Modular Representation Theory of Finite Groups
Jun.-Prof. Dr. Caroline Lassueur TU Kaiserslautern
Skript zur Vorlesung, WS 2020/21 (Vorlesung: 4SWS // Übungen: 2SWS)
Version: 4. Dezember 2021
Foreword iii
Conventions iv
Chapter 1. Foundations of Representation Theory 6
1 (Ir)Reducibility and (in)decomposability . . . 6
2 Schur’s Lemma . . . 7
3 Composition series and the Jordan-Hölder Theorem . . . 8
4 The Jacobson radical and Nakayama’s Lemma . . . 10
5 Indecomposability and the Krull-Schmidt Theorem . . . 11
Chapter 2. The Structure of Semisimple Algebras 15 6 Semisimplicity of rings and modules . . . 15
7 The Artin-Wedderburn structure theorem . . . 18
8 Semisimple algebras and their simple modules . . . 22
Chapter 3. Representation Theory of Finite Groups 26 9 Linear representations of finite groups . . . 26
10 The group algebra and its modules . . . 29
11 Semisimplicity and Maschke’s Theorem . . . 33
12 Simple modules over splitting fields . . . 34
Chapter 4. Operations on Groups and Modules 36 13 Tensors, Hom’s and duality . . . 36
14 Fixed and cofixed points . . . 39
15 Inflation, restriction and induction . . . 39
Chapter 5. The Mackey Formula and Clifford Theory 45 16 Double cosets . . . 45
17 The Mackey formula . . . 47
18 Clifford theory . . . 48
i
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 ii
Chapter 6. Projective Modules over the Group Algebra 51
19 Radical, socle, head . . . 51
20 Projective modules . . . 54
21 Projective modules for the group algebra . . . 54
22 The Cartan matrix . . . 58
23 Symmetry of the group algebra . . . 59
24 Representations of cyclic groups in positive characteristic . . . 62
Chapter 7. Indecomposable Modules 64 25 Relative projectivity . . . 64
26 Vertices and sources . . . 71
27 The Green correspondence . . . 74
28 p-permutation modules . . . 77
29 Green’s indecomposability theorem . . . 80
Chapter 8. p-Modular Systems 82 30 Complete discrete valuation rings . . . 82
31 Splittingp-modular systems . . . 85
32 Lifting idempotents . . . 88
33 Brauer Reciprocity . . . 91
Chapter 9. Brauer Characters 93 34 Brauer characters . . . 94
35 Back to decomposition matrices of finite groups . . . 98
Chapter 10. Blocks 104 36 The blocks of a ring . . . 104
37 p-Blocks of finite groups . . . 106
38 Defect groups . . . 108
39 Brauer’s 1st and 2nd Main Theorems . . . 110
Appendix 1: Background Material Module Theory 1000 A Modules, submodules, morphisms . . . 1000
B Free modules and projective modules . . . 1003
C Direct products and direct sums . . . 1005
D Exact sequences . . . 1006
E Tensor products . . . 1008
F Algebras . . . 1011
Appendix 2: The Language of Category Theory 1013 G Categories . . . 1013
H Functors . . . 1016
Index of Notation 1018
This text constitutes a faithful transcript of the lecture Modular Representation Theory held at the TU Kaiserslautern during the Winter Semester 2020/21 (14 Weeks, 4SWS Lecture + 2SWS Exercises).
Together with the necessary theoretical foundations the main aims of this lecture are to:
‚ provide students with a modern approach tofinite group theory;
‚ learn about the representation theory of finite-dimensional algebras and in particular of the group algebra of a finite group;
‚ establish connections between the representation theory of a finite group over a field ofpositive characteristic and that over a field of characteristiczero;
‚ consistently work with universal properties and get acquainted with the language of category theory.
We assume as pre-requisites bachelor-level algebra courses dealing withlinear algebraandelementary group theory, such as the standard lecturesGrundlagen der Mathematik,Algebraische Strukturen, and Einführung in die Algebra. It is also strongly recommended to have attended the lecturesCommutative AlgebraandCharacter Theory of Finite Groupsprior to this lecture. Therefore, in order to complement these pre-requisites, but avoid repetitions, the Appendix deals formally with some background material on module theory, but proofs are omitted.
