Prove the following statements

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RepresentationTheory— ExerciseSheet5 TU Kaiserslautern

Jun.-Prof. Dr. CarolineLassueur FB Mathematik

BernhardB ¨ohmler

Due date:Wednesday, 20th of January 2021, 10 a.m. WS 2020/21

Throughout, K denotes a commutative ring and G a finite group. Furthermore, allKG- modules considered are assumed to beleftmodules and free of finite rank overK.

A. Exercises for the tutorial.

Exercise1.

Let H ≤ J ≤ G. LetU be a KG-module and letV be a KJ-module. Prove the following statements.

(a) IfUisH-projective thenUisJ-projective.

(b) IfUis a direct summand ofV↑^G

J andVisH-projective, thenUisH-projective.

(c) For anyg∈G,UisH-projective if and only if ^gUis ^gH-projective.

(d) IfUisH-projective andWis anyKG-module, thenU⊗_KWisH-projective.

[Hint: use part (f) of Proposition 25.4.]

Exercise2.

AssumeKis a field of characteristcp>0 and letA,B,C,U,VbeKG-modules. Prove that:

(a) Any direct summand of aV-projectiveKG-module isV-projective;

(b) IfU∈Proj(V), then Proj(U)⊆Proj(V);

(c) Ifp-dimK(V) then anyKG-module isV-projective;

(d) Proj(V)=Proj(V^∗);

(e) Proj(U⊕V)=Proj(U)⊕Proj(V);

(f) Proj(U)∩Proj(V)=Proj(U⊗_KV);

(g) Proj(Ln

j=1V)=Proj(V)=Proj(Nm

j=1V) ∀m,n∈Z_>0;

(h) CA⊕BisV-projective if and only if bothAandBareV-projective;

(i) Proj(V)=Proj(V^∗⊗_KV).

Hint: Use Lemma 13.8 and Exercise 4(c) on Sheet 3. Proceed in the given order.

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B. Exercises to hand in.

Assume now thatKis a field of positive characteristicp.

Exercise3.

LetMbe aKG-module with dimension coprime top. Prove that the vertices ofMare the Sylowp-subgroupsG.

Exercise4.

LetQ≤Gbe ap-subgroup and letL≤G. Prove the following assertions.

(a) IfUis an indecomposableKG-module with vertexQandL≥Q, then there exists an indecomposable direct summand ofU↓^G

L with vertexQ.

(b) IfL≥N_G(Q), then the following assertions hold.

(i) IfVis an indecomposableKL-module with vertexQandUis a direct summand ofV↑^G

L such thatV|U↓^G

L, thenQis also a vertex ofU.

(ii) IfVis an indecomposableKL-module which isQ-projective and there exists an indecomposable direct summandUofV↑^G

L with vertexQ, thenQis also a vertex ofV.

Exercise5.

(a) Verify that modules corresponding to each other via the Green correspondence have a source in common.

(b) Compute the Green correspondent of the trivial module.