statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
deg_ne_zero [nontrivial S] (h : is_adjoin_root_monic S f) : nat_degree f ≠ 0 | h.deg_pos.ne' | lemma | is_adjoin_root_monic.deg_ne_zero | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_basis (h : is_adjoin_root_monic S f) : power_basis R S | { gen := h.root,
dim := nat_degree f,
basis := h.basis,
basis_eq_pow := h.basis_apply } | def | is_adjoin_root_monic.power_basis | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"basis",
"is_adjoin_root_monic",
"power_basis"
] | If `f` is monic, the powers of `h.root` form a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_repr (h : is_adjoin_root_monic S f) (x : S) (i : fin (nat_degree f)) :
h.basis.repr x i = (h.mod_by_monic_hom x).coeff (i : ℕ) | begin
change (h.mod_by_monic_hom x).to_finsupp.comap_domain coe (fin.coe_injective.inj_on _) i = _,
rw [finsupp.comap_domain_apply, polynomial.to_finsupp_apply]
end | lemma | is_adjoin_root_monic.basis_repr | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"finsupp.comap_domain_apply",
"is_adjoin_root_monic",
"polynomial.to_finsupp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_one (h : is_adjoin_root_monic S f) (hdeg : 1 < nat_degree f) :
h.basis ⟨1, hdeg⟩ = h.root | by rw [h.basis_apply, fin.coe_mk, pow_one] | lemma | is_adjoin_root_monic.basis_one | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"fin.coe_mk",
"is_adjoin_root_monic",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_polyₗ {T : Type*} [add_comm_group T] [module R T] (h : is_adjoin_root_monic S f)
(g : R[X] →ₗ[R] T) : S →ₗ[R] T | g.comp h.mod_by_monic_hom | def | is_adjoin_root_monic.lift_polyₗ | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"add_comm_group",
"is_adjoin_root_monic",
"module"
] | `is_adjoin_root_monic.lift_polyₗ` lifts a linear map on polynomials to a linear map on `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff (h : is_adjoin_root_monic S f) : S →ₗ[R] (ℕ → R) | h.lift_polyₗ
{ to_fun := polynomial.coeff,
map_add' := λ p q, funext (polynomial.coeff_add p q),
map_smul' := λ c p, funext (polynomial.coeff_smul c p) } | def | is_adjoin_root_monic.coeff | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"polynomial.coeff",
"polynomial.coeff_add",
"polynomial.coeff_smul"
] | `is_adjoin_root_monic.coeff h x i` is the `i`th coefficient of the representative of `x : S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_apply_lt (h : is_adjoin_root_monic S f) (z : S) (i : ℕ) (hi : i < nat_degree f) :
h.coeff z i = h.basis.repr z ⟨i, hi⟩ | begin
simp only [coeff, linear_map.comp_apply, finsupp.lcoe_fun_apply, finsupp.lmap_domain_apply,
linear_equiv.coe_coe, lift_polyₗ_apply, linear_map.coe_mk, h.basis_repr],
refl
end | lemma | is_adjoin_root_monic.coeff_apply_lt | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"finsupp.lmap_domain_apply",
"is_adjoin_root_monic",
"linear_equiv.coe_coe",
"linear_map.coe_mk",
"linear_map.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_apply_coe (h : is_adjoin_root_monic S f) (z : S) (i : fin (nat_degree f)) :
h.coeff z i = h.basis.repr z i | h.coeff_apply_lt z i i.prop | lemma | is_adjoin_root_monic.coeff_apply_coe | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_apply_le (h : is_adjoin_root_monic S f) (z : S) (i : ℕ) (hi : nat_degree f ≤ i) :
h.coeff z i = 0 | begin
simp only [coeff, linear_map.comp_apply, finsupp.lcoe_fun_apply, finsupp.lmap_domain_apply,
linear_equiv.coe_coe, lift_polyₗ_apply, linear_map.coe_mk, h.basis_repr],
nontriviality R,
exact polynomial.coeff_eq_zero_of_degree_lt ((degree_mod_by_monic_lt _ h.monic).trans_le
(polynomial.degree_... | lemma | is_adjoin_root_monic.coeff_apply_le | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"finsupp.lmap_domain_apply",
"is_adjoin_root_monic",
"linear_equiv.coe_coe",
"linear_map.coe_mk",
"linear_map.comp_apply",
"polynomial.coeff_eq_zero_of_degree_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_apply (h : is_adjoin_root_monic S f) (z : S) (i : ℕ) :
h.coeff z i = if hi : i < nat_degree f then h.basis.repr z ⟨i, hi⟩ else 0 | begin
split_ifs with hi,
