Derivations and relation modules for inverse semigroups

We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentati...

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Дата:	2011
Автор:	Gilbert, N.D.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2011
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154764
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Цитувати:	Derivations and relation modules for inverse semigroups / N.D. Gilbert// Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 1–19. — Бібліогр.: 23 назв. — англ.

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spelling	irk-123456789-1547642019-06-16T01:32:11Z Derivations and relation modules for inverse semigroups Gilbert, N.D. We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps in the Gruenberg resolution obtained from a group presentation. We give a new proof of the characterization of inverse monoids of cohomological dimension zero, and find a class of examples of inverse semigroups of cohomological dimension one. 2011 Article Derivations and relation modules for inverse semigroups / N.D. Gilbert// Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 1–19. — Бібліогр.: 23 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20M18,20M50,18G20. http://dspace.nbuv.gov.ua/handle/123456789/154764 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps in the Gruenberg resolution obtained from a group presentation. We give a new proof of the characterization of inverse monoids of cohomological dimension zero, and find a class of examples of inverse semigroups of cohomological dimension one.
format	Article
author	Gilbert, N.D.
spellingShingle	Gilbert, N.D. Derivations and relation modules for inverse semigroups Algebra and Discrete Mathematics
author_facet	Gilbert, N.D.
author_sort	Gilbert, N.D.
title	Derivations and relation modules for inverse semigroups
title_short	Derivations and relation modules for inverse semigroups
title_full	Derivations and relation modules for inverse semigroups
title_fullStr	Derivations and relation modules for inverse semigroups
title_full_unstemmed	Derivations and relation modules for inverse semigroups
title_sort	derivations and relation modules for inverse semigroups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/154764
citation_txt	Derivations and relation modules for inverse semigroups / N.D. Gilbert// Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 1–19. — Бібліогр.: 23 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT gilbertnd derivationsandrelationmodulesforinversesemigroups
first_indexed	2025-07-14T06:52:23Z
last_indexed	2025-07-14T06:52:23Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 12 (2011). Number 1. pp. 1 – 19 c© Journal “Algebra and Discrete Mathematics” Derivations and relation modules for inverse semigroups N. D. Gilbert Communicated by V. I. Sushchansky Abstract. We define the derivation module for a homo- morphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps in the Gruenberg resolution obtained from a group presentation. We give a new proof of the characterization of inverse monoids of coho- mological dimension zero, and find a class of examples of inverse semigroups of cohomological dimension one. 1. Introduction A cohomology theory for inverse semigroups was established in the funda- mental work of Lausch [10] and Loganathan [14] but since then, there have been limited applications of homological algebra to inverse semigroups. However, important contributions have been made in closely related ar- eas. These include an approach, due to Steinberg [18], to the study of homomorphisms of inverse semigroups based on derived categories, and developed in [18] in the wider context of morphisms of ordered groupoids; the topology of 2–complex models of inverse semigroup presentations and its application to amalgams, also due to Steinberg [19]; homotopy theory in the category of ordered groupoids [12], leading to an alternative proof I am very grateful for constructive email discussions with Benjamin Steinberg and with Jonathon Funk about this work. 2000 Mathematics Subject Classification: 20M18,20M50,18G20. Key words and phrases: inverse semigroup, cohomology, derivation, relation module. 