Full and convex linear subcategories are incompressible
aa r X i v : . [ m a t h . R A ] O c t Full and convex linear subcategories areincompressible
Claude Cibils, Maria Julia Redondo and Andrea Solotar ∗ Abstract
Consider the intrinsic fundamental group `a la
Grothendieck of a linearcategory, introduced in [5] and [6] using connected gradings. In this articlewe prove that any full convex subcategory is incompressible, in the sense thatthe group map between the corresponding fundamental groups is injective.We start by proving the functoriality of the intrinsic fundamental group withrespect to full subcategories, based on the study of the restriction of connectedgradings. Introduction
In two recent papers [5, 6] we have considered a new intrinsic fundamental groupattached to a linear category. In [6] we have made several explicit computations:for instance the fundamental group of kC p where C p is a group of prime order p and k is a field of characteristic p is the direct product of C p with the infinite cyclicgroup.We have obtained this group using methods inspired by the definition of thefundamental group in other mathematical contexts. We briefly recall the definitionof the intrinsic fundamental group as the automorphism group of a fibre functor.Previously a non canonical fundamental group has been introduced by R. Mart´ı-nez-Villa and J.A. de la Pe˜na in [11] and K. Bongartz and P. Gabriel in [2] and [8].This group varies considerably according to the presentation of the linear category,see for instance [1, 3]. See also [9, 10] as a first approach to solve this variabilityproblem.The main tool we use are connected gradings of linear categories. The fun-damental group that we consider is a group which is derived from all the groupsgrading the linear category in a connected way. More precisely each connectedgrading provides a Galois covering through the smash product, see [4]. Consideringthe category of Galois coverings of this type we define the intrinsic fundamentalgroup as the automorphism group of the fibre functor over a chosen object. ∗ This work has been supported by the projects UBACYTX212 and 475, PIP-CONICET112- 200801-00487, PICT-2007-02182 and MATHAMSUD-NOCOMALRET. The second andthird authors are research members of CONICET (Argentina). ne of the main purposes of this paper is to prove that full and convex subcat-egories are incompressible, in the sense used in algebraic topology where a subspaceis called incompressible if the group map between the corresponding fundamentalgroups is injective.Another main purpose is to prove that the intrinsic fundamental group is func-torial with respect to full subcategories, answering in this way a question by AlainBrugui`eres. Note that this is not automatic from the definition of the fundamentalgroup. We provide a description of elements of the intrinsic fundamental group ascompatible families of elements lying in each group which grades the linear categoryin a connected way. The other main ingredient for proving the functoriality is theconnected component of the base object for a restricted grading, and the associatedconnected grading considered in [4] depending on some choices which we prove tobe irrelevant at the intrinsic fundamental group level.Finally we consider convex subcategories. Recall that a linear subcategory ofa linear category is convex if morphisms of the subcategory can only be factorizedthrough morphisms in the subcategory. Our results enables to prove that full andconvex subcategories are incompressible.We thank the referee for useful comments. Coverings and elements of the intrinsic fundamentalgroup
In this section we recall some definitions and results from [5] and [6] that we will usethroughout this paper. A main purpose of this article is to prove that the intrinsicfundamental group of a k -category is functorial with respect to full subcategories.In order to do so we will need a concrete interpretation of elements of this groupthat we provide below.Let k be a commutative ring. A k -category is a small category B with set ofobjects B such that each morphism set y B x from x ∈ B to y ∈ B is a k -module,the composition of morphisms is k -bilinear and the identity at each object is centralin its endomorphism ring. Definition 2.1
The star St b B of a k -category B at an object b is the direct sum ofall k -modules of morphisms with source or target b : St b B = M y ∈B y B b ⊕ M y ∈B b B y Definition 2.2
Let C and B be k -categories. A k -functor F : C → B is a coveringof B if it is surjective on objects and if F induces k -isomorphisms between allcorresponding stars. More precisely, for each b ∈ B and each x in the non-emptyfibre F − ( b ) , the map F xb : St x C −→ St b B . induced by F is a k -isomorphism. bserve that a covering is a faithful functor. Note also that Definition 2.2coincides with the one given by K. Bongartz and P. Gabriel in [2]. Definition 2.3