The main results of the lectureCharacter Theory of Finite Groups will be recovered through a different and more general approach, thus it is formally not necessary to have attended this lecture already, but it definitely brings you some intuition.
Acknowledgement: I am grateful to Gunter Malle who provided me with the Skript of his lecture
"Darstellungstheorie" hold at the TU Kaiserslautern in the WS 12/13, 13/14, 15/16 and 16/17, which I used as a basis for the development of this lecture. I am also grateful to Niamh Farrell, who shared with me her text from 2019/20 to the second part of the lecture, which I had never taken myself prior to the WS 20/21.
I am also grateful to Kathrin Kaiser, Helena Petri and Bernhard Böhmler who mentioned typos to me in the preliminary version of these notes. Further comments, corrections and suggestions are of course welcome.
Conventions
Unless otherwise stated, throughout these notes we make the following general assumptions:
¨ all groups considered arefinite;
¨ all rings considered are associative and unital (i.e. possess a neutral element for the multiplication, denoted 1);
¨ all modules considered areleft modules;
¨ if K is a commutative ring and G a finite group, then all KG-modules considered are assumed to be free of finite rank when regarded as K-modules.
Chapter 1. Foundations of Representation Theory
In this chapter we review four important module-theoretic theorems, which lie at the foundations of representation theory of finite groups:
1. Schur’s Lemma: about homomorphisms between simple modules.
2. The Jordan-Hölder Theorem: about "uniqueness" properties of composition series.
3. Nakayama’s Lemma: about an essential property of the Jacobson radical.
4. The Krull-Schmidt Theorem: about direct sum decompositions into indecomposable submodules.
Notation: throughout this chapter, unless otherwise specified, we letR denote an arbitrary unital and associative ring.
References:
[Ben98] D. J. Benson. Representations and cohomology. I. Vol. 30. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998.
[CR90] C. W. Curtis and I. Reiner.Methods of representation theory. Vol. I. John Wiley & Sons, Inc., New York, 1990.
[Dor72] L. Dornhoff. Group representation theory. Part B: Modular representation theory. Marcel Dekker, Inc., New York, 1972.
[NT89] H. Nagao and Y. Tsushima. Representations of finite groups. Academic Press, Inc., Boston, MA, 1989.
[Rot10] J. J. Rotman. Advanced modern algebra. 2nd ed. Providence, RI: American Mathematical Society (AMS), 2010.
[Web16] P. Webb. A course in finite group representation theory. Vol. 161. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016.
1 (Ir)Reducibility and (in)decomposability
Submodules and direct sums of modules allow us to introduce the two main notions that will enable us to break modules inelementarypieces in order to simplify their study: simplicityandindecomposability.
6
Definition 1.1 (simple/irreducible module / indecomposable module / semisimple module)
(a) An R-module M is called reducible if it admits an R-submodule U such that 0 Ĺ U ĹM. AnR-module M is calledsimple, orirreducible, if it is non-zero and not reducible.
(b) An R-module M is called decomposable if M possesses two non-zero proper submodules M1, M2 such thatM“M1‘M2. AnR-moduleM is called indecomposableif it is non-zero and not decomposable.
(c) An R-module M is called completely reducible or semisimple if it admits a direct sum decomposition into simpleR-submodules.
Our primary goal in Chapter 1 and Chapter 2 is to investigate each of these three concepts in details.
Remark 1.2
Clearly any simple module is also indecomposable, resp. semisimple. However, the converse does not hold in general.
Exercise 1.3
Prove that ifpR,`,¨qis a ring, thenR˝:“Ritself maybe seen as anR-module via left multiplication in R, i.e. where the external composition law is given by
RˆR˝ÝÑR˝,pr, mq ÞÑr¨m .
We callR˝ theregular R-module.
Prove that:
(a) the R-submodules of R˝ are precisely the left ideals ofR;
(b) ICR is a maximal left ideal of R ô R˝{I is a simpleR-module, and ICR is a minimal left ideal of R ô I is simple when regarded as an R-submodule ofR˝.
