Stabilizer Conjecture for Representation-Tame Galois Coverings of Algebras

Journal of Algebra 239, 112᎐149 Ž2001. doi:10.1006rjabr.2000.8635, available online at http:rrwww.idealibrary.com on

Stabilizer Conjecture for Representation-Tame Galois Coverings of Algebras Piotr Dowbor 1 Faculty of Mathematics and Informatics, Nicholas Copernicus Uni¨ ersity, Chopina 12 r 18, 87-100 Torun, ´ Poland E-mail: [email protected] Received December 28, 1999

INTRODUCTION For more than 20 years Galois coverings have been one of the most efficient techniques in contemporary representation theory of algebras over a field and matrix problems and have been used successfully in solutions of various classification and theoretical problems. They were originally invented for studying representation-finite algebras Žsee w1, 15, 17, 20x.. Later, the Galois covering techniques were adopted for the representation-infinite case Žsee w10᎐12x; also w3, 4, 6x. and were effectively applied in w16, 19, 28, 30, 31x. In the meantime, they also became a useful tool for studying matrix problems w9, 14, 23᎐25x. The Galois covering method, also in the infinite-representation case, often allows us to reduce the description of the module category of an algebra A, in particular, the classification of indecomposables or at least determining the representation type of A, to the analogous problems for ˜ which are usually simpler. Therefore especially its cover category A, important, because of these applications, are results which are a specialization of the conjecture Žunder some extra assumptions. saying that, for a given Galois covering F: A˜ ª A with a group G, A is representation-tame ˜ ŽIt is shown in w11x that if A is representation-tame provided that so is A. ˜ . This conjecture has been stimulating research in this topic then so is A. area for many years. Its positive solution was announced by Drozd and 1

The author was supported by Polish KBN Grant 2 P03A 007 12. 112

0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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Ovsienko several years ago, but to the author’s knowledge a written version of the proof is still not available.2 One can find the above-mentioned results in w3, 4, 6, 10᎐12, 19x. Almost all of them refer to a deeper understanding of the structure of modules over the algebra A in terms of A˜ and are based on the description of indecomposable A-modules of the second kind Žw.r.t. F ., i.e., those indecomposables which are not in the image of the so-called ‘‘push-down’’ functor associated with F. A closer analysis of Galois covering functors leads to the concept of the G-atom Žsee 1.2. which in this respect plays an essential role. This description strongly uses the fact that the G-atom B usually yields an embedding of the representation category of the stabilizer GB of B into the module category of A. This is especially interesting if the stabilizers of infinite G-atoms are infinite cyclic groups. Then we deal in fact with the module categories of a Laurent polynomial algebra in one variable and in this way we obtain one-parameter families of indecomposable A-modules which in good situations exhaust the whole class of indecomposable A-modules of the second kind. In the quoted theorems the conditions in assumptions are usually phrased in terms of G-atoms; also, the proofs depend heavily on this notion. In this context several natural questions concerning G-atoms, which seemed to be very important, arose. The so-called ‘‘stabilizer conjecture,’’ strongly related to the previous one, was one of them. It was formulated almost 15 years ago by A. Skowronski ´ and the present author during the Durham symposium ‘‘Representations of Algebras’’ Ž1985. and said that the stabilizers of infinite G-atoms Žcalled there weakly-G-periodic modules w12x. for the representation-tame Galois covering A˜ were cyclic. The problem of determining the groups occurring as stabilizers of G-atoms for locally bounded representation-tame categories remained open until now. The main goal of this paper is to prove the ‘‘stabilizer conjecture’’ for torsion-free groups, precisely formulated below Žsee Section 1 for the definitions.. The problem for finite G-atoms was solved in w15x. Note that the restriction imposed on the group is natural because of w15, 3.5x and w16x. The considered case is very general since it is expected w29, Problem 5x that if a representation-tame algebra A admits a Galois covering by a simply-connected A˜ then its group G is torsion-free. MAIN THEOREM. Let R be a locally bounded representation-tame category o¨ er an algebraically closed field k and G : Aut k Ž R . a torsion-free group of k-linear automorphisms acting freely on R. Then the stabilizer GB of any infinite G-atom B is an infinite cyclic group. 2

In March 2000 Žseveral months after the submission of this paper. the author obtained the preprint by Yu. A. Drozd and S. A. Ovsienko, ‘‘Coverings of tame boxes,’’ Max-PlanckInstitute, Bonn, which contains a proof of the conjecture.

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This theorem was announced in w7x. Most of the results contained in this paper were presented during seminar lectures at Paderborn University in July 1997 and at Torun ´ University in October 1997. The paper is organized as follows. Section 1 is devoted to a short review of basic definitions and facts concerning Galois coverings. Here also the notation used in the paper is fixed. In Section 2 the concept of the n-flower is introduced, and the first embedding theorem for flowers Žsee Theorem 2.3., implying in particular that the supports of indecomposable locally finite-dimensional modules over tame categories cannot be infinite n-flowers, for n G 5, is formulated. Section 3 is devoted to studying the extension embeddings Žsee Theorem 3.3.. The concept of a neighborhood with respect to an indecomposable locally finite-dimensional module and techniques of excision and gluing of indecomposable modules over flowers are introduced in Section 4. In Section 5 the second embedding theorem for flowers Žsee Theorem 5.1. is proved and then applied in the proof of Theorem 2.3. The Section 6 is devoted to the proof of the Main Theorem. Here a useful characterization of the infinite cyclic group is given Žsee Theorem 6.1.. The major part of this section is occupied by the proof of Theorem 6.6 which is the most important ingredient of the proof of our main result.

1. BASIC DEFINITIONS AND NOTATION Now we briefly recall the situation we deal with. Throughout the paper we use in principle the notation and definitions established in w4, 7x. 1.1. Let k be a field Žnot necessarily algebraically closed. and R be a locally bounded k-category, i.e., all objects of R have local endomorphism rings, different objects are nonisomorphic, and both sums Ý y g R dim k RŽ x, y . and Ý y g R dim k RŽ y, x . are finite for each x g R, where RŽ x, y . is the k-linear space of morphisms from x to y in R. By an R-module we mean a contravariant k-linear functor from R to the category of all k-vector spaces. An R-module M is locally finite-dimensional Žresp., finite-dimensional. if dim k M Ž x . is finite for each x g R Žresp., the dimension dim k M s Ý x g R dim k M Ž x . of M is finite.. We denote by MOD R the category of all R-modules, by Mod R Žresp., mod R . the full subcategory of all locally finite-dimensional Žresp. finite-dimensional . Rmodules, and by Ind R Žresp., ind R . the full subcategory of all indecomposable R-modules in Mod R Žresp. mod R .. By the support of an object M in MOD R we mean the full subcategory supp M of R formed by the set  x g R : M Ž x . / 04 . We denote by JR the Jacobson radical of the category Mod R.

115

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1.2. Let G be a group of k-linear automorphisms of R acting freely on objects of R. Then G acts on the category MOD R by translations g Žy., which assign to each M in MOD R the R-module g M s M ( gy1 and to each f : M ª N in MOD R the R-homomorphism g f : g M ªg N given by the family Ž f Ž gy1 Ž x ... x g R of k-linear maps. Given M in MOD R, the subgroup GM s  g g G : g M , M 4 of G is called the stabilizer of M. Let R s RrG be the orbit category which is again the locally bounded k-category Žsee w15x.. We can study the module category mod R in terms of the category Mod R. The tool at our disposal is the pair of functors F␭

MOD R ¡ MOD R , F䢇

where F : MOD R ª MOD R is the ‘‘pull-up’’ functor associated with the canonical Galois covering functor F : R ª R, assigning to each X in MOD R the R-module X ( F, and the ‘‘push-down’’ functor F␭ : MOD R ª MOD R is the left adjoint to F . Denote by ModGf R the category consisting of pairs Ž M, ␮ ., where M is a locally finite-dimensional R-module such that supp M is contained in a union of a finite number of G-orbits of G in R and ␮ is an R-action of G on M Žsee w3, 15x.. The functor F associating with any X in mod R the R-module F X endowed with the trivial R-action of G yields the equivalence 䢇

䢇

䢇

䢇

mod R , ModGf R. Now it is not difficult to observe that an important role in an understanding of the nature of objects from ModGf R, and consequently from mod R, is played by the G-atoms. Recall from w3x that an indecomposable R-module B in Mod R Žwith local endomorphism ring. is called a G-atom Žover R . provided supp B Žis GB-stable . consists of finitely many GB-orbits in R. The G-atom B is said to be finite Žresp. infinite. if GB Žequivalently supp B . is finite Žresp. infinite .. Denote by A a fixed set of representatives of isoclasses of all G-atoms in Mod R, by Ao a fixed set of representatives of G-orbits of the induced action of G on A, and for any B g A denote by S B a fixed set of representatives of left cosets of G modulo GB , containing the unit e s id R of the group G. One can show Žrepeating the arguments from w12, Proof of Proposition 2.3; 12, Proof of Lemma 2.4x, and applying the uniqueness of decomposition into a direct sum of indecomposables in Mod R . that the category mod R is equivalent via F to the full subcategory of ModGf R 䢇

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formed by all possible pairs Ž Mn , ␮ ., where n s Ž n B .B g Ao is a sequence of natural numbers, such that almost all n B are zeros, Mn is an R-module given by the formula Mn s

[ [ gŽ Bn .

Bg Ao

ž

B

ggS B

/,

and ␮ is an arbitrary R-action of G on Mn . Suppose that a G-atom B admits an R-action ␯ of GB on itself Žthis is always the case if the group GB is free.; then F␭ B carries the structure of the kGB-R-bimodule, which is finitely generated free as a left kGB-module, where kGB is the group algebra of GB over k Žsee w12, 3.6x.. This bimodule induces a functor ⌽ B s ymk G B F␭ B : mod kGB ª mod R , which is a representation embedding in the sense of w26x Žsee w4, Proposition 2.3x., provided the factor field of End R Ž B . is equal to k. Note that if GB is an infinite cyclic group then kGB is isomorphic to the Laurent polynomial algebra k w T, Ty1 x. If all infinite G-atoms have cyclic stabilizers then the functors  ⌽ B 4B g Ao⬁ , where Ao⬁ consists of all infinite G-atoms in Ao , induce the representation embedding functor ⬁

⌽ Ao :

@

⬁ o

mod k w T , Ty1 x ª mod R

Bg A

Žsee w4, 2.2x., which in good situations Žsee w3, 4, 6, 12x. yields a description of all indecomposable R-modules being out of the image Im F␭ Žthe so-called second kind modules with respect to F .. The above considerations clearly show that the shape of stabilizers of G-atoms supplies important and deep information on the category mod R. 1.3. The following notation is used in the paper. Given a full subcategory C of R and an R-module M we denote by M N C the C-module which is the restriction of M to C. We say that a full subcategory C of R is nontrivial Žresp. trivial. provided the set ob C of all objects of C is nonempty Žresp. empty.. Let C1 and C2 be full subcategories of a locally bounded k-category R. We denote by C1 j C2 Žresp. C1 l C2 and C1 _ C2 . the full subcategory of R formed by the union Žresp. intersection and difference. of the sets ob C1 and ob C2 . The notation C1 ; C2 means that ob C1 is contained in ob C2 . The subcategories C1 and C2 are called disjoint Žresp. orthogonal. if ob C1 l ob C2 s ⭋ Žresp., RŽ x, y . s 0 s RŽ y, x . for all x g ob C1 , y g ob C2 .. The union C1 j C2 is said to be a disjoint union and then denoted by C1 k C2 , provided C1 and C2 are disjoint.

STABILIZER CONJECTURE

117

Let A be a k-algebra. Then the Jacobson radical of A is denoted by J Ž A.. For any m, n g ⺞ we denote by M m= nŽ A. the set of all m = nmatrices with coefficients in A and by M nŽ A. the algebra of all square n = n-matrices with coefficients in A. Let H be a group. Then for any subset X of H we denote by ² X : the subgroup of H generated by X. For any subgroup H⬘ of H the index Žresp. centralizer, normalizer. of H⬘ in H is denoted by w H : H⬘x Žresp. ZH Ž H⬘., NH Ž H⬘... For any set X we denote by < X < the cardinality of X.

2. THE FIRST EMBEDDING THEOREM FOR FLOWERS 2.1. Concerning infinite G-atoms, an important role is played by a certain class of locally bounded k-categories. DEFINITION. A connected locally bounded k-category R is called an n-flower, for n g ⺞, provided there exists a family of full subcategories R 0 , R1 , . . . , R n of R satisfying the following conditions: ŽF0. ŽF1. ŽF2.

R s R 0 k R1 k ⭈⭈⭈ k R n , R 0 is finite and nontrivial, R1 , . . . , R n are pairwise orthogonal.

An n-flower is called infinite Žresp. finite. provided that ŽF3.

R1 , . . . , R n are infinite Žresp. finite..

