Furthermore, all modules considered are assumed to beleftmodules and finitely generated

RepresentationTheory— ExerciseSheet6 TU Kaiserslautern

Jun.-Prof. Dr. CarolineLassueur FB Mathematik

BernhardB ¨ohmler

Due date:Wednesday, 3rd of February 2021, 10 a.m. WS 2020/21

Throughout,Gdenotes a finite group. Furthermore, all modules considered are assumed to beleftmodules and finitely generated.

A. Exercises for the tutorial.

Exercise1.

Let (F,O,k) be ap-modular system and writep:= J(O). LetLbe anOG-module. Verify that

· settingL^F:=F⊗_OLdefines anFG-module, and

· reduction modulopofL, that isL:=L/pLk⊗_OLdefines akG-module.

Exercise2.

LetObe a complete discrete valuation ring and letF :=Frac(O) be the fraction field ofO. LetVbe a finitely generatedFG-module and let{v₁, . . . ,vn}be anF-basis ofV. Prove that L:=OGv₁+· · ·+OGv_n⊆Vis anO-form ofV.

Exercise3.

LetObe a commutative ring. LetAbe a finitely-generatedO-algebra of finiteO-rank and lete∈Abe an idempotent element. LetVbe anA-module. Prove that

Hom_A(Ae,V)eV as End_A(V)-modules.

B. Exercises to hand in.

Exercise4.

LetObe a local commutative ring with unique maximal idealp:= J(O) and residue field k:=O/J(O).

(a) LetM,Nbe finitely generated freeO-modules.

(i) Let f : M −→ N is anO-linear map and f : M −→ N its reduction modulo p.

Prove that if f is surjective (resp. an isomorphism), then fis surjective (resp. an isomorphism).

(ii) Prove that if elements x₁, . . . ,xn ∈ M (n ∈ Z^≥1) are such that their images x₁, . . .x₂∈Mform ak-basis ofM, then{x₁, . . . ,x_n}is anO-basis ofM.

In particular, dim_k(M)=rkO(M).

Deduce that any direct summand of a finitely generated freeO-module is free.

(b) Prove that ifM is a finitely generatedO-module, then the following conditions are equivalent:

(i) Mis projective;

(ii) Mis free.

[Hint: Use Nakayama’s Lemma.]

Exercise5.

LetObe a complete discrete valuation ring. LetAandBbe a finitely generatedO-algebras of finiteO-rank and let f :ABbe a surjectiveO-algebra homomorphism. Prove that:

(a) f mapsJ(A) ontoJ(B); and (b) f mapsA^×ontoB^×.

Exercise6.

LetObe a complete discrete valuation ring and writep:=J(O). LetAbe a finitely generated O-algebra of finiteO-rank. SetA:=A/pAand fora∈Awritea:=a+pA. Prove that:

(a) For every idempotentx∈A, there exists an idempotente∈Asuch thate=x.

(b) A^×={a∈A|a∈A^×}.

(c) Ife₁,e₂ ∈ Aare idempotents such thate₁ = e₂ then there is a unit u ∈ A^× such that e₁=ue2u⁻¹.

(d) The quotient morphismA→Ainduces a bijection between the central idempotents ofAand the central idempotents ofA.