Given k -categories B , C , D , the set of morphisms Mor ( F, G ) from acovering F : C → B to a covering G : D → B is the set of pairs of k -linear functors ( H, J ) where H : C → D , J : B → B are such that J is an isomorphism, J is theidentity on objects and GH = JF . We also say that H is a J -morphism.We will consider within the group of automorphisms of a covering F : C → B ,the subgroup
Aut F of invertible endofunctors G of C such that F G = F . For any covering F it is known that Aut F acts freely on each fibre, see [10, 5].A k -category B is connected if any two objects can be joined by a non-zerowalk, see [6, Section 2] for details. Definition 2.4
A covering F : C −→ B of k -categories is a Galois covering if C is connected and Aut F acts transitively on some fibre. As expected in similar Galois theories, the automorphism group acts transitivelyat every fibre as soon as it acts transitively on a particular one, see [10, 5].Recall that a grading Z of a k -category B by a group Γ Z is a direct sumdecomposition of each k -module of morphisms from b to c c B b = M s ∈ Γ Z Z sc B b such that for s, t ∈ Γ X Z td B c Z sc B b ⊂ Z tsd B b . A morphism from b to c is called homogeneous of degree s if it belongs to Z sc B b .A grading is connected if given any two objects they can be joined by a non-zerohomogeneous walk of arbitrary degree. For precise definitions see [6, Section 2]. Definition 2.5 [4] Let B be a k -category and let Z be a grading of B . The smashproduct category B Z has set of objects B × Γ Z , the vector spaces of morphismsare homogeneous components as follows: ( c,t ) ( B Z ) ( b,s ) = Z t − s c B b . Note that for a connected grading X , the evident functor F X : B X −→ B isa Galois covering.Let B be a connected k -category with a fixed object b . The category Gal ( B , b ) has as objects the Galois coverings of B . A morphism in Gal ( B , b ) from F : C → B to G : D → B is a morphism of coverings ( H, J ) , see Definition 2.3.Since any Galois covering F of B is isomorphic to a smash product Galoiscovering by considering the natural grading of B by Aut F , we consider as in [6] thefull subcategory Gal ( B , b ) whose objects are the smash product Galois coveringsprovided by connected gradings of B . It can be proved that this full subcategory s equivalent to Gal ( B , b ) , see [6]. The fibre functor Φ : Gal ( B , b ) → Groupsgiven by Φ ( F X ) = F − X ( b ) = Γ X is the main ingredient for the definition of the fundamental group, namely Π ( B , b ) = Aut Φ . Next we will consider in detail this group, and we will prove that an elementof Π ( B , b ) is determined by a family of elements belonging to the groups ofconnected gradings related through canonical surjective morphisms µ : Γ X → Γ X ′ obtained as soon as X and X ′ are connected gradings admitting some morphismfrom B X to B X ′ . Remark 2.6
For any smash product Galois covering F X we identify the isomorphicgroups Aut F X and Γ X through left multiplication, that is, by the correspondence s : F X → F X with s ( x ) = sx for any s ∈ Γ X . Proposition 2.7
Let X and X ′ be connected gradings of a k -category B , and let F X and F X ′ be the corresponding smash product Galois coverings with groups Γ X and Γ X ′ . Let ( H, J ) be a morphism from F X to F X ′ in Gal ( B , b ) , where H : B X −→ B X ′ is given on objects by H ( b, s ) = ( b, H b ( s )) . Then thereexists a unique surjective morphism of groups λ H : Γ X → Γ X ′ verifying H b ( sx ) = λ H ( s ) H b ( x ) for all x ∈ Γ X . Moreover the complete list of J -morphisms from F X to F Y is given by { qH } q ∈ Γ Y ,and λ qH = q ( λ H ) q − . For the proof see [6, Section 2].
Definition 2.8
In case of existence of a J -morphism H : B X → B X ′ weconsider the normalized J -morphism N = H b (1) − H . Observe that N does notdepend on H since its value is the same when H is replaced by qH . Moreover N ( b ,
1) = ( b , .We set µ = λ N and we call µ the canonical group map associated to theexistence of a J -morphism from the smash product with X to the one with X ′ . For H : B X → B X ′ a morphism we have µ = H b (1) − λ H H b (1) . We are now able to describe the elements of the intrinsic fundamental group, namelythe automorphisms of the fibre functor.Recall that σ ∈ Aut Φ is a family { σ X : Γ X → Γ X } where X is any connectedgrading of B making commutative the following diagram for any morphism H inGal ( B , b ) : X σ X / / H b (cid:15) (cid:15) Γ XH b (cid:15) (cid:15) Γ X ′ σ X ′ / / Γ X ′ Lemma 2.9
The map σ X is the right product by an element g X ∈ Γ X . Proof.