An n-flower is called nondegenerate provided ŽF4. R1 , . . . , R n are nontrivial. Remark. Ž1. If R is a nondegenerate n-flower then it is a nondegenerate m-flower for every m g ⺞ such that m F n. Ž2. Any connected subcategory S of an n-flower R such that S l R 0 is nontrivial is equipped with the standard induced n-flower structure defined by subcategories S i s S l R i , i s 0, 1, . . . , n. 2.2. Given a set X we denote by FŽ X . a free group generated by the set X. The following fact yields a natural class of examples of infinite n-flowers. LEMMA. Let R be a locally bounded k-category and G : Aut k Ž R . a noncommutati¨ e free group of k-linear automorphisms acting freely on R such that G has only a finite number of orbits in R. Then R has a structure of an infinite n-flower for e¨ ery n g ⺞.

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Proof. Suppose that G s FŽ X . for some X. For any w g G denote by l Ž w . the length of the noncancelable word representing w and set Gw s  w¨ : l Ž w¨ . s l Ž w . q l Ž ¨ .4 . For any s g ⺞ set Gs s  ¨ g G : l Ž ¨ . s s4 , G- s s  ¨ g G : l Ž ¨ . - s4 , and GG s s  ¨ g G : l Ž ¨ . G s4 . We assume first that < X < s 2. Fix a Žfinite. set R o of representatives of G-orbits in R. Let m be the maximum of length l Ž w . for all w g G such that RŽ x, wy . / 0 for some x, y g R o . The existence of m follows from the fact that R o is finite and G acts freely on the locally bounded category R. Note that m G 1 since R is connected. We prove that for every s g ⺞, s G 1, the subcategories R o s G- m q sy 1 R o and R w s Ž Gw l GG mqsy1 . R o , w g Gs , define on R a structure of an infinite 4 ⭈ 3 sy1-flower. Observe first that G splits into a disjoint union G s G- mqsy1 j

D Ž Gw l GG mqsy1 .

wgG s

of subsets since we have the obvious splittings G s G- mqsy1 j GG mqsy1 and G s G- s j Dw g G s Gw . Consequently, R splits into a disjoint union R s R0 k

E

Rw

wgG s

of subcategories. It is not hard to check that < Gw < s 4 ⭈ 3 sy 1. Since R 0 is nontrivial and all R w , w g Gs , are infinite it remains only to show that R w are pairwise orthogonal. Fix any x, y g R o ; w 1 , w 2 g Gs ; w 1 / w 2 ; and w 1¨ 1 g Gw 2 l GG mqsy1 , w 2¨ 2 g Gw 2 l GG mqsy1. Then l Ž ¨ 1 ., l Ž ¨ 2 . G m y 1, and in the canonical presentation w 1 s wu1 , w 2 s wu 2 , where w is a maximal common divisor of w 1 and w 2 from the left, both u1 and u 2 are y1 . nontrivial. Now we obtain that RŽ w 1¨ 1 x, w 2¨ 2 y . , RŽ x, ¨ y1 1 u1 u 2 ¨ 2 y s 0 y1 y1 y1 y1 since l Ž ¨ 1 u1 u 2¨ 2 . s l Ž ¨ 1 . q l Ž u1 . q l Ž u 2 . q l Ž ¨ 2 . G 2Ž m y 1. q 2 G m q 1. In this way R carries the required structure. To prove a general case it is enough to fix a splitting X s X ⬘ j X ⬙ into a disjoint union of two nonempty subsets and then repeat the arguments from the first part of the proof. Remark. By Lemma 6.9 it follows that under assumptions of Lemma 2.2 G is isomorphic to F Ž x 1 , . . . , x n4. for some n g ⺞. 2.3. For any n g ⺞ we denote by ⌺ n the path category of the quiver below:

6

1

6

2

0 6

n

STABILIZER CONJECTURE

119

THEOREM. Let R be an infinite n-flower o¨ er an algebraically closed field k for some n g ⺞. Suppose that R admits a sincere indecomposable locally finite dimensional R-module. Then there exists a faithful representation embedding functor Ž in the sense of w26x. E : mod ⌺ n ª mod R. In particular, R is wild if n G 5. By w13, Lemma 7 and Theoremx we obtain immediately COROLLARY. Let R be a representation-tame locally bounded category o¨ er an algebraically closed field k. Then R admits no representation M in Ind R such that supp M is an infinite n-flower, for n G 5. The proof needs some preparation and will be given in Section 5 Žsee 5.5.. 2.4. By the same arguments Theorem 2.3 and Lemma 2.2 imply a weaker form of the main result Žcf. Theorem 6.10.. COROLLARY. Let R be a locally bounded category o¨ er an algebraically closed field k and G : Aut k Ž R . a group of k-linear automorphisms acting freely on R. Suppose there exist an infinite G-atom B such that the stabilizer GB is a noncommutati¨ e free group. Then R is wild. In particular, if R is representation-tame and G is a noncommutati¨ e free group then the stabilizer GB of any infinite G-atom B is an infinite cyclic group. Remark. It is well-known Žsee w11, Proposition 2x. that if R is wild then so is the orbit category RrG. Assume that G acts freely on Žind R .r, , then in fact the category mod1Ž RrG . is already wild Žsee w12x for a definition of mod1Ž RrG . and mod 2 Ž RrG ... In case there exists a G-atom B such that GB is a noncommutative free group, not only mod1Ž RrG . but also the category mod 2 Ž RrG . is wild since then the functor ⌽ B s ymk G B F␭ B : mod kGB ª modŽ RrG . induces a representation embedding Žsee w4, Proposition 2.3x.. 2.5. The following two examples of representation-tame 5-flowers show the importance of assumptions in Theorem 2.3. EXAMPLE 1. Let R be the canonical Galois covering with the group F Ž x, y4. of the local special biserial algebra k w X, Y xrŽ XY, X 2 , Y 3 ., where by k w X, Y x is the polynomial algebra in two commuting variables. R is a connected locally bounded k-category equipped with the free action of F Ž x, y4.. By Lemma 2.2 R carries a structure of an infinite 5-flower. The category R is special biserial, consequently R is representation-tame. One

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PIOTR DOWBOR

can show that R does not admit a sincere indecomposable representation in Mod R. EXAMPLE 2. Let R s kQrI, where KQ is the path category of the quiver Q, ⴢ ⴢ ⴢ ⴢ c d

ⴢ

e

ⴢ

6

6 6 ⴢ

6

a

6

ⴢ

g

f

6 6

b

and I is the ideal in kQ generated by the path ad. The category R is in an obvious way a finite nondegenerate 5-flower. One can easily construct a sincere indecomposable representation of R. The category R is a representation-tame regular one-point extension of a hereditary Euclidean ˜ 5 Žsee w22x.. category of type ⺔ For further discussion of the question of when a finite 5-flower is wild we refer the reader to Theorem 5.1.

3. THE EXTENSION EMBEDDING THEOREM In this section, using the extension group technique we will construct certain representation embeddings of the category mod ⌺ r into the category mod R. 3.1. Let M1 , . . . , Mm ; N1 , . . . , Nn be nonzero R-modules, M s n N s [js1 Nj , and

m [is1 Mi ,

e: 0 ª N ª E ª M ª 0 be an exact sequence in MOD R with the extension class m

n

w e x s Ž w e i , j x . 1FiFm , 1FjFn g Ext 1R Ž M, N . s [ [ Ext 1R Ž Mi , Nj . . is1 js1

In our proofs we will apply the following simple observation. Remark. If the middle term E of the exact sequence e is an indecomposable R-module then for every i g  1, . . . , m4 Žresp. j g  1, . . . , n4. there exists j g  1, . . . , n4 Žresp. i g  1, . . . , m4. such that w e i, j x g Ext 1R Ž Mi , Nj . is nonzero. 3.2. Let M, N be R-modules. Then following w7, 4.1x the extension group Ext 1R Ž M, N . can be interpreted in terms of Hochschild cohomology, Ext 1R Ž M, N . , Derk Ž R , Hom k Ž M, N . . rDerk0 Ž R, Hom k Ž M, N . . , Ž * .

121

STABILIZER CONJECTURE

where by Hom k Ž M, N . we mean the R op = R-bimodule defined by the mapping Ž x, y . ¬ Hom k Ž M Ž x ., N Ž y .. and by Derk Ž R, Hom k Ž M, N .. Žresp. Derk0 Ž R, Hom k Ž M, N ... we mean the k-linear space consisting of all k-derivations Žresp. inner derivations. from R to Hom k Ž M, N . Žsee also w2, Corollary 4.4x and w18, 2.7x in the classical situation of modules over k-algebras.. Recall that by a k-deri¨ ation from R to Hom k Ž M, N . we mean a ‘‘diagonal’’ linear map

␦:

[

R Ž x, y . ª

x, ygob R

[

Hom Ž M Ž y . , N Ž x . .

x , ygob R

such that ␦ Ž sr . s N Ž r . ␦ Ž s . q ␦ Ž r . M Ž s . for all r g RŽ x, y ., s g RŽ y, z ., and x, y, z g ob R. The derivation ␦ is inner provided ␦ s innŽ␩ . for some family ␩ s Ž␩ : M Ž x . ª N Ž x .. x g ob R of k-linear maps, where innŽ␩ .Ž r . s ␩x M Ž r . y N Ž r .␩ y for x, y g ob R and r g RŽ x, y .. The k-linear isomorphism Ž*. is induced by the mapping ␦ ¬ e ␦ associating with any ␦ g Derk Ž R, Hom k Ž M, N .. the canonical exact sequence e ␦ : 0 ª N ª E ␦ ª M ª 0, where the R-module E ␦ is defined as E␦ Ž x. s NŽ x. [ MŽ x. for x g ob R and as E␦ Ž r . s

ž

NŽ r . 0

␦Ž r. MŽ r.

/

for r g RŽ x, y .. r Let N0 , N1 , . . . , Nr be R-modules, N s [is1 Ni , and r

w e x s Ž w e1 x , . . . , w e r x . g Ext 1R Ž N, N0 . s [ Ext 1R Ž Ni , N0 . is1

be a fixed extension class with components w e i x g Ext 1R Ž Ni , N0 ., i s 1, . . . , r. We fix a derivation ␦ g Derk Ž R, Hom k Ž N, N0 .. with components ␦ i g Derk Ž R, Hom k Ž Ni , N0 .., i s 1, . . . , r, such that w e x s w e ␦ x. Then following w7x we can define a k-linear functor E s E ␦ : mod ⌺ r ª mod R as follows Žcomp. w21, 27x..

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We fix a set of representatives of all isoclasses in mod ⌺ r consisting of matrix representations of Sr i.e., ⌺ r-modules of the form A s Ž A i ⴢ : k n i ª k n 0 . is1, . . . , r , where A i g M n 0=n iŽ k . for every i s 1, . . . , r Žall matrix representations form a set.. Denote by mod 0 ⌺ r the full subcategory of mod ⌺ r formed by this set. First we construct a functor E0 : mod 0 ⌺ r ª mod R and then we extend it to E : mod ⌺ r ª mod R by composing E0 with a fixed quasi-inverse functor to the embedding mod 0 ⌺ r ¨ mod ⌺ r . Let A s Ž A i ⴢ : k n i ª k n 0 . is1, . . . , r be in mod 0 ⌺ r . Denote by A ␦ a kr derivation in Derk Ž R, Hom k Ž[is1 Ni n i , N0n 0 .. corresponding under the standard identification r

ž

Derk R , Hom k

ž

[ Ni n , N0n i

r

0

is1

//

(

[ M n =n Ž Derk Ž R, Hom k Ž Ni , N0 . . . i

0

is1

to the family A i ␦ i g M n 0=n iŽDerk Ž R, Hom k Ž Ni , N0 ..., i s 1, . . . , r, of matrices, where the Ž s, t .-component Ž ␦ i A i .Ž s, t . of the matrix ␦ i A i is equal to Ž A i .Ž s, t .␦ i for any s g  1, . . . , n 0 4 , t g  1, . . . , n i 4 . Then we set E0 Ž A . s E Ž A ␦ . , where E Ž A ␦ . is the middle term of the canonical exact sequence e Ž A ␦ . : 0 ª N0n 0 ª E Ž A ␦ . ª

r

[ Ni n ª 0. i

is1

Let ␸ : A ª A⬘ be a ⌺ r-homomorphism in mod 0 ⌺ r given by the family X of k-linear Ž ␸ i ⴢ : k n i ª k n i . is0, . . . , r of k-linear maps, where A s X X Ž A i ⴢ : k n i ª k n 0 . is1, . . . , n , A⬘ s Ž AXi ⴢ : k n i ª k n 0 . is1, . . . , r , and ␸ i g M nXi=n iŽ k . for every i g  0, 1, . . . , r 4 . It is easy to check that the family r

␸ ˜s

ž

r

[ ␸i id N Ž x . : [ Ž Ni Ž x . . is0

i

ni

r

ª

is0

[ Ž Ni Ž x . . is0

X

ni

/

xgob R

of k-linear maps defines an R-homomorphism from E0 Ž A. to E0 Ž A⬘.. Then we set E0 Ž ␸ . s ␸ ˜. It is easy to check the following. PROPOSITION. The mapping E0 defines a k-linear functor E0 : mod 0 ⌺ r ª mod R Ž consequently, also E : mod ⌺ r ª mod R .. 3.3. Now we can formulate the main result of this section.