2 Schur’s Lemma
Schur’s Lemma is a basic result, which lets us understand homomorphisms between simple modules, and, more importantly, endomorphisms of such modules.
Theorem 2.1 (Schur’s Lemma)
(a) Let V , W be simple R-modules. Then:
(i) EndRpVqis a skew-field, and (ii) if V flW, then HomRpV , Wq “0.
(b) IfK is an algebraically closed field,Ais aK-algebra, andV is a simpleA-module such that dimKV ă 8, then
EndApVq “ tλIdV |λPKu –K .
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 8 Proof :
(a) First, we claim that everyf PHomRpV , Wqzt0uadmits an inverse in HomRpW , Vq.
Indeed, f ‰ 0 ùñ kerf Ĺ V is a proper R-submodule of V and t0u ‰ Imf is a non-zero R- submodule ofW. But then, on the one hand, kerf “ t0u, becauseV is simple, hencef is injective, and on the other hand, Imf “W because W is simple. It follows thatf is also surjective, hence bijective. Therefore, by Example A.4(d),f is invertible with inversef´1 PHomRpW , Vq.
Now, (ii) is straightforward from the above. For (i), first recall that EndRpVq is a ring, which is obviously non-zero as EndRpVq QIdV and IdV ‰0 becauseV ‰0 since it is simple. Thus, as any f PEndRpVqzt0uis invertible, EndRpVqis a skew-field.
(b) Let f P EndApVq. By the assumptions onK, f has an eigenvalue λ P K. Let v P Vzt0u be an eigenvector off forλ. Thenpf´λIdVqpvq “0. Therefore,f´λIdV is not invertible and
f´λIdV PEndApVq ùñpaq f´λIdV “0 ùñ f“λIdV .
Hence EndApVq Ď tλIdV |λPKu, but the reverse inclusion also obviously holds, so that EndApVq “ tλIdVu –K .
3 Composition series and the Jordan-Hölder Theorem
From Chapter 2 on, we will assume that all modules we work with can be broken intosimple modules in the sense of the following definition.
Definition 3.1 (Composition series / composition factors / composition length) LetM be an R-module.
(a) A series (orfiltration) ofM is a finite chain of submodules
0“M0ĎM1 Ď. . .ĎMn“M pnPZě0q. (b) A composition seriesof M is a series
0“M0 ĎM1Ď. . .ĎMn “M pnPZě0q
where Mi{Mi´1 is simple for each 1 ďiďn. The quotient modules Mi{Mi´1 are called the composition factors (or the constituents) of M and the integer n is called the composition length ofM.
Notice that, clearly, in a composition series all inclusions are in fact strict because the quotient modules are required to be simple, hence non-zero.
Next we see that the existence of a composition series implies that the module is finitely generated. However, the converse does not hold in general. This is explained through the fact that the existence of a composition series is equivalent to the fact that the module is bothNoetherian and Artinian.
Definition 3.2 (Chain conditions / Artinian and Noetherian rings and modules)
(a) An R-module M is said to satisfy the descending chain condition (D.C.C.) on submodules (or to be Artinian) if every descending chain M “ M0 Ě M1 Ě . . . Ě Mr Ě . . . Ě t0u of submodules eventually becomes stationary, i.e. D m0 such thatMm “Mm0 for every měm0. (b) AnR-moduleM is said to satisfy theascending chain condition(A.C.C.) on submodules (or to beNoetherian) if every ascending chain 0“M0ĎM1 Ď. . .ĎMr Ď. . .ĎM of submodules eventually becomes stationary, i.e. D m0 such thatMm“Mm
0 for every měm0.
(c) The ringR is calledleft Artinian(resp. left Noetherian) if the regular moduleR˝ is Artinian (resp. Noetherian).
Theorem 3.3 (Jordan-Hölder)
Any series of R-submodules 0 “M0 ĎM1 Ď. . .ĎMr “M (r PZě0) of an R-module M may be refined to a composition series ofM. In addition, if
0“M0 ĹM1Ĺ. . .ĹMn“M pnPZě0q and
0“M01 ĹM11 Ĺ. . .ĹMm1 “M pmPZě0q
are two composition series of M, then m “ n and there exists a permutation π P Sn such that Mi1{Mi´1
1 –Mπpiq{Mπpiq´1for every 1ďiďn. In particular, the composition length is well-defined.
