Modular representation theory 1 Denitions for the study group Denition 1.1. Let A be a ring and let FA be the category of all left A-modules. The Grothendieck group of FA is the abelian group dened by generators and relations as follows: 1. Generators [E] for E 2 FA. 0 00 0 00 0 00 2. Relations: for E; E ;E 2 FA if 0 ! E ! E ! E ! 0, then [E ] = [E] + [E ]. Denition 1.2. A left A-module P is called projective if any of the following hold: 1. There exists a free A-module F such that F = P ⊕ Q for Q 2 FA 2. P 9! ~ E / E0 / 0 3. The functor HomA(P; −) is exact. Denition 1.3. An A-module homomorphism f : M ! M 0 is called essential if f(M) = M 0 and f(M 00) 6= M 0 for any proper submodule M 00 ⊆ M. Denition 1.4. The projective envelope of a module M is a projective module P and an essential homomorphism f : P ! M. Setup Let G be a nite group, K be a complete eld of characteristic 0 with a discrete valuation v, with residue eld k of characteristic > 0. Let A be the ring of integers of K. We are going to look at K[G]; k[G]and A[G]. Denition 1.5. Let L be a eld, denote by RL(G) the Grothendieck group of nitely generated L[G]-modules. 0 0 We make it into a ring by setting [E] · [E ] = [E ⊗L E ]. Let SL be the set of isomorphism classes of simple L[G]-modules. Proposition 1.6. The element of SL form a basis for RL[G]. Denition 1.7. Let Pk[G] be the Grothendieck ring of the category of left k[G]-modules which are projective. Let PA[G] be the Grothendieck ring of the category of the left A[G]-modules which are projective. Fact. Any k[G]-module has a projective envelope If E; P are k[G]-modules and P is projective, then E ⊗k P is also projective ) we have an action of Rk[G] on Pk[G]. Proposition 1.8. Each projective k[G]-module can be written uniquely as a direct sum of indecomposable projective k[G]-modules Let P and P 0 be projective k[G]-modules, then they are isomorphic if and only if [P ] = [P 0]. 1 Dualities: Let be -modules. Set (i.e, ). This is E; F K[G] hE; F i = dim HomG(E; F ) h ; i : RK [G] × RK [G] ! Z bilinear with respect to short exact sequences. ( 0 E =6∼ E0 Proposition 1.9. If E; E0 2 S then hE; E0i = . K ∼ 0 dE = dim EndG(E) E = E Denition 1.10. Say E is absolutely simple if dE = 1. We dene by . Since is projective, is h ; i : Pk[G] × Rk[G] ! Z (E; F ) 7! hE; F i = dim HomG(E; F ) E h ; i bilinear. 0 0 0 Proposition 1.11. If E; E 2 Sk then HomG(PE;E ) = HomG(E; E ). ( 0 E =6∼ E0 If E; E0 2 S then hP ;E0i = k E ∼ 0 dE E = E Proposition 1.12. Let K be suciently large such that k contains all m-th roots of unity (m = lcm(jgj)g2G). Then dE = 1 for E 2 Sk and hence h ; i is non-degenerate and the base [E] ; [PE] are dual to each other. 2 The cde Triangle We will want to dene three maps for the following triangle to commute c Pk(G) / Rk(G) O d e $ RK (G) Denition 2.1. Cartan Homomorphism. To each k[G]-module P , we associate the class of P , [P ] in Rk(G). We get a homomorphism c : Pk(G) ! Rk(G). We express c in terms of the canonical basis [PS]; [S] for S 2 Sk (basis for Pk(G) and Rk(G) respectively). We obtain a square matrix C, of type Sk × Sk, the Cartan Matrix of G with respect to k. The (S; T ) coecient, CST of C is the number of times that S appears in a composition series of the projective envelope PT of T . X [PT ] = CST [S] in Rk(G) S2Sk Denition 2.2. Decomposition Homomorphism. Let E be a K[G]-module. Choose a lattice E1 in E (nitely generated A-module which generates E as a K-module). We may assume that E1 is G-stable (replace E1 by the sum of its images under G). Denote the reduction E1 = E1=mE1, a k[G]-module. ∼ Theorem 2.3. The image E1 in Rk(G) is independent of the choice of E1. (Warning! You can have that E1 =6 E2 but with the same composition factors) Proof. Let E2 be another G-stable lattice, we want to show E1 = E2 in Rk(G). Replace E2 by a scalar multiple, n we may assume E2 ⊂ E1. There exists n ≥ 0 such that m E1 ⊂ E2 ⊂ E1. If n = 1, we have mE1 ⊂ E2 ⊂ E1. We have 0 ! E2=mE1 ! E1=mE1 ! E1=E2 ! 0 implies [E1=mE1] = [E2=mE1] + [E1=E2]. And 0 ! mE1=mE2 ! E2=mE2 ! E1=mE1 ! 0 implies [E2=mE2] = [mE1=mE2] + [E2=mE1]. But [mE1=mE2] = [E1=E2] hence [E1] = [E2]. This covers our base case. n−1 n−1 For general n, let E3 = m m + E2. We have m E1 ⊂ E3 ⊂ E1, mE3 ⊂ E2 ⊂ E3 ) [E2] = [E3] ) E1 = E2 = E3 2 The map E ! E1 extends to a ring homomorphism d : RK (G) ! Rk(G). The corresponding matrix is denoted D, type Sk × SK . For F 2 Sk;E 2 SK ;DFE is the number of times that F appears in the reduction mod m of E1 X E1 = DFE[F ] in Rk(G): F Denition 2.4. e. ∼ There exists a canonical isomorphism PA(G) ! Pk(G), i.e., • Let E be an A[G]-module, for E to be a projective A[G]-module then (only then) E is a to be free on A and E is a projective k[G]-module • If F is a projective k[G]-module, then there exists a unique (up to isomorphism) projective A[G]-module whose reduction modulo m is isomorphic to F . The functor tensor product with K denes a homomorphism from PA(G) into RK (G). Combining both, we get a map e : Pk(G) ! RK (G) and an associated matrix E of type SK × Sk. 2.1 Basic properties of cde triangle 1. Commutativity: c = d ◦ e, i.e., C = DE 2. d and e are adjoint of one another with respect to bilinear forms on Rk(G);Pk(G). I.e., pick x 2 Pk(G) and then . y 2 RK (G) hx; d(y)ik = he(x); yiK Assume x = X where X is a projective A[G]-module, y = [K ⊗A Y ] where Y is an A[G]-module which is free. Then the A-module HG(X; Y ) is free. Let r be its rank: we have the canonical isomorphism K ⊗ HomG(X; Y ) = HomG(K ⊗ X; K ⊗ Y ) and k ⊗ HomG(X; Y ) = HomG(k ⊗ X; k ⊗ Y ). These both implies that . he(x); yiK = t = hx; d(y)ik 3. Assume that K is suciently large. The canonical bases of Pk(G) and Rk(G) are dual to each other with respect to . (Similarly and ) Hence can be identied with the transpose of . h ; ik RK (G);RK (G) h ; iK e d E = tD, C = DE ) C is symmetric. 3 Examples Notation. • K is a complete eld • A is the valuation ring of K, m its maximal ideal • k the residue eld of K, with characteristic p • K is suciently large, that is µn ⊆ K with n = lcmg2G(o(g)) We have the cde triangle c Pk(G) / Rk(G) O d e $ RK (G) • c : P ! [P ] • d : E ! E1 = E1=mE1 where E1 is a stable lattice in E under G • e: is the inverse of the canonical isomorphism PA(G) ! Pk(G) tensor with K Denition 3.1. A projective envelope P of a L[G] module M is a projective module P with an essential homo- morphism. 3 3.1 Example, p0-groups Proposition 3.2. Assume that the order of G is coprime to p. Then 1. Each k[G]-module (respectively each A-free A[G]-module) is projective 2. The operation of reduction mod m denes a bijection between SK and Sk 3. If we identify SK with Sk, then C; D; E are all identity matrices. Proof. 1. Let E be an A-free A[G]-module, then there is a free A[G]-module L such that E is L=R. Since E is A-free there exists a A-linear projection π of L onto R. If g = #G, we know gcd(g; p) = 1, hence we can replace with 1 P −1 and we get an -projection. By the fact -linear projection and the fact that π g s2G sπs A[G] A[G] L=R =∼ E, we get that E is projective. Moreover C = Id 2. and 3. Let E 2 Sk, the projective envelope E1 of E relative to A[G] is a projective A[G]-module, whose reduction E1 = E1=mE1 is E. If F = K ⊗ E1, then d([F ]) = [E]. Since E is simple, that implies that F is simple, thus F 2 SK . So, we obtain a map E ! F of Sk to SK which is the inverse of d.