123

STABILIZER CONJECTURE

THEOREM. Let N0 , N1 , . . . , Nr be indecomposables in mod R and w e x g r Ext 1R Ž[is1 Ni , N0 . an extension class with components w e i x g Ext 1R Ž Ni , N0 ., i s 1, . . . , r. Then the functor E : mod ⌺ r ª mod R is a faithful representation embedding; in particular, R is wild for r G 5, pro¨ ided the list ŽL. s ŽL1, L2, L3, L4. or ŽL⬘. s ŽL1, L2⬘, L3, L4⬘. of conditions is satisfied, where L1, L2, L2⬘, L3, L4, L4⬘ are listed below. ŽL1 . End R Ž Ni .rJ ŽEnd R Ž Ni .. , k for e¨ ery i s 0, 1, . . . , r. ŽL2 . w e i x / 0 for e¨ ery i s 1, . . . , r. ŽL2⬘. The middle term E of the extension e is indecomposable. ŽL3 . Hom R Ž N0 , Ni . s 0 for e¨ ery i s 1, . . . , r. ŽL4 . N1 , . . . , Nr are pairwise orthogonal. ŽL4⬘. J ŽEnd R Ž N0 .. s 0 and N1 , . . . , Nr are pairwise nonisomorphic. Denote by ⺕ the full subcategory of mod R formed by the set of all R-modules of the form E0 Ž A., where A is in mod 0 ⌺ r . For the proof of the theorem we will construct the functor ␲ : ⺕ ª mod 0 ⌺ r with the following two properties: Ži. ␲ E0 s id mod ⌺ , 0 r Žii. Ker ␲ contains no nonzero idempotents. The construction of ␲ needs some preparation. 3.4. Suppose that the condition ŽL1. is satisfied. For any subset I ;  0, 1, . . . , r 4 and sequences Ž n i . i g I , Ž nXi . i g I of natural numbers, the isomorphisms End R Ž Ni . , k ⭈ id N i [ J ŽEnd R Ž Ni .., i g I, yield the standard isomorphism Hom R

ž [N igI

i

ni

,

[ Ni n / , [ M n =n Ž k . [ [ M n =n Ž JR Ž Ni , Nj . . , X i

igI

X i

igI

i

X j

i , jgI

i

Ž *. I where JR is the Jacobson radical of the category mod R. Assume that also the condition ŽL3. holds. Then for any R-homomor. ª E0 Ž A⬘., where A s Ž A i ⴢ : k n i ª k n 0 . is1, . . . , r , A⬘ s phism Xf : E0 Ž A X Ž AXi ⴢ : k n i ª k n 0 . is1, . . . , r are in mod 0 ⌺ r , we obtain the commutative diagram 0

6

i

is1

f[

f

6

6

r

is1

X i

0.

6

[ Ni n

6

E A⬘␦

6

6

X

N0n 0

6

e A⬘␦ : 0

r

[ Ni n

6

f0

E A␦

6

N0n 0

6

eA ␦ : 0

124

PIOTR DOWBOR

r Note that the homomorphisms f 0 and f [ exist since Hom R Ž N0n 0 , [is1 X ni . Ni s 0 by ŽL3.. They are uniquely determined and therefore they depend functorially on f. By the above diagram each linear map f Ž x . : E A ␦ Ž x . ª E A⬘␦ Ž x ., x g ob R, has a triangular matrix form

f Ž x. s

ž

f 0Ž x.

␩Ž x.

0

f [Ž x.

/

,

r niŽ . where ␩ Ž x . : [is1 N x ª N0n 0 Ž x . is a k-linear map with components X i n n 0 i ␩i Ž x . : Ni Ž x . ª N0 Ž x ., i s 1, . . . , r. We denote by ␩ s ␩ Ž f . Žresp. ␩i , i s 1, . . . , r . the collection Ž␩ Ž x .. x g ob R Žresp. Ž␩i Ž x .. x g ob R , i s 1, . . . , r .; by r Ž f i .1s 1, . . . , r g [is Ž . and by Ž f j,X i . i, js 1, . . . , r g [i,r js 1 M nX = n X 1 M n i= n i k j i Ž JR Ž Ni , Nj .. the collections of matrices corresponding to f [ via Ž*.1, . . . , r4; X and by f 0 g M nX0=n 0Ž k . and f 0 g M nX0=n 0Ž JR Ž N0 , N0 .. the pair of matrices corresponding to f 0 via Ž*.04. X

LEMMA. Let A, A⬘ be in mod 0 ⌺ r . Assume that the assumptions ŽL. or ŽL⬘. Ž see Theorem 3.3. are satisfied. Then, Ži. if f : E0 Ž A. ª E0 Ž A⬘. is an R-homomorphism then the family X Ž f i ⴢ :k n i ª k n i . is0, 1, . . . , r of k-linear maps defines a morphism f : A ª A⬘ in mod 0 ⌺ r ; Žii. if A and A⬘ are different then E0 Ž A. and E0 Ž A⬘. are nonisomorphic. Proof. Ži. Assume first that ŽL. are satisfied. Fix an R-homomorphism f : E0 Ž A. ª E0 Ž A⬘.. We prove that f 0 ⭈ A i s AXi ⭈ f i for every i s 1, . . . , r, or equivalently, that all components c iŽ s⬘, t ., s⬘ g  1, . . . , nX0 4 , t g  1, . . . , n i 4 , of each matrix f 0 ⭈ A i y AXi ⭈ f i g M nX =n Ž k . are 0 i zero. We use for this purpose the equalities f Ž x . E A ␦ Ž r . s E A⬘␦ Ž r . f Ž y . where x, y g ob R and r g RŽ x, y ., which by definition in a matrix form appear as follows:

ž

f 0Ž x.

␩Ž x.

0

f [Ž x. X

s



N0n 0 Ž r .

/

N0n 0 Ž r .

Ž A␦ . Ž r .

0

[ Ni n Ž r .

r

Ž A⬘␦ . Ž r . r

0

i

is1

[ Ni Ž r . is1

X ni

0ž

0

f 0Ž y.

␩Ž y.

0

f [Ž y.

/

.

125

STABILIZER CONJECTURE

Looking at the Ž1, 2.th components we obtain the equality f 0 ) Ž A ␦ . y Ž A⬘␦ . ) f [s inn Ž ␩ .

Ž a.

r in Derk Ž R, Hom k Ž[is1 Ni n i , N0n 0 .., where innŽ␩ . is the inner derivation X defined by ␩ s ␩ Ž x .4x g ob R and ) refers to the standard End R Ž NX0n 0 . y r r End R Ž[is1 Ni n i .-bimodule structure on Derk Ž R, Hom k Ž[is1X Ni n i , N0n 0 ... In n i fact, Ža. is given by the equalities in Derk Ž R, Hom k Ž Ni , N0n 0 .., X

r

f 0 ) Ž A i ␦i . y

Ý Ž AXj ␦ j . ) f j,[i s inn Ž ␩i . ,

Ž a. i

js1 X

i s 1, . . . , r, where f j,[i : Ni n i ª Nj n j is theX Ž j, i .th component of the R-hor r momorphism f [ : [is1 Ni n i ª [js1 Nj n j and innŽ␩i . is the inner derivation defined by ␩i . Writing f 0 s f 0 ⭈ id N 0 q f 0X , f i,[i s f i ⭈ id N i q f i,X i , i s 1, . . . , r, by basic properties of the multiplicative structures of k-derivations and ŽL4. we have

Ž f 0 ⭈ A i y AXi ⭈ fi . ␦i s Ž AXi ␦i . ) fiX, i y f 0X ) Ž A i ␦i . q inn Ž ␩i .

Ž b. i

for every i s 1, . . . , r. It is not hard to see that for any s⬘ g  1, . . . , nX0 4 and t g  1, . . . , n i 4 , Žb. i yields the equality of the form c iŽ s⬘, t . w e i x s w e i x uŽi s⬘, t . y ¨ iŽ s⬘, t . w e i x

Ž b . i s⬘, t Ž

.

in Ext 1R Ž Ni , N0 ., where ¨ iŽ s⬘, t . g J ŽEnd R Ž N0 .. and uŽi s⬘, t . g J ŽEnd R Ž Ni ... Fix for a moment i g  1, . . . , r 4 . Since Ji s J ŽEnd R Ž Ni .. is a nilpotent ideal and the class w e i x is nonzero Žsee ŽL2.., there exists l s l Ž i . g ⺞ such that w e i x Jil / 0 and w e i x Jilq1 s 0. Therefore, by multiplying the equalities Žb.Ži s⬘, t . by Jil we obtain that Jil annihilates the elements Ž c iŽ s⬘, t . ⭈ id N q 0 ¨ iŽ s⬘, t . .w e i x on the right. This implies that each c iŽ s⬘, t . is zero since otherwise c iŽ s⬘, t . ⭈ id N 0 q ¨ iŽ s⬘, t . g End R Ž N0 . is invertible, and this contradicts the choice of l. Consequently, c iŽ s⬘, t . s 0 for all X i g  1, . . . , r 4 , s⬘ g  1, . . . , nX0 4 , t g  1, . . . , n i 4 , and the family Ž f i ⴢ : k n i ª k n i . is0, 1, . . . , r defines a homomorphism f : A ª A⬘ in mod 0 ⌺ r . To prove Ži. when ŽL⬘. are satisfied we proceed in the same way. The essential difference occurs in the analysis of the equalities Ža. i , i s 1, . . . , r. Now by ŽL4⬘. we obtain first the equalities r

Ž f 0 ⭈ A i y AXi ⭈ fi . ␦i s Ý Ž AXj ␦ j . ) f j,[i q Ž AXi ␦i . ) fiX, i q inn Ž ␩i . , js1, j/i

Ž b⬘ . i

126

PIOTR DOWBOR

and next the equalities of the form c iŽ s⬘, t . w e i x s

r

Ý w e i x wjŽ s⬘, t . ,

Ž b⬘ . i s⬘, t Ž

.

js1

where wjŽ s⬘, t . g Hom R Ž Ni , Nj . Žs JR Ž Ni , Nj .., j s 1, . . . , r, j / i, and wiŽ s⬘, t . g Ji . r Suppose that c iŽ s⬘, t . / 0 for some triple i, s⬘, t. Let w g End R Ž[js1 Nj . be the automorphism with components wj, j⬘ : Nj⬘ ª Nj , 1 F j, j⬘ F r, given as follows:

¡id

wj, j⬘ s

~c

if j s j⬘, j / i

Nj

Ž s⬘, t . i

¢yw 0

⭈ id N i y wiŽ s⬘, t .

Ž s⬘, t . j

if j s j⬘, j s i if j⬘ s i , j / i otherwise.

r Then the ith component of the extension class w e x w g Ext 1R Ž[js1 Nj , N0 . Ž s⬘, t . is zero by Eq. Žb⬘. i . But by Remark 3.1 this contradicts ŽL2⬘. since the middle term of the extension ew is isomorphic to E. Consequently, all coefficients c iŽ s⬘, t . are zero and also in this case the family Ž f i ⴢ : k n i ª X ni . k is0, 1, . . . , r defines a morphism f : A ª A⬘ in mod 0 ⌺ r . Žii. Since the distinct objects in mod 0 ⌺ r are nonisomorphic, it is enough to show that if f : E0 Ž A. ª E0 Ž A⬘. is an R-isomorphism then f : A ª A⬘ is a ⌺ r-isomorphism, where A, A⬘ are in mod 0 ⌺ r . Suppose that f is an R-isomorphism. Then both f 0 and f [ are R-isomorphisms since they depend functorially on f. Consequently, by w6, Lemma 2.4x f is a ⌺ r-isomorphism.

3.5. Assume that the assumptions of Theorem 3.3 are satisfied. Now by applying Lemma 3.4 we define a functor ␲ : ⺕ ª mod 0 ⌺ r Žsee 3.2.. Given an object E0 Ž A. in ⺕, where A is in mod 0 ⌺ r , we set

␲ Ž E0 Ž A . . s A. Given an R-homomorphism f : E0 Ž A. ª E0 Ž A⬘. Ža morphism from E0 Ž A. to E0 Ž A. in the category ⺕., where A and A⬘ are in mod 0 ⌺ r , we set

␲ Ž f . s f : A ª A⬘. It is easy to check the following. PROPOSITION.

The mapping ␲ defines a k-linear functor ␲ : ⺕ ª mod 0 ⌺ r .