2 Derivations and relation modules of Steinberg’s Factorization Theorem; and Funk’s [4] study of the topos of representations of an inverse semigroup. A key step in [14] was the reformulation of the module theory for an inverse semigroup S, due to Lausch [10], as the module theory of a left cancellative category L(S). An S–module is then a functor L(S) → Ab from L(S) to the category of abelian groups. This enables the application of the (co)homology theory of categories to be applied directly to the study of inverse semigroups: for example, Loganathan [14, Theorem 4.5(i)] shows that free inverse monoids have cohomological dimension one. Recent work of Webb [22, 23] on the cohomology of categories has developed ideas from group representation theory, such as relation modules and the Schur multiplier, and formulated them for categories. Webb’s results, when applied to Loganathan’s category L(S), give us an entrée to the ideas of augmentation and relation modules for inverse semigroups: the starting point in [23] is the Gruenberg resolution, whose construction we now recall. Let P = 〈X : R〉 be a group presentation of a group G. Gruenberg [6] (and see also [7, chapter 3]) gave a functorial construction from P , of a free resolution of the trivial G–module Z. This Gruenberg resolution has a number of interesting properties and we outline the construction, referring to [6, 7] for details. Let F = F (X) be the free group on X, let N be the normal closure of R in F (X), and let θ : F → G be the natural map with kernel N . Then θ induces a map θ : ZF → ZG of integral group rings, and we let r be its kernel. The augmentation ideal f of F is the kernel of the augmentation map ZF → Z induced by mapping w 7→ 1 for all w ∈ F : clearly r and f are two-sided ideals. Theorem (Gruenberg, [6]). The complex of ZG–modules . . .→ r2/r3 → fr/fr2 → r/r2 → f/fr → ZG→ Z → 0 is a G–free resolution of Z, and this construction gives a functor from the category of free presentations of G to the category of G–free resolutions of Z. Gruenberg [8] went on to show that his resolution gave rise to explicit formulae for the homology groups of G in terms of r and f, generalising the Hopf formula for H2(G,Z). The kernel of the map f/fr → ZG is r/fr and Gruenberg [6] shows that this is isomorphic to the relation module Nab. (Indeed, more is true, as shown in [6]: the kernel of frn−1/frn → rn−1/rn is isomorphic to the tensor product Nab⊗· · ·⊗Nab of n copies of the relation module, with the diagonal G–action.) Defining the relation module in this way permits the introduction of the concept in other algebraic settings where the operation N. D. Gilbert 3 of abelianisation has no obvious counterpart. For a small category C, the integral category algebra is the free abelian group ZC having the morphisms of C as a basis. Multiplication is defined by the Z–linear extension of the composition of morphisms in C, with undefined compositions being set equal to zero. An inverse semigroup S therefore determines the category algebra ZL(S). However, Loganathan defines ZS as a L(S)–module determined by the free abelian groups on Green’s L–classes in S (see section 2 below). Beginning with Loganathan’s definition leads to definitions of the augmentation ideal and relation modules differing from those that result from following the constructions of [23] for the category algebra ZL(S). We shall begin with Loganathan’s ZS, and blend some of the ideas of [14] with Webb’s approach to the cohomology of categories, so that we are able to define the relation module of an inverse semigroup presentation. The key ingredient is the construction of an L(S)–module D that corre- sponds to the module f/fr in Gruenberg’s resolution. Our construction generalises Crowell’s derived module of a group homomorphism [3], and is based on the derived module of a morphism of groupoids as defined by Brown and Higgins in [2]. After some preliminaries on Loganathan’s category L(S) and L(S)–modules in section 2, we define the derivation module Dφ of an inverse semigroup homomorphism φ in section 3 and establish its adjoint relationship to the semidirect products of S with L(S)–modules. For subsequent applications, the most important result of section 3 is that if S is an inverse monoid presented by 〈X : R〉, and if θ : FIM(X) → S is the presentation map from the free inverse monoid on X to S, then the derivation module Dθ is a projective L(S)–module. There is a canonical L(S)–map from Dθ to the augmentation module of S, and its kernel is defined to be the relation module of the presentation 〈X : R〉. We show that the set of relations R gives rise to a natural generating set for the relation module, and show that the relation module may also be interpreted as the first homology group of the Schützenberger graph of S with respect to the generating set X. This naturally leads to consideration of the class of inverse monoids whose Schützenberger graph is a forest: we call such an inverse monoid arboreal. The analogue of the Gruenberg resolution in this case shows that an arboreal inverse monoid has cohomological dimension one. 2. Modules and augmentation Let S be an inverse semigroup, with