{ exact h.coeff_apply_lt z i hi },
{ exact h.coeff_apply_le z i (le_of_not_lt hi) },
end | lemma | is_adjoin_root_monic.coeff_apply | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_root_pow (h : is_adjoin_root_monic S f) {n} (hn : n < nat_degree f) :
h.coeff (h.root ^ n) = pi.single n 1 | begin
ext i,
rw coeff_apply,
split_ifs with hi,
{ calc h.basis.repr (h.root ^ n) ⟨i, _⟩
= h.basis.repr (h.basis ⟨n, hn⟩) ⟨i, hi⟩
: by rw [h.basis_apply, fin.coe_mk]
... = pi.single ((⟨n, hn⟩ : fin _) : ℕ) (1 : (λ _, R) _) ↑(⟨i, _⟩ : fin _) :
by rw [h.basis.repr_self, ← finsupp.single_eq_... | lemma | is_adjoin_root_monic.coeff_root_pow | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"fin.coe_injective",
"fin.coe_mk",
"finsupp.single_apply_left",
"finsupp.single_eq_pi_single",
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_one [nontrivial S] (h : is_adjoin_root_monic S f) :
h.coeff 1 = pi.single 0 1 | by rw [← h.coeff_root_pow h.deg_pos, pow_zero] | lemma | is_adjoin_root_monic.coeff_one | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"nontrivial",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_root (h : is_adjoin_root_monic S f) (hdeg : 1 < (nat_degree f)) :
h.coeff h.root = pi.single 1 1 | by rw [← h.coeff_root_pow hdeg, pow_one] | lemma | is_adjoin_root_monic.coeff_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_algebra_map [nontrivial S] (h : is_adjoin_root_monic S f) (x : R) :
h.coeff (algebra_map R S x) = pi.single 0 x | begin
ext i,
rw [algebra.algebra_map_eq_smul_one, map_smul, coeff_one, pi.smul_apply, smul_eq_mul],
refine (pi.apply_single (λ _ y, x * y) _ 0 1 i).trans (by simp),
intros,
simp
end | lemma | is_adjoin_root_monic.coeff_algebra_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"is_adjoin_root_monic",
"nontrivial",
"pi.smul_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_elem (h : is_adjoin_root_monic S f) ⦃x y : S⦄
(hxy : ∀ i < (nat_degree f), h.coeff x i = h.coeff y i) : x = y | equiv_like.injective h.basis.equiv_fun $ funext $ λ i,
show h.basis.equiv_fun x i = h.basis.equiv_fun y i,
by rw [basis.equiv_fun_apply, ← h.coeff_apply_coe, basis.equiv_fun_apply, ← h.coeff_apply_coe,
hxy i i.prop] | lemma | is_adjoin_root_monic.ext_elem | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"basis.equiv_fun_apply",
"equiv_like.injective",
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_elem_iff (h : is_adjoin_root_monic S f) {x y : S} :
x = y ↔ ∀ i < (nat_degree f), h.coeff x i = h.coeff y i | ⟨λ hxy i hi, hxy ▸ rfl, λ hxy, h.ext_elem hxy⟩ | lemma | is_adjoin_root_monic.ext_elem_iff | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_injective (h : is_adjoin_root_monic S f) : function.injective h.coeff | λ x y hxy, h.ext_elem (λ i hi, hxy ▸ rfl) | lemma | is_adjoin_root_monic.coeff_injective | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_root (h : is_adjoin_root_monic S f) : is_integral R h.root | ⟨f, h.monic, h.aeval_root⟩ | lemma | is_adjoin_root_monic.is_integral_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_self_apply (h : is_adjoin_root S f) (x : S) :
h.lift (algebra_map R S) h.root h.aeval_root x = x | by rw [← h.map_repr x, lift_map, ← aeval_def, h.aeval_eq] | lemma | is_adjoin_root.lift_self_apply | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_self (h : is_adjoin_root S f) :
h.lift (algebra_map R S) h.root h.aeval_root = ring_hom.id S | ring_hom.ext (h.lift_self_apply) | lemma | is_adjoin_root.lift_self | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root",
"ring_hom.ext",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv (h : is_adjoin_root S f) (h' : is_adjoin_root T f) : S ≃ₐ[R] T | { to_fun := h.lift_hom h'.root h'.aeval_root,
inv_fun := h'.lift_hom h.root h.aeval_root,
left_inv := λ x, by rw [← h.map_repr x, lift_hom_map, aeval_eq, lift_hom_map, aeval_eq],
right_inv := λ x, by rw [← h'.map_repr x, lift_hom_map, aeval_eq, lift_hom_map, aeval_eq],
.. h.lift_hom h'.root h'.aeval_root } | def | is_adjoin_root.aequiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"inv_fun",
"is_adjoin_root"
] | Adjoining a root gives a unique ring up to algebra isomorphism.