STABILIZER CONJECTURE

127

3.6. Proof of Theorem 3.3. To prove that the functor E : mod ⌺ r ª mod R is a representation embedding it is enough to show that 3.3Ži. and 3.3Žii. hold. Indeed, 3.3Ži. is trivially satisfied by the construction ␲ , and 3.3Žii. follows from the nilpotency of the ideal Ker ␲ . Note that the nilpotency degree of Ker ␲ is bounded by the maximum of the nilpotency degrees of r J ŽEnd R Ž N0 .. and J ŽEnd R Ž[js1 Nj ... To complete the proof observe that by its very construction the functor E preserves parameterizing k ² x, y :-families in the sense of w26x, where k ² x, y : is a free associative algebra in two noncommuting variables. Therefore if r G 5 then R is wild. 3.7. We finish this section with a formulation of some straightforward Žcf. Remark 3.1. but useful consequences of Theorem 3.3. COROLLARY. Let M be an indecomposable R-module and M0 be an indecomposable submodule of M such that Hom R Ž M0 , MrM0 . s 0. Suppose r that MrM0 decomposes into a direct sum MrM0 s [is1 Mi of nonisomorphic indecomposable R-modules Mi and End R Ž Mi .rJ ŽEnd R Ž Mi .. , k for e¨ ery i s 0, 1, . . . , r. If R-modules M1 , . . . , Mr are pairwise orthogonal or End R Ž M0 . , k then there exists a faithful representation embedding E : mod ⌺ r ª mod R, in particular, R is wild for r G 5. 4. GLUING AND EXCISION OF INDECOMPOSABLE MODULES AND NEIGHBOURHOODS 4.1. We recall first a basic decomposition property for locally finitedimensional modules over a locally bounded k-category R Žsee w12, 6x.. PROPOSITION. Ži. A module M in Mod R is indecomposable if and only if its endomorphism algebra End R Ž M . is local. Žii. Each M in Mod R has a unique Ž up to isomorphism. decomposition into a direct sum of indecomposable R-modules. 4.2. The following elementary consequence of the uniqueness of decomposition into a direct sum of indecomposables in Mod R will play an essential role in further considerations. LEMMA. Let C be a nontri¨ ial full subcategory of R and M an R-module in Mod R such that M N C is nonzero. Suppose that for a full subcategory D of R containing C there exists an indecomposable direct summand M D of M N D such that M N C s M D N C . Then there exists an indecomposable direct summand M0 of M such that M0 N C s M N C .

128

PIOTR DOWBOR

Proof. Take D as above and fix an indecomposable direct summand M D of M N D such that M D N C s M N C . Since M N C is nonzero, there exists an indecomposable direct summand M0 of M such that M0 N C / 0. Note that M D is isomorphic to a direct summand of M0 N D . Otherwise, M D is isomorphic to a direct summand of M NX D , where M⬘ is a complementaryto-M0 direct summand of M, and this contradicts the assumption M0 N C / 0. Consequently M D N C Žs M N C . is a direct summand of M0 N C and M0 N C s MN C . 4.3. We recall the notion introduced in w7x which is essential for a study of the objects of the category Ind R. DEFINITION. Let M be in Ind R and C a full subcategory of R. The full subcategory U of R containing C is called an M-neighbourhood of C Žequivalently, a neighbourhood of C with respect to M . provided there exists an indecomposable U-module M U satisfying the following two conditions: ŽN1. ŽN2.

M U is isomorphic to a direct summand of M NU , M UN C s M N C .

The M-neighbourhood U of C is called finite Žresp. infinite, connected . if the category U is finite Žresp. infinite, connected.. Remark. If U is an M-neighbourhood then M NU / 0. If M N C s 0 then any subcategory U containing C such that M NU / 0 is an M-neighbourhood of C. LEMMA.

Let M be in Ind R and C be a full nontri¨ ial subcategory of R.

Ža. If C is contained in supp M then supp M is an M-neighbourhood of C. Žb. C is an M-neighbourhood of C if and only if M N C is indecomposable. Žc. Suppose that D, U, and V are nontri¨ ial subcategories of R such that D ; C ; U ; V. Then V is an M-neighbourhood of D pro¨ ided U is an M-neighbourhood of C. Proof. Assertions Ža. and Žb. are straightforward; Žc. follows from Lemma 4.2. 4.4. The following fact is crucial for the remaining part of the paper. PROPOSITION. Let R be a connected locally bounded k-category and M be an R-module in Ind R. Then any finite full subcategory of R Ž which intersects nontri¨ ially supp M . admits a finite, connected M-neighbourhood.

STABILIZER CONJECTURE

129

Proof. The proof follows from w12, Proposition 4.2 and Corollary 4.2x Žsee w8x for the full proof of w12, Lemma 4.2x for the arbitrary field k .. COROLLARY. Let R and M be as abo¨ e. Then for any finite full subcate˜ C in ind R such that M˜ CN C s M N C . gory C of R there exists an R-module M Proof. Take any finite M-neighbourhood U of C and fix an indecomposable direct summand M U of M NU such that M UN C s M N C . Denote by e␭U : MOD U ª MOD R the left adjoint functor to the restriction functor ˜ C s e␭U Ž M U .. It is clear that M˜ C eU : MOD R ª MOD U and set M defined above satisfies the required condition. 䢇

4.5. Now we present a construction method of indecomposable modules from a given one Žso-called excision.. LEMMA. Let M be in Ind R, and let Ui 4i g I and  Vi 4i g I be two families of finite full subcategories of R with unions U s Di g I Ui and V s Di g I Vi respecti¨ ely. Suppose that the following conditions are satisfied: Ža. U ; supp M; Žb. Vi is an M-neighbourhood of Ui for e¨ ery i g I; Žc. for any i, i⬘ g I there exists a sequence i 0 , i1 , . . . , i n g I such that i 0 s i, i n s i⬘, and ob Ui j l ob Ui jy 1 / ⭋ for e¨ ery j s 1, . . . , n. Then V is an M-neighbourhood of U; in particular, there exists an M V in Ind V which is a direct summand of M N V such that M VNU s M NU . Proof. For every i g I fix an indecomposable direct summand M i s M V i of M N V i such that M iNUi s M NUi Žthere exists by Žb... Consider a decomposition M N V s [j g J M j into a direct sum of indecomposable Vsubmodules. By the uniqueness of decomposition into a direct sum of indecomposables in Mod R, for every i g I there exists jŽ i . g J such that M i is isomorphic to a direct summand of M jŽ i.N V i . It is clear that M j NUi s

½

M NUi

if j s j Ž i .

0

if j / j Ž i . ,

for j g J. Consequently, by Ža. we have jŽ i1 . s jŽ i 2 . provided Ui1 l Ui 2 is nontrivial. Then by Žc. there exists j g J such that j s jŽ i . for all i g I. It is clear that the indecomposable V-module M V s M j satisfies the required conditions. 4.6. Following w11x for any full subcategory C of R we denote by Cˆ the full subcategory of R formed by the set consisting of all objects x g R such that RŽ x, y . or RŽ y, x . is nonzero for some y g C. By MODC R we denote the full subcategory of MOD R formed by the set of all R-modules M such that supp M is contained in C.

130

PIOTR DOWBOR

LEMMA. Let S be a full subcategory of R. Then for any full subcategory C of S the restriction functor e SN MOD C R : MODC R ª MODC S is full and faithful. If Cˆ is contained in S then e SN MOD C R and the extension by zero’s functor Žy; 0. : MODC S ª MODC R induces an equi¨ alence 䢇

䢇

MODC R , MODC S. In particular, for a gi¨ en M in MOD R there exists a bijection induced by the abo¨ e functors between the set of all submodules Ž resp. direct summands . N $ of M such that supp N; S and $ the set of all submodules Ž resp. direct summands. N⬘ of M N S such that supp N⬘; S. Proof. The first two assertions are easy to check; for the proof of the last one we refer the reader to w11, Lemma 2x. COROLLARY. Let R be an n-flower, n g ⺞, defined by subcategories R 0 , R1 , . . . , R n ; M an R-module; and N a submodule of M such that NN R 0 s M N R 0 . Then N is a direct summand of M pro¨ ided NNŽ R 0 j R i . is a direct summand of M NŽ R 0 j R i . for e¨ ery i s 1, . . . , n. Proof. Fix modules Ni in MOD Ž R 0 j R i ., i s 1, . . . , n, such that Ž M NŽ R 0 j R i . s N$ NŽ R 0 j R i . [ Ni for every i. Then we have supp Ni ; R i NN R 0 s M N R 0 . and R i; R 0 j R i Žsee ŽF2. in Definition 2.1.. Consequently, by the lemma each Ni can be regarded as an R-submodule of M and n M s N [ [is1 Ni . 4.7. The second technique of constructing indecomposables used in the paper is gluing of modules. PROPOSITION. Let R be an n-flower with fixed full subcategories  S i 4i g 1, . . . , n4 a R 0 , R1 , . . . , R n satisfying the conditions of Definition 2.1 and $ family of full connected subcategories of R containing R 0. Suppose that  Mi 4i g 1, . . . , n4 is a family of modules, and Mi is in Mod S i, such that Mi N $ s M j N $, for all i, j s 1, . . . , n. Then there exists a unique S-module M R0

R0

in Mod S such that M NŽ R 0 j S ii . s Mi NŽ R 0 j S ii . for e¨ ery i s 1, . . . , n, where n S s R 0 j Dis1 Sii Ž Sij s S j l R i , i, j s 1, . . . , n; cf. Remark 2.1Ž2... Moreo¨ er, assume that each module Mi is indecomposable, for i s 1, . . . , n. Suppose further that one of the following two conditions is satisfied: n Ža. there exists a neighbourhood U ; Fis1 S i of R 0 with respect to all Mi ’s, such that Mi NU s M j NU for each pair i, j s 1, . . . , n; Žb. R 0-module Mi N R is nonzero and Sii j R 0 is an Mi-neighbourhood 0 $ of R 0 l Sii for each i s 1, . . . , n.

Then the S-module M is indecomposable.

131

STABILIZER CONJECTURE

Proof. By the obvious reasons we assume that n G 2. All of the values M Ž x ., x g ob S, and M Ž s ., s g SŽ x, y ., x, y g ob S, are uniquely determined by the conditions in assumptions of the proposition. To check that M defined above is an S-module we have to show that M Ž sr . s M Ž r . M Ž s . for any r g SŽ x, y . and s g SŽ y, z ., where x, y, z g ob S. We may assume that r and s are both nonzero since for any morphism t in S, M Ž t . s 0 provided t s 0. Consequently x, y belongs to R 0 j Sii and y, z belongs to R0$ j S jj for some i, j s 1, . . . , n. Then either y is in R 0 and then x, z are in R 0 or y is in Sii and then x, z are in R 0 j S ii. In both cases the required equality holds by the assumption and definition of M. Assume that for a given family  Mi 4is1, . . . , n of indecomposables the condition Ža. holds. To prove that M is indecomposable fix a common indecomposable direct summand N of all Mi NU ’s such that NN R 0 s Mi N R 0 for every i s 1, . . . , n. We consider two cases. First assume that M N R 0 s 0. By the indecomposability of Mi and ŽF2. in Definition 2.1, for each i g  1, . . . , n4 there exists exactly one jŽ i . g  1, . . . , n4 such that supp Mi ; S jŽi i. . Since all supp Mi contain a nontrivial subcategory supp N, there exists j such that j s jŽ i . for all i. Then M is isomorphic to M j regarded as an S-module, therefore M is indecomposable Žsee Lemma 4.6.. Assume now that M N R 0 / 0. Note that M NU s Mi NU for every i s n Ž 1, . . . , n, since U s R 0 j Dis1 U l S ii . ŽU l Sii s U l R i .. Therefore N is a direct summand of M NU such that NN R 0 s M N R 0 and Lemma 4.2 forces a decomposition M s M0 [ M⬘ such that M0 is indecomposable and M0 N R 0 s M N R 0 . Consequently, by ŽF2. in Definition 2.1 M⬘ admits a n decomposition M⬘ s [is1 MiX such that supp MiX ; Sii for every i s 1, . . . , n. To show the indecomposability of M it suffices to prove that MiX s 0 for all i s 1, . . . , n. Note that each MiXNŽ R 0 j S ii . is a direct summand of Mi NŽ R 0 j S ii . since n

Mi NŽ R 0 j S ii . s M NŽ R 0 j S ii . s M0 NŽ R 0 j S ii . [

[ MiXNŽ R j S . . is1

0

i i

Then, by Lemma 4.6, each MiX regarded as an S$i-module is isomorphic to a direct summand of Mi Žsupp Mi ; Sii and S ii l S i ; S ii j R 0 .. Consequently by the indecomposability of Mi , MiX s 0 for every i s 1, . . . , n; otherwise Mi is isomorphic to MiX and M N R 0 s 0, a contradiction. In this way M is indecomposable provided that Ža. holds. Assume that for a family  Mi 4is1, . . . , n of indecomposables the condition Žb. holds. First we prove that if X is a nonzero direct summand of the $ n Ž i S-module M such that supp X ; R 0 j Dis1 Si _ R 0. then supp X is contained in R 0 and then X s M Žhence M is indecomposable.. Fix any

132

PIOTR DOWBOR

indecomposable direct summand Y of X as above. Then supp Y is con$ i tained either in R 0 or in S i _ R 0 for some i s 1, . . . , n. In the first case by the indecomposability of Mi and Lemma 4.6, Y regarded as an S i-module is isomorphic to Mi for each i s 1, . . . , n, since X N $ is a direct summand R0 of Mi N $ Žs M N $.. Consequently, we have Y s X s M. In the second case R0

R0

Y N S ii is a direct summand of Mi N S ii for some i s 1, . . . , n, and by analogous arguments we obtain the same isomorphism, which contradicts the assumption Mi N R 0 / 0. To prove that M is indecomposable we show that for any decomposition M s M⬘ [ M⬙ into a direct sum of submodules either M⬘ s 0 or M⬙ s 0. Take any M⬘ and M⬙ as above, and for every i s 1, . . . , n fix an indecomposable direct summand Ni of Mi NŽ R 0 j S ii . such that N i N Ž $ s M i N Ž$ . R 0 l S ii . R 0 l S ii . X Y Each Ni is a direct summand of M NŽ R 0 j S ii . or M NŽ R 0 j S ii . since M N Ž $ i R 0 j Si . s M i N Ž$ for every i s 1, . . . , n. Suppose first that all Ni ’s are direct R 0 j S ii .