Proof : SeeCommutative Algebra. Corollary 3.4
IfM is anR-module, then TFAE:
(a) M has a composition series;
(b) M satisfies D.C.C. and A.C.C. on submodules;
(c) M satisfies D.C.C. on submodules and every submodule of M is finitely generated.
Proof : SeeCommutative Algebra. Theorem 3.5 (Hopkins’ Theorem)
IfM is a module over a left Artinian ring, then TFAE:
(a) M has a composition series;
(b) M satisfies D.C.C. on submodules;
(c) M satisfies A.C.C. on submodules;
(d) M is finitely generated.
Proof : Without proof.
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 10
4 The Jacobson radical and Nakayama’s Lemma
The Jacobson radical is one of the most important two-sided ideals of a ring. As we will see in the next sections and Chapter 2, this ideal carries a lot of information about the structure of a ring and that of its modules.
Proposition-Definition 4.1 (Annihilator / Jacobson radical)
(a) Let M be an R-module. Then annRpMq:“ tr PR |rm “0 @mPMu is a two-sided ideal of R, called annihilatorof M.
(b) TheJacobson radical of R is the two-sided ideal JpRq:“
č
Vsimple R-module
annRpVq “ tx PR |1´axbPRˆ @a, bPRu.
(c) If V is a simple R-module, then there exists a maximal left ideal ICR such that V –R˝{I (as R-modules) and
JpRq “ č
ICR, Imaximal
left ideal
I .
Proof : SeeCommutative Algebra. Exercise 4.2
(a) Prove that any simple R-module may be seen as a simple R{JpRq-module.
(b) Conversely, prove that any simpleR{JpRq-module may be seen as a simple R-module.
[Hint: use a change of the base ring via the canonical morphism R ÝÑR{JpRq.]
(c) Deduce that R and R{JpRqhave the same simple modules.
Theorem 4.3 (Nakayama’s Lemma)
IfM is a finitely generatedR-module andJpRqM“M, then M“0.
Proof : SeeCommutative Algebra. Remark 4.4
One often needs to apply Nakayama’s Lemma to a finitely generated quotient moduleM{U, where U is an R-submodule ofM. In that case the result may be restated as follows:
M “U`JpRqM ùñ U“M
5 Indecomposability and the Krull-Schmidt Theorem
We now consider the notion of indecomposability in more details. Our first aim is to prove that inde- composability can be recognised at the endomorphism algebra of a module.
Definition 5.1
A ringR is said to belocal :ðñRzRˆ is a two-sided ideal of R. Example 1
(a) Any field K is local becauseKzKˆ“ t0u by definition.
(b) Exercise: Letpbe a prime number andR:“ tab PQ|p-bu. Prove thatRzRˆ“ tab PR |p|au and deduce that R is local.
(c) Exercise: Let K be a field and let R :“
! A “
¨
˝
a1a2 ... an
0 a1 ... an´1
... ... ...
0 0 ... a1
˛
‚P MnpKq )
. Prove that RzRˆ “ tAPR |a1“0u and deduce that R is local.
Proposition 5.2
LetR be a ring. Then TFAE:
(a) R is local;
(b) RzRˆ “JpRq, i.e. JpRqis the unique maximal left ideal ofR; (c) R{JpRq is a skew-field.
Proof : SetN:“RzRˆ.
(a)ñ(b): Clear: ICR proper left ideal ñI ĎN. Hence, by Proposition-Definition 4.1(c), JpRq “ č
ICR, Imaximal
left ideal
I ĎN .
Now, by (a) N is an ideal of R, hence N must be a maximal left ideal, even the unique one. It follows thatN“JpRq.
(b)ñ(c): If JpRq is the unique maximal left ideal of R, then in particular R ‰ 0 and R{JpRq ‰ 0. So let rPRzJpRq pbq“ Rˆ. Then obviously r`JpRq P pR{JpRqqˆ. It follows thatR{JpRqis a skew-field.
