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I-SequencesInTheCategoryOfModules

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DOI:10.1006/jabr.2001.9111

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Journal of Algebra 250, 450–472 (2002)doi:10.1006/jabr.2001.9111, available online at http://www.idealibrary.com on

I-Sequences in the Category of Modules

Ahmed A. Khammash

Department of Mathematical Sciences, Umm Algura University,P.O. Box 7296, Makkah, Saudi Arabia

E-mail: khamm@uqu.edu.sa

Communicated by Michel Broue

Received February 28, 2001

The notion of I-sequences (introduced by Green) in the category of modules overfinite-dimensional algebras is developed in this paper and particularly used to givean alternative proof for the results of Knorr concerning the existence and proper-ties of relative projective covers in the category of modules over group algebras. 2002 Elsevier Science (USA)

Key Words: I-sequences; relative projective covers; almost split sequences.

0. INTRODUCTION

Auslander–Reiten theory indicates that category theory and homologicalalgebra can be powerful tools in representation theory of algebras (see, forinstance, [AR1, AR2, AR3]). This fact is also emphasized by the role playedby some other similar sequences such as the relative projective sequencesin the representation theory of finite groups.

This paper is an attempt to give a unifying notion for such sequences(or diagrams) in the category of modules called I-sequences (Section 1),where I is an ideal in the category of modules (see Definition 1.2). Inparticular this notion generalizes the notion of almost split (or Auslander–Reiten) sequences. It turns out that most results that have been provedfor almost split sequences can be proved in a slight generalization forI-sequences once they exist. In fact the map which assigns each end of aminimal I-sequence to the start of that I-sequence turns out to be an equiv-alence of certain categories (Theorem 6) generalizing the known Helleroperator and Auslander–Reiten translate for group algebra modules. As anapplication (Theorem 8), one can deduce the existence of the projective

450

0021-8693/02 $35.00 2002 Elsevier Science (USA)All rights reserved.

I-sequences 451

cover relative to a set of subgroups which is proved originally by Knorr [K].Finally the Appendix deals with ideals in the category of group algebras. Itis proved that if an ideal I satisfies a certain hypothesis (Hypothesis A.3)then the start and end modules of minimal I-sequences have the same ver-tices (Theorem A.4). This generalizes Knorr’s theorem (Theorem A.8) onrelative projective covers and also provides a proof of the known (parallel)result for almost split sequences.

1. PRELIMINARIES

Let � be a finite-dimensional k-algebra, where k is a field. Denote bymod� the category of finite-dimensional left �-modules and by M mod�[resp., M◦mod�] the category whose objects are all contravariant [resp.,covariant] k-linear functors F � mod�→ modk (see [JG1, p. 265]), wheremodk is the category of k-vector spaces and the morphisms betweenobjects in M mod� are the natural transformations. We also denote byindec� the full subcategory of mod� whose objects are all indecompos-able �-modules. If X�Y ∈ mod�, we denote by �X�Y ��, or often �X�Y �,the k-space Hom��X�Y �. If F�G ∈M mod�, we use Mor�F�G� to denotethe set of all morphisms between F and G. If M ∈ mod�, then the homfunctor � �M� ∈M mod� is given by

� �M�� X �→ �X�M�� ∀X ∈ mod�

and if g ∈ �M�V � then � � g� ∈Mor�� �M�� � � V �� is given by � � g��X� �=�X�g� for all X ∈ mod�. An object F ∈ M mod� is said to be finitelygenerated if F is a homomorphic image of � �M� for some M ∈ mod� andis finitely presented if there is an exact sequence in M mod�

� �N� → � �M� → F → 0(1.1)

and we say that (1.1) is a finite presentation of F . If F�G ∈ M mod�, weshall use the notation F ≤ G to denote F as a subfunctor of G. We usemmod� to denote the full subcategory of M mod� whose objects are allfinitely presented functors.

Definition 1.2. If for all X�Y ∈ mod�� I�X�Y � is a k-subspace of�X�Y �� such that

I�Y�Z��X�Y �� ⊆ I�X�Z� and �Y�Z��I�X�Y � ⊆ I�X�Z�for all Z ∈ mod�, then we say that I is an ideal in mod� and writeI � mod�. We also write X ∈ I or X ≡ 0 mod I, to mean that 1X ≡0 mod I�X�X�, i.e., that I�X�X� = �X�X��.

452 ahmed a. khammash

Examples. (1) Let ��X�Y � = �f ∈ �X�Y ��� fg ∈ J�E�Y ��� ∀ g ∈�Y�X���, where E�Y � = End��Y � and J is the Jacobson radical. Then itis known that ��Y�Z��X�Y �� ⊆ ��X�Z� and �Y�Z����X�Y � ⊆ ��X�Z�for all Z ∈ mod�, and so � � mod�. If X ∈ mod� then X is �-trivial(i.e., ��X�X� = �X�X�) if and only if X = 0.

(2) Let

P�X�Y � = �f ∈ �X�Y ��� f factors through a projective �-module��

Then P�Y�Z��X�Y �� ⊆ P�X�Z� and �Y�Z��P�X�Y � ⊆ P�X�Z� andhence P �mod�.

Let ��−� Y � ∈ M mod� be the object defined by: X �→ ��X�Y � forall X ∈ mod�. Then ��−� Y � is a (the radical) subfunctor of �−� Y �. Thefunctor �� � V � ∈ M mod� is the intersection of all the maximal subfunc-tors of � � V � ([PG, p. 2]). A proof of this important fact is sketched in[JG1, Appendix]. This subfunctor shares many properties with the radicalof modules such as the following, which will be needed later.

If L is a subfunctor of �−� Y � satisfying(1.3)L+��−� Y � = �−� Y � then L = �−� Y ��

Yoneda’s Lemma 1.4. For every M ∈ mod��F ∈ mmod�, the mapα → α�M��1M� defines a k-isomorphism between ��−�M�� F�mmod� andF�M�. Consequently

(i) For every M ∈ mod�� � �M� is a projective object in M mod�,

(ii) If M�M ′ ∈ mod�, then every morphism α� � �M� → � �M ′� inM mod� has the form α = � � a� for some a ∈ �M�M ′� (see [JG1, pp. 267and 268, Prop. 1.3]).

The following consequence of Lemma 1.4 will be needed.

Lemma 1.5. If α ∈ End� � Y � is surjective, then it is an automorphism of� � Y �.

Proof. By Lemma 1.4(ii), α = � � a� for some a ∈ End�Y �. If α� � � Y � →� � Y � is surjective, we have that �X�a�� �X�Y � → �X�Y � is surjective, forall X ∈ mod�. In particular, �Y� a�� �Y�Y � → �Y�Y � is surjective. There-fore there is some η ∈ �Y�Y � such that aη = 1Y . But a�η are linear trans-formations on the finite-dimensional k-space Y , so they must be bijective.Hence a ∈ Aut�Y �, which implies that α = � � a� ∈ Aut� � Y �.