This is the converse of `is_adjoin_root.of_equiv`: this turns an `is_adjoin_root` into an
`alg_equiv`, and `is_adjoin_root.of_equiv` turns an `alg_equiv` into an `is_adjoin_root`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aequiv_map (h : is_adjoin_root S f) (h' : is_adjoin_root T f) (z : R[X]) :
h.aequiv h' (h.map z) = h'.map z | by rw [aequiv, alg_equiv.coe_mk, lift_hom_map, aeval_eq] | lemma | is_adjoin_root.aequiv_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_equiv.coe_mk",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv_root (h : is_adjoin_root S f) (h' : is_adjoin_root T f) :
h.aequiv h' h.root = h'.root | by rw [aequiv, alg_equiv.coe_mk, lift_hom_root] | lemma | is_adjoin_root.aequiv_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_equiv.coe_mk",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv_self (h : is_adjoin_root S f) : h.aequiv h = alg_equiv.refl | by { ext a, exact h.lift_self_apply a } | lemma | is_adjoin_root.aequiv_self | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_equiv.refl",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv_symm (h : is_adjoin_root S f) (h' : is_adjoin_root T f) :
(h.aequiv h').symm = h'.aequiv h | by { ext, refl } | lemma | is_adjoin_root.aequiv_symm | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aequiv {U : Type*} [comm_ring U]
(h : is_adjoin_root S f) (h' : is_adjoin_root T f) (i : R →+* U) (x hx z) :
(h'.lift i x hx (h.aequiv h' z)) = h.lift i x hx z | by rw [← h.map_repr z, aequiv_map, lift_map, lift_map] | lemma | is_adjoin_root.lift_aequiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"comm_ring",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_aequiv {U : Type*} [comm_ring U] [algebra R U]
(h : is_adjoin_root S f) (h' : is_adjoin_root T f) (x : U) (hx z) :
(h'.lift_hom x hx (h.aequiv h' z)) = h.lift_hom x hx z | h.lift_aequiv h' _ _ hx _ | lemma | is_adjoin_root.lift_hom_aequiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra",
"comm_ring",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv_aequiv {U : Type*} [comm_ring U] [algebra R U]
(h : is_adjoin_root S f) (h' : is_adjoin_root T f) (h'' : is_adjoin_root U f) (x) :
(h'.aequiv h'') (h.aequiv h' x) = h.aequiv h'' x | h.lift_hom_aequiv _ _ h''.aeval_root _ | lemma | is_adjoin_root.aequiv_aequiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra",
"comm_ring",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv_trans {U : Type*} [comm_ring U] [algebra R U]
(h : is_adjoin_root S f) (h' : is_adjoin_root T f) (h'' : is_adjoin_root U f) :
(h.aequiv h').trans (h'.aequiv h'') = h.aequiv h'' | by { ext z, exact h.aequiv_aequiv h' h'' z } | lemma | is_adjoin_root.aequiv_trans | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra",
"comm_ring",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_equiv (h : is_adjoin_root S f) (e : S ≃ₐ[R] T) : is_adjoin_root T f | { map := ((e : S ≃+* T) : S →+* T).comp h.map,
map_surjective := e.surjective.comp h.map_surjective,
ker_map := by rw [← ring_hom.comap_ker, ring_hom.ker_coe_equiv, ← ring_hom.ker_eq_comap_bot,
h.ker_map],
algebra_map_eq := by ext;
simp only [alg_equiv.commutes, ring_hom.comp_apply, alg_eq... | def | is_adjoin_root.of_equiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_equiv.coe_ring_equiv",
"alg_equiv.commutes",
"is_adjoin_root",
"ring_equiv.coe_to_ring_hom",
"ring_hom.comap_ker",
"ring_hom.comp_apply",
"ring_hom.ker_coe_equiv",
"ring_hom.ker_eq_comap_bot"
] | Transfer `is_adjoin_root` across an algebra isomorphism.
This is the converse of `is_adjoin_root.aequiv`: this turns an `alg_equiv` into an `is_adjoin_root`,
and `is_adjoin_root.aequiv` turns an `is_adjoin_root` into an `alg_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equiv_root (h : is_adjoin_root S f) (e : S ≃ₐ[R] T) :
(h.of_equiv e).root = e h.root | rfl | lemma | is_adjoin_root.of_equiv_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aequiv_of_equiv {U : Type*} [comm_ring U] [algebra R U]
(h : is_adjoin_root S f) (h' : is_adjoin_root T f) (e : T ≃ₐ[R] U) :
h.aequiv (h'.of_equiv e) = (h.aequiv h').trans e | by ext a; rw [← h.map_repr a, aequiv_map, alg_equiv.trans_apply, aequiv_map, of_equiv_map_apply] | lemma | is_adjoin_root.aequiv_of_equiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_equiv.trans_apply",
"algebra",
"comm_ring",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_equiv_aequiv {U : Type*} [comm_ring U] [algebra R U]
(h : is_adjoin_root S f) (h' : is_adjoin_root U f) (e : S ≃ₐ[R] T) :
(h.of_equiv e).aequiv h' = e.symm.trans (h.aequiv h') | by ext a; rw [← (h.of_equiv e).map_repr a, aequiv_map, alg_equiv.trans_apply, of_equiv_map_apply,
e.symm_apply_apply, aequiv_map] | lemma | is_adjoin_root.of_equiv_aequiv | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_equiv.trans_apply",
"algebra",
"comm_ring",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_eq [is_domain R] [is_domain S] [no_zero_smul_divisors R S] [is_integrally_closed R]
