Universal deformation rings and tame blocks with two simple modules. - - PowerPoint PPT Presentation

Universal deformation rings and tame blocks with two simple modules.

Jennifer Schaefer, Dickinson College Joint work with F. Bleher and G. Llosent Conference on Geometric Methods in Representation Theory University of Missouri, Columbia, MO November 19, 2012

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

In the 1980’s, Mazur, using work of Schlessinger, introduced deformations of Galois representations to study lifts of Galois representations over finite fields to p-adic representations.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

In the 1980’s, Mazur, using work of Schlessinger, introduced deformations of Galois representations to study lifts of Galois representations over finite fields to p-adic representations. In the 1990’s, Wiles and Taylor used Mazur’s deformation theory in the proof of Fermat’s Last Theorem.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

In the 1980’s, Mazur, using work of Schlessinger, introduced deformations of Galois representations to study lifts of Galois representations over finite fields to p-adic representations. In the 1990’s, Wiles and Taylor used Mazur’s deformation theory in the proof of Fermat’s Last Theorem. The main motivation for determining universal deformation rings for finite groups is to test or verify conjectures about the ring structure of universal deformation rings for arbitrary Galois groups.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let k be an algebraically closed field of characteristic 2.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let k be an algebraically closed field of characteristic 2. Let G be a finite group and let B be a block of the group algebra kG with semi-dihedral or generalized quaternion defect groups and precisely two isomorphism classes of simple B-modules.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let k be an algebraically closed field of characteristic 2. Let G be a finite group and let B be a block of the group algebra kG with semi-dihedral or generalized quaternion defect groups and precisely two isomorphism classes of simple B-modules. Determine all finitely generated kG-modules V which belong to B and whose endomorphism ring is isomorphic to k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let k be an algebraically closed field of characteristic 2. Let G be a finite group and let B be a block of the group algebra kG with semi-dihedral or generalized quaternion defect groups and precisely two isomorphism classes of simple B-modules. Determine all finitely generated kG-modules V which belong to B and whose endomorphism ring is isomorphic to k. First calculate the universal deformation ring modulo 2 and then calculate the universal deformation ring R(G, V ) for each of these modules.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Definitions and Background Let k be an algebraically closed field of positive characteristic p. Let G be a finite group. Let kG be the group algebra of G with coefficients in k. All modules will be finitely generated left modules.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Universal Deformation Rings Let V be a finitely generated kG-module. Let ˆ C be the category of all complete local commutative Noetherian rings with residue field k where the morphisms are the homomorphisms of complete local rings inducing the identity on k. Let R ∈ Ob( ˆ C).

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Definition: (i) A lift of V over R is a finitely generated RG-module M which is free over R together with a kG-module isomorphism φ : k ⊗R M − → V . Notation: (M, φ).

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Definition: (i) A lift of V over R is a finitely generated RG-module M which is free over R together with a kG-module isomorphism φ : k ⊗R M − → V . Notation: (M, φ). (ii) Two lifts (M, φ) and (M′, φ′) are isomorphic if there exists an RG-module isomorphism f : M − → M′ with φ′ ◦ (id ⊗ f ) = φ.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Definition: (i) A lift of V over R is a finitely generated RG-module M which is free over R together with a kG-module isomorphism φ : k ⊗R M − → V . Notation: (M, φ). (ii) Two lifts (M, φ) and (M′, φ′) are isomorphic if there exists an RG-module isomorphism f : M − → M′ with φ′ ◦ (id ⊗ f ) = φ. (iii) A deformation of V over R is an isomorphism class of a lift (M, φ) of V over R. Notation: [(M, φ)].

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Theorem: (Mazur; Bleher-Chinburg) Suppose EndkG(V ) ∼ = k. Then there exists an R(G, V ) in ˆ C and a lift (U(G, V ), φU) of V

• ver R(G, V ) such that for all A ∈ ˆ

C and for all lifts (M, φ) of V

• ver A, there exists a unique α : R(G, V ) −

→ A in ˆ C such that the lift (M, φ) is isomorphic to the lift (A ⊗R(G,V ),α U(G, V ), φ′) where φ′ is the composition k ⊗A (A ⊗R(G,V ),α U(G, V )) ∼ = k ⊗R(G,V ) U(G, V )

φU

→ V . The pair (R(G, V ), [(U(G, V ), φU)]) is unique up to isomorphism.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Definition: The ring R(G, V ) is called the universal deformation ring of V . [(U(G, V ), φU)] is called the universal deformation of V over R(G, V ).

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let W = W (k) be the ring of infinite Witt vectors over k. Since k is assumed to be algebraically closed of characteristic p, W is the unique (up to isomorphism) complete discrete valuation ring with residue field k such that p generates the maximal ideal.

• Ex. k = Fp implies W =completed p-adic integers

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Theorem: (Mazur) (i) If dimk(Ext1

kG(V , V ))=m, then there exists a surjective

homomorphism Φ : W [[t1, t2, ..., tm]] − → R(G, V ) in ˆ C, and m is minimal with this property.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Theorem: (Mazur) (i) If dimk(Ext1

kG(V , V ))=m, then there exists a surjective

homomorphism Φ : W [[t1, t2, ..., tm]] − → R(G, V ) in ˆ C, and m is minimal with this property. (ii) If dimk(Ext2

kG(V , V ))=s, then s is an upper bound for the

minimal number of generators for Ker(Φ).

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

For the remainder of this talk, we assume char(k) = 2. Let G be a finite group and let B be a block of the group algebra kG such that B has semi-dihedral or generalized quaternion defect groups and precisely two isomorphism classes of simple modules. Then B is Morita equivalent to the algebra SD(2A)1, SD(2A)2, SD(2B)1, SD(2B)2, or SD(2B)3 if B has semi-dihedral defect groups or to the algebra Q(2A), Q(2B)1, or Q(2B)2 if B has generalized quaternion defect groups. [Erdmann]

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

SEMI-DIHEDRAL DEFECT GROUPS Let SD(2A)1 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• 1

and relations α2 = c(γβα)t, βγβ = βα(γβα)t−1, γβγ = αγ(βαγ)t−1, and α(γβα)t = 0 where t 2, t = 2n−1 and c ∈ k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple SD(2A)1-modules corresponding to the vertices 0 and 1 which we denote by S0 and S1, and there are two indecomposable projective SD(2A)1-modules up to isomorphism, which can be described using the following diagrams: P0 =

∗

1 1

∗∗

1 1 1

and P1 =

1 1 1 1 1

where the line ∗ and ∗∗ in P0 corresponds to the relation α2 = c(γβα)t and βγβ = βα(γβα)t−1, respectively.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let SD(2A)2 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• 1

and relations α2 = γβ(αγβ)t−1 + c(γβα)t, (αγβ)t = (γβα)t, and βγ = 0 where t 2, t = 2n−2 and c ∈ k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple SD(2A)2-modules corresponding to the vertices 0 and 1 which we denote by S0 and S1, and there are two indecomposable projective SD(2A)2-modules up to isomorphism, which can be described using the following diagrams: P0 =

∗

1 1 1 1 1

and P1 =

1 1 1 1

where the line ∗ in P0 corresponds to the relation α2 = γβ(αγβ)t−1 + c(γβα)t.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let SD(2B)1 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• η
• 1

and relations ηβ = 0 = γη = βγ, α2 = γβ + c(γβα), γβα = αγβ, βαγ = ηt where t 2, t = 2n−2, and c ∈ k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple SD(2B)1-modules corresponding to the vertices 0 and 1 which we also denote by S0 and S1, and there are two indecomposable projective SD(2B)1-modules up to isomorphism, which can be described using the following diagrams: P0 =

∗

1 1

and P1 =

1 1 1 1

where the line ∗ in P0 corresponds to the relation α2 = γβ + c(γβα).

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let SD(2B)2 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• η
• 1

and relations η2β = 0 = γη2, α2 = c(γβα)2, βγ = ηt−1, γη = αγ(βαγ), ηβ = βα(γβα) where t 4, t = 2n−2, and c ∈ k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple SD(2B)2-modules corresponding to the vertices 0 and 1 which we also denote by S0 and S1, and there are two indecomposable projective SD(2B)2-modules up to isomorphism, which can be described using the following diagrams: P0 =

0 ∗ 1

∗∗

1 1 1

and P1 =

1

• 1
• 1

1 1

where the lines ∗ and ∗∗ in P0 corresponds to the relations α2 = c(γβα)2 and ηβ = βα(γβα) and the lines • and •• in P1 corresponds to the relations βγ = ηt−1 and γη = αγ(βαγ), respectively.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let SD(2B)3 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• η
• 1

and relations α2 = γβ, βα = ηβ, γη = αγ, βγ = η2 + cηt+1, βαt = αtγ = αt+2 = ηt+2 = 0, γηt = ηtβ = 0 where t 2, t = 2n−2 − 1, and c ∈ k [Holm].