(c)ñ(a): Since R{JpRq is a skew-field by (c), R{JpRq ‰ 0, so that R ‰ 0 and there exists a P RzJpRq. Moreover, again by (c),a`JpRq P pR{JpRqqˆ, so thatDbPRzJpRqsuch that
ab`JpRq “1`JpRq PR{JpRq
Therefore,DcPJpRqsuch thatab“1´c, which is invertible inRby Proposition-Definition 4.1(b).
HenceDdPR such thatabd“ p1´cqd“1ñaPRˆ. ThereforeRzJpRq “Rˆ, and it follows thatRzRˆ“JpRqwhich is a two-sided ideal ofR.
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 12 Proposition 5.3 (Fitting’s Lemma)
LetM be an R-module which has a composition series and let φPEndRpMq be an endomorphism of M. Then there existsnPZą0 such that
(i) φnpMq “φn`ipMq for every iě1;
(ii) kerpφnq “kerpφn`iq for every iě1; and (iii) M“φnpMq ‘kerpφnq.
Proof : By Corollary 3.4 the moduleMsatisfies both A.C.C. and D.C.C. on submodules. Hence the two chains of submodules
φpMq Ěφ2pMq Ě. . . , kerpφq Ďkerpφ2q Ď. . .
eventually become stationary. Therefore we can find an indexnsatisfying both (i) and (ii).
Exercise: Prove thatM “φnpMq ‘kerpφnq.
Proposition 5.4
LetM be an R-module which has a composition series. Then:
M is indecomposable ðñ EndRpMq is a local ring.
Proof :“ñ”: Assume that M is indecomposable. Let φ PEndRpMq. Then by Fitting’s Lemma there exists n P Zą0 such that M “ φnpMq ‘kerpφnq. As M is indecomposable either φnpMq “ M and kerpφnq “0 or φnpMq “0 and kerpφnq “M.
¨ In the first caseφis bijective, hence invertible.
¨ In the second caseφ is nilpotent.
Therefore,N:“EndRpMqzEndRpMqˆ“ tnilpotent elements of EndRpMqu. Claim: N is a two-sided ideal of EndRpMq.
LetφPNand mPZą0 minimal such thatφm“0. Then
φm´1pφρq “0“ pρφqφm´1 @ρPEndRpMq. Asφm´1‰0,φρ andρφcannot be invertible, henceφρ, ρφPN.
Next let φ, ρPN. If φ`ρ “:ψ were invertible in EndRpMq, then by the previous argument we would haveψ´1ρ, ψ´1φPN, which would be nilpotent. Hence
ψ´1φ“ψ´1pψ´ρq “IdM´ψ´1ρ would be invertible.
(Indeed,ψ´1ρ nilpotentñ pIdM´ψ´1ρqpIdM`ψ´1ρ` pψ´1ρq2` ¨ ¨ ¨ ` pψ´1ρqa´1q “IdM, where ais minimal such thatpψ´1ρqa “0.)
This is a contradiction. Thereforeφ`ρPN, which proves thatN is an ideal.
Finally, it follows from the Claim and the definition that EndRpMqis local.
“ð”: Assume M is decomposable and letM1, M2 be proper submodules such thatM“M1‘M2. Then consider the two projections
π1:M1‘M2 ÝÑM1‘M2,pm1, m2q ÞÑ pm1,0q
ontoM1 alongM2 and
π2:M1‘M2 ÝÑM1‘M2,pm1, m2q ÞÑ p0, m2q
ontoM2 alongM1. Clearlyπ1, π2PEndRpMqbutπ1, π2REndRpMqˆsince they are not surjective by construction. Now, asπ2 “IdM´π1 is not invertible it follows from the characterisation of the Jacobson radical of Proposition-Definition 4.1(b) thatπ1RJpEndRpMqq. Therefore
EndRpMqzEndRpMqˆ‰ JpEndRpMqq and it follows from Proposition 5.2 that EndRpMqis not a local ring.
Next, we want to be able to decomposeR-modules into direct sums of indecomposable submodules. The Krull-Schmidt Theorem will then provide us with certain uniqueness properties of such decompositions.
Proposition 5.5
LetM be an R-module. IfM satisfies either A.C.C. or D.C.C., thenM admits a decomposition into a direct sum of finitely many indecomposable R-submodules.