X summands of M NŽ by the first part of the R 0 j S ii . . Then we have M⬘ / 0 and $ n Ž i proof M⬙ s 0 since supp M⬙ ; R 0 j Dis1 Si _ R 0.. Suppose now that there exists m g ⺞, 1 F m - n, such that N1 , . . . , Nm are direct sumX Y mands of M NŽ R 0 j S ii . and Nmq1 , . . . , Nn are direct summands of M NŽ R 0 j S ii . X $ Y $ Žclearly hence M⬘, M⬙ / 0.. Then M < Ž R l D n i s 0, M < m i Ž R 0 l D is 0 is mq 1 S i . 1 Si . m s 0, and by the first part of the proof supp M⬘ ; R 0 j Dis1 S ii and n supp M⬙ ; R 0 j Dismq1 Sii, otherwise M⬘ Žresp. M⬙ .$ contains a nonzero n Ž Sii _ R 0. Žresp. supp X ; direct summand X such that supp X ; Dismq1 $ m Ž i Dis1 Si _ R 0... Fix j g  m q 1, . . . , n4 and present S j in the 2-flower form

Sj s

m

n

is1

ismq1

D Sij k R 0 k D

$

$

Sij .

$

Note that we have R 0 l Sij s R 0 l R i s R 0 l Sii, hence M X< Ž $ m j s R 0 l Dis 1 Si . X $ Y $ Ž M j N Ž$ M s 0 resp. M m j , n j n j < Ž R 0 l D is m q 1 S i . < Ž R 0 l D is m q 1 S i . s R 0 l D is 1 S i . Y $ $ . M j N Ž R 0 l Dis n j , M < m j s 0 . Therefore the conditions Ž R 0 l Dis mq 1 S i . 1 Si . M jXN Dmis1S ij s M j N Dmis1S ij , M jYN Dmis1S ij s 0,

M jXN R 0 s M NX R 0 ,

M jYN R 0 s M NY R 0 ,

M jXN Dnismq1S ij s 0;

M jYN Dnismq1S ij s M j N Dnismq1S ij ;

define two submodules M jX and M jY of M j . It is clear that M j s M jX [ M jY and that M jY / 0 since Nj is a direct summand of M Y< Ž $ R 0 j S jj . which is Y $ contained in M j N Ž R 0 j S jj . . Then from the indecomposability of M j we have $ n Ž i M jX s 0 and hence supp M⬘ is contained in Dis1 Si _ R 0. since M NX $ s R0

133

STABILIZER CONJECTURE

M jXN $. Consequently, this contradicts by the first part of the proof the R0

existence of m as above. In conclusion, the proof of indecomposability of M under the assumption Žb. is complete. Repeating arguments from the above proof one can show the following. Remark. Let R,  R i 4i g 0, 1, . . . , n4 ,  S i 4i g 1, . . . , n4 , and S be such as in Proposition 4.7. Suppose that the family  Mi 4i g 1, . . . , n4 , Mi in Mod S i, satisfies the following Žweaker. conditions: Žc.

Mi N R 0 s M j N R 0 , for all i, j s 1, . . . , n, $

Žd. there exists an R 0-module X such that Mi N $ is a submodule of R0

X, for every i s 1, . . . , n.

Then also in this case there exists a unique S-module M in Mod S such that M NŽ R 0 j S ii . s Mi NŽ R 0 j S ii . for every i s 1, . . . , n. Moreover, M is indecomposable provided all Mi ’s are indecomposable and the condition Ža. holds. 4.8. We will use in this paper also a more general version of the gluing of modules over flowers. Let R,  S i 4i g 1, . . . , n4 , and S be as in the statement of Proposition 4.7. Suppose that we are given a family of modules  Mi 4i g 1, . . . , n4 , Mi in Mod S i, together with a family of U-homomorphisms  f j, i : Mi NU ª M j NU 4i, j g 1, . . . , n4 , such that f i, i s id M i NU and f l, j f j, i s f l, i for all i, j, l g $ n  1, . . . , n4 , where U is a subcategory of Fis1 S i containing R 0. Denote by n M the U-submodule of [is1 Mi NU defined as n

½

M Ž x . s Ž mi . g

[ Mi Ž x . : ᭙i , j g 1 , . . . , n4 is1

f j, i Ž x . Ž m i . s m j

5

for x g ob U and by ␲ i : M ª Mi NU , i g  1, . . . , n4 , the isomorphism which n is the restriction of the canonical projection ␲ i : [is1 Mi NU ª Mi NU . i For any x g ob Ž Si _ U ., i g  1, . . . , n4 , we set M Ž x . s Mi Ž x .. For any s g SŽ x, y ., where x, y g ob ŽU j S ii . are not simultaneously in U, i g  1, . . . , n4 , we set

¡M Ž s . ␲ Ž y .

~

i

i

M Ž s . s ␲y1 i Ž x . Mi Ž s .

¢M Ž s . i

if x g ob U, y g ob Ž Sii _ U . if y g ob U, x g ob Ž Sii _ U . if x, y g ob Ž S ii _ U . .

For any s g SŽ x, y ., where x g ob S ii, y g ob S jj, i, j g  1, . . . , n4 , i / j, we set M Ž s . s 0 Ži.e. the zero map from M Ž y . to M Ž x ...

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PIOTR DOWBOR

LEMMA. Let R,  S i 4i g 1, . . . , n4 , S, and M s Ž Mi 4i g 1, . . . , n4 ,  f j, i 4i, j g 1, . . . , n4 . be as abo¨ e. Then the collection D M s Ž M Ž x .4x g ob S ,  M Ž s .4s g S . define an S-module. The pair Ž␲ i , id M N Ž S i _ U . . yields the isomorphism ŽD M . NŽ S ii j U . , i Mi NŽ S ii j U . for e¨ ery i s 1, . . . , n. Proof. We have to show that M Ž sr . s M Ž r . M Ž s . for all r g SŽ x, y ., s g SŽ y, z ., where x, y, z g ob S. We may assume that r and s are both nonzero, and hence x, y, z g ob ŽU j S ii . for some i s 1, . . . , n Žby an analysis similar to that in the proof of Proposition 4.7.. If all x, y, z belong simultaneously to U or to Sii _ U then the required equality is straightforward; in the remaining six cases the proof is an easy check on definitions. The second assertion follows immediately from the definition of D M. Remark. Let M1 be in Mod S2 , M2 be in Mod S2 , and n s 2. Then  f j, i 4i, js1, 2 ¬ f 2, 1 yields a bijective correspondence between ‘‘compatible’’ families f# s  f j, i : Mi NU ª M j NU 4i, js1, 2 and U-isomorphisms f : M1 NU ª M2 NU . For a U-isomorphism f : M1 NU ª M2 NU we set DŽ M1 , M2 , f . s DŽ M1 , M2 4 , f˜., where f˜ corresponds to f via the above bijection. 4.9. Now by slight modification of the above arguments one can prove the following generalization of Proposition 4.7. PROPOSITION. Let R,  S i 4i g 1, . . . , n4 , S, and M s Ž M i 4i g 1, . . . , n4 ,  f j, i 4i, j g 1, . . . , n4 . be as in Lemma 4.8. Assume that each module Mi is indecomposable. Suppose that U is a common neighbourhood of R 0 with respect to all Mi ’s or that the condition Žb. from Proposition 4.7 is satisfied. Then the R-module D M is indecomposable.

5. THE SECOND EMBEDDING THEOREM FOR FLOWERS 5.1. In the proof of the main result we will need a stronger result then Theorem 2.3. THEOREM. Let R be a locally bounded category o¨ er an algebraically closed field k and M be an object in Ind R. Suppose that S s supp M is a nondegenerate n-flower Ž maybe finite . with fixed full subcategories S0 , S1 , . . . , S n satisfying the conditions of Definition 2.1 and that there exist M-neighborhoods C of S0 and D of C contained in S such that each category ˆ is nontri¨ ial, for i g 1, . . . , n4. Then there exists a faithful representaSi _ D tion embedding functor E : mod ⌺ n ª mod S. In particular, R is wild if n G 5.

135

STABILIZER CONJECTURE

5.2. In the proof of Theorem 5.1 the following fact will be used. LEMMA. Let M be in Mod R and N an R-submodule of M. Suppose that the set T s  x g supp N : M Ž x . / N Ž x .4 is finite. Then there exists an R-submodule N⬘ of M satisfying the following conditions: Ža. N ; N⬘ and supp N s supp N⬘, Žb. Hom R Ž N⬘, MrN⬘. s 0. Proof. Fix M, N as above; for simplicity set N s MrN; and denote by ␲ : M ª N the canonical projection. We prove first that if Hom R Ž N, N . / 0 then there exists an R-submodule N1 of M such that N n N1 and supp N1 s supp N Žconsequently, T1 s  x g supp N1 : M Ž x . / N1Ž x .4 ; T .. For this purpose take any nonzero R-homomorphism f : N ª N, denote by ␲ f : N ª NrIm f the canonical projection, and consider the following obvious commutative diagram with exact rows: ␲

N

0

␲f

s

6 NrIm f

0.

6

␲f ␲

6

6

6

KerŽ␲ f ␲ . ¨ M

6

0

¨M

6

N

6

6

0

We show that N1 s KerŽ␲ f ␲ . satisfies the required conditions. Clearly N ; N1 , and by the ‘‘Snake Lemma’’ we have the isomorphism N1rN , Im f Ž/ 0.. This immediately implies the equality supp N1 s supp N since supp N1 s supp N j supp N1rN and supp Im f ; supp N. To complete the proof of the lemma suppose for a moment that for M and N there exists no submodule N⬘ of M satisfying Ža. and Žb. Žequivalently, Hom R Ž N⬘, MrN⬘. / 0 for any N⬘ satisfying Ža... Then by applying the first part of the proof we can construct inductively an ascending chain N n N1 n N2 n ⭈⭈⭈ n Nn of R-submodules of M with the same support Žs supp N . for an arbitrarily large n g ⺞. This leads to a contradiction since the length of such a sequence has to be bounded by Ý x g T dim k M Ž x .. Therefore, for given M and N the R-submodule N⬘ of M with the required properties always exists. 5.3. LEMMA. Let M, X be R-modules and Y be an R-submodule of M $ s M N $. Then Y can be regarded as an R-submodule of such that X N supp Y supp Y X and Hom R Ž Y, XrY . s Hom R Ž Y, MrY ..

136

PIOTR DOWBOR

Proof. The first assertion follows immediately by Lemma 4.6. The second is a consequence of the formula Hom R Ž N⬘, N⬙ . s Hom S Ž NNX Sˆ, NNYSˆ. for any R-modules N⬘, N⬙, where S s supp N⬘ l supp N⬙. 5.4. Proof of Theorem 5.1. Fix an R-module M in Ind R satisfying the assumptions. Treating M as object in Ind S Žsee Lemma 4.6. we will construct from M a configuration of n q 1 indecomposable finite-dimensional S-modules satisfying the conditions ŽL. in Theorem 3.3. In the first step we construct an indecomposable finite-dimensional S-module X and an S-submodule Y satisfying the following conditions: Ža. Hom S Ž Y, XrY . s 0. Žb. Y contains an indecomposable direct summand Y 0 such that 0 Y N C s X N C s MN C . n Žc. XrY s [is1 Zi , where each Zi is a nonzero S-module of XrY with supp Zi ; Si , i g  1, . . . , n4 . We start by constructing Y. By the assumption there exists an indecomposable direct summand M C of M N C Žresp., M D of M N D . such that M C N S 0 s M N S 0 Žresp. M D N C s M N D .. Denote by N the S-submodule of M generated by all k-linear spaces M Ž x ., x g D. The S-module N is finite-dimensional ˆ Therefore by Lemma 5.2 there exists an S-submodbecause supp N ; D. ule N⬘ of M such that N ; N⬘, supp N s supp N⬘, and Hom S Ž N⬘, MrN⬘. ˆ Now we s 0. We set Y s N⬘. Note that Y N D s M N D and supp Y ; D. ˜ E in ind S such that M˜ E N E s M N E , define X. By Corollary 4.4 there exists M ˆ l S. We set X s M˜ E . where E s D By Lemma 5.3 Y can be regarded as an S-submodule of X and the condition Ža. holds. The existence Y 0 satisfying Žb. follows now from the fact that M D is an indecomposable direct summand of Y N D . For checking Žc. observe that supp XrY ; S1 k ⭈⭈⭈ k Sn , since Y N S s M N S . Then by 0 0 ŽF2. Žsee Definition 2.1. XrY admits a canonical decomposition XrY s n [is1 Zi into a direct sum of S-submodules Zi such that supp Zi ; Si for every i s 1, . . . , n. To prove that all Zi ’s are nonzero it suffices to show ˆ is nontrivial Žnote that Si l E _ that, for every i g  1, . . . , n4 , Si l E _ D ˆ ; supp Zi since supp Y ; Dˆ and X N E s M N E .. Suppose that Si l E _ Dˆ D ˆ Žis nontrivial. is trivial for i g  1, . . . , n4 . Then we have Si _ E s S i _ D and S splits into a disjoint union of two orthogonal subcategories Si _ E ˆ. j Dnjs0, j/ i S j . This contradicts the indecomposability of M, and Ž S i l D ˆ is nontrivial. consequently S i l E _ D

STABILIZER CONJECTURE

137

Fix X, Y, Y 0 , Z1 , . . . , Zn satisfying Ža., Žb., and Žc.. In the second step we prove that the following holds: Žd. Zi contains an indecomposable direct summand Zi0 such that 1Ž 0 Ext S Zi , Y 0 . / 0, for every i g  1, . . . , n4 . For every i s 1, . . . , n, denote by Yi the S-module defined by the conditions Yi NŽ S 0 j S j . s

~¡

Y NŽ S 0 j S j .