I-sequences 453

2. I-SEQUENCES

In this section the notion of left, right, and two-sided (bilateral)I-sequences are introduced for an ideal I � mod�. The notion of I-sequences as well as the main results of this section are taken from anunpublished work of J. A. Green, and I am grateful to him for allowingme to include it in this paper. The aim is to apply this theory in Sections 3and 4 to give an alternative proof to that given by Knorr [K] for the exis-tence and properties of the projective cover of a kG-module relative to acollection of subgroups of G. Suppose that

�1� 0→ Kf→M

g→V → 0

is a short exact sequence in mod�.

Definition 2.1. (Green). The sequence (2.1) is said to be right I-sequence if for all X ∈ mod� and θ ∈ �X�V ��� θ factors through g if andonly if θ ∈ I�X�V �; that is, if and only if Im�X�g� = I�X�V � ∀X ∈ mod�.This means that (1) is right I-sequence if and only if the sequence

0→ �X�K� �X�f �→ �X�M� �X�g�→ �X�V � →�X�V �/I�X�V � → 0

is exact for all X ∈ mod�, where �X�g��f � = gf ∀ f ∈ �X�M�. Equiva-lently, (1) is a right I-sequence if and only if the sequence

�2� 0→ � �K� � �f �→ � �M� � �g�→ � � V � →� � V �/I� � V � → 0

is the exact sequence in M mod�. This in particular implies that the func-tors I�−� V �� �V �I = �−� V �/I�−� V � are finitely presented.

Examples 2.2. (i) If we take I = �, and V ∈ indec� then (1) is an�-sequence in mod� if and only if (1) is an almost split (or Auslander–Reiten) sequence (see [PG]). M. Auslander and I. Reiten [AR3] provedthat if V ∈ mod� is nonprojective indecomposable (hence V /∈ �) thenthere exists an almost split sequence ending with V . The quotient functor�V �� is the simple functor SV in the notation of [JG1].

(ii) If we take I = P then (1) is a P-sequence if and only if (1) is aprojective presentation of V .

Definition 2.3. The map g in (1) is said to be right minimal ifKer�X�g� ≤ ��X�M� for all X ∈ mod�, in which case (1) is said to be aminimal right I-sequence. This is equivalent to saying that the morphism�−� g� is right minimal in the category M mod�.

In some cases one is able to grasp an I-sequence (see Section 4). Theexistence of such a sequence ensures existence of a minimal sequence asthe following theorem proves.

454 ahmed a. khammash

Theorem 1. Suppose that I � mod� and V ∈ mod�. If there exists aright I-sequence (1) ending with V then there exists a minimal right I-sequenceending with V .

Proof. Suppose that 0 → Kf→M

g→V → 0 is an I-sequence.Then we have an exact sequence in M mod�

0→ � �K� � �f �→ � �M� � �g�→ � � V � →� � V �/I� � V � → 0�

It follows that the functor I� � V � is a homomorphic image of � �M�,hence finitely generated. By ([PG, p. 3]), I� � V � has a projective cover� �M1� and a minimal projective presentation � �M1�

θ−→ I� � V � → 0 inM mod�; hence Ker θ ≤ �� �M1�. But we also have the inclusion mor-

phism I� � V � i−→ � � V �. If we let η = ioθ ∈ Mor�� �M1�� � � V ��, thenKerη = Ker θ and Imη = Im θ. Moreover η has the form � � g1�, whereg1 ∈ �M1� V �. Now Im� � g1� = Im θ = I� � V �; hence Im�X�g1� = I�X�V �for all X ∈ mod�. By taking X = M , we get Im�M�g1� = I�M�V �. Butsince g ∈ I�M�V �, there exists h ∈ �M�M1� such that �M�g1��h� = g, sog1h = g. Since g is an epimorphism, g1 is also an epimorphism and so wehave a short exact sequence

�3� 0→ K1f1→M1

g1→V → 0

in mod�, which is a right I-sequence since Im� � g1� = I� � V �. More-over Ker� � g1� = Kerη = Ker θ ≤ �� �M1�; therefore (2) is a minimalI-sequence.

Analogously we can define the left I-sequence in the category mod� asfollows.

Definition 2.4. (Green). The short exact sequence (1) is said to be aleft I-sequence if for all X ∈ mod� and α ∈ �K�X��� α factors through fif and only if α ∈ I�K�X�; that is, if and only if Im�f�X� = I�K�X� ∀X ∈mod�. The map f in (1) is called left minimal if Ker�f�X� ≤ ��M�X� forall X ∈ mod� and in this case (1) is said to be minimal left I-sequence.

It is clear that (1) is a left I-sequence if and only if the sequence

0→ �V�X� �g�X�→ �M�X� �f�X�→ �K�X� → �K�X�/I�K�X� → 0

is exact for all X ∈ mod�, which is equivalent to the fact that the sequence

�4� 0→ �V� � �g� �→ �M� � �f� �→ �K� � → �K� �/I�K� � → 0

I-sequences 455

is exact in M◦mod�. Using the same idea as in the proof of Theorem 1we can deduce the following

Theorem 2. Suppose that I � mod� and K ∈ mod�. If there exists aleft I-sequence beginning with K, then there exists a minimal left I-sequencebeginning with K.

Lemma 2.5. Let (1) 0 → Kf−→ M

g−→ V → 0 be a s.e.s. in mod�.Then

(i) If (1) is right (left) I-sequence, then g ∈ I�M�V ��f ∈ I�K�M��;(ii) If V = 0 �K = 0�, then (1) is a right (left) I-sequence, and

(iii) If (1) is a left (right) I-sequence and V = 0 �K = 0�, then K andM are isomorphic and belong to I (M and V are isomorphic and belong to I).

Proof. These are easy consequences of the definitions and are left as anexercise for the reader.

The following lemma on the minimal maps will be used later.

Lemma 2.6. Let X�Y ∈ mod� and let g ∈ �X�Y �.(1) If g is right minimal then, for every a ∈ �X�X�, ga = g⇒ a is an

isomorphism.

(2) If g is left minimal then, for all every b ∈ �Y�Y �, bg = g⇒ b is anisomorphism.

Proof. (1) Let g ∈ �X�Y � be right minimal and take a ∈ �X�X� withga = g. Then �−� g��−� a� = �−� g� and so �−� g��Im�−� a�� = Im�−� g�,which implies that Im�−� a� + Ker�−� g� = �−�X�. Since g is mini-mal we then have Im�−� a� + ��−� g� = �−�X� and so, by Defini-tion 1.2, Im�−� a� = �−�X�. In particular Im�X�a� = �X�X�; i.e.,a�X�X� = �X�X�, which implies that a is an automorphism. The proof of(2) is quite similar.

Definition. If θ� X → Y is a �-epimorphism, then θ is said to besplit epimorphism if there exists a �-map η� Y → X such that θη = 1Y .Similarly if θ is monomorphism then θ is said to be split monomorphism ifthere exists a �-map η� Y → X such that ηθ = 1X . The notions of splitepimorphism and split monomorphism in the category M mod� are definedsimilarly.