semilattice of idempotents E(S). Recall that the natural partial order on S may be defined as follows: for s, t ∈ S we have s 6 t if only if s = et for some e ∈ E(S). We have the following useful alternative characterisations of the natural partial order (see [11, Lemma 1.4.6] for example): 4 Derivations and relation modules Lemma 2.1. Let S be an inverse semigroup. Then the following are equivalent for s, t ∈ S: • s 6 t, • s = tf for some f ∈ E(S), • s = ss−1t, • s = ts−1s. Loganathan’s category L(S) is now defined as follows. Its set of objects is E(S), and the set of arrows is {(e, s) : e ∈ E(S), s ∈ S, e > ss−1}. The identity arrow at e ∈ E(S) is (e, e), and the arrow (e, s) has domain d(e, s) = e and range r(e, s) = s−1s. If (e, s), (f, t) ∈ L(S) and s−1s = f then their composite is (e, s)(f, t) = (e, st). It is now straightforward to check that L(S) is a left-cancellative category. An L(S)–module is then a functor A : L(S) → Ab to the category Ab of abelian groups. Modules for the category algebra ZL(S), as defined in [22], will be L(S)–modules in our sense, but the converse will be true if and only if E(S) is finite. An L(S)–module A determines an abelian group Ae for each e ∈ E(S) and a homomorphism α(e,s) : Ae → As−1s for each arrow (e, s), such that α(e,e) is the identity map on Ae, and such that α(e,s)α(f,t) = α(e,st) whenever s−1s = f . If A is an L(S)–module and a ∈ Ae, we shall write a⊳ (e, s) for aα(e,s) ∈ As−1s. We may regard the poset E(S) as a category in the usual way, with a unique morphism (e, f) : e→ f whenever e 6 f in E(S). Then E(S)op is a subcategory of L(S), and the restriction of an L(S)–module to E(S)op determines a presheaf of abelian groups A over E(S). Hence an L(S)– module A may be considered as a presheaf of abelian groups over E(S) with an S–action. Moreover, for any L(S)–module A, the disjoint union A⊔ = ⊔e∈E(S)Ae is a commutative inverse semigroup, with the product a ∗ b of a ∈ Ae and b ∈ Af defined by a ∗ b = (a⊳ (e, ef)) + (b⊳ (f, ef)). The natural partial order on A⊔ then coincides with that induced by the action of E(S)op: for a ∈ Ae and b ∈ Af , a 6 b⇐⇒ e 6 f and a = b⊳ (f, e). Moreover, A⊔ also admits a semigroup action of S by endomorphisms, defined for a ∈ Ae and s ∈ S by a · s = a⊳ (e, es). A homomorphism of L(S)–modules is just a natural transformation of functors. We call such a natural transformation an L(S)–map. Hence an L(S)–map µ : A → B is determined by a family of abelian group homomorphisms µe : Ae → Be indexed by E(S), such that the following N. D. Gilbert 5 square commutes: Ae ⊳(e,s) �� µe // Be ⊳(e,s) �� As−1s µ s−1s // Bs−1s We denote the category of L(S)–modules and L(S)–maps by ModS . The kernel of an L(S)–map µ is the L(S)–module kerµ defined by (kerµ)e = ker(µe : Ae → Be). Clearly kerµ is then an L(S)–submodule of A. A fixed abelian group A determines a trivial L(S)–module A, with A e = A for all e ∈ E(S), and α(e,s) = id for all (e, s). In particular, the group of integers determines the trivial module Z (denoted by ∆Z in [14]). Given an L(S)–module A we can define the semidirect product S ⋉A, which will again be an inverse semigroup. The construction is a special case of both Billhardt’s λ–semidirect product of inverse semigroups [1] and of Steinberg’s semidirect product of ordered groupoids, and so as an instance of [18, proposition 3.3] we have the following result. Proposition 2.2. Let S be an inverse semigroup and A an L(S)–module. Define S ⋉A = {(s, a) : s ∈ S, a ∈ As−1s}. Then S ⋉A is an inverse semigroup, with multiplication (s, a)(t, b) = (st, (a⊳ (s−1s, s−1st)) + (b⊳ (t−1t, t−1s−1st))) = (st, (a · t) ∗ b). Clearly the projection (s, a) 7→ s is a homomorphism of inverse semi- groups, and the construction A → S⋉A is a functor from the category of L(S)–modules to the slice category (IS ↓ S) of inverse semigroups over S. In section 3 we shall construct a left adjoint, modifying a construc- tion originally due to Crowell [3] and then generalised in [2]. This left adjoint will then give us the term corresponding to f/fr in the Gruenberg resolution. Now Loganathan [14] constructs the L(S)–module ZS as follows. For each idempotent e ∈ E(S), let Le be the L–class of e and let ZSe be the free abelian group with basis Le. Now if a ∈ Le and (e, s) ∈ L(S) we define a⊳ (e, s) = as. Since e = a−1a > ss−1, it follows that (as)−1(as) = s−1a−1as = s−1s, so that as ∈ Ls−1s and the basis transformation a 7→ as induces a homomorphism ZSe → ZSs−1s. Loganathan notes [14, Remark 4.2] that if S is an inverse monoid then ZS is a free L(S)–module. In this setting, a basis for a free L(S)–module is an E(S)–set, that is a family of disjoint sets Xe indexed by e ∈ E(S). The