Projective modules over finite p-adic group algebras

For a finite monoid G, in particular a finite group, multiplication by p lies in the Jacobson radical of the monoid algebra \mathbb Z_p[G]. Consequently, finitely generated projective \mathbb Z_p[G]-modules are determined by their reductions modulo p.

The radical statement is valid even when p divides the order of G. To prove it, represent left multiplication on the finite free \mathbb Z_p-module \mathbb Z_p[G] by a matrix. Every entry of the matrix of multiplication by p x is topologically nilpotent, so the matrix of 1 - p x is invertible. Faithfulness of the left regular representation then gives invertibility in the group algebra itself.

The lifting theorem is the integral part of detecting projective lattices by a field or residue field calculation. Uniqueness of projective covers supplies an isomorphism between the integral projective modules; the isomorphism itself is not asserted to be unique.

Since \mathbb Z_p[G] ⧸ (p) ≃ \mathbb F_p[G] is finite, so is the radical quotient of \mathbb Z_p[G]. Therefore TauCeti.nonempty_linearEquiv_of_projective_of_natCard_linearMap_eq applies: finitely generated projective \mathbb Z_p[G]-modules are also determined by the numbers of their maps to the simple \mathbb Z_p[G]-modules.

Main results

• TauCeti.span_p_le_jacobson_padicInt_monoidAlgebra: the ideal generated by p is contained in the Jacobson radical of \mathbb Z_p[G].
• TauCeti.jacobson_padicInt_monoidAlgebra_eq_span_p: when p does not divide the order of the finite group G, the Jacobson radical of \mathbb Z_p[G] is generated by p.
• TauCeti.instFiniteQuotientJacobsonPadicIntMonoidAlgebra: the radical quotient of \mathbb Z_p[G] is finite.
• TauCeti.nonempty_linearEquiv_of_projective_of_reduction: finitely generated projective \mathbb Z_p[G]-modules with isomorphic reductions modulo p are isomorphic.
• TauCeti.exists_projective_reduction_linearEquiv: conversely, every direct summand of the reduction modulo p of a finitely generated projective \mathbb Z_p[G]-module is the reduction of a finitely generated projective \mathbb Z_p[G]-module.
• TauCeti.exists_projective_reduction_bijective_of_projective: every finitely generated projective \mathbb F_p[G]-module is the reduction of a finitely generated projective \mathbb Z_p[G]-module.

References

See J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, second edition, Proposition (5.6.10)(iii).

theorem TauCeti.span_p_le_jacobson_padicInt_monoidAlgebra (p : ℕ) [Fact (Nat.Prime p)] (G : Type u) [Monoid G] [Finite G] : Ideal.span {↑p} ≤ Ring.jacobson (MonoidAlgebra ℤ_[p] G)

In the monoid algebra of a finite monoid (for instance a finite group) over ℤ_[p], the ideal generated by p lies in the Jacobson radical. No hypothesis relating p to the order of G is needed.

instance TauCeti.instFiniteQuotientJacobsonPadicIntMonoidAlgebra (p : ℕ) [Fact (Nat.Prime p)] (G : Type u) [Monoid G] [Finite G] : Finite (MonoidAlgebra ℤ_[p] G ⧸ Ring.jacobson (MonoidAlgebra ℤ_[p] G))

The radical quotient of ℤ_p[G] is finite for a finite monoid G. The ideal generated by p lies in the Jacobson radical, and ℤ_p[G] ⧸ (p) ≃ 𝔽_p[G] is finite.

theorem TauCeti.jacobson_padicInt_monoidAlgebra_eq_span_p (p : ℕ) [Fact (Nat.Prime p)] {G : Type u} [Group G] [Finite G] (hG : ¬p ∣ Nat.card G) : Ring.jacobson (MonoidAlgebra ℤ_[p] G) = Ideal.span {↑p}

The Jacobson radical of ℤ_p[G] for p ∤ |G|. When the order of the finite group G is prime to p, the Jacobson radical of ℤ_p[G] is generated by p: it contains p for every finite monoid, and conversely the reduction ℤ_p[G] ⧸ (p) ≃ 𝔽_p[G] is a semisimple ring by Maschke's theorem, so it has zero radical.

theorem TauCeti.nonempty_linearEquiv_of_projective_of_reduction (p : ℕ) [Fact (Nat.Prime p)] (G : Type u) [Monoid G] [Finite G] (M : Type v) (N : Type w) [AddCommGroup M] [Module (MonoidAlgebra ℤ_[p] G) M] [Module.Finite (MonoidAlgebra ℤ_[p] G) M] [Module.Projective (MonoidAlgebra ℤ_[p] G) M] [AddCommGroup N] [Module (MonoidAlgebra ℤ_[p] G) N] [Module.Finite (MonoidAlgebra ℤ_[p] G) N] [Module.Projective (MonoidAlgebra ℤ_[p] G) N] (h : Nonempty ((M ⧸ Ideal.span {↑p} • ⊤) ≃ₗ[MonoidAlgebra ℤ_[p] G] N ⧸ Ideal.span {↑p} • ⊤)) : Nonempty (M ≃ₗ[MonoidAlgebra ℤ_[p] G] N)

Projectives over ℤ_p[G] are detected modulo p (NSW (5.6.10)(iii)). Two finitely generated projective modules over the monoid algebra of a finite monoid (in particular the group algebra of a finite group) are isomorphic when their reductions modulo p are isomorphic.

theorem TauCeti.exists_projective_reduction_linearEquiv (p : ℕ) [Fact (Nat.Prime p)] (G : Type u) [Monoid G] [Finite G] (N : Type v) [AddCommGroup N] [Module ℤ_[p] N] [Module (MonoidAlgebra ℤ_[p] G) N] [IsScalarTower ℤ_[p] (MonoidAlgebra ℤ_[p] G) N] [Module.Finite (MonoidAlgebra ℤ_[p] G) N] [Module.Projective (MonoidAlgebra ℤ_[p] G) N] {Y : Type w} [AddCommGroup Y] [Module (MonoidAlgebra ℤ_[p] G) Y] (i : Y →ₗ[MonoidAlgebra ℤ_[p] G] N ⧸ Ideal.span {↑p} • ⊤) (q : N ⧸ Ideal.span {↑p} • ⊤ →ₗ[MonoidAlgebra ℤ_[p] G] Y) (hqi : q ∘ₗ i = LinearMap.id) : ∃ (X : Submodule (MonoidAlgebra ℤ_[p] G) N), Module.Finite (MonoidAlgebra ℤ_[p] G) ↥X ∧ Module.Projective (MonoidAlgebra ℤ_[p] G) ↥X ∧ Nonempty ((↥X ⧸ Ideal.span {↑p} • ⊤) ≃ₗ[MonoidAlgebra ℤ_[p] G] Y)

Projectives lift from 𝔽_p[G] to ℤ_p[G]. Let N be a finitely generated projective module over the monoid algebra ℤ_p[G] of a finite monoid (in particular the group algebra of a finite group). Every direct summand Y of its reduction N ⧸ p • N, given by maps i : Y → N ⧸ p • N and q : N ⧸ p • N → Y with q ∘ i = id, is the reduction X ⧸ p • X of a finitely generated projective submodule X of N.

For N a finitely generated free module, this applies to every finitely generated projective 𝔽_p[G]-module; see TauCeti.exists_projective_reduction_bijective_of_projective.

theorem TauCeti.exists_projective_reduction_bijective_of_projective (p : ℕ) [Fact (Nat.Prime p)] (G : Type u) [Monoid G] [Finite G] (Y : Type v) [AddCommGroup Y] [Module (MonoidAlgebra (ZMod p) G) Y] [Module.Finite (MonoidAlgebra (ZMod p) G) Y] [Module.Projective (MonoidAlgebra (ZMod p) G) Y] : ∃ (n : ℕ) (X : Submodule (MonoidAlgebra ℤ_[p] G) (Fin n → MonoidAlgebra ℤ_[p] G)), Module.Finite (MonoidAlgebra ℤ_[p] G) ↥X ∧ Module.Projective (MonoidAlgebra ℤ_[p] G) ↥X ∧ ∃ (f : ↥X ⧸ Ideal.span {↑p} • ⊤ →ₛₗ[MonoidAlgebra.mapRingHom G PadicInt.toZMod] Y), Function.Bijective ⇑f

Projective 𝔽_p[G]-modules lift to ℤ_p[G]. Every finitely generated projective module Y over the monoid algebra 𝔽_p[G] of a finite monoid (in particular the group algebra of a finite group) is the reduction modulo p of a finitely generated projective ℤ_p[G]-module X: there is a bijection X ⧸ p • X → Y which is semilinear along the coefficient reduction ℤ_p[G] → 𝔽_p[G]. The module X is realised as a submodule of some ℤ_p[G]ⁿ, and it is unique up to isomorphism by TauCeti.nonempty_linearEquiv_of_projective_of_reduction.