In case X = X ′ , the vertical arrows in the diagram can be specializedby any element in Aut F X = Γ X . By Remark 2.6, this vertical morphisms areleft product by some g ∈ Γ X . We infer σ X ( g ) = σ X ( g
1) = gσ X (1) and we set g X = σ X (1) . ⋄ Proposition 2.10
The automorphisms in Π ( B , b ) are in one to one correspon-dence with families of group elements { g X } verifying µ ( g X ) = g X ′ for each canon-ical group map µ corresponding to the existence of a morphism from B X to B X ′ . Proof.
Let H be a J -morphism from F X to F ′ X where X and X ′ are connectedgradings of B , and let µ : Γ X → Γ X ′ be the corresponding canonical surjectivegroup map. The previous diagram becomes Γ X .g X / / H b (1) µ (cid:15) (cid:15) Γ XH b (1) µ (cid:15) (cid:15) Γ Y ′ .g Y ′ / / Γ Y ′ Hence H b (1) µ ( gg X ) = H b (1) µ ( g ) g Y ′ for any g . Since µ is a group homomorphism we infer µ ( g X ) = g Y ′ . Reciprocally afamily of elements with the stated property clearly defines an automorphism of thefibre functor. ⋄ We say that a family of group elements is compatible if it satisfies the conditionin the proposition above. Functoriality and incompressible subcategories
An important tool for proving the functoriality of the intrinsic fundamental groupis the restriction of a grading to full subcategories that we will consider below.Let Z be a non necessarily connected grading of a connected k -category B . Let b (Γ Z ) b be the set of walk’s degrees from b to b , that is, the set of elements in Γ Z which are degrees of homogeneous non-zero walks from b to b . Note that if = b this set is a subgroup of Γ Z which we denote Γ Z,b . Let s be any walk’sdegree from b to b . Then b (Γ Z ) b = s [Γ Z,b ] = [Γ Z,b ] s. Recall that by definition, a grading X is connected if and only if for any objects b , b ∈ B b (Γ X ) b = Γ X . This is equivalent to b (Γ X ) b = Γ X for some pair of objects, and in particular to Γ X,b = Γ X for a given object b , see [6].Let B be a k -category and let X be a connected grading of B by the group Γ X .Let D be a connected full subcategory of B . The restricted grading by the samegroup Γ X is denoted X ↓ D . Note that since D is full, each k -module of morphismsin D has the same direct sum decomposition from the original grading. Clearly thegrading X ↓ D is not connected in general.In order to consider the corresponding connected grading out of a non con-nected one we will provide some preliminary results. The description of a connectedcomponent will make use of conjugated gradings as defined in [6]. More precisely,let Z be a non necessarily connected grading of a connected k -category B and let ( a b ) b ∈B be a family of elements in Γ Z . The conjugated grading a Z of B by thesame group Γ Z is as follows: we set the a Z -degree of a non-zero homogeneousmorphism from b to c of Z -degree t to be a − c ta b , that is, ( a Z ) sc B b = Z a c sa − b c B b . Note that the underlying homogeneous components remain unchanged, and thereis no difficulty for proving that a Z is indeed a grading. We observe that a Z isconnected if Z is so. Definition 3.1
Let B be a connected k -category with a non necessarily connectedgrading Z with group Γ Z . Let b be a fixed object. The connected component of B Z containing the object ( b , is denoted ( B X ) and is called the connectedcomponent of the base object . Proposition 3.2
The connected component of the base object ( B Z ) is thesmash product of B by a conjugated grading of Z . The group of this Galois covering(or of the corresponding connected grading) is Γ Z,b . Proof.