augmentation map εS : ZS → Z is the L(S)–map defined on the basis Le of ZSe by s 7→ 1 ∈ Ze. It is clear that this is an L(S)–map, and 6 Derivations and relation modules its kernel is the augmentation module s of S. Lemma 2.3. For each e ∈ E(S), the abelian group se is freely generated by the elements s− e with e 6= s ∈ Le. Proof. Let x = ∑ i∈I nisi ∈ se, so that each si ∈ Le. Since xεS = 0, we have ∑ i∈I ni = 0 and hence ∑ i∈I nie = 0. Then x = ∑ i∈I nisi − ∑ i∈I nie = ∑ i∈I ni(si − e). Hence the elements s− e with s ∈ Le generate se, and since ZSe is freely generated by Le, it is clear that the elements s− e with e 6= s ∈ Le are a basis. We remark that essentially the same definitions of ZG and its aug- mentation module were given for a groupoid G by Brown and Higgins [2]. In the next section we shall adapt another definition for groupoids from [2] in order to define the relation module of an inverse semigroup presentation. 3. The derivation module of a homomorphism The second step in Gruenberg’s resolution is the module f/fr, which is G–free on the cosets of the elements 1 − x , (x ∈ X). It is therefore isomorphic to the module f ⊗F ZG, which is Crowell’s derived module of the homomorphism θ : F → G: indeed, Crowell gives the short exact sequence 0 → Nab → f⊗F ZG→ g → 0 in [3]. We shall adapt a generalisation of Crowell’s derived module due to Brown and Higgins [2] and apply it to a homomorphism of inverse semigroups. From a homomorphism φ : T → S of inverse semigroups, we shall construct an L(S)–module Dφ, called the derivation module of φ, with a universal property detailed in Proposition 3.5, and with a canonical L(S)–map to the augmentation module of S. Let A be an L(S)–module. We first define the notion of a (L(T ), φ)– derivation from L(T ) to A, following the definition of a derivation to a module for a category algebra given in [22]. The properties of the composition of an (L(T ), φ)–derivation with the map T → L(T ) that carries t 7→ (tt−1, t) will then lead us to the formulation of the notion of a φ–derivation. Given a functor φ : L(T ) → L(S), we define an (L(T ), φ)– derivation to A to be a function ζ : L(T ) → A⊔, defined on the arrows of L(T ), such that • if (p, a) is an arrow of L(T ) then (p, a)ζ ∈ A(a−1a)φ, N. D. Gilbert 7 • if (p, a), (a−1a, b) are composable arrows in L(T ), so that a−1a > bb−1, then (p, ab)ζ = ((p, a)(a−1a, b))ζ = (p, a)ζ ⊳ (a−1a, b)φ+ (a−1a, b)ζ. Now let φ be a homomorphism of inverse semigroups. Of course, φ induces a functor L(T ) φ → L(S) mapping (p, a) 7→ (pφ, aφ). A φ–derivation η : T → A is a function T → A⊔ such that • if a ∈ T then aη ∈ A(a−1a)φ, • if a, b ∈ T and a−1a > bb−1, then (ab)η = aη ⊳ ((a−1a)φ, bφ) + bη. (3.1) Example 3.1. If φ : T → S is a homomorphism of inverse semigroups then the function η : T → s, t 7→ tφ − (t−1t)φ is a φ–derivation. If t−1t > uu−1 then (tu)η = (tu)φ− (u−1u)φ = (tφ)(uφ) − uφ+ uφ− (u−1u)φ = (tφ− (t−1t)φ) ⊳ ((t−1t)φ, uφ) + uφ− (u−1u)φ = (tη) ⊳ ((t−1t)φ, uφ) + uη. For each e ∈ E(S) we define Xe = {(a, s) : a ∈ T, s ∈ Le, (a −1a)φ > ss−1}. We note that if (a, s) ∈ Xe then (aφ)s ∈ Le. Now the group Dφ,e is defined as the (additive) abelian group generated by Xe subject to all relations of the form (ab, s) − (b, s) = (a, (bφ)s) (3.2) where a, b ∈ T with a−1a > bb−1 and s ∈ S. We denote the image of (a, s) in Dφ,e by 〈a, s〉. For subsequent use, we note the following consequences of the relations in Dφ,e. Lemma 3.2. (a) If (a, s) ∈ Xe then 〈aa−1, s〉 = 0 in Dφ,e. (b) If a, b ∈ T with b > a, and s ∈ S with (a−1a)φ > ss−1, then 〈a, s〉 = 〈b, s〉 in Dφ,e. Proof. For (a) we have 0 = 〈a, s〉 − 〈a, s〉 = 〈aa−1a, s〉 − 〈a, s〉 = 〈a, (a−1a)φs〉 + 〈a−1a, s〉 − 〈a, s〉 8 Derivations and relation modules = 〈a, s〉 + 〈a−1a, s〉 − 〈a, s〉 = 〈a−1a, s〉. For (b), we note that a = ba−1a with b−1b > a−1a, and therefore 〈a, s〉 = 〈ba−1a, s〉 = 〈b, (a−1a)φs〉 + 〈a−1a, s〉 = 〈b, s〉. Lemma 3.3. Let s ∈ S with s−1s = e and suppose that 〈a, s〉 ∈ Dφ,e. If b ∈ T and bφ = s then 〈ab, e〉 ∈ Dφ,e and 〈ab, e〉 = 〈a, s〉 + 〈b, e〉 . Proof. Set f = (a−1a)φ: then f > ss−1 and so ((ab)φ)−1(ab)φ = s−1fs = s−1s = e. Therefore 〈ab, e〉 ∈ Dφ. Since (a−1ab)(a−1ab)−1 = a−1abb−1 6 a−1a we have 〈ab, e〉 = 〈a(a−1ab), e〉 = 〈a, (a−1ab)φ〉 + 〈a−1ab, e〉 (by (3.2)) = 〈a, fs〉 + 〈a−1ab, e〉 = 〈a, s〉 + 〈b, e〉 since fs = s and, by Lemma 3.2(b), 〈a−1ab, e〉 = 〈b, e〉. Lemma 3.4. Suppose that T is generated as an inverse semigroup by a subset X. Then Dφ,e is generated, as an abelian group, by the subset {〈x, s〉 : x ∈ X, (x−1x)φ > ss−1, s−1s = e}. Proof. Suppose that t ∈ T with t = ua for some u ∈ T and a ∈ X ∪X−1. Then t = u(u−1ua) and u−1u > u−1uaa−1 = (u−1ua)(u−1ua)−1. The relations (3.2) then imply that, if (t, s) ∈ Xe , then 〈t, s〉 = 〈u, ((u−1ua)φ)s〉 + 〈u−1ua, s〉. By part (b) of Lemma 3.2, we have 〈u−1ua, s〉 = 〈a, s〉 and so 〈t, s〉 = 〈u, ((u−1ua)φ)s〉 + 〈a, s〉. It now follows, by induction on the minimum length of a product of elements of X ∪ X−1 representing t ∈ T , that Dφ,e is generated as an abelian group by the subset {〈x, s〉 : x ∈ X ∪X−1}. But, by part (a) of Lemma 3.2 and the relations (3.1), for any (a, s) ∈ Xe, 0 = 〈aa−1, s〉 = 〈a, (a−1φ)s〉 + 〈a−1, s〉 N. D. Gilbert 9 and so 〈x−1, s〉 = −〈x, (x−1φ)s〉. We may now define the derivation module of φ, denoted by Dφ. For e ∈ E(S) we have (Dφ)e = Dφ,e , and the action of (e, t) on a generator 〈a, s〉 of Dφ,e is given by 〈a, s〉 ⊳ (e, t) = 〈a, st〉. Since s−1s = e > tt−1, (st)−1(st) = t−1s−1st = t−1t, and so 〈a, st〉 ∈ Dφ,t−1t . It is easy to check that Dφ is indeed an L(S)–module. Proposition 3.5. There exists a canonical φ–derivation δ : T → Dφ such that, given any φ–derivation η : T → A to an L(S)–module A, there is a unique L(S)–map ξ : Dφ → A such that η = δξ. Proof. We define the map δ : T → Dφ by t 7→ 〈t, (t−1t)φ〉. Then tδ ∈ Dφ,(t−1t)φ and δ clearly satisfies the defining property for a φ–derivation given in (3.1). Let A be an L(S)–module. Suppose that an L(S)–map ξ : Dφ → A satisfies η = δξ, and consider 〈a, s〉 ∈ D⊔ φ . Then (a−1a)φ > ss−1 and 〈a, s〉ξ = (〈a, (a−1a)φ〉⊳ ((a−1a)φ, s))ξ = (〈a, (a−1a)φ〉)ξ ⊳ ((a−1a)φ, s) = (aδ)ξ ⊳ ((a−1a)φ, s) = aη ⊳ ((a−1a)φ, s) , so ξ is uniquely determined by η. Given η : T → A, we may define a map ξ : Dφ → A by setting 〈a, s〉ξ = aη ⊳ ((a−1a)φ, s). Firstly, this will be well-defined on D⊔ φ since, if a, b ∈ T with a−1a > bb−1, then (〈ab, s〉 − 〈b, s〉)ξ = ((ab)η ⊳ ((b−1b)φ, s)) − (bη ⊳ ((b−1b)φ, s)) = ((ab)η − bη) ⊳ ((b−1b)φ, s) = (aη ⊳ ((bb−1)φ, bφ)) ⊳ ((b−1b)φ, s) = aη ⊳ ((bb−1)φ, (bφ)s) = 〈a, (bφ)s〉ξ , and is an L(S)–map since (〈a, s〉⊳ (e, x))ξ = 〈a, sx〉ξ = aη ⊳ ((a−1a)φ, sx) = aη ⊳ ((a−1a)φ, s) ⊳ (e, x) = 〈a, s〉ξ ⊳ (e, x). Corollary 3.6. Given any inverse semigroup homomorphism φ : T → S, there is an L(S)–map ∂1 : Dφ → ZS whose image is the augmentation module s. 10 Derivations and relation modules Proof. The map ∂1 is induced by the φ–derivation t 7→ tφ − (t−1t)φ of Example 3.1, and so 〈a, s〉∂1 = (aφ− (a−1a)φ) ⊳ s = (aφ)s− s. Lemma 2.3 then implies that its image is s. Suppose that we have inverse semigroup homomorphisms φ : T → S and ψ : U → S and that λ : T → U is a homomorphism of inverse semigroups such that φ = λψ. Then λ is a morphism in the slice category (IS ↓ S) and induces a mapping λ∗ : Dφ → Dψ given by 〈a, s〉 7→ 〈aλ, s〉. In this way the construction of the derivation module gives a functor (IS ↓ S) → ModS . Proposition 3.7. The derivation module functor (T φ → S) 7→ Dφ is left adjoint to the semidirect product functor A 7→ S ⋉A. Proof. The first part of the proof is the standard verification, adapted to our setting, of the correspondence between derivations and maps to semidirect products. The details are as follows. Let φ : T → S be an inverse semigroup homomorphism and let α : T → S ⋉A be a morphism in the slice category (IS ↓ S), so that for all t ∈ T we have tα = (tφ, tζ) for some function ζ : T → A⊔. Since (tφ, tζ) ∈ S ⋉ A, it follows that tζ ∈ A(t−1t)φ. Now assume that a, b ∈ T with a−1a > bb−1. Then (aα)(bα) = (aφ, aζ)(bφ, bζ) = (aφbφ, aζ ⊳ ((a−1a)φ, (a−1ab)φ) + bζ ⊳ ((b−1b)φ, (b−1a−1ab)φ)) and, since a−1ab = b, = (aφbφ, aζ ⊳ ((a−1a)φ, bφ) + bζ ⊳ ((b−1b)φ, (b−1b)φ)) = (aφbφ, aζ ⊳ ((a−1a)φ, bφ) + bζ). Now (aα)(bα) = (ab)α = ((ab)φ, (ab)ζ), and so if a−1a > bb−1, we have (ab)ζ = aζ ⊳ ((a−1a)φ, bφ) + bζ and ζ is a φ-derivation. By Proposition 3.5, ζ induces a unique L(S)–map α̂ : Dφ → A defined by α̂ : 〈a, s〉 7→ aζ ⊳ ((a−1a)φ, s) . (3.3) Now an L(S)–map γ : Dφ → A induces an inverse semigroup homo- morphism γ† : T → S ⋉A in the slice category (IS ↓ S) given by γ† : t 7→ (tφ, 〈t, (t−1t)φ〉γ). (3.4) N. D. Gilbert 11 We check that γ† is indeed an inverse semigroup homomorphism as follows. For all a, b ∈ T we have (ab)γ† = ((ab)φ, 〈ab, (b−1a−1ab)φ〉γ). Now we compute (aγ†)(bγ†): (aγ†)(bγ†) = (aφ, 〈a, (a−1a)φ〉γ)(bφ, 〈b, (b−1b)φ〉γ) = (aφbφ, 〈a, (a−1a)φ〉γ ⊳ ((a−1a)φ, (a−1ab)φ) + 〈b, (b−1b)φ〉γ ⊳ ((b−1b)φ, (b−1a−1ab)φ)) = (aφbφ, 〈a, (a−1ab)φ〉γ + 〈b, (b−1a−1ab)φ〉γ) = (aφbφ, (〈a, (a−1ab)φ〉 + 〈b, (b−1a−1ab)φ〉)γ). Now it follows from part (b) of Lemma 3.2 that 〈b, (b−1a−1ab)φ〉 = 〈a−1ab, (b−1a−1ab)φ〉 and then by Lemma 3.3 that 〈a, (a−1ab)φ〉 + 〈a−1ab, (b−1a−1ab)φ〉 = 〈ab, (b−1a−1ab)φ〉. Hence (aγ†)(bγ†) = (aφbφ, 〈ab, (b−1a−1ab)φ〉γ) and so γ† is an inverse semigroup homomorphism. The correspondences α 7→ α̂ (defined in (3.3)) and γ 7→ γ† (defined in (3.4)) are mutually inverse bijections, giving a natural isomorphism between the bifunctors ModS(D−,−) and (IS ↓ S)(−, S ⋉−) confirming the adjunction given in the Proposition. Theorem 3.8. Let S be an inverse monoid generated by a set X, let FIM(X) be the free inverse monoid on X, and let θ : FIM(X) → S be the presentation map. (a) Dθ,e is generated, as an abelian group, by the subset {〈x, s〉 : x ∈ X, (x−1x)θ > ss−1, s−1s = e} and hence Dθ is generated as an L(S)–module by the set {〈x, (x−1x)θ〉 : x ∈ X}. (b) Dθ is a projective L(S)–module. Proof. (a) This follows from Lemma 3.4. 