG module Hopf algebra M. In other words, every G group morphism H → G(M) extends by k linearity to a le kG module Hopf algebra morphism kH → M. It is well known that groups can be thought of as Hopf algebras over the eld k with one element ∗. Groups can be axiomatized by the same diagrams as Hopf algebras, where G is taken to be a set instead of a vector space. It su ces to take as mG the multiplication in the group, as uG : k → G the mapping ∗ → 1G, as comultiplication G the diagonal mapping, as counit εG the mapping 1G → ∗, and as antipode the inverse g → g−1. From this point of view, G groups turn out to be le kG module Hopf algebras.
5. Spectral category
Let H bea G group and let SubG(H) be the set of all G subgroups of H. Partially order SubG(H) by set inclusion. It is easily seen that the intersection of any family of G subgroups of H is a G subgroup of H. Hence SubG(H) turns out to be a complete bounded lattice. 1704 M. J. A. PANIAGUA AND A. FACCHINI
If H is a G group and A is a G subgroup of H, we say that A is G essential in H if, for every G subgroup B of H, A ∩ B = 1 implies B = 1. In this case, we will write A ≤G e H.
′ ′ ′ ′ −1 ′ Lemma 5.1. (a) Iff : (H, ϕ) → (H , ϕ ) is a G group morphism and A ≤G e H , then f (A ) ≤G e H. ′ ′ (b) If H is a G group and A, A ≤G e H, then A ∩ A ≤G e H. ′ ′ (c) If A ≤G e A and A ≤G e H, then A ≤G e H.
′ ′ ′ ′ Proof. Let) f : (H, ϕ → (H , ϕ ) be a G group morphism and A ≤G e H . Suppose B ≤G H and f −1(A′) ∩ B = 1. Then A′ ∩ f (B) = 1, because if b ∈ B and f (b) ∈ A′, then b ∈ f −1(A′), so that f −1(A′) ∩ B = 1 implies b = 1. From A′ ∩ f (B) = 1, it follows that f (B) = 1, because A′ is G essential in H′. Then B ⊆ ker f ⊆ f −1(A′) and 1 = f −1(A′) ∩ B = B. This proves (a). The proofs of (b) and (c) are trivial.
Now we will present a construction similar to the construction of the spectral category of a Grothendieck category. Let G be a group and G Grp the category of all G groups. De ne a new category Spec(G Grp), called the spectral category of G Grp, as follows. The objects of Spec(G Grp) are the same ′ ′ as the objects of G Grp. Let H , H be two G groups. By Lemma 5.1(b), the set { Hi | i ∈ I} of all ′ ′ ′ ′ G essential G subgroups of H is downward directed. If Hi, Hj are G essential G subgroup of H and ′ ′ ′ ′ ′ ′ Hi ⊆ Hj, the embedding Hi ֒→ Hj induces a mapping HomG(Hj, H) → HomG(Hi, H). Thus we ′ ′ ′ obtain an upward directed family of sets HomG(Hi, H) and mappings HomG(Hj, H) → HomG(Hi, H). Set
Hom (H′, H) := lim Hom (H′, H), Spec(G Grp) −→ G i ′ ′ where Hi ranges in the set of all G essential G subgroups of H . The composition in Spec(G Grp) is ′ → ℓ ′′ → ′ Grp de ned as follows. If f : H H and : H H are morphisms in the category Spec(G ), then ′ ′ f is represented by a G group morphism f : Hf → H and ℓ is represented by a G group morphism ′ ′′ ′ ′ ′ ′′ ′′ ′−1 ′ ℓ : Hℓ → H for suitable Hf ≤G e H and Hℓ ≤G e H . By Lemma 5.1(a), ℓ (Hf ) is a G essential ′′ ′−1 ′ ′′ G subgroup of Hℓ . By Lemma 5.1(c), ℓ (Hf ) is a G essential G subgroup of H . The composite morphism f ℓ in Spec(G Grp) is the image in Hom (H′′, H) := lim Hom (H′′, H) of the Spec(G Grp) −→ G i composite homomorphism
′ ′ ′−1 ′ ℓ ′ f ℓ (Hf ) −→ Hf −→ H. There is a canonical functor P: G Grp → Spec (G Grp), which is the identity on objects and maps any G group morphism f : H′ → H to its canonical image in Spec(G Grp).
Lemma 5.2. A G group H is a null object in G Grp if and only if P(H) is a null object in Spec(G Grp).
Proof. As we have already remarked, the null objects in G Grp are exactly the trivial groups. It is also easily seen that if H is a G group with one element, then P(H) is a null object in Spec(G Grp). Conversely, let H be a G group with P(H) a null object. Then the identity mapping ι: H → H and the trivial morphism 0: H → H are G group morphisms that coincide in the direct limit, hence there exists a G essential G subgroup H′ of H such that the embedding H′ ֒→ H and the trivial morphism H′ → H coincide. This implies that H′ = 1. Thus 1 is a G essential G subgroup of H. So 1 ∩ H = 1 implies H = 1, as we wanted to prove.