Given two s.e.s Ei� 0 → Xi

fi−→ Yi

gi−→ Zi → 0� i = 1� 2 in mod�.Define their direct sum

E = E1∐

E2� 0→ X1∐

X2f→Y1

∐Y2

g→Z1∐

Z2 → 0�

456 ahmed a. khammash

where f �x1� x2� = �f1�x1�� f2�x2�� and g�y1� y2� = �g1�y1�� g1�y2��. It isclear that E is exact. We shall see later that every I-sequence containsa minimal sequence as a direct summand. To reach that end we need thefollowing

Proposition 2.7. Let F�G ∈M mod� and suppose that

�1� � � �ρ2→A1

ρ1→A0ε→F → 0

t1↓↑s1 t0↓↑s0 θ↓�2� � � �

d2→ B1d1→ B0

e→G→ 0

are two projective presentations of F and G, respectively, with (2) minimal;i.e., Ker�di� ≤ �Bi, for all i. If θ ∈ �F�G� is a split epimorphism, then thereare morphisms ti ∈ �Ai�Bi�� si ∈ �Bi�Ai�, which make the diagram commuteand such that tisi ∈ Aut�Bi� for all i.

Proof. Let β ∈ �G�F� be such that θβ = 1G. The existence of ti� sicomes from the fact that Ai�Bi are projective in M mod�. From the com-mutativity of the diagram we have

di�tisi� = �diti�si = �tiρi�si = �ti−1si−1�di ∀ i�

In particular we have e�t0s0� = �θβ�e = 1e = e and so e�Im t0s0� = e�B0�.Therefore Im t0s0 + Ker e = B0. But Ker�e� ≤ ��B0�; hence Im t0s0 = B0and so t0s0 ∈ Aut�B0� (see (1.3) and Lemma 1.5). Inductively for all i � 0we have

di�Im tisi� = ti−1si−1�Imdi� = Imdi = di�Bi��

Hence Im tisi + Ker�di� = Bi and since Ker�di� ≤ ��Bi�, we have Bi =Im�tisi� and so tisi ∈ Aut�Bi�.Corollary 2.8. If S1 is a projective presentation of F ∈ M mod� then

S1∼= S2

∐N, where S2 is minimal projective presentation of F and N is a

projective presentation of the form � � �→ P2 → P1 → 0→ 0.

Proof. Apply Proposition 2.7, taking �1� = S1, �2� = S2, and θ = 1F .We have tisi ∈ Aut�Bi�; hence we get Ai = Ker ti ⊕ Im si. It is easy to seethat ρi�Ker ti� ⊆ Ker ti, and hence U � � � � → Ker t1 → Ker t0 → F → 0is a subpresentation of (1). We have also ρi�Im si� ⊆ Im si−1, which showsthat N � � � → Im s1 → Im s0 → 0 → 0 is also a subpresentation of (1). So�1� = U

∐N and U ∼= �2�.

The following is another consequence of Proposition 2.7.

I-sequences 457

Proposition 2.9. Suppose that

E1� 0→ K′f ′→M ′

g′→V ′ → 0

t↓↑t1 s↓↑s1 r↓↑r1E2� 0→ K

f→Mg→V → 0

are two right I-sequences in mod� with E2 minimal. Let r ∈ �V ′� V � be asplit epimorphism. Then there exist maps r1� s� s1� t� t1 as shown in the abovediagram such that

ft = sf ′� gs = rg′� f ′t1 = s1f� g′s1 = r1g�(2.10)

and tt1� ss1� rr1 are all automorphisms. In particular s and t are split epimor-phisms.

Proof. The map r ∈ �V ′� V � gives rise to a morphism θ� �V ′�I → �V �I ,which is a split epimorphism in M mod�. By Proposition 2.7, we have acommutative diagram

�S1� 0→ �−�K′� �−�f′�→ �−�M ′� �−�g

′�→ �−� V ′� → �V ′�I → 0

β3↓↑α3 β2↓↑α2 β1↓↑α1 ↓θ�S2� 0→ �−�K� �−�f �→ �−�M� �−�g�→ �−� V � → �V �I → 0�

with S2 as a minimal projective presentation of �V �I , since E2 is a minimalI-sequence. Also the morphisms αi� βi� i = 1� 2� 3 shown in the diagramare such that βiαi� i = 1� 2� 3 are automorphisms. Now apply Yoneda’slemma 1.4 to the morphisms βiαi� i = 1� 2� 3 we get the proposed mapsr1� s� s1� t� t1 in the theorem.

Now Corollary 2.8 implies the following

Corollary 2.11. If E1 is a right I-sequence ending with V . Then E1∼=

E2∐N , where E2 is minimal right I-sequence and N is an I-sequence of the

form 0→ S→ S′ → 0→ 0.

Analogously to 2.9 for left I-sequences we have the following.

Proposition 2.12. Suppose that

E1� 0→ K′f ′→M ′

g′→V ′ → 0

r↑↓r1 s↑↓s1 t↑↓t1E2� 0→ K

f→Mg→V → 0

are two left I-sequences in mod� with E2 minimal. Let r ∈ �K�K′� be a splitmonomorphism. Then there exist maps r1� s� s1� t� t1 as shown in the abovediagram such that sf = f ′r� fr1 = s1f

′� gs1 = t1g′� tg = g′s, and t1t� s1s� r1r

are all automorphisms.

458 ahmed a. khammash

And just as in Corollary 2.11 this implies the following

Corollary 2.13. If E1 is any left I-sequence starting with K, then EI∼=

E1∐N , where E2 is minimal left I-sequence and N is an I-sequence of the

form 0→ 0→ T → T ′ → 0.

We now concentrate on the direct �-summands of the starting (ending)modules for a left (right) I-sequence.

Proposition 2.14. Let

0→ Kf→M

g→V → 0(2.15)

be a right I-sequence and let α� V → V1 be a split epimorphism, where V1 ∈mod�. Then

0→ kerαginc�→M

αg→V1 → 0(2.16)

is a right I-sequence.

Proof. By the hypothesis, there is a �-map µ� V1 → V such that αµ =1V1

. So if X ∈ mod� and u1 ∈ I�X�V1� then µu1 ∈ I�X�V � and since (2.15)is a right I-sequence there exists h ∈ �X�M� such that gh = µu1. Henceu1 = αµu1 = �αg�h. Conversely if u1 = �αg�h1, where h1 ∈ �X�E� then,since g ∈ I�M�V �, it follows that u1 ∈ I�X�V1�. This proves that (2.15) isa right I-sequence.

In a similar fashion to 2.14 we may prove

Proposition 2.17. Let 0 → Kf−→ M

g−→ V → 0 be a left I-sequenceand let β� K1 → K be a split monomorphism, where K1 ∈ mod�. Then

0→ K1fβ−→M−→Co ker fβ→ 0 is a left I-sequence.

Remark 2.18. Propositions 2.14 and 2.17 say that if there exists a right

(resp., left) I-sequence 0→ Kf−→ M

g−→ V → 0 ending (resp. starting)with V (resp. K), then there is a right (resp. left) I-sequence ending (resp.starting) with any given direct �-summand of V (resp. K).

We now look at the direct sum of left and right I-sequences.

Theorem 3. Let Ei� 0→ Ki

fi−→ Mi

gi−→ Vi → 0� i = 1� 2 be two shortexact sequences in mod�. Then

(1) E1 and E2 are right (left) I-sequences if and only if E1∐E2 is a

right (left) I-sequence.(2) E1 and E2 are minimal right (left) I-sequences if and only if E1

∐E2

is minimal right (left) I-sequence.