¢Y

0

NŽ S 0 j S j .

if j s i if j / i

and by X i the S-module defined by the conditions X i NŽ S 0 j S j . s

~¡

X NŽ S 0 j S j .

¢Y

0

NŽ S 0 j S j .

if j s i if j / i

where j s 1, . . . , n. The S-modules X i and Yi , i s 1, . . . , n, are well defined by Proposition 4.7 and Remark 4.7, and all X i ’s are indecomposable Žapply Žb. and Proposition 4.7Ža... Moreover, Yi is an S-submodule of X i , X irYi is isomorphic to Zi , and by Corollary 4.6 Y 0 is isomorphic to a direct summand of Yi for every i. Now Žd. follows easily from Žc. and Remark 3.1 applied to the exact sequences 0 ª Yi ª X i ª Z i ª 0, i s 1, . . . , n. To complete the proof fix indecomposable direct summands Zi0 of Zi satisfying Žd. and nonzero elements w e i x g Ext 1S Ž Zi0 , Y 0 ., i s 1, . . . , n. Then the modules Y 0 , Z10 , . . . , Zn0 and the extension class Žw e i x. is1, . . . , n g n Ext 1S Ž[is1 Zi0 , Y 0 . satisfy the condition list ŽL. in Theorem 3.3 Ž k is algebraically closed!.. Consequently, there exists the representation embedding functor E : mod ⌺ n ª mod S. The last assertion follows now by w13, Lemma 7x. 5.5. Proof of Theorem 2.3. Let M be a sincere R-module in Ind R, where R is an infinite n-flower with fixed full subcategories R 0 , R1 , . . . , R n satisfying ŽF0. ᎐ ŽF3. in Definition 2.1. The by Proposition 4.4 we can construct subcategories C and D of R such that C is a finite M-neighbourhood of R 0 and D is a finite neighbourhood of C. Since R is locally ˆ is nontrivial by ŽF3. for each i s 1, . . . , n. Now the bounded, R i _ D assertion follows from Theorem 5.1.

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6. THE PROOF OF THE MAIN RESULT Throughout this section we will assume that R is a locally bounded category over an algebraically closed field k and G : Aut k Ž R . is a group of k-linear automorphisms acting freely on R. 6.1. We start with the characterization of an infinite cyclic group which is crucial for the proof of the main result and motivates further considerations. THEOREM. Let H be a torsion-free finitely generated group. The group H is cyclic if and only if H satisfies the following two conditions: Ža. for any nontri¨ ial subgroup H⬘ of H the index w NH Ž H⬘. : H⬘x of H⬘ in the normalizer NH Ž H⬘. is finite, Žb. for any two cyclic subgroups H1 , H2 of H the intersection H1 l H2 is a nontri¨ ial subgroup of H. The proof needs some preparation. 6.2. The following fact is crucial for the proof of the above theorem. PROPOSITION. Let H be a torsion-free Ž finitely generated . group. Suppose that H contains an infinite cyclic normal subgroup C such that the factor group HrC is finite. Then H is an infinite cyclic group. Proof. Fix H as above. The proof that H satisfies the assertion consists in three steps. First we prove that the centralizer ZH Ž C . of C in H is equal to H. For any element h g H the inner automorphism hŽy. hy1 g AutŽ G . induces the automorphism ␴ h g AutŽ C . , ⺪ 2 Ž C , ⺪ and C is a normal subgroup of H .. We show that ␴ h s id C for every h g H. Suppose for a moment that ␴ h s Žy.y1 for h g H Žin fact, h g H _ C .. Fix any positive m g ⺞ such that h m belongs to C Žthis exists by the assumptions.. Then we have h m s hh m hy1 s ␴ hŽ h m . s hym and consequently h is a torsion Žnontrivial. element in H, a contradiction. In this way we have proved that ZH Ž C . s H. Next we prove that the factor group H s HrC is abelian. We apply here elementary properties of cohomology groups of a finite group. Note first that since ZH Ž C . s H the class of the canonical extension eªCªHªHªe of H by C can be treated as an element of the second cohomology group H 2 Ž H, C . of the group H in coefficients in the trivial H-module C.

STABILIZER CONJECTURE

139

Observe also that the connecting homomorphism in the long exact sequence of cohomology groups of H, induced by the short exact sequence of Žtrivial. H-modules 0 ª ⺪ ª ⺡ ª ⺡r⺪ ª 0, yields under the identification C , ⺪ the isomorphism H 1 Ž H , ⺡r⺪ . , H 2 Ž H , C . .

Ž ).

This follows from the fact that all groups H i Ž H, ⺡. s Ext ⺪i H Ž⺪, ⺡., i g ⺞, are zero. Note that the multiplication by the natural number < H < < H < ⴢ s Ext ⺪i H Ž ⺪, < H < ⴢ . s Ext ⺪i H Ž < H < ⴢ , ⺡ . : Ext ⺪i H Ž ⺪, ⺡ . ª Ext ⺪i H Ž ⺪, ⺡ . is simultaneously an isomorphism of abelian groups Ž< H < ⴢ : ⺡ ª ⺡ is an isomorphism. and a zero map Ž< H < ⴢ : ⺪ ª ⺪ is a composition of the standard ⺪ H-homomorphisms ‘‘diagonal’’ ⌬ : ⺪ ª ⺪ H and ‘‘codiagonal’’ ⵜ : ⺪ H ª ⺪.. The first cohomology group H 1 Ž H, ⺡r⺪. is equal to the homomorphism group HomŽ H, ⺡r⺪. since ⺡r⺪ is a trivial H-module. Then interpreting the second cohomology group H 2 Ž H, C . as the extension class group, the isomorphism Ž*. is given by the mapping which to f g HomŽ H, ⺡r⺪. assigns the class of the extension ␣

␤

␩f : e ª C ª E f ª H ª e, where E f s ⺡ =⺡ r ⺪ H Žs Ž q, h. g ⺡ = H : f Ž h. s q q ⺪4., ␣ is the map induced by the canonical embedding C ª ⺡ ª ⺡ = H, and ␤ is the map induced by the canonical projection ⺡ = H ª H. As a result of the above considerations H is isomorphic to E f for some f g HomŽ H, ⺡r⺪.. Ker f is in a natural way a subgroup of E f , therefore the derived subgroup w H, H x of H can be regarded as a subgroup of H Ž⺡r⺪ is abelian, hence w H, H x ; Ker f .. Consequently, H s HrC is abelian since H is torsion free. Finally, we prove that if H is abelian then H is an infinite cyclic group. We apply an induction on the number of cyclic direct factors of the finite abelian group H s HrC. We start by observing that if H is abelian then by the structure theorem for finitely generated abelian groups H is an infinite cyclic group. Suppose now that H is a cyclic Žnontrivial. group. Then H is generated by two elements c and h, where c is a generator of C and the coset h of h

140

PIOTR DOWBOR

is a generator of H. By the first assertion c and h commute Ž ZH Ž C . s H . and H is abelian. Thus by the previous remark H is an infinite cyclic group. Suppose next that H s H1 = H2 , where H1 and H2 are nontrivial direct factors of H. Denote by ␲ : H ª H and by ␲ 1 : H ª H1 the canonical projections, set H⬘ s KerŽ␲ 1␲ ., and consider the following obvious commutative diagram of groups with exact rows: e

6

6

H1

e.

6

␲ 1␲

6

6

6

H⬘ ¨ H

H ␲1

s

6

e

␲

6

C ¨ H

6

e

Clearly, C is a normal subgroup of H⬘ and by the ‘‘Snake Lemma’’ H⬘rC is isomorphic to H2 . Then by the inductive assumption Žfor H2 . H⬘ is an infinite cyclic group. H⬘ is a normal subgroup of H with the factor group H1 , hence the inductive assumption Žfor H1 . implies that H is an infinite cyclic group, and the proof of the proposition is complete. 6.3. The following consequence of the above proposition takes on special importance in the context of w6, Theorem Bx. COROLLARY. Let H be a torsion-free Ž finitely generated . group. Suppose that H contains an infinite cyclic subgroup of C such that the index w H : C x is finite. Then H is an infinite cyclic group. Proof. Fix H as above. Then C l hChy1 is nontrivial for every h g H; otherwise all cosets hchy1 C, c g C, are pairwise different Žfor c, c⬘ g C, hchy1 C s hc⬘hy1 C implies hc⬘cy1 hy1 g C l hChy1 and consequently c s c⬘.. Fix representatives h 0 s e, h1 , . . . , h n g H of the cosets set HrC. Then the largest normal subgroup H⬘ of H contained in C looks like H⬘ s

F hChy1 s

hgH

.y1

F

is0, 1, . . . , n

h i Chy1 s i

F

C l h i Chy1 i

is1, . . . , n

y1

Ž hcC Ž hc s hCh for c g C, h g H .. H⬘ as an intersection of a finite number of nontrivial Žinfinite cyclic. subgroups of the infinite cyclic group C is a nontrivial Žinfinite cyclic. subgroup of C, hence w C : H⬘x is finite. Consequently, w H : H⬘x is finite since so is w H : C x, and the assertion follows immediately from the previous proposition. 6.4. Proof of Theorem 6.1. It is clear that an infinite cyclic group satisfies both conditions Ža. and Žb.. To prove the implication in the opposite direction we show that if a finitely generated group H satisfies Ža. and Žb. then H contains an infinite cyclic normal subgroup C such that HrC is finite.

STABILIZER CONJECTURE

141

Fix a set of generators  h1 , . . . , h n4is1, . . . , n of H. For every i s 1, . . . , n, denote by ² h i : the infinite cyclic group generated by h i . Then by Žb. the intersection ² h1 : l ⭈⭈⭈ l ² h n : is a nontrivial Žinfinite cyclic. subgroup of H. Fix a generator c of ² h1 : l ⭈⭈⭈ l ² h n :. We show that the subgroup ² c : is centralized Žso normalized. by all elements of H. For every i s 1, . . . , n, there exists n i g ⺞ such that h ni i s c. Then h i chy1 s h i h ni i hy1 s h ni i s c i i for all i’s, and consequently ZH Ž² c :. s H. Now by Ža. the factor group Hr² c : is finite. Thus C s ² c : satisfies the required conditions, and consequently by Proposition 6.2 H is an infinite cyclic group. 6.5. The following result plays an essential role in the proof of the main theorem. THEOREM. Let B be an R-module in Ind R and H be a subgroup of G stabilizing supp B Ž i.e., h Žsupp B . s supp B for e¨ ery h g H . such that supp B consists of finitely many H-orbits in R. Suppose that H contains a subgroup H⬘ satisfying the following two conditions: the index w NH Ž H⬘. : H⬘x is infinite, H⬘ contains an infinite finitely generated subgroup H0 .

Ža. Žb.

Then R is wild. Proof. Fix B and H, H⬘, H0 satisfying the above assumptions. Our aim is to construct an indecomposable module B over some subcategory of supp B such that supp B is an infinite 5-flower. Set S s supp B and N s NH Ž H⬘. for simplicity. From now on we will treat B as an indecomposable S-module Žsee Lemma 4.6.. Fix a Žfinite. set So of representatives of H-orbits in S, generators h1 , . . . , h s of the subgroup H0 , and a finite, connected B-neighbourhood V of the subcategory U s So j h1 So j ⭈⭈⭈ j h s So Žsee Proposition 4.4.. Set W s  h g H : ob hSo l ob Vˆ / ⭋4 . Then for g , h g H the subcategories gV and hV are orthogonal provided gW l hW s ⭋, $

Ž *.

since hVs hVˆ ; hWSo and gV ; gWSo . Note also that for any finite subset A of G there exists an element g g N such that H⬘gW l H⬘ AW s ⭋ ŽWWy1 is finite and the set of right cosets N modulo H⬘ is infinite .. Applying the above remark we inductively construct elements g 1 s e, g 2 , . . . , g 5 g N such that the sets H⬘g iW, i s 1, . . . , 5, are pairwise disjoint. Consequently, by Ž*. the subcategories g i H0 V, i s 1, . . . , 5, are pairwise orthogonal since g i H0 V ; g i H⬘V s H⬘g i V.

142

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For every i s 2, . . . , 5 fix a finite, connected B-neighbourhood Vi of Ui s So j g i So Žsee Proposition 4.4.. It is not hard to check that the fam ilies  Ui 4 is 2 , . . . , 5 j  g i hU 4 h g H 0 , is 1 , . . . , 5 and  V i 4 is 2 , . . . , 5 j  g i hV 4h g H , is1, . . . , 5 of subcategories of S satisfy the assumptions of Lemma 0 4.5. Consequently, the subcategory V s D5is1 g i H0 V j D5is1 Vi is a Bneighbourhood of U s D5is1 g i H0 U j D5is1 Ui . Fix an indecomposable direct summand B V of B N V such that B VNU s B NU and set B s B V. To show that S s supp B V is an infinite 5-flower observe that the subcategories V0 s D5is2 Vi and Vi s g i H0 V _ V0 , i s 1, . . . , 5, define on V a structure of an infinite 5-flower. The category S inherits from V this structure Žsee Remark 2.1. since S contains U Žnote that each g i H0 U _ V0 is infinite since H0 is infinite .. Now the assertion follows immediately from Theorem 2.3 and w13, Lemma 7x Ž S is a wild full subcategory of R .. 6.6. Our next result constitutes the most important part of the proof of the stabilizer conjecture. THEOREM. Let B be an infinite G-atom. Suppose the stabilizer GB of B contains two infinite cyclic subgroups X, Y such that X l Y s  e4 . Then the category R is wild. The proof of Theorem 6.6 needs some preparatory facts. 6.7. Given an infinite cyclic group H with a fixed generator h we set Hn s  h l : l g wyn, n x4 , for n g ⺞, where w n1 , n 2 x s  l g ⺪ : n1 F l F n 2 4 for any n1 , n 2 g ⺪ j  ⬁, y⬁4 , n1 F n 2 . LEMMA. Let W be a finite subset of a group G and X, Y be cyclic subgroups of G Ž with generators x, y respecti¨ ely . such that X l Y s  e4 . Ži. For e¨ ery i g ⺪ _ wym 0 , m 0 x Ž resp. j g ⺪ _ wyn 0 , n 0 x. there exists m 0 g ⺞ Ž resp. n 0 g ⺞. such that x i W l YW s ⭋ Ž resp. y j W l XW s ⭋.. Consequently, for any p g ⺞, x i W l x p YW s ⭋ and x p y j W l XW s ⭋ for all i g ⺪ _ w p y m 0 , p q m 0 x and j g ⺪ _ wyn 0 , n 0 x. Žii. Suppose that A is an infinite subset of ⺞. Then for any n g ⺞, there exists an arbitrarily large p g A such that x p YnW l YnW s ⭋. Proof. Ži. Note that by assumptions all elements x i y j, i, j g ⺞, are pairwise different. The existence of m 0 , n 0 is a consequence of the fact that x i W l y j W / ⭋ if and only if x i yyj belongs to the finite set WWy1 . The remaining assertions now follow trivially. Žii. Fix n g ⺞. Observe first that any finite subset A⬘ ; ⺞ with < A⬘ < ) Ž2 n q 1.< W < 2 contains an element p⬘ s p⬘Ž n. such that x p⬘ YW l YnW s ⭋ Žnote that x i y j W l YnW / ⭋ if and only if x i y j g YnWWy1 and that < YnWWy1 < F Ž2 n q 1.< W < 2 ..

STABILIZER CONJECTURE

143

Now take an infinite subset A of ⺞ and fix a splitting A s Dl g ⺞ A l into a union of pairwise disjoint subsets such that < A l < s 1 q Ž2 n q 1.< W < 2 for all l g ⺞. One can construct an infinite sequence Ž pl . l g ⺞ of pairwise different elements of A Ž pl g A l . such that x p l YnW l YnW s ⭋. In this way the proof is complete. 6.8. Proof of Theorem 6.6. Let B be an infinite G-atom whose stabilizer GB contains infinite cyclic groups X and Y such that X l Y s  e4 . Set S s supp B, H s GB for simplicity, and fix a Žfinite. set So of representatives of H-orbits in S and generators x of X, y of Y, respectively. From now on we treat B as an S-module Žsee Lemma 4.6.. Given a subcategory T of S and i, i⬘, j, j⬘ g ⺪ j  ⬁, y⬁4 , i F i⬘, j F j⬘, we denote by Ti, i⬘ Žresp. T j, j⬘, Ti,j,i⬘j⬘ . the subcategory Dl g w i, i⬘x x l T Žresp. Dl g w j, j⬘x y l T, Dl g w i, i⬘x x l T j Dl g w j, j⬘x y l T .. Fix a finite connected B-neighbourhood V of the subcategory U s So j xSo j ySo Žsee Proposition 4.4.. Given p g ⺞ and m, n g ⺞ j  ⬁4 we ,n p yn, n denote by K Ž p, m, n. the category Vyn ym , 0 j V0, p j x V0, m . Note that all j, j⬘ j, j⬘ categories of the type Vi, i⬘ , V , Vi, i⬘ , and K Ž p, m, n. are connected. Our goal is to construct, for some triple p, m, n, an indecomposable K Ž p, m, n.-module B such that supp B is a 6-flower satisfying the assumptions of Theorem 5.1. Set W s  h g H : ob hSo l ob Vˆ / ⭋4 and fix m 0 , n 0 g ⺞ such that x i W l YW s ⭋ and XW l y j W s ⭋ for all i g ⺪ _ wym 0 , m 0 x and j g ⺪ _ wyn 0 , n 0 x Žsee Lemma 6.7.. Then by Ž*. in the proof of Theorem 6.5 the subcategories XV and V j, ⬁ j Vy⬁, yj Žresp. YV and Vy⬁ , yi j Vi, ⬁ ., and consequently V j, ⬁ j Vy⬁ , yj and Vy⬁ , yi j Vi, ⬁ , are orthogonal for all i, j g ⺞ such that i G m 0 and j G n 0 . Denote by ¨ x Žresp. ¨ y . the greatest i g ⺞ such that V and x i V Žresp. i . y V are not disjoint and by wx Žresp. wy . the greatest i g ⺞ such that V and x i V Žresp. y i V . are not orthogonal. These numbers exist since S is locally bounded and H acts freely on S. They satisfy the inequality wx G ¨ x G 1 Žresp. wy G ¨ y G 1. since U ; V. Fix numbers m1 , n1 g ⺞ such that m1 ) m 0 , 12 wx and n1 ) n 0 , 12 wy . ,⬁ Ž . Then the category Vy⬁ y⬁ , ⬁ s XV j YV , with the structure defined by the n1, ⬁ ⬁, ⬁ Ž subcategories V# s Vy _ V j Vy ⬁, y n 1 ., y ⬁, ⬁ y ⬁, y m 1 j Vm 1 , ⬁ j V n1, ⬁ y⬁, yn 1 Vy⬁ , ym 1, Vm 1 , ⬁ , V , and V , is an infinite 4-flower Žnote that 1q1 , n 1y1 .. In consequence, each connected subcategory V ; V# ; Vyn ym 1 q1, m 1 y1 j⬘  4 Vyj, yi , i⬘ , i, i⬘, j, j⬘ g ⺞ j ⬁ , i, i⬘ G m1 , j, j⬘ G n1 , is equipped with the y⬁ , ⬁ standard nondegenerate, induced from Vy⬁ , ⬁ , 4-flower structure such that yj, j⬘ Ž Vyi , i⬘ . 0 s V# Žsee Remark 2.1.. i i . ,⬁ Ž Note that by Lemma 4.5 the category Vy⬁ y⬁ , ⬁ s Di g ⺪ x V j Di g ⺪ y V i i . y⬁ , ⬁ Ž is a B-neighbourhood of Uy⬁ , ⬁ s Di g ⺪ x U j Di g ⺪ y U . The assumption Žc. is satisfied since x iU l x iq1 U and y iU l y iq1 U are nontrivial for every i g ⺪, Ža. and Žb. follow easily by the definitions of U, V, and S. Fix

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,⬁ such that B y⬁ ,⬁ s an indecomposable direct summand B⬘ of B N Vy⬁ NU y y⬁ ,⬁ ⬁,⬁ ,⬁ . We will use B⬘ for the construction of the announced module B. BXNUy⬁ y⬁ ,⬁ For this purpose we analyse a decomposition BXN X V s [j g J Bj into a direct sum of indecomposable XV-modules. First we prove that < J < F dim k BXN V# l X V . It suffices to show that Bj N V#l X V / 0 for every j g J. Fix j g J and suppose that Bj N V# l X V s 0. Then supp Bj is contained in the union Vy⬁, ym 1 j Vm 1 , ⬁ , which is orthogonal to YV. By Lemma 4.6 applied to the direct summand Bj NŽ X V _ Y V . of BXNŽ X V _ Y V . , the XV-module Bj can be extended by zeros to a nonzero direct summand of the Ž XV j XY .-module B⬘. By indecomposability of B⬘ this summand is isomorphic to B⬘, a contradiction, since XU j YU ; supp B⬘ and U ; V# l XV. Consequently, Bj N V# l X V / 0 for every j g J. Next we show that each XV-module Bj , j g J, is an X-atom. The XV-module BXN X V is a direct summand of B N X V . The group X stabilizes XV and XŽ B N X V . s X. Clearly supp BXN X V is contained in a union of a finite number of X-orbits in S. Now the claim follows by w12, 2.3; 6, Corollary 2.4x. Set J 0 s  j g J : dim k Bj - ⬁4 and J⬘ s J _ J 0 . Then the intersection X 0 s Fj g J ⬘ X B j Žof stabilizers of all infinite X-atoms Bj , j g J . is a nontrivial Žcyclic. subgroup of an infinite cyclic group X, since J⬘ is finite and all X B j’s are nontrivial. Denote by r the positive natural number such that x r generates X 0 , fix m 2 g ⺞, m 2 G m1 , such that the union Dj g J 0 supp Bj j Ž V# l XV . is contained in Vym 2 , m 2 and set m 3 s m 2 q ¨ x q 1, m 4 s m 3 q wx . It is easily seen that for any l g ⺞, l G 2 m 4 , the subcategory V m$ 4 , l y m4 l K Ž l, ⬁, ⬁. is contained in Vm 3 , lym 3 . Note also that for every p g r⺪, p p G 2 m 3 , the modules BXN V m , py m and x BXN V m , py m are isomorphic since 3 3 3 3 BXN V m , ⬁ s [j g J ⬘ Bj N V m , ⬁ , BXN Vy ⬁, y m s [j g J ⬘ Bj N Vy ⬁ , y m , and X 0 ; XŽ[ j g J ⬘ B j . . 3 3 3 3 For any p as above we fix one Žarbitrarily chosen. isomorphism xp X X ␸p : B N V m , py m ª B N V m , py m . 3 3 3 3 Now fix a finite common B⬘-neighbourhood A of Vm 3 , m 4 , Vym 4 , ym 3 , and V# Žsee Proposition 4.4 and Lemma 4.3.. Without loss of generality we ya y , a y may assume that A s Vya for some a x , a y g ⺞, a x G m 4 , a y G n1. x, a x Then for any p g r⺪, p G 2 a x q wx the subcategories K 0 Ž p . s Vm 4 , pym 4 , y⬁, ⬁ y⬁, ⬁ K 1Ž p . s Vy⬁, K 2 Ž p . s Vpym _ Vm 4 , pym 4 Žresp. m 4 y1 _ Vm 4 , pym 4 , 4 q1, ⬁ X X X y⬁, ⬁ K 0 Ž p . s Vm 4 , pym 4 , K 1Ž p . s Vy⬁, m 4y1 , K 2 Ž p . s x p Vym _ Vm 4 , pym 4 . 4 q1, ⬁ define a 2-flower structure on XV j YV Žresp. XV j x p YV . and the two sequences of inclusions

$

y, ay K 0 Ž p . l K 1 Ž p . ; Vm 3 , m 4 ; A ; Vya ya x , pym 4 ,

$

y, ay , K 0 Ž p . l K 2X Ž p . ; Vpym 4 , pym 3 ; x pA ; x p Vmya4yp , ax

hold true.

Ž ). Ž )).

