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Home , Grothendieck group, Simple module

BASIC MODULAR

GYUJIN OH

Contents 1. Algebra 1 1.1. Related randos 2 2. cde triangle 2 3. Brauer characters 4 4. Green correspondence 5 5. Block and defect 7 6. Example: SL2(Fp) 9 6.1. Brauer tree 10 7. Ext 10 8. Auslander-Reiten theory 11

I scratched the surface of Schneider’s Modular representation theory of nite groups.

1. Algebra Here our base is unital and our modules are left modules. ∙ Any artinian and noetherian has Jordan-Holder . ∙ A is a of simple modules. Every module has a unique maximal semisimple submodule which we call as socle. It is just the sum of all simple submodules. ∙ Radical is intersection of maximal submodules. So that the radical of base ring is the usual . Semisimple = artinian + zero radical. ∙ If M is artinian and f ∈ EndR(M) is injective, then f is bijective (Fitting’s lemma). Same for noetherian/surjective. This uses increasing/decreasing chain of im(f n) and ker(f n). In particular this shows that, for any indecomposable finite length R, EndR(M) is local. ∙ Given any idempotent e ∈ R, eRe ⊂ R is a subring with unit element e. Decomposing e = e1 + ⋯ + er as sum of pairwise orthogonal idempotents is the same as decomposing Re = Re1 ⊕ ⋯ ⊕ Rer into sum of nonzero left ideals. Thus Re is indecomposable i e is primitive. ∙ If R is left noetherian, R has only nitely many central idempotents, primitive central idempotents are pairwise orthogonal, and they sum up to 1. Given a central primitive idempotent e, M is said to belong to e-block if eM = M. Then not only eM = M, but ex = x for all x ∈ M. 1 ∙ For a 2-sided I ⊂ R such that either every element in I is (topologically) nilpotent. Then, any idempotent of R/I lifts to an idempotent of R. This operation can be done preserving primitivity and orthogonality. ∙ A denition of is that P such that HomR(P, −) is an exact . A module hom f ∶ M → N is essential if it’s surjective but f (L) ≠ N for any proper L ⊂ M. A projective cover of M is an essential R-module hom f ∶ P → M where P is projective. There is at most one projective cover up to isomorphism. ∙ Conversely if R is complete and R/ Jac(R) is left artinian, then any fg module has projective cover, which mod Jac(R) is an iso. ∙ The Grothendieck R(A) of category of nite length A-modules is the same as the free ℤ-module generated over simple A-modules (denoted as A). The K0(A) of category of nitely generated projective A-modules is generated over ℤ by nitely generated indecomposable projective A-modules (denoted as Až), if A is left noetherian. ∙ In particular, if A is semisimple, then any A-module is projective. ž ∙ If A is left artinian, then K0(A) is freely generated by A. Thus we can consider the com- ž  posed homomorphism cA ∶ ℤ[A] → ℤ[A] called the Cartan homomorphism. ∙ If A is complete and A = A/ Jac(A) is left artinian, then Grothendieck groups for A and A are the same. So, A → A/ Jac(A) is a projective cover, and A = P1 ⊕ ⋯ ⊕ Pr , a decomposi- tion into a sum of nitely generated indecomposable A-modules, and every such module occurs as direct summand. ž  ∙ So if A is left artinian, we have A = {P1, ⋯ ,Pt }, and thus A = {M1, ⋯ ,Mt } where Mi = Pi/ Jac(A)Pi. 1.1. Related randos. ∙ A is a nite-dimensional algebra over a eld with a nondegenerate symmetric bilinear form such that (xy, z) = (x, yz). In particular k[G] is a Frobenius algebra via (x, y) ↦ xy(1) (the coecient of 1 of xy). This I think is also referred as symmetric Frobenius algebra at some places. ∙ A module is projective and indecomposable i P/ rad(P) = soc(P). ∙ Given a ring hom f ∶ A → B, a B-module P is relatively projective wrt f if, for any commutative diagram P

0 ~  M / N

where , are B-module hom’s while 0 is an A-module hom, there is a B-module hom ∶ P → M replacing 0 still making the diagram commute. ∙ For any A-module L0, B ⊗A L0 is relatively projective. More generally P is relatively pro- jective i it is a direct summand of B ⊗A L0 for some A-module L0. ∙ There is criterion for left R[G]-module to be relatively R[H]-projective (for H ≤ G, R any commutative ring).