I-sequences 459

Proof. (1) Assume that E1 and E2 are right I-sequences. For i =1� 2, let πi� V1

∐V2 → Vi be the projection of V1

∐V2 onto Vi and let

µi� Vi → V1∐V2 be the inclusion map. Now let X ∈ mod�, and take

any u ∈ �X�V1∐V2� then u = 1V1

∐V2u = �µ1π1 + µ2π2�u = µ1u1 + µ2u2,

where ui = πiu� i = 1� 2. Hence, if u ∈ I�X�V1∐V2� then ui = πiu ∈

I�X�Vi�� i = 1� 2, and so ui = gihi for some hi ∈ �X�Mi�� i = 1� 2. It fol-lows that u = µ1g1h1 + µ2g2h2 = gh, where h ∈ �X�M1

∐M2� is defined

by h�x� = �h1�x�� h2�x�� and so u factors through g. Conversely if u = ghfor some h ∈ �X�M1

∐M2� then, u = �µ1g1π

′1+µ2g2π

′2�h ∈ I�X�V1

∐V2�,

where π ′i (resp. µ′i) is the projection (resp. injection) of M1∐M2 (resp. Mi)

onto Mi (resp. into M1∐M2); since gi ∈ I�Mi� Vi�� i = 1� 2.

(2) Suppose that E1 and E2 are minimal right I-sequences. We needto show that E = E1

∐E2 is minimal; that is, ker�−� g� ≤ ��−�M� or

equivalently, ker�X�g� ≤ ��X�M� ∀X ∈ mod�. Let λ ∈ ker�X�g� thengλ = 0. Define λi ∈ �X�Mi�� i = 1� 2 by λ�x� = �λ1�x�� λ2�x�� for all x ∈X. Then gλ = 0 implies that giλi = 0� i = 1� 2 and so λi ∈ ��X�Mi�� i =1� 2, since Ei is minimal. Therefore λ = 1M1

∐M2λ = �µ′1π ′1 + µ′2π

′2�λ =

µ′1π′1λ + µ′2π

′2λ ∈ ��X�M1

∐M2�. Conversely suppose that E = E1

∐E2

is minimal I-sequence. We must show that Ker� � g1� ≤ �� �M1�; i.e.,Ker�X�g1� ≤ ��X�M1� for all X ∈ mod�.

Suppose that λ1 ∈ Ker�X�g1�. Then taking

λ =(λ1

0

)∈ (

X�M1∐

M2)�

we have (g1 00 g2

)λ = 0

and so λ ∈ ��X�M1∐M2�, since E = E1

∐E2 is minimal I-sequence. But

this implies that λ1 ∈ ��X�M1�. Hence E1 is minimal I-sequence. Similarlyfor E2.

Corollary 2.19. If for each i = 1� 2 there is a right (left) I-sequenceending with Vi ∈ mod� (starting with Ki ∈ mod�), then there is a right (left)I-sequence ending with V1

∐V2 (starting with K1

∐K2).

Theorem 4. Let 0→ Kf−→ M

g−→ V → 0 be a right I-sequence. Thenthe following conditions on V are equivalent

(1) V ≡ 0 mod I; i.e., 1V ∈ I�V� V �.(2) �−� V � = I�−� V �.(3) 0→ �−�K� �−�f �→ �−�M� �−�g�→ �−� V � → 0 is exact inM mod�.

(4) 0→ Kf−→M

g−→ V → 0 is split in mod�.

460 ahmed a. khammash

Proof. �1� ⇒ �2� since for any θ ∈ �X�V �, we have θ = 1V θ ∈I�X�V �.�2� ⇒ �3� clear.

�3� ⇒ �4� From (3) the morphism �X�g� is epimorphism, for all X ∈mod�. Taking X = V , we get θ = 1V factors through g and so the sequence

0→ Kf−→M

g−→ V → 0 is split in mod�.

�4� ⇒ �1� If 0 → Kf−→ M

g−→ V → 0 is split then 1V = gg′, forsome g′ ∈ �V�M�. But since g ∈ I�M�V �� 1V = gg′ ∈ I�V� V �.

In a similar way we may prove the following

Theorem 5. Let 0 → Kf−→ M

g−→ V → 0 be a left I-sequence. Thefollowing conditions on K are equivalent

(1) K ≡ 0 �mod I�.(2) �K�−� = I�K�−�.(3) 0→ �V�−� �g�−�→�M�−� �f�−�→�K�−� → 0 is exact in M◦mod�.

(4) 0→ Kf−→M

g−→ V → 0 is split in mod�.

3. BILATERAL I-SEQUENCES: THE HELLER FUNCTOR 5I

A short exact sequence 0→ Kf−→M

g−→ V → 0 is mod� is said to bea bilateral (or two-sided) I-sequence if it is both a left and right I-sequence.

Proposition 3.1. If E� 0 → Kf−→ M

g−→ V → 0 is a two-sidedI-sequence in mod�, then any minimal right (left) I-sequence ending (start-ing) with V �K� is also two-sided.

Proof. This follows from 2.11, 2.13, and Theorem 3.

Remark. When we speak of a minimal bilateral I-sequence, we shallalways mean a bilateral I-sequence, which is minimal as a right I-sequence.

Definition 3.2. If 0 → Kf−→ M

g−→ V → 0 is a bilateral, minimalI-sequence, we write K = 51�V �, and note that, by 2.11, 5I�V � is uniquelydefined up to isomorphism. We call 5I a generalized Heller operator. Notethat if I = P (see Section 1, Example (2)) then 5I = 5, the usual Helleroperator (see [PL, p. 35]).

I-sequences 461

Our next aim is to show that 5I may be developed into a functor, whichis in fact an equivalence between certain categories RmodI ��LmodI �,which will be defined later (Definition 3.5). To do this, we need the nextproposition.

Proposition 3.3. Let E and E′ be bilateral I-sequences in mod�, asshown.

E� 0→ Kf→M

g→V → 0

z↓ y↓ ↓xE′� 0→ K′

f ′→M ′g′→V ′ → 0�

(3.4)

Then (i) For each x ∈ �V� V ′� there exist y ∈ �M�M ′� and z ∈ �K�K′� suchthat the diagram (3.4) commutes. Conversely for each z ∈ �K�K′� there existy ∈ �M�M ′� and x ∈ �V� V ′� such that the diagram (3.4) commutes.

(ii) Given any commuting diagram (3.4), then x ∈ I�V� V ′� if and onlyif z ∈ I�K�K′�.

(iii) Say that a pair �x� z� ∈ �V� V ′� × �K�K′� is compatible, if thereexists some y ∈ �M�M ′� such that (3.4) commutes. Then there is a k-linearisomorphism

�3�5� ωI � �V� V ′�/I�V� V ′� → �K�K′�/I�K�K′�

defined by the rule ωI�x + I�V� V ′�� = z + I�K�K′� if and only if the pair�x� z� is compatible.