145

STABILIZER CONJECTURE

Fix finite B⬘-neighbourhoods C of A and D of C, a finite subcategory ˆ l Ž XV j YV ., and a finite B⬘-neighbourE of XV j YV containing D hood F of E Žcf. 5.4 and 5.5.. Without loss of generality we may assume that all of them have a standard 4-flower form yc , c yd , d ye , e yf , f C s Vyc , D s Vyd , E s Vye , F s Vyf , x, cx x, dx x, ex x, fx y

y

y

y

y

y

y

y

where a x F c x F d x Žresp. a y F c y F d y . and d x q 2 wx - e x F f x Žresp. d y q 2 wy - e y F f x . Žsee Lemma 4.3.. We fix now indecomposable direct summands B C of BXN C , B D of BXN D , and B F of BXN F such that B C N D s BXN A , D D N C s BXN C , and B F N E s BXN E . We establish for simplicity the notation yc , c yd , d yf , f Cqs Vyc , Dqs Vyd , Fqs Vyf x, ⬁ x, ⬁ x, ⬁ y

y

y

y

y

y

and yc , c yd , d yf , f Cys Vy⬁ , c x , Dys Vy⬁ , d x , F s Vy⬁ , f x . y

y

y

y

y

y

By Proposition 4.7 and Remark 4.7 we can construct two triples of modules C D F C D in Mod Cq, Bq in Mod Dq, Bq in Mod Fq and By in Mod Cy, By Bq F in Mod Dy, By in Mod Fy, defined by the conditions X C BqN V 0 , ⬁ s BN V 0 , ⬁ , D BqN V0 , ⬁

s

BXN V 0 , ⬁ ,

X F BqN V 0 , ⬁ s BN V 0 , ⬁ ,

X C BqN y , c y s B N V yc y , c y ; V yc yc , 0 yc , 0 x

x

D BqN y, d y V yd yd x , 0

s

BXN Vyd y, d y ; yd x , 0

X F BqN y , f y s B N V yf y , f y ; V yf yf , yf , x 0

x 0

and X C ByN V y⬁ , 0 s B N V y ⬁ , 0 , D ByN V y⬁ , 0

s

BXN Vy ⬁ , 0 ,

X F ByN V y⬁ , 0 s B N V y ⬁ , 0 ,

X C yc , c yc , c ByN V 0 , cy y s B N V 0 , cy y ; x

D yd y , d y ByN V 0, d x

x

s

y, d y ; BXN Vyd 0, d x

X yf , f F yf , f ByN V 0 , fy y s B N V 0 , fy y . x

x

The all newly constructed modules are indecomposable since A is a common with respect to B⬘, B C , B D , B F neighbourhood of V# such that BXN A s B C N D s D D N C s B F N E Žsee Proposition 4.7.. Note that by Lemma C. F F D C Ž Ž . 4.6 Bq resp. By is a direct summand of BqN C q resp. ByN C y and Bq Žresp. D. F F Ž . By is a direct summand of BqN D q resp. ByN D y . Set m s f x and n s f y . Using the two triples of indecomposable modules defined above we will construct for some p g ⺞ an indecomposable K Ž p, m, n.-module B whose support is a 6-flower having the announced properties. Note that a category K Ž p, m, n. does not necessarily have to carry a 6-flower structure for every p g ⺞.

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PIOTR DOWBOR

We show first that there exists p g r⺪, p G 2 a x q wx , such that the category K s K Ž p, ⬁, n. is a 2-flower with respect to the structure defined by the subcategories K 0 s K 0 Ž p ., K 1 s K 1Ž p . l K, and K 2 s K 1X Ž p . l K. Note that by Lemma 6.7 there exists p g r⺪, p G 2 a x q wx , such that YnW l x p YnW s ⭋. Hence, Vyn, n and x p Vyn, n are orthogonal by Ž*. in the proof of Theorem 6.5. In consequence, K 1 and K 2 are orthogonal since so are K 1Ž p . and K 2 Ž p . Žresp. K 1X Ž p . and K 2X Ž p ... Fix any p as above. Then each of the subcategories Cq, Dq, Fq Žresp. x p Cy, x p Dy, x p Fy . inherits from K the structure of a finite 2-flower Žsee Remark 2.1.. The same holds true for C s K Ž p, c x , c y ., D s K Ž p, d x , d y ., and F s K Ž p, m, n.. There 2-flower structures of C, D, and F using the notation convention from Remark 2.1 are described by the splittings C s Ž Cq . 1 k K 0 k Ž x p Cy . 2 , D s Ž D q . 1 k K 0 k Ž x p Dy . 2 , F s Ž Fq . 1 k K 0 j Ž x p Fy . 2 . p

p

C x C We p show now that the collections Ž Bq , By ,␸ ˜p ., Ž BqD, x ByD, ␸˜p ., and x F F Ž Bq , By ,␸ 4.9 ˜p . Žsee Remark 4.8. satisfy the assumptions of Proposition p C x C and in fact Žb. in Proposition 4.7. We check it only for Ž Bq , By ,␸ ˜p .; the remaining two cases follow by analogous arguments. Recall first that X X C C Žresp. ByN ya y, a y s B y a y , a y . from the construction of BqN y , a y s BN Vy a y , a y V ya Vy N Vy ya x , ⬁ ya x, ⬁ ⬁, a x ⬁, a x p C Ž C. C C . Bq resp. By . Consequently, BqN K 0 Žresp. x ByN K 0 is nonzero since K 0 contains the nontrivial category Um 4 , pym 4 . Moreover, by Lemma 4.3, K 0 j C y, cy y, c y Ž Cq .1 s Vyc Ž . is a Bq K 0 j Ž x p Cy . 2 s x p Vmyc4yp, yc x , pym 4$ resp. yc x $ p x C neighbourhood of K 0 l Ž Cq .1 Žresp., is a By-neighbourhood of K 0 l Ž x p Cy . 2 .. One needs to apply here the fact that A is a B⬘-neighbourhood C p Ž of Vm 3 , m 4 , the inclusions Ž*., and the equality BXN A s BqN A resp., that x A xp is a p B⬘-neighbourhood of Vpy m 4 , pym 3 , the inclusions Ž**., and the equalp C p .. Finally, note that ity x BXN x p A s x ByN ␸p can be regarded as an x A $ ; xp C C isomorphism BqN V m , py m ª ByN V m , py m and that K 0; Vm 3 , pym 3 ; Cql 3 3 3 3 x p Cy holds. In this way by Proposition 4.9 we obtain a triple of indep C x C composable modules B C s D Ž B q , By , ␸ p . in mod C, B D s p p D x D F x F DŽ Bq , By , ␸p . in mod D, and F s DŽ Bq , By , ␸p . in mod F. To complete the proof we show that the module B s B F satisfies the assumptions of Theorem 5.1. We start by observing that F carries the natural structure of a 6-flower defined by the subcategories F0 s V# j V0, p j x p V#, F1 s V 0, n _ F0 , F2 s Vym, 0 _ F0 , F3 s Vyn, 0 _ F0 , F4 s x p Vyn, 0 _ F0 , F5 s x p V0, m _ F0 , and F1 s x p V 0, n _ F0 . Next we prove that C is a B-neighbourhood of F0 and D is a B-neighbourhood of C. Note first that from the construction of B C , B D , and B F we have B C N F 0 s B N F 0

STABILIZER CONJECTURE

147

and B D N C s B N C . Moreover, B C can be regarded as submodule of B N C , C and by Corollary 4.6 B C is a direct summand of B N C , since BqN is y, c y V yc yc x , py m 4 F C , c yc , c a direct summand of BqN Vyc and B is a direct summand of y y y y yN V m 4 y yc x , py m 4 p, c x F D ByN is a direct summand of B N D . y , c y . By the analogous arguments B V yc m 4 y p, c x Now we are in position to close the proof. Set S s supp B for simplicity. Then the S i s S l Fi , i s 0, 1, . . . , 6, define the structure of the 6-flower p ye y , e y y, ey on S. Note that S contains Uye by the construction ye x , 0 j U0, p j x U0, e x of B. In particular, each Si contains a subcategory of the form h i U, where h 0 s e, h1 s y e y , h 2 s x e x , h 3 s yye y , h 4 s x p yye y , h 5 s x pq e x , and h 6 s ˙ s D l S. x p yye y Ž e x ) m1 q ¨ x , e y ) n1 q ¨ y .. Set C˙ s C l S and D ˙ is a B-neighbourhood of C Clearly C˙ is B-neighbourhood of S0 and D ˆ ˙ is nontrivial, observe that for contained in S. To show that each Si _ D ˆ contains h U since Dˆ l S ; Vyd yyw y , d yqw y and ˙ every i s 1, . . . , 6, Si _ D i yd xyw x , d xqw x e x ) d x q wx q ¨ x Žresp. e y ) d y q wy q ¨ y .. ˙ and D˙ satisfy the assumptions of Theorem 5.1, the Since F, B, S, C, category F is wild, and consequently so is R Žsee w13, Lemma 7x.. 6.9. We need also a general property of G-atoms. LEMMA. Let H : Aut k Ž S . be a group acting freely on a locally bounded k-category S, So a fixed set of representati¨ es of H-orbits in S, and H⬘ s H⬘Ž So . the set consisting of all h g H such that SŽ x, hy . / 0 for some x, y g So . If S is a connected category then ² H⬘: s H. In particular, the group H is finitely generated pro¨ ided H has only finitely many orbits in S. Proof. Fix any h g H. By the connectedness of S there exist elements h 0 , h1 , . . . , h n g H, h 0 s e, h n s h, and objects x 1 , . . . , x n ; y 1 , . . . , yn in So such that either SŽ h iy1 x i , h i yi . / 0 or SŽ h i yi , h iy1 x i . / 0 for every i s .Ž y1 . Ž y1 . ² H⬘: 1, . . . , n. Consequently, h s Ž hy1 0 h1 h1 h 2 ⭈⭈⭈ h ny1 h n belongs to y1 y1 since either h iy1 h i g H⬘ or h i h iy1 g H⬘, for each i s 1, . . . , n. Thus, ² H⬘: s H. The remaining assertion follows now immediately from the general assumptions on S and H. Remark. If So is connected and H⬘ generates H then S is connected. In the case S is connected one can construct So connected. COROLLARY. The stabilizer GB of any G-atom B is a finitely generated subgroup of G. 6.10. Now the Main Theorem follows immediately from the result below. THEOREM. Let R be a locally bounded representation-tame category o¨ er an algebraically closed field k and G : Aut k Ž R . be a group of k-linear automorphisms acting freely on R. Suppose that B is an infinite G-atom such

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that the stabilizer GB is a torsion-free group. Then GB is an infinite cyclic group. Proof. Take B as above. By Corollary 6.9, GB is a finitely generated group. Therefore to prove that GB is an infinite cyclic group it is enough to show that GB satisfies conditions Ža. and Žb. in Theorem 6.1. By the assumption R is not wild Žsee w13, Theoremx.. Then Ža. follows by Theorem 6.5 since any nontrivial subgroup of torsionfree group contains an infinite cyclic subgroup; Žb. follows immediately from Theorem 6.6. In this way Theorem 6.10 and consequently the Main Theorem are proved.

ACKNOWLEDGMENT Most of the results contained in this paper were presented during seminar lectures at Paderborn University in July 1997 and at Torun ´ University in October 1997. The author expresses his gratitude to Grzegorz Gromadzki for his assistance during the author’s work on the group theoretical part of this paper, which considerably influenced the final shape of the main result. He also thanks Stanisław Kasjan and Jolanta Słominska for ´ their comments concerning the proof of Proposition 6.2.

REFERENCES 1. K. Bongartz and P. Gabriel, Covering spaces in representation theory, In¨ ent. Math. 65 Ž1982., 331᎐378. 2. H. Cartan and S. Eilenberg, ‘‘Homological Algebra,’’ Princeton Mathematical Series, Vol. 19, Princeton Univ. Press, Princeton, NJ, 1956. 3. P. Dowbor, On modules of the second kind for Galois coverings, Fund. Math. 149 Ž1996., 31᎐54. 4. P. Dowbor, Galois covering reduction to stabilizers, Bull. Polish Acad. Sci. Math. 44 Ž1996., 341᎐352. 5. P. Dowbor, The pure projective ideal of a module category, Colloq. Math. 71 Ž1996., 203᎐214. 6. P. Dowbor, Properties of G-atoms and full Galois covering reduction to stabilizers, Colloq. Math. 83 Ž2000., 231᎐265. 7. P. Dowbor, On stabilizers of G-atoms of representation-tame categories, Bull. Polish Acad. Sci. Math. 46 Ž1998., 304᎐315. 8. P. Dowbor, Nonorbicular modules for Galois coverings, preprint, Torun, ´ 1999. 9. P. Dowbor and S. Kasjan, Galois covering technique and non-simply connected posets of polynomial growth, J. Pure Appl. Algebra 147 Ž2000., 1᎐24. 10. P. Dowbor, H. Lenzing, and A. Skowronski, ‘‘Galois Coverings of Algebras by Locally ´ Support-Finite Categories,’’ Lecture Notes in Math., Vol. 1177, pp. 91᎐93, SpringerVerlag, 1986. 11. P. Dowbor and A. Skowronski, On Galois coverings of tame algebras, Arch. Math. ´ Ž Basel . 44 Ž1985., 522᎐529. 12. P. Dowbor and A. Skowronski, Galois coverings of representation-infinite algebras, ´ Comment. Math. Hel¨ . 62 Ž1987., 311᎐337.

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