(h : is_adjoin_root_monic S f) (hirr : irreducible f) :
minpoly R h.root = f | let ⟨q, hq⟩ := minpoly.is_integrally_closed_dvd h.is_integral_root h.aeval_root in
symm $ eq_of_monic_of_associated h.monic (minpoly.monic h.is_integral_root) $
by convert (associated.mul_left (minpoly R h.root) $
associated_one_iff_is_unit.2 $ (hirr.is_unit_or_is_unit hq).resolve_left $
minpoly.not_is_unit R h... | lemma | is_adjoin_root_monic.minpoly_eq | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"associated.mul_left",
"irreducible",
"is_adjoin_root_monic",
"is_domain",
"is_integrally_closed",
"minpoly",
"minpoly.is_integrally_closed_dvd",
"minpoly.monic",
"minpoly.not_is_unit",
"mul_one",
"no_zero_smul_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.adjoin.power_basis'_minpoly_gen [is_domain R] [is_domain S]
[no_zero_smul_divisors R S] [is_integrally_closed R] {x : S} (hx' : is_integral R x) :
minpoly R x = minpoly R (algebra.adjoin.power_basis' hx').gen | begin
haveI := is_domain_of_prime (prime_of_is_integrally_closed hx'),
haveI := no_zero_smul_divisors_of_prime_of_degree_ne_zero
(prime_of_is_integrally_closed hx') (ne_of_lt (degree_pos hx')).symm,
rw [← minpoly_gen_eq, adjoin.power_basis', minpoly_gen_map, minpoly_gen_eq, power_basis'_gen,
← is_adjoin_r... | lemma | algebra.adjoin.power_basis'_minpoly_gen | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra.adjoin.power_basis'",
"irreducible",
"is_domain",
"is_integral",
"is_integrally_closed",
"minpoly",
"no_zero_smul_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_tensor_product : Prop | function.bijective (tensor_product.lift f) | def | is_tensor_product | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"tensor_product.lift"
] | Given a bilinear map `f : M₁ →ₗ[R] M₂ →ₗ[R] M`, `is_tensor_product f` means that
`M` is the tensor product of `M₁` and `M₂` via `f`.
This is defined by requiring the lift `M₁ ⊗[R] M₂ → M` to be bijective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tensor_product.is_tensor_product : is_tensor_product (tensor_product.mk R M N) | begin
delta is_tensor_product,
convert_to function.bijective linear_map.id using 2,
{ apply tensor_product.ext', simp },
{ exact function.bijective_id }
end | lemma | tensor_product.is_tensor_product | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"linear_map.id",
"tensor_product.ext'",
"tensor_product.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_tensor_product.equiv (h : is_tensor_product f) : M₁ ⊗[R] M₂ ≃ₗ[R] M | linear_equiv.of_bijective _ h | def | is_tensor_product.equiv | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"linear_equiv.of_bijective"
] | If `M` is the tensor product of `M₁` and `M₂`, it is linearly equivalent to `M₁ ⊗[R] M₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_tensor_product.equiv_to_linear_map (h : is_tensor_product f) :
h.equiv.to_linear_map = tensor_product.lift f | rfl | lemma | is_tensor_product.equiv_to_linear_map | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"tensor_product.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_tensor_product.equiv_symm_apply (h : is_tensor_product f) (x₁ : M₁) (x₂ : M₂) :
h.equiv.symm (f x₁ x₂) = x₁ ⊗ₜ x₂ | begin
apply h.equiv.injective,
refine (h.equiv.apply_symm_apply _).trans _,
simp
end | lemma | is_tensor_product.equiv_symm_apply | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_tensor_product.lift (h : is_tensor_product f) (f' : M₁ →ₗ[R] M₂ →ₗ[R] M') : M →ₗ[R] M' | (tensor_product.lift f').comp h.equiv.symm.to_linear_map | def | is_tensor_product.lift | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"tensor_product.lift"
] | If `M` is the tensor product of `M₁` and `M₂`, we may lift a bilinear map `M₁ →ₗ[R] M₂ →ₗ[R] M'`
to a `M →ₗ[R] M'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_tensor_product.lift_eq (h : is_tensor_product f) (f' : M₁ →ₗ[R] M₂ →ₗ[R] M')
(x₁ : M₁) (x₂ : M₂) : h.lift f' (f x₁ x₂) = f' x₁ x₂ | begin
delta is_tensor_product.lift,
simp,
end | lemma | is_tensor_product.lift_eq | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"is_tensor_product.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_tensor_product.map (hf : is_tensor_product f) (hg : is_tensor_product g)
(i₁ : M₁ →ₗ[R] N₁) (i₂ : M₂ →ₗ[R] N₂) : M →ₗ[R] N | hg.equiv.to_linear_map.comp ((tensor_product.map i₁ i₂).comp hf.equiv.symm.to_linear_map) | def | is_tensor_product.map | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"tensor_product.map"
] | The tensor product of a pair of linear maps between modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_tensor_product.map_eq (hf : is_tensor_product f) (hg : is_tensor_product g)
(i₁ : M₁ →ₗ[R] N₁) (i₂ : M₂ →ₗ[R] N₂) (x₁ : M₁) (x₂ : M₂) :
hf.map hg i₁ i₂ (f x₁ x₂) = g (i₁ x₁) (i₂ x₂) | begin
delta is_tensor_product.map,
simp