Since P(1) isthenullobjectinSpec(G Grp), the zero morphism P(H) → P(H′) in Spec(G Grp) P(0) P(0) is the composite morphism P(H) −→ P(1) −→ P(H′), i.e., it is P(0): P(H) → P(H′), where 0 now denotes the zero morphism of H into H′. COMMUNICATIONS IN ALGEBRA® 1705
Hence we can consider kernels in Spec(G Grp), where the kernel of a morphism f : P(H) → P(H′) in Spec(G Grp) is the equalizer of f : P(H) → P(H′) and P(0): P(H) → P(H′). We leave to the reader to ′ ′ ′ check that if f : P(H) → P(H ) is a morphism in Spec(G Grp) represented by a morphism f : Hf → H ′ ′ ′ in G Grp, with H ≤G e H, and ε : ker(f ) → H is the embedding, then the kernel of f : P(H) → P(H ) in Spec(G Grp) is P(ε). In the special case in which Hf = H, we get that:
Proposition 5.3. The functor P: G Grp → Spec(G Grp) preserves kernels, that is, is le exact.
The next lemma is easy.
Lemma 5.4. Let G be a group and H be a non trivial G group. The following two conditions are equivalent: (a) The intersection of any two non trivial G subgroups of H is non trivial. (b) Every non trivial G subgroup of H is G essential.
We say that a non trivial G group H is uniform if it satis es the equivalent conditions of Lemma 5.4. Recall that a division monoid is a monoid M = 1 with zero such that U(M) = M \{0} [4]. For instance, if H is a G group that is a nite simple group, then the endomorphism monoid of H in the category G Grp and the endomorphism monoid of H in the category Grp are division monoids.
Proposition 5.5. If H is a uniform G group, then EndSpec(G Grp)(P(H)) is a division monoid.
Proof. It su ces to show that every morphism
f ∈ EndSpec(G Grp)(P(H)) is either the zero morphism or an invertible element in EndSpec(G Grp)(P(H)). Now if f ∈ EndSpec(G Grp)
′ (P(H)), f is the image in the direct limit of a G group morphism f : Hf → H for a suitable Hf ≤G e H. The morphism f ′ is either injective or non injective. If f ′ is non injective, it has a non trivial kernel ′ ′ ker(f ), which is a G subgroup of Hf , hence of H. But H is a uniform G group, so that ker(f ) ≤G e H. Thus the restriction of f ′ to the G essential G subgroup ker(f ′) of H is zero, so that f = 0 in EndSpec(G Grp)(P(H)). ′ Hence, it remains to show that if f : Hf → H is injective, then f is invertible in EndSpec(G Grp)(P(H)). ′ ′ f (Hf ) Now if f : Hf → H is injective, then its corestriction f | : Hf → f (Hf ) is an isomorphism. Let ′ f (Hf ) −1 (f | ) be its inverse and ε : Hf → H be the embedding. Then it is easily seen that the inverse of f in EndSpec(G Grp)(P(H)) is represented by the composite homomorphism of the isomorphism ′ f (Hf ) −1 (f | ) : f (Hf ) → Hf and the embedding ε : Hf → H.
6. The dual construction It is possible to dualize the construction of the spectral category of a Grothendieck category [3]. Recall that in a category C, a monomorphism f : A → B is essential if for every object C and every morphism g : B → C, if gf is a monomorphism, then g is a monomorphism. Dually, an epimorphism f : A → B is super uous if for every object C and every morphism g : C → A, if fg is an epimorphism, then g is an epimorphism. Let G be a group. In the category of G groups, a monomorphism f : H → H′ is essential if and only if its image is a G essential G subgroup of H′. Dually, an epimorphism f : H → H′ is super uous if and only if its kernel ker f is a G super uous G subgroup of H, in the following sense. Let H be a G group and A a G subgroup of H. We say that A is G super uous in H if, for every G subgroup B of H, AB = H implies B = H. In this case, we will write A ≤G s H. For any group H, the group of all inner automorphisms of H will be denoted by Inn(H). The group Inn(H) is a subgroup of Aut(H). 1706 M. J. A. PANIAGUA AND A. FACCHINI
′ ′ Lemma 6.1. (a) Iff : H → H is a G group morphism and A ≤G s H, then f (A) ≤G s H . (b) If A ≤G s B and B ≤G s H, then A ≤G s H. (c) If (H, ϕ) is a G group, ϕ(G) ⊇ Inn(H),N ≤G M ≤G H are G subgroups of H, N ≤G s H and M/N ≤G s H/N, then M ≤G s H.