12 Derivations and relation modules (b) Let µ : A → B be an epimorphism of L(S)–modules. For any e ∈ E(S), the evaluation functor evale : ModS → Ab given by A 7→ Ae is exact, and hence µ : Ae → Beµ is surjective. Suppose that β : Dθ → B is an L(S)–map. By Proposition 3.7, β induces a inverse monoid homomorphism β† : FIM(X) → S ⋉ B, and there is a surjection µ∗ : S ⋉ A → S ⋉ B of inverse monoids, given by (s, a) 7→ (s, aµ). The freeness of FIM(X) then implies that we can lift β† to α† : FIM(X) → S ⋉A, with α†µ∗ = β†. For w ∈ FIM(X), it follows that wα† = (wθ,wη), where η : FIM(X) → A is a θ–derivation such that wηµ = 〈w, (w−1w)θ〉β. By Proposition 3.5 η induces an L(S)–map α : Dθ → A defined by α = (̂α†) : 〈w, s〉 7→ wη ⊳ ((w−1w)θ, s), and 〈w, s〉αµ = (wη ⊳ ((w−1w)θ, s))µ = wηµ⊳ ((w−1w)θ, s) = 〈w, (w−1w)θ〉β ⊳ ((w−1w)θ, s) = (〈w, (w−1w)θ〉⊳ ((w−1w)θ, s))β = 〈w, s〉β. This shows that Dθ is projective. Proposition 3.7 is based on the adjunction result for groupoid actions given in [2], but may also be seen as a special case of a result of Nico [16] for the semidirect product of categories. This work is elaborated by Steinberg and Tilson in [20], and by Steinberg in [18] for the category of ordered groupoids. The results of [18] are of particular relevance since the category of inverse semigroups is isomorphic to the subcategory of inductive groupoids in the category of ordered groupoids. Steinberg constructs a left adjoint Der(φ) to the semidirect product, where the latter is regarded as a functor from the category of ordered groupoid actions to the category of ordered functors: from a pair (G,H) of ordered groupoids withH acting on G, the semidirect product H ⋉G is an ordered groupoid with a projection map H ⋉G→ H. We have restricted attention to L(S)–modules, and so have replaced the category of ordered groupoid actions by ModS , and the category of ordered functors by the slice category (IS ↓ S). Steinberg’s adjunction then tells us that morphisms T → S⋉A in (IS ↓ S) correspond bijectively to a certain subclass of morphisms Der(φ) → A in the category of ordered groupoid actions. Following through the details of Steinberg’s construction, it is readily seen that the subclass of morphisms are those that arise from φ–derivations T → A. N. D. Gilbert 13 4. Presentations and relation modules Let S be an inverse semigroup given by a presentation P = Inv[X : R], and let θ : FIS(X) → S be the associated presentation map. Corollary 3.1 gives an L(S)–map ∂1 : Dθ → s: its kernel M is an L(S)–module called the relation module of P . Our first task is to find a convenient generating set for the relation module: we approach this in stages in our next result. Theorem 4.1. (a) The relation module Me at e ∈ E(S) is generated, as an abelian group, by all elements of the form 〈u, s〉− 〈v, s〉, where uθ = vθ and s−1s = e. (b) A smaller generating set for Me, as an abelian group, is given by the set of all elements of the form 〈pr1q, s〉 − 〈pr2q, s〉, where p, q ∈ FIS(X), (r1, r2) ∈ R and s−1s = e. (c) The relation module M is generated as an L(S)–module by the set of all elements of the form 〈r1, e〉−〈r2, e〉 where (r1, r2) ∈ R, e ∈ E(S), and (r−1 1 r2)θ = e. Proof. (a) Let Pe be the subgroup of Me generated by the set of elements specified in part (a) of the theorem. Now suppose that α = ∑ i∈I εi〈ui, si〉 is an element of the relation module Me at e (where εi = ±1), so that α∂1 = ∑ i∈I εi((uiθ)si − si) = 0. (4.1) Since ZS is (additively) free on the elements of the L–class Le, the terms in the sum in (4.1) must cancel in pairs. We fix such a pairing of cancelling terms, and for each si occurring in the given expression for α we choose wsi ∈ FIS(X) with wsiθ = si. Suppose that in our pairing of cancelling terms, (uiθ)si is paired with si. Then (uiθ)si = si and, by Lemma 3.2, 〈ui, si〉 = 〈uiwsiw −1 si , si〉 and 〈uiwsiw −1 si , si〉 = 0. It follows that 〈ui, si〉 = 〈uiwsiw −1 si , si〉 − 〈uiwsiw −1 si , si〉 ∈ Pe. Now suppose that (uiθ)si is paired with (ujθ)sj : that is (uiθ)si = (ujθ)sj and −εi = εj . Then by Lemma 3.3, 〈uiwsi , e〉 = 〈ui, si〉 + 〈wsi , e〉 and 〈ujwsj , e〉 = 〈uj , sj〉 + 〈wsj , e〉, with 〈uiwsi , e〉 − 〈ujwsj , e〉 ∈ Pe. Therefore, we may write εi〈ui, si〉+ εj〈uj , sj〉 = εi(〈uiwsi , e〉 − 〈ujwsj , e〉)− εi〈wsi , e〉 − εj〈wsj , e〉, 14 Derivations and relation modules and modulo Pe, we see that εi〈ui, si〉+ εj〈uj , sj〉 is equal to −εi〈wsi , e〉 − εj〈wsj , e〉. Finally, if (ui)θsi is paired with sj , then 〈uiwsi , e〉 − 〈wsj , e〉 ∈ Pe and, again by Lemma 3.3, 〈ui, si〉 = (〈uiwsi , e〉 − 〈wsj , e〉) + 〈wsj , e〉 − 〈wsi , e〉. These considerations show that modulo Pe, the element α ∈ Me is equal to a sum β = ∑ kmk〈wsk , e〉 with mk ∈ Z and the sk distinct. But then β∂1 = ∑ kmk(sk− e) = 0 is a sum of Z–independent elements of ZS, and so each mk = 0. Therefore β = 0, and we conclude that α is in Pe. (b) Let Qe be the subgroup of Me generated by the set of elements specified in part (b) of the theorem. Let 〈u, s〉 − 〈v, s〉, as in part (a), be a generator of Pe. Now S is presented by Inv[X : R] and so, since uθ = vθ, there exists a finite sequence of elements w0, w1, . . . , wm ∈ FIM(X) with w0 = u, wm = v and such that for each i, 0 6 i < m, we may write wi = piriqi and wi+1 = pir ′ iqi with pi, qi ∈ FIM(X) and with either (ri, r ′ i) or (r′i, ri) in R. We show by induction on m that the element 〈u, s〉−〈v, s〉 of Dθ is in the subgroup Qe. If m = 1 then u = p0r0q0 and v = p0r ′ 0q0 and so 〈u, s〉 − 〈v, s〉 = 〈p0r0q0, s〉 − 〈p0r ′ 0q0, s〉 ∈ Qe. If m > 1 we have u = p0r0q0 and w1 = p0r ′ 0q0 〈u, s〉 − 〈v, s〉 = 〈p0r0q0, s〉 − 〈p0r ′ 0q0, s〉 + 〈w1, s〉 − 〈v, s〉. Now we have w1θ = vθ and the elements w1, v are linked by the sequence w1, . . . , wm ∈ FIM(X), and by induction 〈w1, s〉 − 〈v, s〉 ∈ Qe. It follows that 〈u, s〉 − 〈v, s〉 ∈ Qe. (c) We show that a generator 〈pr1q, s〉− 〈pr2q, s〉 is an L(S)–translate of the element 〈r1, f〉 − 〈r2, f〉, where f = (r−1 1 r2)θ. Now by Lemma 3.3, 〈pr1q, s〉 = 〈p(p−1p)r1q, s〉 = 〈p, (p−1pr1q)θs〉 + 〈p−1pr1q, s〉 and similarly 〈pr2q, s〉 = 〈p, (p−1pr2q)θs〉 + 〈p−1pr2q, s〉 . Now (p−1pr1q)θ = (p−1pr2q)θ, and so 〈pr1q, s〉 − 〈pr2q, s〉 = 〈p−1pr1q, s〉 − 〈p−1pr2q, s〉 = 〈r1q, s〉 − 〈r2q, s〉 by part (b) of Lemma 3.2. We see that 〈pr1q, s〉−〈pr2q, s〉 does not depend on p. In the same manner, 〈r1q, s〉 = 〈r1(r −1 1 r1q), s〉 = 〈r1, (r −1 1 r1q)θs〉 + 〈r−1 1 r1q, s〉 N. D. Gilbert 15 = 〈r1, (r −1 1 r1q)θs〉 + 〈q, s〉 since 〈r−1 1 r1q, s〉 = 〈q, s〉 by part (b) of Lemma 3.2. It follows that 〈r1q, s〉 − 〈r2q, s〉 = 〈r1, (r −1 1 r1q)θs〉 − 〈r2, (r −1 2 r2q)θs〉 = 〈r1, s ′〉 − 〈r2, s ′〉 where s′ = (r−1 1 r1q)θs = (r−1 2 r2q)θs. Then finally we have 〈r1, s ′〉 − 〈r2, s ′〉 = (〈r1, e〉 − 〈r2, e〉) ⊳ (e, s′). Corollary 4.2. Let S be an inverse monoid presented by Inv[X : R] with presentation map θ : FIM(X) → S. Then there is an exact sequence of L(S)–modules F(R) → Dθ → ZS → Z → 0 where F(R) is the free L(S)–module on the E(S)–set X = {Xe : e ∈ E(S)}, with Xe = {(r1, r2) ∈ R : (r−1 1 r2)θ = e}. 4.1. The Schützenberger representation Suppose that S is an inverse monoid presented by Inv[X : R] with pre- sentation map θ : FIM(X) → S. The (left) Schützenberger graph of S with respect to X, denoted by SchL(S,X), has vertex set S and, for each (x, s) ∈ X × S with (x−1x)θ > ss−1, has a directed edge labelled by (x, s) with initial vertex s and terminal vertex (xθ)s. Equivalently, there is a directed edge from s to xs labelled by (x, s) whenever sL(xθ)s in S. The connected components of SchL(S,X) are the full subgraphs spanned by the L–classes of S, and the connected component containing e ∈ E(S) is denoted by SchL(S,X, e). The usual convention for Schützenberger graphs [19, 21] is to have a directed edge (s, x) from s to sx whenever sRsx. To get a natural right L(S)–module, we have to use the left Schützenberger graphs. Suppose that s−1s = e and that (e, t) ∈ L(S). Then by defining s⊳ (e, t) = st and (a, s) ⊳ (e, t) = (a, st) we get a graph map SchL(S,X, e) → SchL(S,X, t−1t) and in this way we obtain a functor from L(S) to the category of directed graphs. The cellular chain complex CL(S,X) of SchL(S,X) is then a complex of L(S)– modules, with CL 0 (S,X) = ZS and with the group CL 1 (S,X)e free abelian on the set {(x, s) : x ∈ X, s ∈ S, (x−1x)θ > ss−1, s−1s = e}. The boundary map CL 1 (S,X)e → CL 0 (S,X)e maps (x, s) 7→ (xθ)s− s. Theorem 4.3. Suppose that S is an inverse monoid presented by Inv[X : 16 Derivations and relation modules R] with presentation map θ : FIM(X) → S. Then the derivation module Dθ is isomorphic to CL 1 (S,X) and the relation module M is isomorphic to the first homology group of the Schützenberger graph SchL(S,X). Proof. By Corollary 3.6 we have a commutative square of L(S)–maps CL 1 (S,X) κ �� // CL 0 (S,X) ∼= �� Dθ ∂1 // ZS in which the left-hand map κ is the L(S)–map induced (as a homo- morphism of abelian groups) by (x, s) 7→ 〈x, s〉, and κ restricts to a surjection H1(SchL(S,X)) → M. Now the function X → S ⋉ CL 1 (S,X) that maps x 7→ (xθ, (x, (x−1x)θ) induces