2. cde triangle We do modular representation theory. Let R be a (0, p)-ring for k, where k is an algebraically closed eld of characteristic p > 0, R is a complete local commutative domain such that mR is 2 principal, residue eld is k and K = Frac(R) is of characteristic zero. Then there is the ramication e index e ≥ 1 such that Rp = mR. The trinity of the cde triangle is ∙ RK (G) ∶= R(K[G]) = K0(K[G]), ∙ Rk(G) ∶= R(k[G]), ∙ K0(k[G]). Here G is a nite group.

∙ The c. This is the Cartan homomorphism cG ∶ K0(k[G]) → Rk(G). This makes sense because k[G] is a nite-dimensional k- so of course this is left artinian. ∙ The d. This map dG ∶ RK (G) → Rk(G) is something like “mod p reduction.” Of course as usual this is not a priori well-dened. Namely, one chooses a G-invariant lattice inside a G-representation over K (which is possible b/c of averaging) and then reduce mod p (or R), but the resulting representation might not be isomorphic. But considering Rk(G) is like we’re in a “semisimplied world” and in particular the resulting module in Rk(G) is always the same. This is almost the same as Brauer-Nesbitt theorem. This particular statement can be proved by passing to a common renement (after mutiplying by suciently high power of R). – Perhaps a more classical statement of Brauer-Nesbitt is that a semisimple represen- tation of G over k is completely determined by characteristic polynomials of g ∈ G’s. ¤ ¤ ¡ ∙ The e. By the Jacobson radical theory we did before, K0(R[G]) ≅ ℤ[R[G]], R[G] = k[G] ¡ and K0(k[G]) ≅ ℤ[k[G]]. Thus we can dene eG ∶ K0(k[G]) → RK (G) by ∼ K0(k[G]) ←ãããããããã K0(R[G]) → K0(K[G]). Thus the triangle is of form

dG RK (G) / Rk(G) e 9

eG cG

K0(k[G]) Quite obviously the triangle commutes. ∙ If p does not divide |G|, then cde are all isomorphisms. ∙ If G is a p-group, then Jac(k[G]) = Ik[G], the augmentation ideal. In particular, k[G] is a , and the trivial module is the only simple k[G]-module (“p-group lemma” says that there is always a xed vector). Thus Rk(G) and K0(k[G]) are both ℤ, and the diagram becomes [V ]↦dimK V RK (G) / ℤ b ?

1↦K[G] ⋅|G| ℤ I am not doing Cliord theory. ∙ Induction gives a well-dened map G indH ∶ RF (H) → RF (G),

[W ] ↦ [F[G] ⊗F[H] W ], 3 for any eld F and H ≤ G. This is not a homomorphism but rather the is an ideal in RF (G). ∙ Induction commutes with d. ∙ Brauer induction theorem says that G Rk(G) = ∑ indH (Rk(H)), H∈e where e is the set of elementary of G. From this one sees that d is surjective. m m ∙ c is injective and has nite cokernel. Furthermore p Rk(G) ⊂ im(c), where p ‖|G|. Proof uses Brauer induction to reduce to case of elementary subgroups. ∙ e is injective and its image is a direct summand of RK (G). This uses two bilinear forms ([V ], [W ]) ↦ dimK HomK[G](V,W ) RK (G) × RK (G) ããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããã→ ℤ,

([P], [V ]) ↦ dimk Homk[G](P,V ) K0(k[G]) × Rk(G) ããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããã→ ℤ. The second bilinear form is perfect, and the rst one is also perfect if K is a splitting eld for G. ∙ Furthermore this duality shows that, over suciently large eld, so that K0(k[G]) and Rk(G) are free over ℤ, the matrices representing d and e are transposes to each other. Therefore, the Cartan matrix corresponding to the natural bases is symmetric!