Proof : Let us assume thatM is not expressible as a finite direct sum of indecomposable submodules. Then in particular M is decomposable, so that we may write M “ M1‘W1 as a direct sum of two proper submodules. W.l.o.g. we may assume that the statement is also false for W1. Then we also have a decompositionW1 “M2‘W2, whereM2andW2 are proper sumbodules ofW1 with the statement being false forW2. Iterating this argument yields the following infinite chains of submodules:
W1 ĽW2ĽW3Ľ ¨ ¨ ¨,
M1ĹM1‘M2ĹM1‘M2‘M3 Ĺ ¨ ¨ ¨.
The first chain contradicts D.C.C. and the second chain contradicts A.C.C.. The claim follows.
Theorem 5.6 (Krull–Schmidt)
LetM be an R-module which has a composition series. If
M“M1‘ ¨ ¨ ¨ ‘Mn“M11 ‘ ¨ ¨ ¨ ‘Mn11 pn, n1 PZą0q
are two decomposition ofM into direct sums of finitely many indecomposable R-submodules, then n“n1, and there exists a permutation π PSn such thatMi –Mπpiq1 for each 1ďiďnand
M“Mπp1q1 ‘ ¨ ¨ ¨ ‘Mπ1prq‘
n
à
j“r`1
Mj for every 1ďr ďn.
Proof : For each 1ďiďnlet
πi:M “M1‘ ¨ ¨ ¨ ‘MnÑMi, m1`. . .`mnÞÑmi
be the projection on thei-th factor of first decomposition, and for each 1ďj ďn1 let ψj :M “M11‘ ¨ ¨ ¨ ‘Mn11ÑMj1, m11`. . .`m1n1ÞÑm1j be the projection on thej-th factor of second decomposition.
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 14 Claim: ifψPEndRpMqis such thatπ1˝ψ|M
1:M1ÑM1 is an isomorphism, then M“ψpM1q ‘M2‘ ¨ ¨ ¨ ‘Mn andψpM1q –M1. Indeed: By the assumption of the claim, bothψ|M
1 :M1ÑψpM1qandπ1|ψpM1q:ψpM1q ÑM1 must be isomorphisms. ThereforeψpM1q Xkerpπ1q “0, and for everymPM there existsm11PψpM1qsuch that π1pmq “π1pm11q, hencem´m11Pkerpπ1q. It follows that
M “ψpM1q `kerpπ1q “ψpM1q ‘kerpπ1q “ψpM1q ‘M2‘ ¨ ¨ ¨ ‘Mn. Hence the Claim holds.
Now, we have IdM “řn1
j“1ψj, and so IdM1 “řn1
j“1π1˝ψj|M
1PEndRpM1q. But asM has a composition series, so has M1, and therefore EndRpM1q is local by Proposition 5.4. Thus if all the π1˝ψj|M
1 P
EndRpM1q are not invertible, they are all nilpotent and then so is IdM1, which is in turn not invertible.
This is not possible, hence it follows that there exists an indexj such that π1˝ψj|M
1 :M1 ÑM1
is an isomorphism and the Claim implies thatM“ψjpM1q ‘M2‘ ¨ ¨ ¨ ‘Mn andψjpM1q –M1. We then setπp1q:“j. By definitionψjpM1q ĎMj1 asMj1 is indecomposbale, so that
ψjpM1q –Mj1 “Mπp1q1 . Finally, an induction argument (Exercise!) yields:
M “Mπp1q1 ‘ ¨ ¨ ¨ ‘Mπprq1 ‘
n
à
j“r`1
Mj,
mitMπpiq1 –Mi (1ďiďr). In particular, the caser“nimplies the equalityn“n1.
In this chapter we study an important class of rings: the class of rings R which are such that any R- module can be expressed as a direct sum ofsimple R-submodules. We study the structure of such rings through a series of results essentially due to Artin and Wedderburn. At the end of the chapter we will assume that the ring is a finite dimension algebra over a field and start the study of its representation theory.
Notation: throughout this chapter, unless otherwise specified, we letR denote a unital and associative ring.