Proof. For (a), let A be a G super uous G subgroup of H. Let B′ be a G subgroup of H′ with f (A)B′ = H′. We must show that B′ = H′. We have that Af −1(B′) = H, because if h ∈ H, then f (h) = f (a)b′ for suitable a ∈ A and b′ ∈ B′, so that a−1h ∈ f −1(B′). Thus h ∈ Af −1(B′). From Af −1(B′) = H and the fact that A is G super uous in H, it follows that f −1(B′) = H, hence f (f −1(B′)) = f (H). Thus B′ ⊇ f (H) ⊇ f (A). It follows that B′ = f (A)B′ = H′. This concludes the proof of (a). The proofs of (b) and (c) are easy and le to the reader. Notice that the hypothesis ϕ(G) ⊇ Inn(H) implies that every G subgroup of H is normal in H, so that it is possible to construct the quotient G group H/N.
We now introduce two more categories. The rst category is the full subcategory CG of G Grp whose objects are all the G groups (H, ϕ) with ϕ(G) ⊇ Inn(H), where Inn(H) is the subgroup of Aut(H) of all ′ ′ inner automorphisms of H. So, if H is an object of CG and H is a G subgroup of H, then H is a normal subgroup of H and it is possible to construct the quotient G group H/H′.
′ ′ ′ Lemma 6.2. If H is an object of CG and H is a G subgroup of H, then H and H/H are objects of CG.
C′ C The second category G we now introduce has the same objects as G, but its morphisms are de ned as follows. The set of all G super uous G subgroups of a G group H is upward directed, because the product of any two G super uous G subgroups of H is a G super uous G subgroup of H. Let H′ and H be two G groups. For any two G super uous G subgroups N1, N2 of H with N1 ⊆ N2, composition with ′ ′ the canonical projection H/N1 → H/N2 induces a mapping HomG(H , H/N1) → HomG(H , H/N2). ′ ′ Thus we get an upward directed family of sets HomG(H , H/N) and mappings HomG(H , H/N1) → ′ HomG(H , H/N2). Set ′ ′ HomC′ (H , H) := lim HomG(H , H/N), G −→ C′ where N ranges in the set of all G super uous G subgroup of H. The composition in G is de ned as ′ ′′ ′ C′ follows. If f : H → H and g : H → H are two morphisms in G, then f is represented by a G ′ ′ ′ ′′ ′ group morphism f : H → H/Nf and g is represented by a G group morphism g : H → H /Ng ′ ′ for suitable G super uous G subgroups Nf ≤G s H and Ng ≤G s H . Set f (Ng) := Mf ,g/Nf . Then ′ ′ Mf ,g is G super uous in H (Lemma 6.1 ). The G group morphism f : H → H/Nf induces a G ′′ ′ C′ group morphism f : H /Ng → H/Mf ,g, and the composite morphism fg in G is the image, in ′′ ′′ HomC′ (H , H) := lim HomG(H , H/N), of the composite mapping G −→ ′ ′′ ′′ g ′ f H −→ H /Ng −→ H/Mf ,g. C C′ There is a canonical functor Q: G → G which is the identity on objects and maps any G group ′ ′ morphism f ∈ HomC (H , H) to its canonical image in HomC′ (H , H). G G
C C C′ Lemma 6.3. An object H of G is a null object in G if and only if Q(H) is a null object in G.
Proof. Clearly, the null objects of G Grp are the trivial groups. It is easily seen that if H is a trivial G C′ group, then Q(H) is a null object in G. Conversely, let H be a G group such that Q(H) is a null object. Then the identity mapping ι: H → H and the trivial morphism 0: H → H are G group morphisms that coincide in the direct limit, hence there exists a G super uous G subgroup N of H such that the canonical projection H → H/N and the trivial morphism H → H/N coincide. Thus their kernels N and H are equal. Hence H is G super uous in H, so H = 1. COMMUNICATIONS IN ALGEBRA® 1707
Lemma 6.4. The following conditions are equivalent for a G group H = 1: (a) The product of any two proper G subgroups of H is a proper subgroup of H. (b) Every proper G subgroup of H is G super uous in H.
A non trivial G group H is couniform if it satis es the equivalent conditions of Lemma 6.4.
Lemma 6.5. The following conditions are equivalent for an object H of CG: (a) H is a couniform G group. (b) For every N ≤G H,N = H, the quotient G group H/N is indecomposable as a direct product in the category G Grp.
Proposition 6.6. bIf( HC ∈ O ) is a couniform G group, then EndC′ (Q(H)) is a division monoid. G G
Proof. We must show that every morphism f ∈ EndC′ (Q(H)) is either the zero morphism or invertible. G ′ Now f ∈ EndC′ (Q(H)) is the image in the direct limit of a G group morphism f : H → H/N for a G f ′ suitable G super uous G subgroup Nf of H. The morphism f is either surjective or non surjective. ′ ′ If f is non surjective, then f (H) = M/Nf for some Nf ≤G M