end | lemma | is_tensor_product.map_eq | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"is_tensor_product.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_tensor_product.induction_on (h : is_tensor_product f) {C : M → Prop}
(m : M) (h0 : C 0) (htmul : ∀ x y, C (f x y)) (hadd : ∀ x y, C x → C y → C (x + y)) : C m | begin
rw ← h.equiv.right_inv m,
generalize : h.equiv.inv_fun m = y,
change C (tensor_product.lift f y),
induction y using tensor_product.induction_on,
{ rwa map_zero },
{ rw tensor_product.lift.tmul, apply htmul },
{ rw map_add, apply hadd; assumption }
end | lemma | is_tensor_product.induction_on | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_tensor_product",
"tensor_product.induction_on",
"tensor_product.lift",
"tensor_product.lift.tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change : Prop | is_tensor_product
(((algebra.of_id S $ module.End S (M →ₗ[R] N)).to_linear_map.flip f).restrict_scalars R) | def | is_base_change | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra.of_id",
"is_tensor_product",
"module.End",
"restrict_scalars"
] | Given an `R`-algebra `S` and an `R`-module `M`, an `S`-module `N` together with a map
`f : M →ₗ[R] N` is the base change of `M` to `S` if the map `S × M → N, (s, m) ↦ s • f m` is the
tensor product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_base_change.lift (g : M →ₗ[R] Q) : N →ₗ[S] Q | { map_smul' := λ r x, begin
let F := ((algebra.of_id S $ module.End S (M →ₗ[R] Q))
.to_linear_map.flip g).restrict_scalars R,
have hF : ∀ (s : S) (m : M), h.lift F (s • f m) = s • g m := h.lift_eq F,
change h.lift F (r • x) = r • h.lift F x,
apply h.induction_on x,
{ rw [smul_zero, map_zero, s... | def | is_base_change.lift | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra.of_id",
"module.End",
"restrict_scalars",
"smul_add",
"smul_zero"
] | Suppose `f : M →ₗ[R] N` is the base change of `M` along `R → S`. Then any `R`-linear map from
`M` to an `S`-module factors thorugh `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_base_change.lift_eq (g : M →ₗ[R] Q) (x : M) : h.lift g (f x) = g x | begin
have hF : ∀ (s : S) (m : M), h.lift g (s • f m) = s • g m := h.lift_eq _,
convert hF 1 x; rw one_smul,
end | lemma | is_base_change.lift_eq | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.lift_comp (g : M →ₗ[R] Q) : ((h.lift g).restrict_scalars R).comp f = g | linear_map.ext (h.lift_eq g) | lemma | is_base_change.lift_comp | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"linear_map.ext",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.induction_on (x : N) (P : N → Prop)
(h₁ : P 0)
(h₂ : ∀ m : M, P (f m))
(h₃ : ∀ (s : S) n, P n → P (s • n))
(h₄ : ∀ n₁ n₂, P n₁ → P n₂ → P (n₁ + n₂)) : P x | h.induction_on x h₁ (λ s y, h₃ _ _ (h₂ _)) h₄ | lemma | is_base_change.induction_on | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.alg_hom_ext (g₁ g₂ : N →ₗ[S] Q) (e : ∀ x, g₁ (f x) = g₂ (f x)) :
g₁ = g₂ | begin
ext x,
apply h.induction_on x,
{ rw [map_zero, map_zero] },
{ assumption },
{ intros s n e', rw [g₁.map_smul, g₂.map_smul, e'] },
{ intros x y e₁ e₂, rw [map_add, map_add, e₁, e₂] }
end | lemma | is_base_change.alg_hom_ext | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.alg_hom_ext' [module R Q] [is_scalar_tower R S Q] (g₁ g₂ : N →ₗ[S] Q)
(e : (g₁.restrict_scalars R).comp f = (g₂.restrict_scalars R).comp f) :
g₁ = g₂ | h.alg_hom_ext g₁ g₂ (linear_map.congr_fun e) | lemma | is_base_change.alg_hom_ext' | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_scalar_tower",
"linear_map.congr_fun",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_product.is_base_change : is_base_change S (tensor_product.mk R S M 1) | begin
delta is_base_change,
convert tensor_product.is_tensor_product R S M using 1,
ext s x,
change s • 1 ⊗ₜ x = s ⊗ₜ x,
rw tensor_product.smul_tmul',
congr' 1,
exact mul_one _,
end | lemma | tensor_product.is_base_change | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_base_change",
"mul_one",
"tensor_product.is_tensor_product",
"tensor_product.mk",
"tensor_product.smul_tmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.equiv : S ⊗[R] M ≃ₗ[S] N | { map_smul' := λ r x, begin
change h.equiv (r • x) = r • h.equiv x,
apply tensor_product.induction_on x,
{ rw [smul_zero, map_zero, smul_zero] },
{ intros x y, simp [smul_tmul', algebra.of_id_apply] },
{ intros x y hx hy, rw [map_add, smul_add, map_add, smul_add, hx, hy] },
end,
..h.equiv } | def | is_base_change.equiv | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra.of_id_apply",
"smul_add",
"smul_zero",
"tensor_product.induction_on"
] | The base change of `M` along `R → S` is linearly equivalent to `S ⊗[R] M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_base_change.equiv_tmul (s : S) (m : M) : h.equiv (s ⊗ₜ m) = s • (f m) | tensor_product.lift.tmul s m | lemma | is_base_change.equiv_tmul | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"tensor_product.lift.tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.equiv_symm_apply (m : M) : h.equiv.symm (f m) = 1 ⊗ₜ m | by rw [h.equiv.symm_apply_eq, h.equiv_tmul, one_smul] | lemma | is_base_change.equiv_symm_apply | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.of_lift_unique
