Frobenius.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Frobenius elements

In algebraic number theory, if L/K is a finite Galois extension of number fields, with rings of integers 𝓞L/𝓞K, and if q is prime ideal of 𝓞L lying over a prime ideal p of 𝓞K, then there exists a Frobenius element Frob p in Gal(L/K) with the property that Frob p x ≡ x ^ #(𝓞K/p) (mod q) for all x ∈ 𝓞L.

Following RingTheory/Invariant.lean, we develop the theory in the setting that there is a finite group G acting on a ring S, and R is the fixed subring of S.

Main results

Let S/R be an extension of rings, Q be a prime of S, and P := R ∩ Q with finite residue field of cardinality q.

AlgHom.IsArithFrobAt: We say that a φ : S →ₐ[R] S is an (arithmetic) Frobenius at Q if φ x ≡ x ^ q (mod Q) for all x : S.
AlgHom.IsArithFrobAt.apply_of_pow_eq_one: Suppose S is a domain and φ is a Frobenius at Q, then φ ζ = ζ ^ q for any m-th root of unity ζ with q ∤ m.
AlgHom.IsArithFrobAt.eq_of_isUnramifiedAt: Suppose S is noetherian, Q contains all zero-divisors, and the extension is unramified at Q. Then the Frobenius is unique (if exists).

Let G be a finite group acting on a ring S, and R is the fixed subring of S.

IsArithFrobAt: We say that a σ : G is an (arithmetic) Frobenius at Q if σ • x ≡ x ^ q (mod Q) for all x : S.
IsArithFrobAt.mul_inv_mem_inertia: Two Frobenius elements at Q differ by an element in the inertia subgroup of Q.
IsArithFrobAt.conj: If σ is a Frobenius at Q, then τστ⁻¹ is a Frobenius at σ • Q.
IsArithFrobAt.exists_of_isInvariant: Frobenius element exists.

Imports (4)

Imported by 2 files

lemma IsArithFrobAt.exists_of_isInvariant_of_profinite Let G be a finite group acting on S, and R be the fixed subring line 68 source
Source documentation
Let G be a finite group acting on S, and R be the fixed subring. If Q is a prime of S with finite residue field, then there exists a Frobenius element σ : G at Q.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, Group, MulSemiringAction, SMulCommClass, Ideal, TopologicalSpace, CompactSpace, TotallyDisconnectedSpace, IsTopologicalGroup, Algebra.IsInvariant, ContinuousSMulDiscrete, Ideal.IsMaximal, Finite, HasQuotient.Quotient, Ideal.under, Exists, IsArithFrobAt
Proof uses: Bot.bot, DiscreteTopology, DiscreteTopology.mk, rfl, Algebra.IsIntegral, Algebra.IsInvariant.isIntegral_of_profinite, Ideal.isMaximal_under_of_isIntegral_of_isMaximal, Nat, CharP, CharP.exists, Fintype, Nonempty, Eq, nonempty_fintype, Nonempty.intro, PNat, And, Nat.Prime, Fintype.card, HPow.hPow and 44 more
114 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
lemma exists_of_isInvariant_of_profinite
    [Q.IsMaximal] [Finite (R ⧸ Q.under R)] : ∃ σ : G, IsArithFrobAt R σ Q := by
…
```
lemma IsArithFrobAt.exists_primesOver_isConj_of_profinite line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, Group, MulSemiringAction, SMulCommClass, TopologicalSpace, CompactSpace, TotallyDisconnectedSpace, IsTopologicalGroup, Algebra.IsInvariant, ContinuousSMulDiscrete, Ideal, Finite, HasQuotient.Quotient, Ideal.IsPrime, Exists, Ideal.primesOver, And, IsArithFrobAt, Membership.mem and 2 more
Proof uses: Bot.bot, DiscreteTopology, DiscreteTopology.mk, rfl, Ideal.IsMaximal, Ideal.Quotient.maximal_of_isField, Finite.isField_of_domain, Algebra.IsIntegral, Algebra.IsInvariant.isIntegral_of_profinite, Eq, EmptyCollection.emptyCollection, Set.Nonempty, Or, Set.eq_empty_or_nonempty, of_eq_true, Eq.trans, True, congrArg, funext, congr and 30 more
79 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
def IsArithFrobAt._root_.arithFrobAt' line 119 source
lemma IsArithFrobAt.arithFrobAt' line 122 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, Group, MulSemiringAction, SMulCommClass, Ideal, TopologicalSpace, CompactSpace, TotallyDisconnectedSpace, IsTopologicalGroup, Algebra.IsInvariant, ContinuousSMulDiscrete, Ideal.IsPrime, Finite, HasQuotient.Quotient, Ideal.under, IsArithFrobAt, arithFrobAt'
Proof uses: And.left, Ideal.primesOver, Exists.choose, And, Membership.mem, Set, IsConj, IsArithFrobAt.exists_primesOver_isConj_of_profinite, Exists.choose_spec, Subtype.mk, And.intro, Ideal.LiesOver, Ideal.LiesOver.mk, rfl
40 instances, coercions and auxiliary constants left out.
lemma IsArithFrobAt._root_.isConj_arithFrobAt' line 126 source