Proof. (i) Given x ∈ �V� V ′�, the existence of y� z making (3.4) com-mute follows as in the proof of Proposition 2.9 (take r = x in that proof;the existence of s �=y� and t �=z� does not depend on the hypothesis thatr is a split epimorphism). Conversely if z is given, the existence of y� xmaking (3.4) commute follows as in the proof of 2.12.

(ii) We now assume that x� y� z are any �-homomorphisms such that(3.4) commutes. Suppose that x ∈ I�V� V ′�. Since E′ is a right I-sequence,there exists some a ∈ �V�M ′� such that g′a = x. Then g′�y − ag� = g′y −xg = 0. Therefore Im�y − ag� ⊆ ker g′ = Im f ′. Define b ∈ �M�K′� to bethe composite of y − ag with the inverse of the bijective map f ′� K′ → Im f ′

(which is legitimate because Im�y − ag� ⊆ Im f ′). We have now f ′b = y −ag; i.e., y = ag + f ′b. Therefore f ′z = yf = agf + f ′bf = f ′bf , and sincef ′ is a monomorphism, f ′z = f ′bf implies z = bf . But since f ∈ I�K�M�(Lemma 2.5 (i)), this shows that z ∈ I�K�K′�. An exactly similar argumentshows that z ∈ I�K�K′� implies x ∈ I�V� V ′�.

462 ahmed a. khammash

(iii) By (i) and (ii), ωI�x + I�V� V ′�� is defined uniquely for everyx ∈ I�V� V ′�. The map ωI so defined is monomorphic, again by (ii). It is sur-jective, since (i) shows that for every z there exists x such that the pair �x� z�is compatible. Finally ωI is linear, because if, for each i = 1� 2� xi� yi� zi aresuch that xig = g′yi and yif = f ′zi, then clearly for any λ1� λ2 ∈ k� �λ1x1 +λ2x2�g = g′�λ1y1+λ2y2�, and �λ1y1+λ2y2�f = f ′�λ1z1+λ2z2�. This showsthat ��λ1x1 + λ2x2�� �λ1z1 + λ2z2�� is a compatible pair, whenever �x1� z1�and �x2� z2� are both compatible. Therefore (using notation such as x1 =x1 + I�V� V ′�), we get ωI�λ1x1 + λ2x2� = λ1ωI�x1� + λ2ωI�x2�.

Given any ideal I in the category mod�, we construct the quotient cate-gory modI � as follows: The objects of modI � are the same as the objectsof mod�, but the morphism set between given objects X�X ′ is defined tobe the k-space

MormodI����X�X ′� = �X�X ′�/I�X�X ′��

Thus to every map f ∈ �X�X ′� in mod� corresponds a morphism f =f + I�X�X ′� in the category modI �. The product of morphisms in modI �is given by the rule �g + I�X ′�X ′′���f + I�X�X ′�� = gf + I�X�X ′′�, forf ∈ �X�X ′�� g ∈ �X ′�X ′′�. The identity morphism for the object X is 1X +I�X�X�; this is the identity element of the k-algebra �X�X�/I�X�X�.Definition 3.5. Let Rmod� �Lmod�� be the full subcategory of

mod�, whose objects are all V (all K), which end (start) some bilateralI-sequence 0→ K→M → V → 0 in mod�. Let RmodI����LmodI����be the corresponding subcategory of modI �.

Theorem 6. There is a k-additive functor 5I � RmodI � → LmodI �,which takes each V ∈ RmodI � to K = 5I�V � and maps �V� V ′�/I�V� V ′� →�K�K′�/I�K�K′� according to the map ωI of (3.5). This functor is an equiv-alence of categories.

Definition. 5I is called the Heller functor for the ideal I of mod�.

Proof of Theorem 6. This follows from Proposition 3.3 (for the defini-tion of k-additive functor, see, e.g., [JG2, p. 373, footnote]). The fact thatthe functor 5I is an equivalence follows because (i) the map ωI is bijec-tive for each pair V� V ′ in RmodI �, and (ii) 5I is dense; i.e., each objectK in LmodI � is isomorphic to 5I�V � for some V ∈ RmodI � (see [JG2,p. 374]).

Corollary 3.6. A module V ∈ Rmod� is indecomposable and not inI, if and only if K = 5I�V � is indecomposable and not in I.

I-sequences 463

Proof. For any object X ∈ mod��X is indecomposable and not inI, if and only if EndmodI ��X� = �X�X�/I�X�X� is a local algebra, i.e.,is nonzero and has no nonzero idempotent except the identity. Nowlet V ∈ Rmod� and K = 5I�V �. From Theorem 6 (or direct fromProposition 3.3) we know that EndRmodI ��V � = �V� V �/I�V� V � andEndLmodI ��K� = �K�K�/I�K�K� are isomorphic. So if one of thesealgebras is local, then so is the other.

The following characterization for the minimal bilateral I-sequences fol-lows from 2.7 and 3.6.

Proposition 3.7. Any two minimal bilateral I-sequences ending with V ∈indec� are isomorphic as short exact sequences.

Finally Theorem 3 implies the following

Corollary 3.8. The functor 5I is additive.

4. RELATIVE PROJECTIVE COVER

Now we take � = kG, the group algebra of a finite group G over k andK given set of subgroups of G. In this section we shall use the results ofSections 2 and 3 to prove the existence of a minimal relative projectivecover for any object in modkG. If X�Y ∈ modkG and H ≤ G we write�X�Y �H for �X�Y �kH . If H ≤ K ≤ G, let tKH � �X�Y �H → �X�Y �K be thetrace map given by tKH�f � =

∑s∈≺K/H� sfs−1, where ≺ K/H � denotes a left

transversal of H in K and sfs−1� x �→ sf �s−1x� ∀x ∈ X (see [PL, Chap. II]).

Definition. Write PH�X�Y � = Im tGH and let PK�X�Y � =∑

H∈K ×PH�X�Y �. It is clear that PK�X�Y � is a k-subspace of �X�Y �G. Thefollowing shows that PK �modkG.

Proposition 4.1. Suppose that X�Y�Z ∈ modkG, f ∈ �X�Y �G, andg ∈ �Y�Z�G. Then gf ∈ PK�X�Z� if either f ∈ PK�X�Y � or g ∈ PK�Y�Z�.That is, PK �modkG.

Proof. If f ∈ PK�X�Y � then f = ∑H∈K fH and fH ∈ PH�X�Y �� ∀H ∈

K, and so gf = ∑H∈K gfH . But gfH ∈ PH�X�Y � since PH � modkG.

Therefore gf = ∑H∈K gfH ∈ PK�X�Y �. Similarly we can prove that gf ∈

PK�X�Z� if g ∈ �Y�Z�G.

Definition. V ∈ modkG is said to be K-projective if 1V ∈ PK�V� V �.Since PK�V� V � � EkG�V �; it follows that V is K-projective if and only ifPK�V� V � = EkG�V �.

464 ahmed a. khammash

Definition. f ∈ �X�Y �G is said to be K-split epimorphism if for eachH ∈ K there is δH ∈ �Y�X�H such that fδH = 1Y and K-split monomor-phism if for each H ∈ K there is θH ∈ �Y�X�H such that θHf = 1X .