(h : ∀ (Q : Type (max v₁ v₂ v₃)) [add_comm_monoid Q], by exactI ∀ [module R Q] [module S Q],
by exactI ∀ [is_scalar_tower R S Q], by exactI ∀ (g : M →ₗ[R] Q),
∃! (g' : N →ₗ[S] Q), (g'.restrict_scalars R).comp f = g) : is_base_change S f | begin
obtain ⟨g, hg, -⟩ := h (ulift.{v₂} $ S ⊗[R] M)
(ulift.module_equiv.symm.to_linear_map.comp $ tensor_product.mk R S M 1),
let f' : S ⊗[R] M →ₗ[R] N := _, change function.bijective f',
let f'' : S ⊗[R] M →ₗ[S] N,
{ refine { to_fun := f', map_smul' := λ s x,
tensor_product.induction_on x _ (λ s' y... | lemma | is_base_change.of_lift_unique | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"add_comm_monoid",
"fun_like.ext_iff",
"is_base_change",
"is_scalar_tower",
"linear_equiv.of_linear",
"linear_map.cancel_left",
"linear_map.comp_apply",
"linear_map.restrict_scalars_apply",
"module",
"one_smul",
"smul_add",
"smul_assoc",
"smul_zero",
"tensor_product.induction_on",
"tenso... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.iff_lift_unique :
is_base_change S f ↔
∀ (Q : Type (max v₁ v₂ v₃)) [add_comm_monoid Q], by exactI ∀ [module R Q] [module S Q],
by exactI ∀ [is_scalar_tower R S Q], by exactI ∀ (g : M →ₗ[R] Q),
∃! (g' : N →ₗ[S] Q), (g'.restrict_scalars R).comp f = g | ⟨λ h, by { introsI,
exact ⟨h.lift g, h.lift_comp g, λ g' e, h.alg_hom_ext' _ _ (e.trans (h.lift_comp g).symm)⟩ },
is_base_change.of_lift_unique f⟩ | lemma | is_base_change.iff_lift_unique | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"add_comm_monoid",
"is_base_change",
"is_base_change.of_lift_unique",
"is_scalar_tower",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.of_equiv (e : M ≃ₗ[R] N) : is_base_change R e.to_linear_map | begin
apply is_base_change.of_lift_unique,
introsI Q I₁ I₂ I₃ I₄ g,
have : I₂ = I₃,
{ ext r q,
rw [← one_smul R q, smul_smul, ← smul_assoc, smul_eq_mul, mul_one] },
unfreezingI { cases this },
refine ⟨g.comp e.symm.to_linear_map, by { ext, simp }, _⟩,
rintros y (rfl : _ = _),
ext,
simp,
end | lemma | is_base_change.of_equiv | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_base_change",
"is_base_change.of_lift_unique",
"mul_one",
"one_smul",
"smul_assoc",
"smul_eq_mul",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_base_change.comp {f : M →ₗ[R] N} (hf : is_base_change S f) {g : N →ₗ[S] O}
(hg : is_base_change T g) : is_base_change T ((g.restrict_scalars R).comp f) | begin
apply is_base_change.of_lift_unique,
introsI Q _ _ _ _ i,
letI := module.comp_hom Q (algebra_map S T),
haveI : is_scalar_tower S T Q := ⟨λ x y z, by { rw [algebra.smul_def, mul_smul], refl }⟩,
haveI : is_scalar_tower R S Q,
{ refine ⟨λ x y z, _⟩,
change (is_scalar_tower.to_alg_hom R S T) (x • y) •... | lemma | is_base_change.comp | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"alg_hom.map_smul",
"algebra.smul_def",
"algebra_map",
"is_base_change",
"is_base_change.lift_comp",
"is_base_change.lift_eq",
"is_base_change.of_lift_unique",
"is_scalar_tower",
"is_scalar_tower.to_alg_hom",
"module.comp_hom",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.is_pushout : Prop | (out : is_base_change S (to_alg_hom R R' S').to_linear_map) | class | algebra.is_pushout | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"is_base_change"
] | A type-class stating that the following diagram of scalar towers
R → S
↓ ↓
R' → S'
is a pushout diagram (i.e. `S' = S ⊗[R] R'`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_pushout.symm
(h : algebra.is_pushout R S R' S') :
algebra.is_pushout R R' S S' | begin
letI := (algebra.tensor_product.include_right : R' →ₐ[R] S ⊗ R').to_ring_hom.to_algebra,
let e : R' ⊗[R] S ≃ₗ[R'] S',
{ refine { map_smul' := _, ..(tensor_product.comm R R' S).trans $ h.1.equiv.restrict_scalars R },
intros r x,
change
h.1.equiv (tensor_product.comm R R' S (r • x)) = r • h.1.eq... | lemma | algebra.is_pushout.symm | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra.is_pushout",
"algebra.smul_def",
"algebra.tensor_product.include_right",
"is_base_change.of_equiv",
"ring",
"ring_hom.algebra_map_to_algebra",
"smul_add",
"smul_zero",
"tensor_product.comm",
"tensor_product.induction_on",
"tensor_product.is_base_change",
"tensor_product.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.is_pushout.comm :
algebra.is_pushout R S R' S' ↔ algebra.is_pushout R R' S S' | ⟨algebra.is_pushout.symm, algebra.is_pushout.symm⟩ | lemma | algebra.is_pushout.comm | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra.is_pushout"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_product.is_pushout {R S T : Type*} [comm_ring R] [comm_ring S] [comm_ring T]
[algebra R S] [algebra R T] :
algebra.is_pushout R S T (tensor_product R S T) | ⟨tensor_product.is_base_change R T S⟩ | instance | tensor_product.is_pushout | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra",