References:
[CR90] C. W. Curtis and I. Reiner.Methods of representation theory. Vol. I. John Wiley & Sons, Inc., New York, 1990.
[Dor72] L. Dornhoff. Group representation theory. Part B: Modular representation theory. Marcel Dekker, Inc., New York, 1972.
[NT89] H. Nagao and Y. Tsushima. Representations of finite groups. Academic Press, Inc., Boston, MA, 1989.
[Rot10] J. J. Rotman. Advanced modern algebra. 2nd ed. Providence, RI: American Mathematical Society (AMS), 2010.
6 Semisimplicity of rings and modules
To begin with, we prove three equivalent characterisations for the notion of semisimplicity.
Proposition 6.1
IfM is anR-module, then the following assertions are equivalent:
(a) M is semisimple, i.e. M“ ‘iPISi for some familytSiuiPI of simpleR-submodules ofM; (b) M“ř
iPISi for some familytSiuiPI of simpleR-submodules of M;
(c) every R-submodule M1 Ď M admits a complement in M, i.e. D an R-submodule M2 Ď M such thatM“M1‘M2.
15
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 16
Proof :
(a)ñ(b): is trivial.
(b)ñ(c): Write M “ř
iPISi, where Si is a simple R-submodule of M for eachi PI. Let M1 Ď M be an R-submodule of M. Then consider the family, partially ordered by inclusion, of all subsetsJ ĎI such that
(1) ř
iPJSi is a direct sum, and (2) M1Xř
iPJSi“0.
Clearly this family is non-empty since it contains the empty set. Thus Zorn’s Lemma yields the existence of a maximal elementJ0. Now, set
M1:“M1`ÿ
iPJ0
Si “M1‘ÿ
iPJ0
Si,
where the second equality holds by (1) and (2). Therefore, it suffices to prove that M “M1, i.e.
that Si ĎM1 for every i PI. But if j P I is such thatSj ĘM1, the simplicity of Sj implies that SjXM1“0 and it follows that
M1`Sj “M1‘
˜ ÿ
iPJ0
Si
¸
‘Sj
in contradiction with the maximality ofJ0. The claim follows.
(b)ñ(a): follows from the argument above withM1“0.
(c)ñ(b): LetM1 be the sum of all simple R-submodules inM. By (c) there exists a complementM2 ĎM to M1, i.e. such that M “M1‘M2. If M2 “0, we are done. IfM2 ‰0, then M2 must contain a simple R-submodule (Exercise: prove this fact), say N. But then N ĎM1 by definition of M1, a contradiction. ThusM2 “0 and soM “M1.
Example 2
(a) The zero module is completely reducible.
(b) IfS1, . . . , Snare simpleR-modules, then their direct sumS1‘. . .‘Snis completely reducible by definition.
(c) The following exercise shows that there exists modules which are not completely reducible.
Exercise (Sheet 1): LetK be a field and letA be theK-algebra `a
1 a 0 a1
˘|a1, aPK( . Con- sider the A-moduleV :“K2, where Aacts by left matrix multiplication. Prove that:
(1) tpx0q |x PKu is a simple A-submodule ofV; but (2) V is not semisimple.
(d) Exercise (Sheet 1): Prove that any submodule and any quotient of a completely reducible module is again completely reducible.
Theorem-Definition 6.2 (Semisimple ring)
A ringR satisfying the following equivalent conditions is called semisimple. (a) All short exact sequences of R-modules split.
(b) AllR-modules are semisimple.
(c) All finitely generatedR-modules are semisimple.
(d) The regular leftR-moduleR˝ is semisimple, and is a direct sum of a finite number of minimal left ideals.
Proof : First, (a) and (b) are equivalent as a consequence of Lemma D.4 and the characterisation of semisimple modules given by Proposition 6.1(c). The implication (b) ñ (c) is trivial, and it is also trivial that (c) implies the first claim of (d), which in turn implies the second claim of (d). Indeed, ifR˝ “À
iPILi for some familytLiuiPI of minimal left ideals. Then, by definition of a direct sum, there exists a finite number of indicesi1, . . . , inPI such that 1R “xi1 `. . .`xin withxij PLij for each 1 ďj ďn. Therefore each aPR may be expressed in the form
a“a¨1R “axi1`. . .`axin
and henceR˝“Li1`. . .`Lin.
