I-Sequences in the Category of Modules

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: [email protected] Communicated by Michel Brou´e Received February 28, 2001

The notion of I-sequences (introduced by Green) in the category of modules over finite-dimensional algebras is developed in this paper and particularly used to give an alternative proof for the results of Kn¨ orr concerning the existence and properties 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 homological algebra can be powerful tools in representation theory of algebras (see, for instance, [AR1, AR2, AR3]). This fact is also emphasized by the role played by some other similar sequences such as the relative projective sequences in 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). In particular this notion generalizes the notion of almost split (or Auslander– Reiten) sequences. It turns out that most results that have been proved for almost split sequences can be proved in a slight generalization for I-sequences once they exist. In fact the map which assigns each end of a minimal I-sequence to the start of that I-sequence turns out to be an equivalence of certain categories (Theorem 6) generalizing the known Heller operator and Auslander–Reiten translate for group algebra modules. As an application (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 Kn¨ orr [K]. Finally the Appendix deals with ideals in the category of group algebras. It is 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 vertices (Theorem A.4). This generalizes Knorr’s theorem (Theorem A.8) on relative 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 by mod  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  → mod k (see [JG1, p. 265]), where mod k is the category of k-vector spaces and the morphisms between objects in M mod  are the natural transformations. We also denote by indec  the full subcategory of mod  whose objects are all indecomposable -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 MorF G to denote the set of all morphisms between F and G. If M ∈ mod , then the hom functor   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   gX = X g for all X ∈ mod . An object F ∈ M mod  is said to be finitely generated if F is a homomorphic image of   M for some M ∈ mod  and is finitely presented if there is an exact sequence in M mod  (1.1)

  N →   M → F → 0

and we say that (1.1) is a finite presentation of F. If F G ∈ M mod , we shall use the notation F ≤ G to denote F as a subfunctor of G. We use m mod  to denote the full subcategory of M mod  whose objects are all finitely presented functors. Definition 1.2. If for all X Y ∈ mod  IX Y  is a k-subspace of X Y  such that IY ZX Y  ⊆ IX Z

and

Y Z IX Y  ⊆ IX Z

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

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Examples. (1) Let
X Y  = f ∈ X Y   fg ∈ JEY  ∀ g ∈ Y X , where EY  = End Y  and J is the Jacobson radical. Then it is known that
Y ZX 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

PX Y  = f ∈ X Y   f factors through a projective -module Then PY ZX Y  ⊆ PX Z and Y Z PX Y  ⊆ PX Z and hence P mod . Let
− Y  ∈ M mod  be the object defined by: X →
X Y  for all X ∈ mod . Then
− Y  is a (the radical) subfunctor of − Y . The functor
  V  ∈ M mod  is the intersection of all the maximal subfunctors of   V  ([PG, p. 2]). A proof of this important fact is sketched in [JG1, Appendix]. This subfunctor shares many properties with the radical of modules such as the following, which will be needed later. (1.3)

If L is a subfunctor of − Y  satisfying L +
− Y  = − Y  then L = − Y 

Yoneda’s Lemma 1.4. For every M ∈ mod  F ∈ m mod , the map α → αM1M  defines a k-isomorphism between − M Fm mod  and FM. 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   in M mod  has the form α =   a for some a ∈ M M   (see [ JG1, pp. 267 and 268, Prop. 1.3]). The following consequence of Lemma 1.4 will be needed. Lemma 1.5.   Y .

If α ∈ End  Y  is surjective, then it is an automorphism of

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

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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 Isequences as well as the main results of this section are taken from an unpublished work of J. A. Green, and I am grateful to him for allowing me to include it in this paper. The aim is to apply this theory in Sections 3 and 4 to give an alternative proof to that given by Kn¨ orr [K] for the existence and properties of the projective cover of a kG-module relative to a collection of subgroups of G. Suppose that 1

0→K

f

→M

g

→V → 0

is a short exact sequence in mod . Definition 2.1. (Green). The sequence (2.1) is said to be right Isequence if for all X ∈ mod  and θ ∈ X V   θ factors through g if and only if θ ∈ IX V ; that is, if and only if ImX g = IX V  ∀ X ∈ mod . This means that (1) is right I-sequence if and only if the sequence 0 → X K

Xf 

→ X M

Xg

→ X V 

→ X V /IX V  → 0

is exact for all X ∈ mod , where X gf  = gf ∀ f ∈ X M. Equivalently, (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 functors 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] proved that if V ∈ mod  is nonprojective indecomposable (hence V ∈ /
) then there 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 a projective presentation of V . Definition 2.3. The map g in (1) is said to be right minimal if KerX g ≤
X M for all X ∈ mod , in which case (1) is said to be a minimal 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). The existence of such a sequence ensures existence of a minimal sequence as the following theorem proves.

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

g

Proof. Suppose that 0 → K →M → 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 θ

  M1  and a minimal projective presentation   M1  −→ I  V  → 0 in M mod ; hence Ker θ ≤
  M1 . But we also have the inclusion mori

phism I  V  −→   V . If we let η = ioθ ∈ Mor  M1    V , then Ker η = Ker θ and Im η = Im θ. Moreover η has the form   g1 , where g1 ∈ M1  V . Now Im  g1  = Im θ = I  V ; hence ImX g1  = IX V  for all X ∈ mod . By taking X = M, we get ImM g1  = IM V . But since g ∈ IM V , there exists h ∈ M M1  such that M g1 h = g, so g1 h = g. Since g is an epimorphism, g1 is also an epimorphism and so we have a short exact sequence 3

0 → K1

f1

→ M1

g1

→V → 0

in mod , which is a right I-sequence since Im  g1  = I  V . Moreover Ker  g1  = Ker η = Ker θ ≤
  M1 ; therefore (2) is a minimal I-sequence. Analogously we can define the left I-sequence in the category mod  as follows. Definition 2.4. (Green). The short exact sequence (1) is said to be a left I-sequence if for all X ∈ mod  and α ∈ K X  α factors through f if and only if α ∈ IK X; that is, if and only if Imf X = IK X ∀ X ∈ mod . The map f in (1) is called left minimal if Kerf X ≤
M X for all 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

gX

→ M X

fX

→ K X → K X/IK 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 /IK  → 0

I-sequences

455

is exact in M ◦ mod . Using the same idea as in the proof of Theorem 1 we can deduce the following Theorem 2. Suppose that I mod  and K ∈ mod . If there exists a left I-sequence beginning with K, then there exists a minimal left I-sequence beginning with K. f

g

Lemma 2.5. Let (1) 0 → K −→ M −→ V → 0 be a s.e.s. in mod . Then (i)

If (1) is right (left) I-sequence, then g ∈ IM V f ∈ IK 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 and M 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 an exercise 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 an isomorphism. Proof. (1) Let g ∈ X Y  be right minimal and take a ∈ X X with ga = g. Then − g− a = − g and so − gIm− a = Im− g, which implies that Im− a + Ker− g = − X. Since g is minimal we then have Im− a +
− g = − X and so, by Definition 1.2, Im− a = − X. In particular ImX a = X X; i.e., aX 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 be split 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 if there exists a -map η Y → X such that ηθ = 1X . The notions of split epimorphism and split monomorphism in the category M mod  are defined similarly. fi

gi

Given two s.e.s Ei  0 → Xi −→ Yi −→ Zi → 0 i = 1 2 in mod . Define their direct sum



f g → Y1 Y2 → Z1 Z2 → 0 E = E1 E2  0 → X1 X2

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ahmed a. khammash

where f x1  x2  = f1 x1  f2 x2  and gy1  y2  = g1 y1  g1 y2 . It is clear that E is exact. We shall see later that every I-sequence contains a minimal sequence as a direct summand. To reach that end we need the following Proposition 2.7. 1 2

Let F G ∈ M mod  and suppose that  

ρ2

→ A1

d2

t1 ↓↑s1

→ B1

ρ1

→ A0

d1

t0 ↓↑s0

→ B0

ε

→F → 0 θ↓

e

→G → 0

are two projective presentations of F and G, respectively, with (2) minimal; i.e., Kerdi  ≤
Bi , for all i. If θ ∈ F G is a split epimorphism, then there are morphisms ti ∈ Ai  Bi  si ∈ Bi  Ai , which make the diagram commute and such that ti si ∈ AutBi  for all i. Proof. Let β ∈ G F be such that θβ = 1G . The existence of ti  si comes from the fact that Ai  Bi are projective in M mod . From the commutativity of the diagram we have di ti si  = di ti si = ti ρi si = ti−1 si−1 di

∀ i

In particular we have et0 s0  = θβe = 1e = e and so eIm t0 s0  = eB0 . Therefore Im t0 s0 + Ker e = B0 . But Kere ≤
B0 ; hence Im t0 s0 = B0 and so t0 s0 ∈ AutB0  (see (1.3) and Lemma 1.5). Inductively for all i  0 we have di Im ti si  = ti−1 si−1 Im di  = Im di = di Bi  Hence Im ti si + Kerdi  = Bi and since Kerdi  ≤
Bi , we have Bi = Imti si  and so ti si ∈ AutBi . 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 ti si ∈ AutBi ; hence we get Ai = Ker ti ⊕ Im si . It is easy to see that ρi Ker ti  ⊆ Ker ti , and hence U    → Ker t1 → Ker t0 → F → 0 is a subpresentation of (1). We have also ρi Im si  ⊆ Im si−1 , which shows that 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

Proposition 2.9.

457

Suppose that

E1 

0 → K t↓↑t1

E2 

0→K

f

g

→ M

f

→V  → 0

s↓↑s1 →M

g

r↓↑r1 →V → 0

are two right I-sequences in mod  with E2 minimal. Let r ∈ V   V  be a split epimorphism. Then there exist maps r1  s s1  t t1 as shown in the above diagram such that (2.10)

ft = sf   gs = rg  f  t1 = s1 f g s1 = r1 g

and tt1  ss1  rr1 are all automorphisms. In particular s and t are split epimorphisms. 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 a commutative diagram S1 

0 → − K  

−f  

→ − M  

β3 ↓↑α3 S2 

0 → − K

−g 

→ − V   → V  I → 0

β2 ↓↑α2 −f 

→ − M

β1 ↓↑α1

↓θ

−g

→ − V  → V I → 0

with S2 as a minimal projective presentation of V I , since E2 is a minimal I-sequence. Also the morphisms αi  βi  i = 1 2 3 shown in the diagram are such that βi αi  i = 1 2 3 are automorphisms. Now apply Yoneda’s lemma 1.4 to the morphisms βi αi  i = 1 2 3 we get the proposed maps r1  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. E1 

Suppose that 0 → K r↑↓r1

E2 

0→K

f

→ M

f

s↑↓s1 →M

g

→V  → 0

g

t↑↓t1 →V → 0

are two left I-sequences in mod  with E2 minimal. Let r ∈ K K   be a split monomorphism. Then there exist maps r1  s s1  t t1 as shown in the above diagram such that sf = f  r fr1 = s1 f   gs1 = t1 g  tg = g s, and t1 t s1 s r1 r 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→K

(2.15)

f

→M

g

→V → 0

be a right I-sequence and let α V → V1 be a split epimorphism, where V1 ∈ mod . Then (2.16)

0 → ker αg

inc

αg

→M

→ V1 → 0

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 ∈ IX V1  then µu1 ∈ IX V  and since (2.15) is a right I-sequence there exists h ∈ X M such that gh = µu1 . Hence u1 = αµu1 = αgh. Conversely if u1 = αgh1 , where h1 ∈ X E then, since g ∈ IM V , it follows that u1 ∈ IX V1 . This proves that (2.15) is a right I-sequence. In a similar fashion to 2.14 we may prove f

g

Proposition 2.17. Let 0 → K −→ M −→ V → 0 be a left I-sequence and let β K1 → K be a split monomorphism, where K1 ∈ mod . Then fβ

0 → K1 −→ 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 f

g

(resp., left) I-sequence 0 → K −→ M −→ 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. fi

gi

Theorem 3. Let Ei  0 → Ki −→ Mi −→ Vi → 0 i = 1 2 be two short exact 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 =  → V be the projection of V onto V and let V V 1 2, let πi  V 1 2 i 1 2 i  µi  Vi → V1  V2 be the inclusion map. Now let X ∈ mod , and take µ2 π2 u = µ1 u1 + µ2 u2 , any u ∈ X V1 V2  then u = 1V1  V2 u = µ1 π1 +  where ui = πi u i = 1 2. Hence, if u ∈ IX V1 V2  then ui = πi u ∈ IX Vi  i = 1 2, and so ui = gi hi for some hi ∈ X Mi  i = 1 2. It follows that u = µ1 g1 h1 + µ2 g2 h2 = gh, where h ∈ X M1 M2  is defined by hx = h1 x h2 x  and so u factors through g. Conversely if u= gh for some h ∈ X M1 M2  then, u = µ1 g1 π1 + µ2 g2 π2 h  ∈ IX V1 V2 , projection (resp. injection) of M M2 (resp. Mi ) where πi (resp. µi ) is the 1  onto Mi (resp. into M1 M2 ); since gi ∈ IMi  Vi  i = 1 2. (2) Suppose thatE1 and E2 are minimal right I-sequences. We need to show that E = E1 E2 is minimal; that is, ker− g ≤
− M or equivalently, kerX g ≤
X M ∀ X ∈ mod . Let λ ∈ kerX g then gλ = 0. Define λi ∈ X Mi  i = 1 2 by λx = λ1 x λ2 x for all x ∈ X. Then gλ = 0 implies that gi λi = 0 i = 1 2 and so λi ∈
X Mi  i = λ = 1M1  M2 λ = µ1 π1 + µ2 π2 λ 1 2, since Ei is minimal. Therefore   =     µ1 π1 λ + µ2 π2 λ ∈
X M1 M2 . Conversely suppose that E = E1 E2 is minimal I-sequence. We must show that Ker  g1  ≤
  M1 ; i.e., KerX g1  ≤
X M1  for all X ∈ mod . Suppose that λ1 ∈ KerX g1 . Then taking  
  λ1 λ= ∈ X M1 M2  0 we have



g1 0

0 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. Similarly for E2 . Corollary 2.19. If for each i = 1 2 there is a right (left) I-sequence ending with Vi ∈ mod  (starting with Ki ∈ mod ),   then there is a right (left) I-sequence ending with V1 V2 (starting with K1 K2 ). f

g

Theorem 4. Let 0 → K −→ M −→ V → 0 be a right I-sequence. Then the following conditions on V are equivalent (1) V ≡ 0 mod I; i.e., 1V ∈ IV V . (2) − V  = I− V . −f 

−g

(3)

0 → − K

(4)

0 → K −→ M −→ V → 0 is split in mod .

f

→ − M g

→ − V  → 0 is exact in M mod .

460

ahmed a. khammash

Proof. 1 ⇒ 2 IX V . 2 ⇒ 3

since for any θ ∈ X V , we have θ = 1V θ ∈

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 f

g

0 → K −→ M −→ V → 0 is split in mod . f

g

4 ⇒ 1 If 0 → K −→ M −→ V → 0 is split then 1V = gg , for some g ∈ V M. But since g ∈ IM V  1V = gg ∈ IV V . In a similar way we may prove the following f

g

Theorem 5. Let 0 → K −→ M −→ V → 0 be a left I-sequence. The following conditions on K are equivalent (1)

K ≡ 0 mod I.

(2)

K − = IK −.

(3)

0 → V −

(4)

0 → K −→ M −→ V → 0 is split in mod .

f

g−

f−

→ K − → 0 is exact in M ◦ mod .

→ M − g

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

g

A short exact sequence 0 → K −→ M −→ V → 0 is mod  is said to be a bilateral (or two-sided) I-sequence if it is both a left and right I-sequence. f

g

Proposition 3.1. If E 0 → K −→ M −→ V → 0 is a two-sided I-sequence in mod , then any minimal right (left) I-sequence ending (starting) 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 shall always mean a bilateral I-sequence, which is minimal as a right I-sequence. f

g

Definition 3.2. If 0 → K −→ M −→ V → 0 is a bilateral, minimal I-sequence, we write K = 51 V , and note that, by 2.11, 5I V  is uniquely defined up to isomorphism. We call 5I a generalized Heller operator. Note that if I = P (see Section 1, Example (2)) then 5I = 5, the usual Heller operator (see [PL, p. 35]).

I-sequences

461

Our next aim is to show that 5I may be developed into a functor, which is in fact an equivalence between certain categories R modI  L modI , which will be defined later (Definition 3.5). To do this, we need the next proposition. Proposition 3.3. shown.

Let E and E  be bilateral I-sequences in mod , as f

E 0 → K z↓

(3.4) 

E0→K



g

→M

f

y↓

→M



→V → 0

g

↓x

→ V  → 0

Then (i) For each x ∈ V V   there exist y ∈ M M   and z ∈ K K   such that the diagram (3.4) commutes. Conversely for each z ∈ K K   there exist y ∈ M M   and x ∈ V V   such that the diagram (3.4) commutes. (ii) Given any commuting diagram (3.4), then x ∈ IV V   if and only if z ∈ IK K  . (iii) Say that a pair x z ∈ V V   × K K   is compatible, if there exists some y ∈ M M   such that (3.4) commutes. Then there is a k-linear isomorphism 35

ωI  V V  /IV V   → K K  /IK K  

defined by the rule ωI x + IV V   = z + IK 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) commute 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 that r is a split epimorphism). Conversely if z is given, the existence of y x making (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 ∈ IV 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 Imy − ag ⊆ ker g = Im f  . Define b ∈ M K   to be the composite of y − ag with the inverse of the bijective map f   K  → Im f  (which is legitimate because Imy − 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 since f  is a monomorphism, f  z = f  bf implies z = bf . But since f ∈ IK M (Lemma 2.5 (i)), this shows that z ∈ IK K  . An exactly similar argument shows that z ∈ IK K   implies x ∈ IV V  .

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ahmed a. khammash

(iii) By (i) and (ii), ωI x + IV V   is defined uniquely for every x ∈ IV V  . The map ωI so defined is monomorphic, again by (ii). It is surjective, 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 are such that xi g = g yi and yi f = f  zi , then clearly for any λ1  λ2 ∈ k λ1 x1 + λ2 x2 g = g λ1 y1 + λ2 y2 , and λ1 y1 + λ2 y2 f = f  λ1 z1 + λ2 z2 . This shows that λ1 x1 + λ2 x2  λ1 z1 + λ2 z2  is a compatible pair, whenever x1  z1  and x2  z2  are both compatible. Therefore (using notation such as x¯ 1 = x1 + IV V  ), we get ωI λ1 x¯ 1 + λ2 x¯ 2  = λ1 ωI x¯ 1  + λ2 ωI x¯ 2 . Given any ideal I in the category mod , we construct the quotient category modI  as follows: The objects of modI  are the same as the objects of mod , but the morphism set between given objects X X  is defined to be the k-space MormodI  X X   = X X  /IX X   Thus to every map f ∈ X X   in mod  corresponds a morphism f¯ = f + IX X   in the category modI . The product of morphisms in modI  is given by the rule g + IX   X  f + IX X   = gf + IX X  , for f ∈ X X   g ∈ X   X  . The identity morphism for the object X is 1X + IX X; this is the identity element of the k-algebra X X/IX X. Definition 3.5. Let R mod  L mod  be the full subcategory of mod , whose objects are all V (all K), which end (start) some bilateral I-sequence 0 → K → M → V → 0 in mod . Let R modI L modI  be the corresponding subcategory of modI . Theorem 6. There is a k-additive functor 5I  R modI  → L modI , which takes each V ∈ R modI  to K = 5I V  and maps V V  /IV V   → K K  /IK K   according to the map ωI of (3.5). This functor is an equivalence 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 definition of k-additive functor, see, e.g., [JG2, p. 373, footnote]). The fact that the functor 5I is an equivalence follows because (i) the map ωI is bijective for each pair V V  in R modI , and (ii) 5I is dense; i.e., each object K in L modI  is isomorphic to 5I V  for some V ∈ R modI  (see [JG2, p. 374]). Corollary 3.6. A module V ∈ R mod  is indecomposable and not in I, 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 in I, if and only if EndmodI  X = X X/IX X is a local algebra, i.e., is nonzero and has no nonzero idempotent except the identity. Now let V ∈ R mod  and K = 5I V . From Theorem 6 (or direct from Proposition 3.3) we know that EndR modI  V  = V V /IV V  and EndL modI  K = K K/IK K are isomorphic. So if one of these algebras is local, then so is the other. The following characterization for the minimal bilateral I-sequences follows 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 and K given set of subgroups of G. In this section we shall use the results of Sections 2 and 3 to prove the existence of a minimal relative projective cover for any object in mod kG. If X Y ∈ mod kG and H ≤ G we write K X Y H for X Y kH . If H≤ K ≤ G, let tH  X Y H → X Y K be the K −1 trace map given by tH f  = s∈≺K/H sfs , where ≺ K/H  denotes a left transversal of H in K and sfs−1  x → sf s−1 x ∀ x ∈ X (see [PL, Chap. II]).  G Definition. Write PH X Y  = Im tH 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 . The following shows that PK mod kG. Proposition 4.1. Suppose that X Y Z ∈ mod kG, f ∈ X Y G , and g ∈ Y ZG . Then gf ∈ PK X Z if either f ∈ PK X Y  or g ∈ PK Y Z. That is, PK mod kG.  Proof. If f ∈ PKX 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 mod kG. Therefore gf = H∈K gfH ∈ PK X Y . Similarly we can prove that gf ∈ PK X Z if g ∈ Y ZG . Definition. V ∈ mod kG 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 if PK V V  = EkG V .

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ahmed a. khammash

Definition. f ∈ X Y G is said to be K-split epimorphism if for each H ∈ K there is δH ∈ Y XH such that fδH = 1Y and K-split monomorphism if for each H ∈ K there is θH ∈ Y XH such that θH f = 1X . Examples. Suppose that V ∈ indec kG. For each subgroup H of G we have the following short exact sequence in mod kG 0 → Ker βH

(4.2)

inc

→ V↓H ↑G

βH

→ V → 0

  where βH  i ti ⊗ vi  = i ti vi . Let δ V → V↓H ↑G be the kH-map defined by δv = 1 ⊗ v. Then it is clear that βH δ = 1V ; therefore the sequence (4.2) is H-split in all cases. Similarly for an kG-module X we have a short exact sequence 0→X

(4.3)

αH 

→ X↓H ↑G

→ Co ker αH → 0

 where αH x = s∈G\H s ⊗ s−1 x for all x ∈ X. It is clear that αH is a kG-map and  that µαH = 1X , where µ X↓H ↑G → X is the kH-map defined as µ s s ⊗ xs  = x1 . This means that Lemma (4.4) H-splits and that 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 then G by the Higman criterion ∃ η ∈ EndkH V  tH η = 1V and so G G G 1M = tH η = tH βH δη = βH tH δη G Since tH δη is a kG-map, it follows that βH (hence the sequence (4.2)) splits. (2) is similar

Lemma 4.5. K-projective.

If H ∈ K and V ∈ mod kG is H-projective, then V is

Proof. It is clear that PH X V  ≤ PK X V  ∀ X V ∈ mod RG. Therefore V is H-projective ⇔ 1V ∈ PH V V  ⇒ 1V ∈ PK V V ; hence V is K-projective. f

g

Definition 4.6. A s.e.s. (1) 0 → K −→ M −→ V → 0 in mod kG is called a K-projective s.e.s. if (a) (b)

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

I-sequences

465

If (1) is a K-projective s.e.s, we say that (1) is a K-projective resolution of V , or a K-projective co-resolution of K. Note that (b) in the previous definition is equivalent to either of the conditions (b ) (b )

f is a K-split monomorphism, or g is a K-split epimorphism.

Examples 4.7. It follows from 4.4 that the sequence
β=βH  (4.8) V↓H ↑G →V → 0 SK V  0 → ker β → H∈K

is a K-projective resolution of V . Since each βH is H-split, it follows that β is K-split. Similarly for any kG-module X we have that a sequence
α=αH  SX K 0 → X (4.9) → X↓H ↑G → Co ker α → 0 H∈K

is a K-projective co-resolution of X. These examples show that every kGmodule 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 SK V  [resp. SX K] splits if and only if V [resp. X] is K-projective kG-module. As a consequence of Lemma 4.10 we have Lemma 4.11. Let M ∈ mod kG. Then M is K-projective if and only if, for  each H ∈G K, there exists a module LH ∈ mod kH such that M! ⊕ H∈K LH↑ . Now Lemma 4.10 implies the following Corollary.  V ∈ mod kG is K-projective if and only if V has a decomposition 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. and bilateral PK -sequences: f

g

Proposition 4.12. Let (1) 0 → K −→ M −→ V → 0 be a s.e.s. in mod kG. 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 prove Take any X ∈ mod kG, and any θ ∈ first that (1) is a right PK -sequence.  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  G (4.13) tH σH  θ= H∈K

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ahmed a. khammash

By Definition 4.6, g is K-split; hence for all H ∈ K there exists δH ∈ V MH such that gδH = 1V . Therefore σH = gδH σH , which G G implies tH σ H  = GgtH δH σH . Putting this into (4.13), we get θ = gλ, where λ = H∈K tH δH σH  ∈ X MG . Therefore θ factors through g. Conversely if θ ∈ X V G factors through g, there is some µ ∈ X MG such that θ = gµ. But by Definition 4.6, M is K-projective; hence 1M ∈ PK M M. Therefore g = g1M ∈ PK M V , and since PK is an ideal in mod kG θ = gµ ∈ PK X V . This proves that (1) is a right PK -sequence. But since Definition 4.6 is right–left symmetric, (1) is also a left PK -sequence; hence (1) is a bilateral PK -sequence. ⇐ Now assume that (1) is a bilateral PK -sequence. By Corollary 2.11, we have 1 ∼ = E ⊕ N, where E 0 → K2 → M2 → V → 0 is a minimal PK -sequence ending with V , and N is a right I-sequence of the form 0 → S → S  → 0 → 0. But since (1) is also a left I-sequence, both E 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 = 48 (which is a 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
β=βH  SK V  0 → ker β → V↓H ↑G →V → 0 H∈K



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 the map g2 inE is K-split, because for each H ∈ K we have a kH-map δH  V → H∈K VH G such that βH δH = 1V ; hence sδH  V → M2 is a kH-map such that g2 sδH  = βH δH = 1V . We have now proved that E is a K-projective s.e.s. But also the s.e.s. N 0 → S → S  → 0 → 0 is K-projective, since S  is a K-projective module, and of course N is split even as a kG-sequence. The direct sum  of two K-projective s.e.s is clearly K-projective. This shows that 1 = E N, N is K-projective. Now in the light of Proposition 4.12, the previous results of Sections 2 and 3 on I-sequences provide an alternative proof of the following theorem which was originally proved by Kn¨ orr [K]. Theorem 8. (1) For any kG-module V there exists a minimal Kprojective resolution EV K 0 → 5K V  → M → V → 0, where 5K V  = 5PK V .

I-sequences

(2)

 SV K ∼ = EV K 0 → P1 → P2 → 0 → 0.

(3)

5K V  has no nonzero K-projective summands.

467

(4) If V is indecomposable not K-projective then 5K V  is indecomposable 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 SV K of V ; (2) follows from (2.10) and 2.12, (3) follows from 3.6 and the fact that 5I W  = 0 for any K-projective W ∈ mod kG, (4) follows from Proposition 3.6, and (5) follows from Proposition 2.14. Remark. Notice that when I = PK , then R mod  = L mod  = mod  (see Examples 4.7); hence R modI  = L modI  = modI ?. So the Heller functor 5PK is an equivalence of modPK kG with itself, this fact was first discovered by R. Kn¨ orr [K]. Definition. If EV 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 direct summand of  V↓H ↑G  H∈K

The following gives a homological characterization for the space PK X Y . Theorem 9. equivalent

If X Y ∈ mod kG and f ∈ X Y G , then the following are

(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 factors through u. (4)

f factors through some K-projective kG-module.

Proof. 1 ⇒ 2 Suppose that f ∈ PK X Y .Then there is a G sequence σH H∈K ; σH ∈ X Y H such that f = H∈K tH σH . Now  if v ∈ Y  Y G is a K-split epimorphism then for all H ∈ K there is a δH ∈ Y Y  H such that νδH = 1Y . Therefore  G  G f = tH νδH σH  = ν tH δH σH  = νh H∈K

where h =



G H∈K tH δH σH 

∈ X Y  , and so f factors through ν.

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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 there   is β ∈ X H∈K Y!H ↑G  such that f = ββ . Since H∈K Y!H ↑G is Kprojective we get (4). 4 ⇒ 1 Assume that M is a K-projective kG-module and that f = gh, where h ∈ X MG  and g ∈ M Y G . Since M is K-projective, 1M ∈ G ηH  for some ηH ∈ M MH . Therefore PK M M and so 1M = H∈K tH  G f = gh = g1M h = H∈K tH gηH h ∈ 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 factors through any K-split monomorphism in X   X. Then f factors  through the map 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 = f 1U = 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 categories m mod kG, m◦ mod kG as well as vertices and sources of the indecomposable objects in these categories. For the rest of this section, we would like to compare the vertices of the indecomposable kG-modules K V forming the two ends of a minimal I-sequence I mod kG with the vertices of the quotient functors V I =   V /I  V  ∈ m mod kG and ◦ I K = K /IK  ∈ m mod kG. The following is an analogue to [JG1, 5.12] for I-sequences. f

g

Theorem 10. If E 0 → K −→ M −→ V → 0 is a bilateral minimal I-sequence with V indecomposable and V #≡ 0 mod I, K #≡ mod I, then (1)

vertexV  ≤ vertexV I 

(2)

vertexV  ≤ vertexI K.

Proof. (1) Follows by applying [JG1, 8.11] to the minimal projective presentation 0 → − K

−f 

→ − M

−g

→ − V  → V I → 0

of the object V I in m mod . Similarly we may deduce (2) by considering the minimal projective presentation 0 → V − in m◦ mod .

g−

→ M −

f−

→ K − →I K → 0

I-sequences

469

Remark. If V ∈ indec kG then, taking I =
, we have V 
= SV . Hence Theorem 10 implies the fact proved in [JG1, 5.12] that if 0→K

f

g

→M

→V → 0

is an almost split sequence then vxV  ≤ vxSV . G

APPENDIX: CONNECTIONS WITH VERTEX THEORY Let I mod kG, and let D be a subgroup of G. Let modkGD be the full subcategory of mod kD, whose objects are those kD-modules which are the restrictions MD to kD of kG-modules M. In other words, the objects M M   M       of modkGD are the same as those of mod kG, but the morphism set M M   in modkGD is M M  kD = HomkD M M  . Since M M  kG is a subset of M M  kD , then IM M   ⊆ M M  kD , for all M M  ∈ modkGD . Define ID to be the ideal of modkGD , which is generated 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 for some N ∈ mod kG there holds α ∈ M NkD  β ∈ N MkD and at least one of β α lies in I. Definition A.2. Let M V ∈ mod  kG and let σ ∈ M  V kD . Define G  V kG by the rule σ G  u∈$G/D% u ⊗ mu  = u∈$G/D% uσmu . σ G ∈ MD 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 mod kG which satisfy the following Hypothesis A.3. For all subgroups D of G there holds ID G ⊆ I; i.e., for all M V ∈ mod kG and all σ ∈ M V kD we have G σ ∈ ID M V  ⇒ σ G ∈ IMD  V 

Theorem A.4. Suppose that I mod kG satisfies Hypothesis A.3, and that V ∈ R mod kG (see Definition 3.5), V indecomposable and V does not lie in I (i.e., 1V #∈ IV V kG = IEndkG V ). Then V and 5I V  have the same vertices. f

g

Proof. Let E 0 → K −→ M −→ V → 0 be a minimal, bilateral Isequence. 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 subgroup D of G, (A.5)

V is D-projective if and only if K is D-projective.

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ahmed a. khammash

Suppose that V is D-projective. Then there exists some ξ ∈ V V kD such G ξ. We shall prove that there exist kD-maps η ζ such that that 1V = tD the diagram

(A.6)

f

g

f

g

0 → K −→M −→ V → 0 ζ ↓ η↓ ξ↓ 0 → K −→M −→ V → 0

commutes. The existence of η follows from Hypothesis A.3. For since g ∈ IM V  (by 2.5(i)), the kD-map σ = ξg ∈ ID M V . Therefore HypothG esis A.3 says that σ G ∈ IMD  V . Because E is a right I-sequence, there G exists a kG-map φ MD → M such that gφ = σ G . Define η M → M by setting ηm = φ1 ⊗ m for all m ∈ M. We verify easily that η is a kD-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 ζp = q, for all p ∈ K. It is clear that ζ is a kD-map and that fζ = ηf . G Now apply the operator t = tD to the equations gη = ξg and fζ = ηf . Since g f are both kG-maps, we get gtη = tξg and ftζ = tηf . By assumption tξ = 1V . Also y = tη z = tζ are kG-maps. Thus we have a commutative diagram such as (3.4), with E = E  and x = 1V . This shows that the map (A.7)

ωI  V V /IV V  → K K/IK K

(see (3.5)), which in the present case is an isomorphism of k-algebras, taking 1V + IV V  to z + IK K. However, 1V ∈ / IV V  by assumption; therefore 1V + IV V  is the identity element of the algebra EndV /IEndK. So z + IK K is the identity element 1K + IK K of G the algebra EndK/IEndK. Now z = tD ζ ∈ PD K K = PD EndK. Therefore 1K , which is a primitive idempotent of EndK because K is indecomposable (see 3.6), lies in the sum of the ideals PD EndK and IEndK of EndK. We know that 1K ∈ / IEndK, and it follows that 1K ∈ PD EndK, 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 Kn¨ orr’s theorem. Theorem A.8 ([K]). Using the notation of Section 4, if V is indecomposable 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 mod kG satisfies Hypothesis ( A.3). Proof. Let D be a subgroup of G, let M V ∈ mod kG, and let σ ∈ G ID M V . We must prove that σ G ∈ IMD  V . By Definition A.1, it will be enough to do this when σ = βα with α ∈ IM N β ∈ N V kD for some N ∈ mod kG. G Ifσ is any element of M V kD , define σ1 ∈ MD  V kD by the rule σ1  u u ⊗ mu  = σm1 . It is then easily verified that (A.10)

G i σ G = tD σ1 

and

ii if λ ∈ M V kL

for some subgroup L of D, then tLD λ1  Since α ∈ IM N = PK M N exists for each H ∈ K a map  there G θH ∈ M NkH , such that α = H tH θH. If we regard α as a kDmap, then Mackey’s formula (see[PL, Theorem 1.7]) gives (A.11)

α=

 H

w

D w tD∩ w  θH H

Here the second sum is over a transversal $D\G/H% of the double cosets DwH 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 NkD . Therefore σ = βα lies in PL M V kD . But then (A.10) shows that G σ G = tD σ1  lies in PK M N, because every element D ∩w H of L lies in a G-conjugate of a subgroup of an element of K. This proves Proposition A.9. As a consequence of Theorem A.8 one can deduce the following known property of almost split sequences. f

g

Proposition A.12. If 0 → K −→ M −→ V → 0 is an almost split sequence, where V ∈ indec kG is nonprojective, then vertexV  = vertexK. Proof. It is known (see [PL, Chap. II, Sect. 9]) that K = 52 V , where 5 = 5PK is the usual Heller operator with K as the collection of subgroups of G consisting of the trivial subgroup. Now from Theorem A.8 we have vertexK = vertex52 V  = vertex5V  = vertexV  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) that the I-sequence theory could be applied to get many of the Kn¨ orr results on relative projective covers. I thank Professor Green for many useful comments on the subject of this paper.

REFERENCES [AR1] [AR2] [AR3] [PG] [JG1] [JG2] [K] [PL]

M. Auslander and I. Reiten, Representation theory of Artin algebras I, Comm. Algebra 1 (1974), 177–268. M. Auslander and I. Reiten, Representation theory of Artin algebras II, Comm. Algebra 1, No. 4 (1974), 269–310. M. Auslander and I. Reiten, Representation theory of Artin algebras III, Comm. Algebra 3 (1977), 239–294. P. Gabriel, Auslander–Reiten sequences and representation-finite algebras, in “Representation Theory I,” Lecture Notes in Mathematics, Vol. 831, Springer-Verlag, Berlin/New York, 1980. J. A. Green, Functors on categories of finite group representations, J. Pure Appl. Algebra 37 (1985), 265–298. J. A. Green, Relative module categories for finite groups, J. Pure Appl. Algebra 2 (1972), 371–393. R. Kn¨ orr, Relative projective covers, in “Proc. of Symp. Mod. Representations of Finite Groups,” Arhus University, 1978; Publication Series No. 24, Mathematics Institute, Arhus University. P. Landrock, “Finite Group Algebras and Their Modules,” LMS Lecture Notes Series 84, Cambridge Univ. Press, Cambridge, UK, 1983.