"algebra.is_pushout",
"comm_ring",
"tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_product.is_pushout' {R S T : Type*} [comm_ring R] [comm_ring S] [comm_ring T]
[algebra R S] [algebra R T] :
algebra.is_pushout R T S (tensor_product R S T) | algebra.is_pushout.symm infer_instance | instance | tensor_product.is_pushout' | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra",
"algebra.is_pushout",
"algebra.is_pushout.symm",
"comm_ring",
"tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.pushout_desc [H : algebra.is_pushout R S R' S']
{A : Type*} [semiring A] [algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A)
(hf : ∀ x y, f x * g y = g y * f x) : S' →ₐ[R] A | begin
letI := module.comp_hom A f.to_ring_hom,
haveI : is_scalar_tower R S A :=
{ smul_assoc := λ r s a, show f (r • s) * a = r • (f s * a), by rw [f.map_smul, smul_mul_assoc] },
haveI : is_scalar_tower S A A :=
{ smul_assoc := λ r a b, mul_assoc _ _ _ },
have : ∀ x, H.out.lift g.to_linear_map (algebra_map ... | def | algebra.pushout_desc | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"alg_hom.of_linear_map",
"algebra",
"algebra.is_pushout",
"algebra_map",
"is_scalar_tower",
"linear_map.map_smul",
"linear_map.restrict_scalars_apply",
"map_one",
"module.comp_hom",
"mul_assoc",
"mul_comm",
"mul_zero",
"restrict_scalars",
"semiring",
"smul_assoc",
"smul_mul_assoc",
"... | If `S' = S ⊗[R] R'`, then any pair of `R`-algebra homomorphisms `f : S → A` and `g : R' → A`
such that `f x` and `g y` commutes for all `x, y` descends to a (unique) homomoprhism `S' → A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.pushout_desc_left [H : algebra.is_pushout R S R' S']
{A : Type*} [semiring A] [algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H) (x : S) :
algebra.pushout_desc S' f g H (algebra_map S S' x) = f x | begin
rw [algebra.pushout_desc_apply, algebra.algebra_map_eq_smul_one, linear_map.map_smul,
← algebra.pushout_desc_apply S' f g H, _root_.map_one],
exact mul_one (f x)
end | lemma | algebra.pushout_desc_left | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra.is_pushout",
"algebra.pushout_desc",
"algebra_map",
"linear_map.map_smul",
"mul_one",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.lift_alg_hom_comp_left [H : algebra.is_pushout R S R' S']
{A : Type*} [semiring A] [algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H) :
(algebra.pushout_desc S' f g H).comp (to_alg_hom R S S') = f | alg_hom.ext (λ x, (algebra.pushout_desc_left S' f g H x : _)) | lemma | algebra.lift_alg_hom_comp_left | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"alg_hom.ext",
"algebra",
"algebra.is_pushout",
"algebra.pushout_desc",
"algebra.pushout_desc_left",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.pushout_desc_right [H : algebra.is_pushout R S R' S']
{A : Type*} [semiring A] [algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H) (x : R') :
algebra.pushout_desc S' f g H (algebra_map R' S' x) = g x | begin
apply_with @@is_base_change.lift_eq { instances := ff },
end | lemma | algebra.pushout_desc_right | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"algebra",
"algebra.is_pushout",
"algebra.pushout_desc",
"algebra_map",
"is_base_change.lift_eq",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.lift_alg_hom_comp_right [H : algebra.is_pushout R S R' S']
{A : Type*} [semiring A] [algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H) :
(algebra.pushout_desc S' f g H).comp (to_alg_hom R R' S') = g | alg_hom.ext (λ x, (algebra.pushout_desc_right S' f g H x : _)) | lemma | algebra.lift_alg_hom_comp_right | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"alg_hom.ext",
"algebra",
"algebra.is_pushout",
"algebra.pushout_desc",
"algebra.pushout_desc_right",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.is_pushout.alg_hom_ext [H : algebra.is_pushout R S R' S']
{A : Type*} [semiring A] [algebra R A] {f g : S' →ₐ[R] A}
(h₁ : f.comp (to_alg_hom R R' S') = g.comp (to_alg_hom R R' S'))
(h₂ : f.comp (to_alg_hom R S S') = g.comp (to_alg_hom R S S')) : f = g | begin
ext x,
apply H.1.induction_on x,
{ simp only [map_zero] },
{ exact alg_hom.congr_fun h₁ },
{ intros s s' e, rw [algebra.smul_def, f.map_mul, g.map_mul, e],
congr' 1, exact (alg_hom.congr_fun h₂ s : _) },
{ intros s₁ s₂ e₁ e₂, rw [map_add, map_add, e₁, e₂] }
end | lemma | algebra.is_pushout.alg_hom_ext | ring_theory | src/ring_theory/is_tensor_product.lean | [
"ring_theory.tensor_product",
"algebra.module.ulift"
] | [
"alg_hom.congr_fun",
"algebra",
"algebra.is_pushout",
"algebra.smul_def",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson (R : Type*) [comm_ring R] : Prop | (out' : ∀ (I : ideal R), I.is_radical → I.jacobson = I) | class | ideal.is_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"ideal"
] | A ring is a Jacobson ring if for every radical ideal `I`,
the Jacobson radical of `I` is equal to `I`.