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple SD(2B)3-modules corresponding to the vertices 0 and 1 which we also denote by S0 and S1, and there are two indecomposable projective SD(2B)3-modules up to isomorphism, which can be described using the following diagrams: P0 =

1 1 1 1 1

and P1 =

1

∗

1 1 1 1 1 1

where the line ∗ in P1 corresponds to the relation βγ = η2 + cηt+1.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

GENERALIZED QUATERNION DEFECT GROUPS Let Q(2A) be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• 1

and relations α2 = γβ(αγβ)s−1 + c(αγβ)s, βγβ = βα(γβα)s−1, γβγ = αγ(βαγ)s−1, βα2 = 0 where s 2, s = 2n−1 and c ∈ k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple Q(2A)-modules corresponding to the vertices 0 and 1 which we denote by S0 and S1, and there are two indecomposable projective Q(2A)-modules up to isomorphism, which can be described using the following diagrams: P0 =

∗

1 1

∗∗

1 1 1

and P1 =

1 1 1 1 1

where the line ∗ and ∗∗ in P0 corresponds to the relations α2 = γβ(αγβ)s−1 + c(αγβ)s and βγβ = βα(γβα)s−1, respectively.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let Q(2B)1 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• η
• 1

and relations α2 = γβ(αγβ) + c(αγβ)2, ηβ = βα(γβα), βγ = ηs−1, γη = αγ(βαγ), βα2 = 0 where s 4, s = 2n−2, and c ∈ k.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple Q(2B)1-modules corresponding to the vertices 0 and 1 which we also denote by S0 and S1, and there are two indecomposable projective Q(2B)1-modules up to isomorphism, which can be described using the following diagrams: P0 =

∗

1

∗∗

1 1 1

and P1 =

1

• 1
• 1

1 1

where the lines ∗ and ∗∗ in P0 corresponds to the relations α2 = γβ(αγβ) + c(αγβ)2 and ηβ = βα(γβα) and the lines • and

• • in P1 corresponds to the relations βγ = ηs−1 and

γη = αγ(βαγ), respectively.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Let Q(2B)2 be the finite dimensional k-algebra with quiver Q =

• α
• β

γ

• η
• 1

and relations βα = ηβ, γη = αγ, γβ = α2 + α3q(α), βγ = η2 + η3q(η) + aηs−1 + cηs, αs+1 = 0 = ηs+1, αs−1γ = 0 = βαs−1 where q(t) ∈ k[t], s 4, s = 2n−2, a, c ∈ k and a = 0.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

There are two simple Q(2B)2-modules corresponding to the vertices 0 and 1 which we also denote by S0 and S1, and there are two indecomposable projective Q(2B)2-modules up to isomorphism, which can be described using the following diagrams: P0 =

1

∗

1 1 1 1

and P1 =

1

• 1

1 1 1 1 1

where the line ∗ in P0 and • in P1 corresponds to the relations γβ = α2 + α3q(α) and βγ = η2 + η3q(η) + aηs−1 + cηs, respectively.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Modules with Endomorphism Rings Isomorphic to k Theorem (BLS): Let Λ be Q(2A). Let M be a Λ-module. Then EndΛ(M) ∼ = k if and only if M ∈ {S0, S1, S0 S1 , S1 S0 , S0 S0 S1 , S1 S0 S0 }.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Sketch of proof: Case (i) Radical length is 1: S0, S1

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Sketch of proof: Case (i) Radical length is 1: S0, S1 Case (ii) Radical length is 2: 0 0 , 0 1 , 1 0 , 1 , 0 1 , 1 1

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Case(iii) Radical length is 3:

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Case(iii) Radical length is 3:

a) Suppose top(M) is a direct sum of copies of S1:

1

,

1 1

,

1 1

,

1 1

,

1 1 1 Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

b) Suppose top(M) is a direct sum of copies of S0:

1

,

1 1

,

1 1

,

1 1

,

1 1

,

1 1 1

,

1 1 1

,

1 1

,

1 1 1

,

1 1 1

,

1 1 1 1 Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Case(iv) Radical length is greater than 3: Consider M/rad3(M) and soc3(M).

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Case(iv) Radical length is greater than 3: Consider M/rad3(M) and soc3(M).

a) Suppose top(M) is a direct sum of copies of S1 and soc(M) is a direct sum of copies of S0.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules

Case(iv) Radical length is greater than 3: Consider M/rad3(M) and soc3(M).

a) Suppose top(M) is a direct sum of copies of S1 and soc(M) is a direct sum of copies of S0. b) Suppose top(M) is a direct sum of copies of S0 and soc(M) is a direct sum of copies of S1.

Schaefer with Bleher and Llosent UDRs and tame blocks with two simple modules