Therefore, it remains to prove that (d)ñ(b). So, assume thatR satisfies (d) and let M be an arbitrary non-zeroR-module. Then writeM“ř
mPMR¨m. Now, each cyclic submodule R¨mofM is isomorphic to anR-submodule ofR˝, which is semisimple by (d). ThusR¨mis semisimple as well by Example 2(d).
Finally, it follows from Proposition 6.1(b) thatM is semisimple.
Example 3
Fields are semisimple. Indeed, ifV is a finite-dimensional vector space over a fieldK of dimensionn, then choosing a K-basis te1,¨ ¨ ¨ , enu of V yields V “ K e1‘. . .‘K en, where dimKpK eiq “ 1, hence K ei is a simple K-module for each 1 ď i ď n. Hence, the claim follows from Theorem- Definition 6.2(c).
Corollary 6.3
LetR be a semisimple ring. Then:
(a) R˝ has a composition series;
(b) R is both left Artinian and left Noetherian.
Proof :
(a) By Theorem-Definition 6.2(d) the regular moduleR˝admits a direct sum decomposition into a finite number of minimal left ideals. Removing one ideal at a time, we obtain a composition series forR˝. (b) Since R˝ has a composition series, it satisfies both D.C.C. and A.C.C. on submodules by Corol-
lary 3.4. In other words,R is both left Artinian and left Noetherian.
Next, we show that semisimplicity is detected by the Jacobson radical. This leads us to introduce a slightly weaker concept: the notion ofJ-semisimplicity.
Skript zur Vorlesung: Modular Representation Theory WS 2020/21 18 Definition 6.4 (J-semimplicity)
A ringR is said to beJ-semisimpleif JpRq “0.
Exercise 6.5 (Sheet 1)
Let R “ Z. Prove that JpZq “ 0, but not all Z-modules are semisimple. In other words, Z is J-semisimple but not semisimple.
Proposition 6.6
Any left Artinian ringR isJ-semisimple if and only if it is semisimple.
Proof :“ñ”: Assume R ‰ 0 and R is not semisimple. Pick a minimal left ideal I0 IJ R (e.g. a minimal element of the family of non-zero principal left ideals ofR). Then 0‰I0 ‰R since I0 seen as an R-module is simple.
Claim: I0 is a direct summand ofR˝. Indeed: since
I0‰0“JpRq “ č
ImaximalICR, left ideal
I
there exists a maximal left idealm0CR which does not containI0. Thus I0Xm0“ t0uand so we must haveR˝“I0‘m0, as R{m0 is simple. Hence the Claim.
Notice that thenm0 ‰0, and pick a minimal left idealI1 inm0. Then 0‰I1 ‰m0, else R would be semisimple. The Claim applied toI1 yields thatI1 is a direct summand ofR˝, hence also inm0. Therefore, there exists a non-zero left idealm1 such thatm0 “I1‘m1. Iterating this process, we obtain an infinite descending chain of ideals
m0Ľm1 Ľm2Ľ ¨ ¨ ¨
contradicting D.C.C.
“ð”: Conversely, if R is semisimple, then R˝ – R{JpRq ‘JpRq by Theorem-Definition 6.2 and so as R-modules,
JpRq “JpRq ¨ pR{JpRq ‘JpRqq “JpRq ¨JpRq so that by Nakayama’s LemmaJpRq “0.
Proposition 6.7
The quotient ring R{JpRq isJ-semisimple.
Proof : Since by Exercise 4.2 the ringsR andR:“R{JpRqhave the same simple modules (seen as abelian groups), Proposition-Definition 4.1(a) yields:
JpRq “ č
Vsimple R-module
annRpVq “ č
Vsimple R-module
annRpVq `JpRq “JpRq{JpRq “0
7 The Artin-Wedderburn structure theorem
The next step in analysing semisimple rings and modules is to sort simple modules into isomorphism classes. We aim at proving that each isomorphism type of simple modules actually occurs as a direct summand of the regular module. The first key result in this direction is the following proposition:
