The quotient of a monoid algebra by the left ideal of two eigenvalue relations

For two elements σ, τ of a monoid G and two scalars a, b of a commutative ring R, consider the left ideal J = R[G]·(σ - a) + R[G]·(τ - b) of the monoid algebra R[G] and the left R[G]-module R[G] ⧸ J. Its elements are the classes [x], and right multiplication by σ and τ acts on them as the scalars a and b: [x σ] = a [x] and [x τ] = b [x].

Consequently the quotient is killed by b ^ orderOf τ - 1, and when σ, τ generate the monoid G every class is an R-multiple of the class of 1: the quotient is a cyclic R-module. It is the largest quotient of R[G] on which σ and τ act through the scalars a and b.

In the computation of the generator rank of the absolute Galois group of a p-adic field (NSW (7.4.1)), R = ℤ_p, G is the group of a tamely ramified finite Galois layer L/K generated by a Frobenius lift σ and a generator τ of tame inertia, and a, b are exponents through which they act on the p-power roots of unity of L; the cyclicity of the quotient is what makes it no larger than the roots of unity themselves.

Main statements

• MonoidAlgebra.mk_mul_single_eq_smul_of_sub_mem: if a left ideal I contains g - c, right multiplication by g acts on R[G] ⧸ I as c.
• MonoidAlgebra.pow_orderOf_sub_one_smul_eq_zero_of_sub_mem: such a quotient is killed by c ^ orderOf g - 1.
• MonoidAlgebra.toSpanSingleton_mk_one_surjective: when σ, τ generate the monoid G, the quotient is generated over R by the class of 1.
• TauCeti.MonoidAlgebra.natCard_quotient_span_one_sub_natCast_ne_two: for a finite nontrivial monoid and odd a, b, the corresponding ℤ₂-monoid-algebra quotient cannot have order two.

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, proof of (7.4.1).

theorem MonoidAlgebra.mk_mul_single_eq_smul_of_sub_mem {R : Type u_1} [CommRing R] {G : Type u_2} [Monoid G] {I : Ideal (MonoidAlgebra R G)} {g : G} {c : R} (h : single g 1 - single 1 c ∈ I) (x : MonoidAlgebra R G) : Submodule.Quotient.mk (x * single g 1) = c • Submodule.Quotient.mk x

If the left ideal I contains g - c, then right multiplication by g acts on R[G] ⧸ I as the scalar c.

theorem MonoidAlgebra.mk_mul_single_pow_eq_pow_smul_of_sub_mem {R : Type u_1} [CommRing R] {G : Type u_2} [Monoid G] {I : Ideal (MonoidAlgebra R G)} {g : G} {c : R} (h : single g 1 - single 1 c ∈ I) (x : MonoidAlgebra R G) (m : ℕ) : Submodule.Quotient.mk (x * single (g ^ m) 1) = c ^ m • Submodule.Quotient.mk x

If the left ideal I contains g - c, then right multiplication by g ^ m acts on R[G] ⧸ I as the scalar c ^ m.

theorem MonoidAlgebra.pow_orderOf_sub_one_smul_eq_zero_of_sub_mem {R : Type u_1} [CommRing R] {G : Type u_2} [Monoid G] {I : Ideal (MonoidAlgebra R G)} {g : G} {c : R} (h : single g 1 - single 1 c ∈ I) (y : MonoidAlgebra R G ⧸ I) : (c ^ orderOf g - 1) • y = 0

If the left ideal I contains g - c, then R[G] ⧸ I is killed by c ^ orderOf g - 1, since g acts on it as c and g ^ orderOf g = 1.

theorem MonoidAlgebra.toSpanSingleton_mk_one_surjective {R : Type u_1} [CommRing R] {G : Type u_2} [Monoid G] {σ τ : G} {a b : R} (hgen : Submonoid.closure {σ, τ} = ⊤) : Function.Surjective ⇑(LinearMap.toSpanSingleton R (MonoidAlgebra R G ⧸ Ideal.span {single σ 1 - single 1 a, single τ 1 - single 1 b}) (Submodule.Quotient.mk 1))

When σ, τ generate the monoid G, the quotient R[G] ⧸ R[G]·(σ - a, τ - b) is a cyclic R-module generated by the class of 1: the class of every element of G is a word in a, b times the class of 1. For a finite group, generation as a group suffices, by Subgroup.closure_toSubmonoid_of_finite.

theorem TauCeti.MonoidAlgebra.natCard_quotient_span_one_sub_natCast_ne_two {G : Type u_1} [Monoid G] [Finite G] [Nontrivial G] {a b : ℕ} (ha : Odd a) (hb : Odd b) : Nat.card (MonoidAlgebra ℤ_[2] G ⧸ Ideal.span {MonoidAlgebra.single 1 1 - ↑a, MonoidAlgebra.single 1 1 - ↑b}) ≠ 2

If G is finite and nontrivial and a, b are odd, then the quotient of ℤ₂[G] by (1 - a, 1 - b) cannot have order two. Indeed, reduction modulo two makes both relations vanish, so the quotient surjects onto 𝔽₂[G], which has at least four elements.