Examples. Suppose that V ∈ indeckG. For each subgroup H of G wehave the following short exact sequence in modkG

0→ KerβH

inc�→�V↓H�↑GβH→V → 0�(4.2)

where βH�∑

i ti ⊗ vi� =∑

i tivi. Let δ� V → �V↓H�↑G be the kH-mapdefined by δ�v� = 1 ⊗ v. Then it is clear that βHδ = 1V ; therefore thesequence (4.2) is H-split in all cases. Similarly for an kG-module X wehave a short exact sequence

0→ XαH�→�X↓H�↑G →Co kerαH → 0�(4.3)

where αH�x� =∑

s∈G\H s ⊗ s−1x for all x ∈ X. It is clear that αH is akG-map and that µαH = 1X , where µ� �X↓H�↑G → X is the kH-mapdefined as µ�∑s s ⊗ xs� = x1. This means that Lemma (4.4) H-splits andthat α is an H-split monomorphism. The following shows when (4.2) and(4.3) kG-split.

Lemma 4.4. (1) (4.2) splits if and only if V is H-projective.(2) (4.3) splits if and only if X is H-projective.

Proof. (1) Suppose that (4.2) splits. Then V is a kG-summand of�V↓H�↑G and so V is H-projective. Conversely if V is H-projective thenby the Higman criterion ∃η ∈ EndkH�V �� tGH�η� = 1V and so

1M = tGH�η� = tGH�βHδη� = βHtGH�δη��

Since tGH�δη� is a kG-map, it follows that βH (hence the sequence (4.2))splits.

(2) is similar

Lemma 4.5. If H ∈ K and V ∈ modkG is H-projective, then V isK-projective.

Proof. It is clear that PH�X�V � ≤ PK�X�V �� ∀X�V ∈ modRG. There-fore V is H-projective ⇔ 1V ∈ PH�V� V � ⇒ 1V ∈ PK�V� V �; hence V isK-projective.

Definition 4.6. A s.e.s. (1) 0 → Kf−→ M

g−→ V → 0 in modkG iscalled a K-projective s.e.s. if

(a) M is K-projective, and(b) For any H ∈ K, (1) is split, when regarded as a s.e.s. in modkH.

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If (1) is a K-projective s.e.s, we say that (1) is a K-projective resolution ofV , or a K-projective co-resolution of K.

Note that (b) in the previous definition is equivalent to either of theconditions

(b′) f is a K-split monomorphism, or(b′′) g is a K-split epimorphism.

Examples 4.7. It follows from 4.4 that the sequence

S�K�V �� 0→ kerβ→ ∐H∈K�V↓H�↑G

β=�βH�����→V → 0(4.8)

is a K-projective resolution of V . Since each βH is H-split, it follows thatβ is K-split. Similarly for any kG-module X we have that a sequence

S�X�K�� 0→ Xα=�αH�����→ ∐

H∈K�X↓H�↑G→ Co kerα→ 0(4.9)

is a K-projective co-resolution of X. These examples show that every kG-module can appear as the start, and also as the end, of a K-projective s.e.s.We also have the following, which follows from Lemma 4.4.

Lemma 4.10. The sequence S�K�V � [resp. S�X�K�] splits if and only ifV [resp. X] is K-projective kG-module.

As a consequence of Lemma 4.10 we have

Lemma 4.11. Let M ∈ modkG. Then M is K-projective if and onlyif, for each H ∈ K, there exists a module L�H� ∈ modkH such thatM!∑⊕H∈K L�H�↑G.

Now Lemma 4.10 implies the following

Corollary. V ∈ modkG is K-projective if and only if V has a decom-position V =∑⊕

i Vi such that Vi is H-projective for some H ∈ K.

Next we give the following connection between K-projective s.e.s. andbilateral PK-sequences:

Proposition 4.12. Let (1) 0 → Kf−→ M

g−→ V → 0 be a s.e.s. inmodkG. Then (1) is a K-projective s.e.s. if and only if it is a bilateral PK-sequence.

Proof. �⇒� Assume that (1) is a K-projective s.e.s. We shall provefirst that (1) is a right PK-sequence. Take any X ∈ modkG, and any θ ∈�X�V �G. If θ ∈ PK�X�V � =

∑H∈K PH�X�V �, then for each H ∈ K there

is some σH ∈ �X�V �H , such that

θ = ∑H∈K

tGH�σH��(4.13)

466 ahmed a. khammash

By Definition 4.6, g is K-split; hence for all H ∈ K there existsδH ∈ �V�M�H such that gδH = 1V . Therefore σH = gδHσH , whichimplies tGH�σH� = gtGH�δHσH�. Putting this into (4.13), we get θ = gλ,where λ =∑

H∈K tGH�δHσH� ∈ �X�M�G. Therefore θ factors through g.

Conversely if θ ∈ �X�V �G factors through g, there is some µ ∈ �X�M�Gsuch that θ = gµ. But by Definition 4.6, M is K-projective; hence1M ∈ PK�M�M�. Therefore g = g1M ∈ PK�M�V �, and since PK is anideal in modkG� θ = gµ ∈ PK�X�V �. This proves that (1) is a rightPK-sequence. But since Definition 4.6 is right–left symmetric, (1) is also aleft PK-sequence; hence (1) is a bilateral PK-sequence.�⇐� Now assume that (1) is a bilateral PK-sequence. By Corol-

lary 2.11, we have �1� ∼= E ⊕ N , where E� 0 → K2 → M2 → V → 0 is aminimal PK-sequence ending with V , and N is a right I-sequence of theform 0 → S → S′ → 0 → 0. But since (1) is also a left I-sequence, bothE and N are also left I-sequences (Theorem 3(2)). Now Lemma 2.5 (iii)shows that S and S′ belong to PK; i.e., they are K-projective kG-modules.Next apply Corollary 2.8, with I = PK , in the case E1 = �4�8� (which isa K-resolution of V ; hence a PK-sequence, by what we proved above),E2 = E and r = 1V . Then Corollary 2.8 gives us a commutative diagram,which may be written

S�K�V �� 0→ kerβ→ ∐H∈K�V↓H�↑G

β=�βH�����→V → 0

p↑↓ ∥∥E 0→ K2 → M2

g2 → V → 0�

where sp is an automorphism of M2. But since∑

H∈K�VH�G is K-projective,then sp ∈ PK�M2�M2�, and this shows that M2 is K-projective. Also themap g2 in E is K-split, because for each H ∈ K we have a kH-mapδH � V →

∑H∈K�VH�G such that βHδH = 1V ; hence sδH � V → M2 is a

kH-map such that g2�sδH� = βHδH = 1V . We have now proved that Eis a K-projective s.e.s. But also the s.e.s. N� 0 → S → S′ → 0 → 0 isK-projective, since S′ is a K-projective module, and of course N is spliteven as a kG-sequence. The direct sum of two K-projective s.e.s is clearlyK-projective. This shows that �1� = E

∐N , N is K-projective.

Now in the light of Proposition 4.12, the previous results of Sections 2and 3 on I-sequences provide an alternative proof of the following theoremwhich was originally proved by Knorr [K].

Theorem 8. (1) For any kG-module V there exists a minimal K-projective resolution E�V�K�� 0→ 5K�V � →M → V → 0, where 5K�V � =5PK�V �.

I-sequences 467

(2) S�V�K� ∼= E�V�K�∐�0→ P1 → P2 → 0→ 0�.(3) 5K�V � has no nonzero K-projective summands.

(4) If V is indecomposable not K-projective then 5K�V � is indecom-posable not K-projective.

(5) Any two minimal K-projective resolutions of V are isomorphic.

Proof. (1) follows from Theorem 1 and Proposition 4.12, since (4.8)ensures the existence of a K-projective presentation S�V�K� of V ; (2) fol-lows from (2.10) and 2.12, (3) follows from 3.6 and the fact that 5I�W � = 0for any K-projective W ∈ modkG, (4) follows from Proposition 3.6, and(5) follows from Proposition 2.14.

Remark. Notice that when I = PK , then Rmod� = Lmod� = mod�(see Examples 4.7); hence RmodI � = LmodI � = modI ?. So the Hellerfunctor 5PK

is an equivalence of modPK kG with itself, this fact was firstdiscovered by R. Knorr [K].

Definition. If E�V�K�� 0 → 5PK�V � → M → V → 0 is a minimal

K-projective resolution we call M the K-projective cover of V .

Note that in light of Theorem (2), the K-projective cover of V is a directsummand of

∑H∈K�V↓H�↑G�

The following gives a homological characterization for the space PK�X�Y �.Theorem 9. If X�Y ∈ modkG and f ∈ �X�Y �G, then the following are

equivalent

(1) f ∈ PK�X�Y �.(2) If v ∈ �Y ′� Y �G is a K-split epimorphism then f factors through v.

(3) If u ∈ �X�X ′�G is a K-split monomorphism then f factorsthrough u.

(4) f factors through some K-projective kG-module.

Proof. �1� ⇒ �2� Suppose that f ∈ PK�X�Y �. Then there is asequence �σH�H∈K; σH ∈ �X�Y �H such that f = ∑

H∈K tGH�σH�. Now

if v ∈ �Y ′� Y �G is a K-split epimorphism then for all H ∈ K there is aδH ∈ �Y�Y ′�H such that νδH = 1Y . Therefore

f = ∑H∈K

tGH�νδHσH� = ν∑

tGH�δHσH� = νh�

where h =∑H∈K t

GH�δHσH� ∈ �X�Y ′�, and so f factors through ν.

468 ahmed a. khammash

�2� ⇒ �4� Assume (2) and take V = Y and ν = β defined in 4.7,which is a K-split epimorphism. Therefore f factors through β; hence thereis β′ ∈ �X�∐H∈K�Y!H�↑G� such that f = ββ′. Since

∐H∈K�Y!H�↑G is K-

projective we get (4).�4� ⇒ �1� Assume that M is a K-projective kG-module and that f =

gh, where h ∈ �X�M�G and g ∈ �M�Y �G. Since M is K-projective, 1M ∈PK�M�M� and so 1M =

∑H∈K t

GH�ηH� for some ηH ∈ �M�M�H . Therefore

f = gh = g1Mh =∑

H∈K tGH�gηHh� ∈ PK�X�Y �.

In a similar fashion we can show that �1� ⇒ �3� ⇒ �4� ⇒ �1� and so�1� ⇔ �3�. It remains to show that �3� ⇒ �4�, so assume that f factorsthrough any K-split monomorphism in �X ′�X�. Then f factors through themap α in (4.8) and hence factors through the kG-module

∐H∈K�X!H�↑G,

which is K-projective and so f ∈ PK�X�Y �.Lemma. If U or V is K-projective then PK�U�V � = �U�V �.Proof. If f ∈ �U�V �, then f = f1U = 1V f and so f factors through a

K-projective RG-module if either U or V is K-projective.

Green [JG1] introduced the notion of relative projectivity in the cate-gories mmodkG, m◦modkG as well as vertices and sources of the inde-composable objects in these categories. For the rest of this section, wewould like to compare the vertices of the indecomposable kG-modulesK�V forming the two ends of a minimal I-sequence �I �modkG� with thevertices of the quotient functors �V �I = � � V �/I� � V � ∈ mmodkG andI�K� = �K� �/I�K� � ∈ m◦modkG. The following is an analogue to [JG1,5.12] for I-sequences.

Theorem 10. If E� 0 → Kf−→ M

g−→ V → 0 is a bilateral minimalI-sequence with V indecomposable and V #≡ 0 mod I, K #≡ mod I, then

(1) vertex�V � ≤ vertex��V �I�(2) vertex�V � ≤ vertex�I�K��.

Proof. (1) Follows by applying [JG1, 8.11] to the minimal projectivepresentation

0→ �−�K� �−�f �→ �−�M� �−�g�→ �−� V � → �V �I → 0

of the object �V �I in mmod�. Similarly we may deduce (2) by consideringthe minimal projective presentation

0→ �V�−� �g�−�→�M�−� �f�−�→�K�−� →I �K� → 0

in m◦mod�.

I-sequences 469

Remark. If V ∈ indeckG then, taking I = �, we have �V �� = SV .Hence Theorem 10 implies the fact proved in [JG1, 5.12] that if

0→ Kf→M

g→V → 0

is an almost split sequence then vx�V �≤Gvx�SV �.

APPENDIX: CONNECTIONS WITH VERTEX THEORY

Let I �modkG, and let D be a subgroup of G. Let mod�kG�D be thefull subcategory of modkD, whose objects are those kD-modules which arethe restrictions MD to kD of kG-modules M . In other words, the objectsM�M ′�M ′′� � � � � of mod�kG�D are the same as those of modkG, but themorphism set �M�M ′� in mod�kG�D is �M�M ′�kD = HomkD�M�M ′�. Since�M�M ′�kG is a subset of �M�M ′�kD, then I�M�M ′� ⊆ �M�M ′�kD, for allM�M ′ ∈ mod�kG�D. Define ID to be the ideal of mod�kG�D, which isgenerated by I. Equivalently, for given kG-modules M�M ′.

Definition A.1. ID�M�M ′� �= the k-subspace of �M�M ′�kD =HomkD�M�M ′� spanned by all products of the form βα, where forsome N ∈ modkG there holds α ∈ �M�N�kD�β ∈ �N�M�kD and at leastone of β�α lies in I.

Definition A.2. Let M�V ∈ modkG and let σ ∈ �M�V �kD. DefineσG ∈ �MG

D� V �kG by the rule σG�∑u∈$G/D% u ⊗ mu� =∑

u∈$G/D% u�σmu�.Here $G/D% is a transversal of the set of cosets uD in G, and mu ∈M .

We shall consider from now on only those ideals I of modkG whichsatisfy the following

Hypothesis A.3. For all subgroups D of G there holds �ID�G ⊆ I; i.e.,for all M�V ∈ modkG and all σ ∈ �M�V �kD we have

σ ∈ ID�M�V � ⇒ σG ∈ I�MGD� V ��

Theorem A.4. Suppose that I � modkG satisfies Hypothesis A.3, andthat V ∈ RmodkG (see Definition 3.5), V indecomposable and V does notlie in I (i.e., 1V #∈ I�V� V �kG = IEndkG�V �). Then V and 5I�V � have thesame vertices.