Firstly we assert that the restriction of F Z : B Z → B to the connectedcomponent of the base object is still surjective on objects. Indeed B is connected,then for any object b ∈ B there exists a walk w from b to b . For each morphismappearing in w we choose an homogeneous non-zero component, obtaining in thisway an homogeneous walk w ′ of some degree s from b to b . Lifting appropriatelythe homogeneous morphisms of w ′ provides a walk in the connected componentof the base object from some ( b, s ) to ( b , . Thus ( b, s ) is an object of theconnected component of the base object lying in the fibre of b of the restriction of F Z . Consequently this restriction is a Galois covering. econdly we know that each Galois covering provides a connected grading of thebase category (with group the automorphism group of the covering) once a familyof objects is chosen in each fibre (see [4]). Let ( b, u b ) be such a choice for therestriction of F Z , with u b = 1 . Since ( B X ) is connected, there is a walk from ( b , to ( b, u b ) providing through F Z an homogeneous walk in B from b to b of Z -degree u b . We set u = ( u b ) b ∈B . The definition of the grading (see [4]) showsthat the grading coincides with u X .Finally in order to show that the group of the grading (namely the automorphismgroup of the Galois covering) is Γ Z,b consider ( b , s ) an object in the fibre of b and a walk from ( b , s ) to ( b , . Its image using F Z provides an homogeneousclosed walk at b in B which degree is precisely s according to the constructionof the grading. Conversely let w be an homogeneous closed walk at b in B . Byappropriate lifting, w provides a walk in B Z between ( b , and some ( b , s ) ,where the degree of w is s . ⋄ Remark 3.3
The connected grading obtained depends on the choice ( u b ) b ∈B ,where u b ∈ b (Γ Z ) b . Any other choice ( u ′ b ) b ∈B with u ′ b = 1 is obtained as ( u b a b ) b ∈B where a b ∈ Γ Z,b and a b = 1 . The group of the connected gradingsremains the same. Theorem 3.4
Let B be a connected k -category with a fixed object b and let D be a connected full subcategory containing b . Then there is a canonical groupmorphism κ : Π ( D , b ) −→ Π ( B , b ) In this way Π becomes a functor from the category of small k -categories witha chosen base object with morphisms the fully faithful functors which are injectiveon objects and preserve base objects to the category of groups. Proof.
According to Proposition 2.10 let σ ∈ Π ( D , b ) be determined by acompatible family { g Y } where Y varies over all the connected gradings of D andwhere g Y is in Γ Y . Recall that µ ( g Y ) = g Y ′ for each pair of connected gradings Y and Y ′ such that there is a morphism from D Y to D Y ′ . In order to define ( κ ( σ )) X for a given connected grading X of B we consider the connected componentof the base object ( D X ↓ D ) . According to the previous proposition this Galoiscovering is given by a smash product with respect to a connected grading u ( X ↓ D ) with group Γ X ↓ D ,b . We define κ ( σ ) X = g u ( X ↓ D ) . In order to check that κ is well defined we have to verify that for any set u ′ = ( u ′ b ) of degrees of homogeneous walks in D from b to b with u ′ b = 1 we have g u ( X ↓ D ) = g u ′ ( X ↓ D ) . This will be insured by the following Lemma, which shows that there is a morphismbetween the corresponding Galois smash coverings whose corresponding canonicalgroup map µ is the identity. e prove now that the obtained family is compatible. Let X and X ′ be twoconnected gradings of B , let ( H, J ) be a morphism of coverings B X → B X ′ and let µ : Γ X → Γ X ′ be the corresponding canonical group map.Since J is the identity on objects it restricts to an isomorphism of D . Hence ( H, J ) restricts to a morphism from the full subcategory ( D X ↓ D ) to the fullsubcategory ( D X ′ ↓ D ) .Note that a morphism between non necessarily connected coverings is faithful,consequently it preserves connected components. Hence the preceding restrictiongives a morphism of Galois coverings ( D X ↓ D ) → ( D X ′ ↓ D ) . The canonical group map arising from this morphism is the restriction of the canoni-cal µ associated to ( H, J ) . This shows that the family defined by κ ( g ) is compatible.As a consequence we note that the image of the restriction of µ to Γ X ↓ D ,b isin Γ X ′ ↓ D ,b , a fact which can also be obtained easily directly.Finally note that κ clearly preserves composition of inclusions of full subcate-gories. ⋄ Remark 3.5
The connected gradings u ( X ↓ D ) and u ′ ( X ↓ D ) are conjugated grad-ings by the family ( a b ) b ∈B with a b = 1 . Lemma 3.6
Let X be a connected grading of B and let ( a b ) b ∈B be a family ofelements in Γ X . There is a covering morphism B X → B a X between conjugatedgradings. The corresponding induced canonical group morphism µ : Γ X → Γ X isconjugation by a b . In particular if a b = 1 then µ = 1 . Proof.