an inverse monoid homomor- phism FIM(X) → S ⋉ CL 1 (S,X) and by Proposition 3.7 there is then an L(S)–map Dθ → CL 1 (S,X) mapping 〈x, (x−1x)θ〉 7→ (x, (x−1x)θ) giving an inverse to κ. 4.2. Cohomological dimension The classification of all small categories of cohomological dimension zero was completed by Laudal [9], following a conjecture of Oberst [17]. Lau- dal’s result was then used by Leech [13] to classify the inverse monoids of cohomological dimension zero: the classification is the expected gener- alisation of the result that a group has cohomological dimension zero if and only if it is trivial. We present another proof here that makes direct use of the fact that, if S has cohomological dimension zero, then Z is a projective L(S)–module. Theorem 4.4 (Leech [13]). An inverse monoid S has cohomological dimension zero if and only if it is a semilattice. Proof. If S is a semilattice then ZS = Z and so Z is a projective L(S)– module, and hence cd(S) = 0. Conversely, if cd(S) = 0 then Z has a projective resolution P0 → Z and hence Z is a projective L(S)–module. In particular, the augmentation map ε : ZS → Z splits as an L(S)–map, and so there is an L(S)–map σ : Z → ZS such that σε = id. Let 1e denote the copy of 1 ∈ Z in Ze (that is, in the copy of Z indexed by e ∈ E(S)) and let 1̂ = 11. We set 1̂σ = w ∈ ZS1. Then wε = 1̂ and w determines σ since 1eσ = (1̂ ⊳ (1, e))σ = w ⊳ (1, e) = we ∈ ZLSe . N. D. Gilbert 17 Now for any s ∈ S, we have ws = w ⊳ (1, s) = (1̂σ) ⊳ (1, s) = (1̂ ⊳ (1, s))σ = 1s−1sσ = ws−1s (4.2) in the free abelian group ZSs−1s. We may write w = ∑ t−1 i ti=1mtiti (with mti ∈ Z and at most finitely many non-zero). Suppose that s ∈ S with ss−1 = 1. Then for each ti with mti 6= 0, there exists a tj with mtj 6= 0 such that tis = tjs −1s. But then tit −1 i = (tis)(tis) −1 = (tjs −1s)(tjs −1s)−1 = tjs −1st−1 j 6 tjt −1 j . Since the collection of such ti is finite, there must exist tk occurring in w with tkt −1 k = tks −1st−1 k , or equivalently with tk = tks −1s. Then 1 = t−1 k tk = s−1st−1 k tk = s−1s . We see that ss−1 = 1 = s−1s, and so Green’s L–class at 1 ∈ E(S) coincides with the H–class H1, and is a group. In particular, ZS1 is the integral group ring ZH1. Now for all h ∈ H1 we have, by (4.2), ∑ t−1 i ti=1 mtitih = ∑ t−1 i ti=1 mtiti . Taking h = t−1 i tj in the group H1, we see that mti = mtj , and since∑ imti = 1, it follows that \|H1\| = 1. In particular w = 1. From (4.2) we see, for any s ∈ S, that s = s−1s and so S is a semilattice. We note the contrast between this result and the theory of monoids of cohomological dimension zero. Laudal [9, Lemma A] shows that if a monoid M has (left) monoid cohomological dimension equal to zero then M has a right zero. See [5] for further results on the cohomological dimension of monoids. Arboreal inverse monoids An arboreal inverse monoid S is an inverse monoid given by a presentation of the form S = Inv[X : ei = fi , (i ∈ I)] where I is some indexing set and ei, fi are idempotents in FIM(X). There- fore ei, fi are Dyck words in (X∪X−1)∗, that is words whose freely reduced form is equal to 1. Free inverse monoids and free groups are arboreal inverse monoids, as is the bicyclic monoid B, presented by Inv[x : xx−1 = 1]. Arboreal inverse monoids are the main concern of [15], wherein it is shown that arboreal inverse monoids with generating set X are pre- 18 Derivations and relation modules cisely the E–unitary quotients of FIM(X) with maximum group image F (X), and are also characterised as the inverse monoids each of whose Schützenberger graphs is a tree. This latter characterization has motivated the use of the adjective ‘arboreal’. A principal result of [15] is that finitely presented arboreal inverse monoids have decidable word problem. The following result may be readily deduced from the work of Steinberg [19] and also follows from Funk’s topos-theoretic approach to inverse semigroup representations [4]. Theorem 4.5. An arboreal inverse monoid S has cohomological dimension equal to 1. Proof. Let S be presented by S = Inv[X : ei = fi , (i ∈ I)], and let θ : FIM(X) → S be the presentation map. 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Webb, Resolutions, relation modules and Schur multipliers for categories. J.Algebra 326 (2011) 245-276.. Contact information N. D. Gilbert School of Mathematical and Computer Sciences & the Maxwell Institute for Mathematical Sci- ences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. E-Mail: N.D.Gilbert@hw.ac.uk URL: www.ma.hw.ac.uk/∼ndg Received by the editors: 23.07.2010 and in final form 10.10.2011.

Derivations and relation modules for inverse semigroups

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