Proof. Let E� 0 → Kf−→M

g−→V → 0 be a minimal, bilateral I-sequence. Then K = 5I�V �, by Definition 3.2. To prove that V and 5I�V �have the same vertices, it will be enough to prove that, for any subgroupD of G,

V is D-projective if and only if K is D-projective.(A.5)

470 ahmed a. khammash

Suppose that V is D-projective. Then there exists some ξ ∈ �V� V �kD suchthat 1V = tGD �ξ�. We shall prove that there exist kD-maps η� ζ such thatthe diagram

0→ Kf−→M

g−→ V → 0ζ ↓ η ↓ ξ ↓

0→ Kf−→M

g−→ V → 0

(A.6)

commutes. The existence of η follows from Hypothesis A.3. For since g ∈I�M�V � (by 2.5(i)), the kD-map σ = ξg ∈ ID�M�V �. Therefore Hypoth-esis A.3 says that σG ∈ I�MG

D� V �. Because E is a right I-sequence, thereexists a kG-map φ� MG

D → M such that gφ = σG. Define η� M → Mby setting η�m� = φ�1 ⊗m� for all m ∈ M . We verify easily that η is akD-map and that gη = σ = ξg.

The existence of ζ is now proved by a familiar argument: let p ∈ K,then gηf �p� = ξgf �p� = 0, since gf = 0. Hence ηf �p� ∈ Kerg = Imf ;therefore there exists a unique element q ∈ K such that f �q� = ηf �p�.Define ζ by setting ζ�p� = q, for all p ∈ K. It is clear that ζ is a kD-mapand that fζ = ηf .

Now apply the operator t = tGD to the equations gη = ξg and fζ = ηf .Since g� f are both kG-maps, we get gt�η� = t�ξ�g and ft�ζ� = t�η�f . Byassumption t�ξ� = 1V . Also y = t�η�� z = t�ζ� are kG-maps. Thus we havea commutative diagram such as (3.4), with E = E′ and x = 1V . This showsthat the map

ωI � �V� V �/I�V� V � → �K�K�/I�K�K�(A.7)

(see (3.5)), which in the present case is an isomorphism of k-algebras,taking 1V + I�V� V � to z + I�K�K�. However, 1V /∈ I�V� V � by assump-tion; therefore 1V + I�V� V � is the identity element of the algebraEnd�V �/IEnd�K�. So z + I�K�K� is the identity element 1K + I�K�K� ofthe algebra End�K�/IEnd�K�. Now z = tGD �ζ� ∈ PD�K�K� = PDEnd�K�.Therefore 1K , which is a primitive idempotent of End�K� because K isindecomposable (see 3.6), lies in the sum of the ideals PDEnd�K� andIEnd�K� of End�K�. We know that 1K /∈ IEnd�K�, and it follows that1K ∈ PDEnd�K�, and this implies that K is D-projective.

The converse implication, namely K is D-projective ⇒ V is D-projective,follows by a similar argument (all our hypothesis are left–right symmetric).This proves Theorem A.4.

A corollary to Theorem A.4 is Knorr’s theorem.

Theorem A.8 ([K]). Using the notation of Section 4, if V is indecompos-able and not K-projective, then V and 5PK

�V � have the same vertices.

I-sequences 471

This will follow from Theorem A.4 as soon as we have

Proposition A.9. For any set K of subgroups of G, the ideal I = PK �modkG satisfies Hypothesis (A.3).

Proof. Let D be a subgroup of G, let M�V ∈ modkG, and let σ ∈ID�M�V �. We must prove that σG ∈ I�MG

D� V �. By Definition A.1, it willbe enough to do this when σ = βα with α ∈ I�M�N�� β ∈ �N�V �kD forsome N ∈ modkG.

If σ is any element of �M�V �kD, define σ1 ∈ �MGD� V �kD by the rule

σ1�∑

u u⊗mu� = σ�m1�. It is then easily verified that

�i� σG = tGD �σ1� and �ii� if λ ∈ �M�V �kL(A.10)

for some subgroup L of D, then �tDL ��1�

Since α ∈ I�M�N� = PK�M�N� there exists for each H ∈ K a mapθ�H� ∈ �M�N�kH , such that α = ∑

H tGH�θ�H��. If we regard α as a kD-map, then Mackey’s formula (see[PL, Theorem 1.7]) gives

α =∑H

∑w

tDD∩wH�wθ�H���(A.11)

Here the second sum is over a transversal $D\G/H% of the double cosetsDwH in G. Let L denote the set of all subgroups D ∩w H �H ∈ K�w ∈ G�of D. Then (A.11) shows that α, regarded as kD-map, lies in PL�M�N�kD.Therefore σ = βα lies in PL�M�V �kD. But then (A.10) shows thatσG = tGD �σ1� lies in PK�M�N�, because every element D ∩w H of Llies in a G-conjugate of a subgroup of an element of K. This provesProposition A.9.

As a consequence of Theorem A.8 one can deduce the following knownproperty of almost split sequences.

Proposition A.12. If 0 → Kf−→ M

g−→ V → 0 is an almost splitsequence, where V ∈ indeckG is nonprojective, then vertex�V � = vertex�K�.

Proof. It is known (see [PL, Chap. II, Sect. 9]) that K = 52�V �, where5 = 5PK

is the usual Heller operator with K as the collection of subgroupsof G consisting of the trivial subgroup. Now from Theorem A.8 we have

vertex�K� = vertex�52�V �� = vertex�5�V �� = vertex�V ��

This proves Proposition A.12.

472 ahmed a. khammash

ACKNOWLEDGMENT

Since writing this paper, I have learned that J. A. Green knew (but did not publish) thatthe I-sequence theory could be applied to get many of the Knorr results on relative projectivecovers. I thank Professor Green for many useful comments on the subject of this paper.

REFERENCES

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[AR2] M. Auslander and I. Reiten, Representation theory of Artin algebras II, Comm.Algebra 1, No. 4 (1974), 269–310.

[AR3] M. Auslander and I. Reiten, Representation theory of Artin algebras III, Comm.Algebra 3 (1977), 239–294.

[PG] P. Gabriel, Auslander–Reiten sequences and representation-finite algebras, in “Rep-resentation Theory I,” Lecture Notes in Mathematics, Vol. 831, Springer-Verlag,Berlin/New York, 1980.

[JG1] J. A. Green, Functors on categories of finite group representations, J. Pure Appl.Algebra 37 (1985), 265–298.

[JG2] J. A. Green, Relative module categories for finite groups, J. Pure Appl. Algebra 2(1972), 371–393.

[K] R. Knorr, Relative projective covers, in “Proc. of Symp. Mod. Representations ofFinite Groups,” Arhus University, 1978; Publication Series No. 24, Mathematics Insti-tute, Arhus University.

[PL] P. Landrock, “Finite Group Algebras and Their Modules,” LMS Lecture Notes Series84, Cambridge Univ. Press, Cambridge, UK, 1983.