The functor H is given on objects by H ( b, s ) = ( b, sa b ) while on morphismsthe functor is the identity since ( c,ta c ) ( B a X ) ( b,sa b ) = ( a X ) a − c t − sa b c B b = X t − sb B c = ( c,t ) ( B X ) ( b,s ) . In order to compute µ we first normalize H by considering N = H b (1) − H .Since H b (1) = a b we infer N ( s ) = a − b sa b . ⋄ We end this section with a general criterion for κ being injective, and we give afamily of cases where the criterion applies. Corollary 3.7
Let B be a connected k -category, let b be a fixed object and let D be a connected full subcategory containing b . Assume any connected grading of D is of the form u ( X ↓ D ) for some connected grading X of B . Then the groupmorphism κ : Π ( D , b ) −→ Π ( B , b ) is injective. roof. Let σ = ( g Y ) be a compatible family defining an element in Π ( D , b ) .Assume κ ( σ ) = 1 which means that for any connected grading X of B we have κ ( σ ) X = 1 . Recall that κ ( σ ) X = g [ u ( X ↓ D )] . Consequently those elements aretrivial. By hypothesis any connected grading Y of D is of this form, then σ = 1 . ⋄ Definition 3.8
A subcategory D of B is said to be convex if any morphism of D only factors through morphisms in D . In case D is full, this condition is equivalentto the fact that any composition of an outcoming morphism (with source in D andtarget not in D ) and an incoming one (reverse conditions) must be zero. Corollary 3.9
Let B be a connected k -category, let b be a fixed object and let D be a connected full convex subcategory containing b . Then κ is injective. Proof.
Let Y be a connected grading of D . We extend Y to B by providing trivialdegree to any morphism whose source or target is not in D . By hypothesis there isno non-zero morphism of the form gf where f has source in D , g has target in D ,and the source of g and the target of f coincide without being in D . We infer thatthis setting indeed provides a grading. The grading is connected since any elementof the group is a walk’s degree, already in D . The preceding result insures that κ is injective. ⋄ References [1] Assem, I.; de la Pe˜na, J. A. The fundamental groups of a triangular algebra.Comm. Algebra (1996), 187–208.[2] Bongartz, K.; Gabriel, P. Covering spaces in representation-theory, Invent.Math. (1981/82) 331-378.[3] Bustamante, J.C.; Castonguay, D. Fundamental groups and presentations ofalgebras. J. Algebra Appl. (2006), 549–562.[4] Cibils, C.; Marcos, E. Skew category, Galois covering and smash product of acategory over a ring. Proc. Amer. Math. Soc. (2006), no. 1, 39–50.[5] Cibils, C.; Redondo M. J.; Solotar, A. The intrinsic fundamental group of alinear category. Algebr. Represent. Theory, DOI: 10.1007/s10468-010-9263-1.[6] Cibils, C.; Redondo M. J.; Solotar, A. Connected gradings and fundamentalgroup. Algebra Number Theory (2010), no. 5, 625–648.[7] Cibils,C.; Solotar, A. Galois coverings, Morita equivalence and smash exten-sions of categories over a field. Documenta Math. (2006), 143–159.[8] Gabriel, P. The universal cover of a representation-finite algebra. Represen-tations of algebras (Puebla, 1980), 68–105, Lecture Notes in Math. ,Springer, Berlin-New York, 1981.
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M.J.R.: Universidad Nacional del Sur,Departamento de Matem´atica, Universidad Nacional del Sur,Av. Alem 12538000 Bah´ıa Blanca, Argentina. [email protected]
A.S.:Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales,Instituto de Matem´atica Luis Santal´o, IMAS-CONICETUniversidad de Buenos Aires,Ciudad Universitaria, Pabell´on 11428, Buenos Aires, Argentina. [email protected]@dm.uba.ar