3. Brauer characters ∙ Usual charcter theory gives a homomorphism

Tr ∶ Rk(G) → Cl(G, k), where Cl(G, k) means the space of class functions. By Artin-Wedderburn,

Tr ∶ k ⊗ℤ Rk(G) → Cl(G, k) is injective. This is why in characteristic p the usual is less useful; you lose information by k⊗ℤ. We use V as the character of V . ∙ Somehow quite surprisingly there is the notion of p-regular(∼semisimple)/p-unipotent elements (e.g. there is Jordan decomposition), and the general slogan is that mod p repre- sentation theory is encoded not in all conjugacy classes but rather in p-regular conjugacy classes. ∙ To be more precise, a p-regular conjugacy class is a conj. class whose elements have prime to p. An element is p-unipotent if it is of order a power of p. Then, there is Jordan decomposition, namely for any g ∈ G, there are unique gr , gu ∈ G such that gr is p- regular, gu is p-unipotent, g = gr gu = gugr . The construction is quite simple, namely you just take appropriate powers of g. – Finite groups are algebraic. Ho ho ho. ∙ Now V (g) = V (gr ), because all roots of characteristic polynomials for g and gr are the same even with multiplicities; as gr is semisimple on V (order prime to p), V decomposes into gr -eigenspaces, and gu acts on each eigenspace (b/c gu and gr commute) with eigen- values 1 (b/c p-group lemma tells that only is a trivial module). Thus if we denote Greg as the set of p-regular elements, then

Trreg ∶ k ⊗ℤ Rk(G) → Cl(G, k) → Cl(Greg, k), 4 is injective. ∙ Let e be the exponent of G and e = e¨ps where (p, e¨) = 1. Then there is Teichmuller character e¨ (k) → e¨ (K). Using this we can now dene Brauer character V of V ,

V (g) = [1(g, V )] + ⋯ + [d (g, V )],

where [−] is Teichmuller representative and i’s are roots of characteristic polynomial of g. This gives a homomorphism

TrB ∶ Rk(G) → Cl(Greg,K). ∙ Brauer characters give linearly independent characters.

∙ If V ∈ RK (G), then V , restricted to Greg, is the same as dG (V )! In particular, TrB ∶ K ⊗ℤ Rk(G) → Cl(Greg,K) is an isomorphism. ∙ This shows that the number of isomorphism classes of simple k[G]-modules is equal to the number of p-regular conjugacy classes, and that Trreg ∶ k ⊗ℤ Rk(G) → Cl(Greg, k) is an isomorphism.

4. Green correspondence ∙ Let R be a commutative unital ring. For an R[G]-module M, let (M) be the set of sub- groups H ≤ G such that M is relatively R[H]-projective. This set contains G, is closed under conjugation. ∙ Let 0(M) ⊂ (M) be the subset consisted of subgroups in (M) minimal with respect to inclusion. This is called the set of vertices. ∙ If every except p is invertible in R, then all vertices are p-groups. ∙ If R is noetherian and R/ Jac(R) is artinian, then for any nitely generated indecompos- able R[G]-module M, 0(M) is consisted of a single conjugacy class of subgroups. This is because by Mackey theory G G G −1 ∗ Ind (Ind (M)) ≅ ⊕ Ind −1 ((g ) M), H0 H1 H0∩giH1gi i for any H0,H1 ∈ 0(M). ∙ For M f.g. indecomposable, V ∈ 0(M), a V -source is a fg indecomposable R[V ]-module G Q such that M is isomorphic to a direct summand of IndV (Q). ∙ If R noetherian complete and R/ Jac(R) artinian, and M fg indecomposable R[G]-module, then for every V ∈ 0(M), there is a V -source, and any two V -sources are conjugates to each other by an element in normalizer. Now we keep assuming that R is noetherian complete and R/ Jac(R) (so that one can use Krull- Remak-Schmidt theorem, i.e. there is unique decomposition into sum of indecomposables). ∙ If M is fg indecomposable R[G]-module and if H ∈ (M), then let M = L1 ⊕ ⋯ ⊕ Lr be G decomposition as R[H]-module. Then as M is a direct summand of IndH M, it is a direct G G summand of IndH Li for some i. If it is a direct summand of IndH Li, then vertex-conjugacy- class of Li (in H) and that of M (in G) are the same (in G). In particular M and Li have a common Vi-source. ∙ A recurring idea of proof is that Li is a direct summand of M as R[H]-modules whereas M G is a direct summand of IndH Li as R[G]-modules, so one can go back and forth. ∙ Now we introduce several sets. 0 IH (M) = {Li ∣ 0(Li) ⊂ 0(M)} 5 1 ⊕ G IH (M) = {Li ∣ M ⊂ IndH Li} H 0 (M) = {V ∈ 0(M) ∣ V ⊂ H} 1 0 The above point says that IH (M) ⊂ IH (M). Also dene a map

M 0 H vH ∶ IH (M) → H-orbits in 0 (M) ,

Li ↦ 0(Li). This map is surjective, which you can see easily if you see things as R[V ]-modules for H each V ∈ 0 (M). ∙ So a possible source of problem is that even if you start with a vertex V ⊂ H, maybe G- conjugacy class of V can introduce other H-conjugacy class. So if NG(V ) ⊂ H there is no M such problem and vH is bijective. – So in this setting we have Green correspondence: there is a bijection ⎧ ⎫ ⎧ ⎫ ⎪ iso-class of fg ⎪ ⎪ iso-class of fg ⎪ ⎪ indecomposable ⎪ ⎪ indecomposable ⎪ Γ ∶ ⎨ ⎬ → ⎨ ⎬ , ⎪ R[G]-module with ⎪ ⎪ R[H]-module with ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ vertex V ⎭ ⎩ vertex V ⎭ with the property that Γ(M) is a direct summand of M. This map is equivalently G characterized by M is a direct summand of IndH (Γ(M)). – The Green correspondence has the following rigidity property. Namely if L = Γ(M), then for any other fg indecopmosable R[G]-module M ¨, if L is a direct summand of M ¨, then M ¨ ≅ M. Furthermore M is a direct summand of a fg R[G]-module M ¨ i L is a direct summand of M ¨. At the other extreme, there is Green’s indecomposability theorem (still under the assumption of Krull-Schmidt). Let R be furthermore be local such that R/ Jac(R) = k is an alg.closed char p eld. Theorem 4.1 (Green’s indecomposability theorem). If N ⊂ G is a normal such that G [G ∶ N ] is a power of p, then for any fg indecomposable R[N ]-module L, IndN L is fg indecomposable R[G]-module.

Proof. By induction we can assume that [G ∶ N ] = p. Let IG(L) be the subgroup of G consisted ∗ G of g such that g L ≅ L. Then either IG(L) = N or G, and if IG(L) = N , then quite easily IndN (L) is G indecomposable. If IG(L) = G, then EG = EndR[G](IndN (L)) is complete local with EG/ Jac(EG) = k, G so 1 is the only idempotent, which means that IndN (L) is indecomposable in this case too. The reason why EG is local and EG/ Jac(EG) = k is as follows. ETS EG/ Jac(EG) = k. Consider EN = G 0 0 EndR[N ](IndN (L)). By Mackey theory, EN ≅ Mp(EN ) where EN = EndR[N ](L) so that EN / Jac(EN ) = −1 Mp(k). The subring EG ⊂ EN is characterized by the condition { ∈ EN ∣   = } where G × G  ∈ EndR(IndN (L)) is the eect of action of a chosen generator ℎ of G/N on IndN (L). Quite easily one sees that conjugation by  restricts to EN , which we denote by Θ. This automorphism Θ on 0 i G Mp(EN ) sends ei to ei+1 where ei is the idempotent to ℎ L-factor in the decomposition IndN (L) = p−1 L ⊕ ℎL ⊕ ⋯ ⊕ ℎ L by Mackey theory. As EG/ Jac(EG) is a quotient of a sub of Q/ Jac(Q) where Q = {x ∈ Mp(k) ∣ Θx = x}, i.e. those xed by the reduction Θ of Θ, ETS Q/ Jac(Q) = k. Θ is necessarily inner as it is an auto. of Mp(k), so Θ is the conjugation by a matrix T0 ∈ GLp(k). Since T0ei = ei+1T0 where ei now here is diag(0, ⋯ , 1, ⋯ , 0) where 1 is the only nonzero entry and is 6 0 ⋯ ⋯ ⋯ tp 0 ⋯ ⋯ ⋯ t t1 0 ⋯ ⋯ ⋯ 1 0 ⋯ ⋯ ⋯ at i-th spot, this forces T0 to be of form ⋯ t2 ⋯ ⋯ ⋯ , which is conjugate to ⋯ 1 ⋯ ⋯ ⋯ for H ⋯ ⋯ ⋯ 0 ⋯ I 0 ⋯ ⋯ ⋯ 0 ⋯ 1 0 ⋯ ⋯ tp−1 0 0 ⋯ ⋯ 1 0 × t = t1 ⋯ tp ∈ k . This has a Jordan normal form consisted of one big Jordan block with eigenvalue t. By explicit computation you get Q as < a ∗ ⋯ ∗ ∗ = 0 a ⋯ ∗ ∗ Q = ⋯ ⋯ ⋯ ⋯ ⋯ , 0 0 0 ⋯ a ∗ 1 0 0 ⋯ 0 a

< 0 ∗ ⋯ ∗ ∗ = 0 0 ⋯ ∗ ∗ Jac(Q) = ⋯ ⋯ ⋯ ⋯ ⋯ . 0 0 0 ⋯ 0 ∗ 1 0 0 ⋯ 0 0 

5. Block and defect

We study k[G]. Let E = {e1, ⋯ , er } be the set of primitive central idempotents of k[G], and blocks in this setting will refer to ei-blocks. ∙ A fg indecomposable k[G]-module belongs to a block. Conversely, we can describe which fg indecomposable k[G]-modules belong to the same block: we can dene an equivalence ¡ relation on k[G] generated by P ∼ Q if either Homk[G](P,Q) ≠ 0 or Homk[G](Q,P) ≠ 0, and the set of equivalence classes = set of blocks. One side is easy, namely equiclass belongs to a block. On the other hand if two dierent equiclass belongs to the same block then this will decompose the block k[G]e into a direct sum of smaller nonzero stus. We denote e ¡  for the block/idempotent corresponding to an equivalence class  ⊂ k[G]. ∙ k[G] V e V ≅ P/ rad(P) P ∈ A simple -module belongs to -block if and only if for some . Thus if V is projective, it is only simple module in its block. ∙ If a simple k[G]-module belongs to the e-block, then the central character is e pr ae ↦ a, ∼ e ∶ Z(k[G]) ãããããããããã→ Ze  Ze/ Jac(Ze) ããããããããããããããããããããããããããããããããããããããããããããããããããã→ k. ∙ (G) G ∈ (G)  ∶= ∑ g ∈ Let  be the set of conjclasses in . Given a conjclass   , let  g∈ k[G]. This forms a k-basis of Z(k[G]). Let    12 = ∑ (1, 2; ). ∈(G) ∙ Now we can apply Green’s theory as follows. Consider the action of G × G on k[G] via (g, ℎ)⋅x = gxℎ−1. Then, 2-sided ideals are k[G ×G]-submodules of k[G], so k[G] = ⊕k[G]e is the decomposition of k[G] into sum of indecomposable k[G × G]-modules. Also, k[G] ≅ G×G IndΔ(G)(k), so k[G] is k[Δ(G)]-projective, where Δ(G) ⊂ G × G is the diagonal G. So the vertices of k[G]e are of form Δ(H) for some H ⊂ G, and all vertices are Δ(G)-conjugate. We call these vertices defect groups. By above these are p-groups and form a single conjugacy class in G. ∙ Given an e-block, and given a defect group D, D contains a vertex of any fg indecompos- able module in this block. ∙ We start from a dierent end. Let p-Sylow subgroups of CG(x) for x ∈  in a conjclass  be called defect groups of . 7 ∙ For a p-subgroup P ⊂ G, let  IP = ∑ k ⊂ Z(k[G]) P contains defect gp of 

(or IP (k[G]) if G is not clear). This depends only on the conjclass of P, and IP = Z(k[G]) if P is p-Sylow of G. ∙ If 1, 2,  ∈ (G) such that (1, 2; ) ≠ 0, then if a p-subgroup P ⊂ G centralizes an I I ⊂ ∑ I −1 element of , then it also centralizes elements of 1 and 2. Thus, P1 P2 g∈G P1∩gP2g , and in particular every IP is an ideal in Z(k[G]). ∙ More generally, let k[G]ad(H) = {x ∈ k[G] ∣ ℎx = xℎ for all ℎ ∈ H}, and let ad(H) trH ∶ k[G] → Z(k[G]) −1 be dened by trH (x) = ∑g∈G/H gxg . Then IP ⊂ im(trP ). ∙ If e is a central primitive idempotent contained in IP , then P contains a defect group of e-block. ∙ Let D be a defect group of the e-block, and N = NG(D). Using the Green correspondence for N ×N and (D), k[G]e has a Green correspondent which is the unique indecomposable direct summand with vertex (D) of k[G]e as a k[N × N ]-module. ∙ We can describe this Green correspondent more explicitly, which is a particular case of Brauer correspondence.

Theorem 5.1 (Brauer’s First Main Theorem). For a p-group D ⊂ G, let ED(G) = {e ∈ E ∣ D is a defect of e-block}. Let H ⊂ G such that NG(D) ⊂ H. Then, there is a bijection ∼ B ∶ ED(G) ãããããããã→ ED(H), such that k[H]B(e) is the Green correspondent of k[G]e. Proof. Note that as a k[H × H]-module there is a decomposition k[G] = ⨁ k[HyH]. y∈H⧵G/H Note that by explicit identication, H×H k[HyH] ≅ Ind(1,y)−1(H∩yHy−1)(1,y)(k), as k[H ×H]-modules. Thus Γ(k[G]e) is a direct summand of k[HyH] for some y ∈ G. Thus any indecomposable direct summand of k[HyH] has a vertex contained in (1, y)−1(H ∩ yHy−1)(1, y). On the other hand (D) is a defect group of Γ(k[G]e), so H × H-conjugate of (D) should be contained in (1, y)−1(H ∩ yHy−1)(1, y). This turns out to be impossible unless y ∈ H. What this tells you is that Γ(k[G]e) is actually a direct summand of k[H], ¨ ¨ ¨ and Γ(k[G]e) = k[H]e for some e ∈ ED(H). Such choice is unique because k[H]e ’s are all dierent even as k[H]-modules. By the bijectivity of Green correspondence it is obviously ¨ ¨ injective. Surjectivity is also easy, you pick any e ∈ ED(H) and k[H]e is a direct summand of k[G], so a direct summand of some k[G]e.  P ∙ IndQ(k) is indecomposable with vertex Q for P, Q p-groups. So if we started with a p-group H then there is only one... ∙ There is Brauer correspondence, dened as follows. Let

E≥D(G) = {e ∈ E ∣ D is contained in a defect of e-block}. 8 Then, if D is a p-group in G and if CG(D) ⊂ H, the Brauer correspondence

∶ E≥D(H) → E≥D(G),

is dened by, for f ∈ E≥D(H), k[H]f is a k[H × H]-direct summand of k[G] (f ). Note that direction is reversed. This is well-dened because this operation f ↦ (f ) only increases defect group. That this map is well-dened (i.e. that there is such central idempotent (f )) can be proved exactly in the same way as the proof of Brauer’s rst main theorem. That the defect groups increase is also clear as defect groups are after all vertices (as k[G × G]-modules though). ∙ In fact defect groups are realized as vertices as k[G]-modules. If e-block has defect group D, you take H = NG(D), and take any simple k[H]-module in B(e)-block of k[H], and it has vertex D. As k[D]-module it is a direct sum of trivial k[D]-modules so vertex D as k[D]- module ⇒ as k[H]-modules (easy) ⇒ as k[G]-modules (Green theory). This belongs to the right block (namely, e-block) by... you do some analysis as k[H]-module. ∙ What this says is that e.g. e-block has trivial defect i k[G]e is simple (so a matrix algebra). Now nally meets rep theory.

∙ Let D ⊂ G be a p-subgroup, and let CG(D) ⊂ H ⊂ NG(D). Then the restriction map

k[G] → k[CG(D)] ⊂ k[H]

(i.e. forget all irrelevant elements outside CG(D)) gives a map

sD,H ∶ Z(k[G]) → Z(k[H]),

because for any G-conjclass ,  ∩ CG(D) is a union of nitely many H-conjclasses. This is called the Brauer homomorphism (i.e. this is a homomorphism of k-algebras). In particular, for any e ∈ E(G), s(e) is either zero or idempotent. ∙ s(e) is nonzero if D is minimal with respect to e ∈ ID(k[G]) (pure group theory). ∙ Now further assume that DCG(D) ⊂ H. Suppose sD,H (e) ≠ 0. Then sD,H (e) = e1 + ⋯ + er for primitive idempotents ei ∈ E(H). Then, k[H]sD,H (e) is isomorphic to a direct summand of k[G]e.

∙ In particular if D0 is the defect group of e-block, then e ∈ ID0 (k[G]), and sD0,NG (D0)(e) ∈

E(NG(D0)), B(e) = sD0,H (e). ∙ By the same group theory reason, for any defect group D of a block, if Q ⊂ G is a p- Sylow group containing D, then D = Q ∩ gQg−1 for some g ∈ G. Also, any normal p-subgroup is contained in D. In particular, if DCG(D) ⊂ H ⊂ NG(D), every defect group contains D. In this case, the Brauer correspondence ∶ E(H) → E≥D(G) is characterized by (f ) = f ◦sD,H .  ∙ For any e ∈ E(G) and defect group D0 of e-block, if e() ≠ 0 for some  ∈ (G), then D0 is contained in the defect group of , and there is some  as above such that D0 is precisely the defect group of .

6. Example: SL2(Fp)

If we count p-regular conjugacy classes in G = SL2(Fp), there are p of them. So there are p many simple k[G]-modules. 9 Proposition 6.1. The p simple modules are given by Vn, the G-representation on the space of ho- a b mogeneous degree n polynomials in 2 variables X and Y (say c d  acts on X and Y via aX + cY and bX + dY , respectively), for 0 ≤ n < p. Proof. Since the dimensions are dierent, ETS they are simple. Let B be the lower triangular Borel and U be its unipotent radical. Note that each Vn has a ltration of k[U ]-submodules, i−1 j n−j 0 = W0 ⊂ W1 ⊂ ⋯ ⊂ Wn ⊂ Wn+1 = Vn where Wi = ∑j=0 kX Y . The graded pieces are all n trivial modules, and sock[U ] Vn = kY , because for any nite p-group the socle is just invariant − n − vectors. In particular it is indecomposable (otherwise socle is dim ≥ 2). As k[U ]Y = Vn (U is the unipotent radical of the opposite Borel, i.e. upper triangular Borel), any nonzero sub has n socle equal to kY , so it should contain all Vn.  What about fg indecomposables? 6.1. Brauer tree.

//,-mm,,,,/7. Ext Recall the notion of projective cover. We can rephrase the denition slightly as follows: given a module M, a projective cover P is a (surjective) map P → M which induces P/ rad(P) ãããããããã→∼ M/ rad(M). This exists if A is artinian. Dually, we can dene the notion of injective hull as an injective module I with an injective map M → I which induces soc(M) ãããããããã→∼ soc(I). This exists if A is a quasi-frobenius artinian algebra. We from now on assume A is so if otherwise noted.

Denition 7.1 (Heller operator). Let PM be the projective cover (unique up to isomorphism) of M. Let ΩM be the of PM  M. For n ∈ ℤ inductively we dene ⎧ ⎪Ω(Ωi−1(M)) if i > 0 ⎪ Ωi(M) ∶= M i = 0 ⎪⎨ if ⎪ −i ∗ ∗ ⎩⎪Ω (M ) if i < 0. This is an additive functor, and one has many properties like −1 ∙Ω (M) is the cokernel of M → IM , where IM is the injective hull of M. ∙Ω −1(Ω(M)) ≅ Ω(Ω−1(M)) ≅ M, ∙ (Heller’s lemma) M ≅ Ω(Ω(M ∗)∗), ∙ M is indecomposable i ΩM is indecomposable. Denition 7.2. For A-modules X,Y , let n n ExtA(X,Y ) ∶= coker (HomA(PΩn−1X ,Y ) → HomA(Ω X,Y )) , where the map is induced from the injection in the n n−1 0 → Ω X → PΩn−1X → Ω X → 0. Note that by denition n n−1 1 n−1 ExtA(X,Y ) = ExtA (ΩX,Y ) = ⋯ = ExtA(Ω X,Y ). ∙ There is a long exact sequence of Ext’s. From this, and from that Ext of projectives (in this case are equal to injectives) is zero, the proof that Ext1 parametrizes equivalence classes of extensions in the setting of modules verbatim works here. 10 1 ∙ If X,Y are simple, then ExtA(X,Y ) = HomA(PX J,Y ) where J is the radical of A. This is because for any ∈ HomA(PX ,Y ), ker ⊃ PX J as Y is simple. ∙ From the above point, one sees the following, that two simple A-modules X,Y lie in the same block if and only if there exists a sequence of simple modules T1 = X,T2, ⋯ ,Tn = Y 1 such that ExtA(Ti,Ti+1) ≠ 0 for all i. 8. Auslander-Reiten theory BℝUℍ

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