See `is_jacobson_iff_prime_eq` and `is_jacobson_iff_Inf_maximal` for equivalent definitions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_iff {R} [comm_ring R] :
is_jacobson R ↔ ∀ (I : ideal R), I.is_radical → I.jacobson = I | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | theorem | ideal.is_jacobson_iff | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson.out {R} [comm_ring R] :
is_jacobson R → ∀ {I : ideal R}, I.is_radical → I.jacobson = I | is_jacobson_iff.1 | theorem | ideal.is_jacobson.out | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_iff_prime_eq : is_jacobson R ↔ ∀ P : ideal R, is_prime P → P.jacobson = P | begin
refine is_jacobson_iff.trans ⟨λ h I hI, h I hI.is_radical, _⟩,
refine λ h I hI, le_antisymm (λ x hx, _) (λ x hx, mem_Inf.mpr (λ _ hJ, hJ.left hx)),
rw [← hI.radical, radical_eq_Inf I, mem_Inf],
intros P hP,
rw set.mem_set_of_eq at hP,
erw mem_Inf at hx,
erw [← h P hP.right, mem_Inf],
exact λ J hJ,... | lemma | ideal.is_jacobson_iff_prime_eq | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ideal"
] | A ring is a Jacobson ring if and only if for all prime ideals `P`,
the Jacobson radical of `P` is equal to `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_iff_Inf_maximal : is_jacobson R ↔
∀ {I : ideal R}, I.is_prime → ∃ M : set (ideal R), (∀ J ∈ M, is_maximal J ∨ J = ⊤) ∧ I = Inf M | ⟨λ H I h, eq_jacobson_iff_Inf_maximal.1 (H.out h.is_radical),
λ H, is_jacobson_iff_prime_eq.2 (λ P hP, eq_jacobson_iff_Inf_maximal.2 (H hP))⟩ | lemma | ideal.is_jacobson_iff_Inf_maximal | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ideal"
] | A ring `R` is Jacobson if and only if for every prime ideal `I`,
`I` can be written as the infimum of some collection of maximal ideals.
Allowing ⊤ in the set `M` of maximal ideals is equivalent, but makes some proofs cleaner. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_iff_Inf_maximal' : is_jacobson R ↔
∀ {I : ideal R}, I.is_prime → ∃ M : set (ideal R),
(∀ (J ∈ M) (K : ideal R), J < K → K = ⊤) ∧ I = Inf M | ⟨λ H I h, eq_jacobson_iff_Inf_maximal'.1 (H.out h.is_radical),
λ H, is_jacobson_iff_prime_eq.2 (λ P hP, eq_jacobson_iff_Inf_maximal'.2 (H hP))⟩ | lemma | ideal.is_jacobson_iff_Inf_maximal' | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_eq_jacobson [H : is_jacobson R] (I : ideal R) : I.radical = I.jacobson | le_antisymm (le_Inf (λ J ⟨hJ, hJ_max⟩, (is_prime.radical_le_iff hJ_max.is_prime).mpr hJ))
(H.out (radical_is_radical I) ▸ jacobson_mono le_radical) | lemma | ideal.radical_eq_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ideal",
"le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_field {K : Type*} [field K] : is_jacobson K | ⟨λ I hI, or.rec_on (eq_bot_or_top I)
(λ h, le_antisymm (Inf_le ⟨le_rfl, h.symm ▸ bot_is_maximal⟩) (h.symm ▸ bot_le))
(λ h, by rw [h, jacobson_eq_top_iff])⟩ | instance | ideal.is_jacobson_field | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"Inf_le",
"bot_le",
"field"
] | Fields have only two ideals, and the condition holds for both of them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_of_surjective [H : is_jacobson R] :
(∃ (f : R →+* S), function.surjective f) → is_jacobson S | begin
rintros ⟨f, hf⟩,
rw is_jacobson_iff_Inf_maximal,
intros p hp,
use map f '' {J : ideal R | comap f p ≤ J ∧ J.is_maximal },
use λ j ⟨J, hJ, hmap⟩, hmap ▸ (map_eq_top_or_is_maximal_of_surjective f hf hJ.right).symm,
have : p = map f (comap f p).jacobson :=
(is_jacobson.out' _ $ hp.is_radical.comap f)... | theorem | ideal.is_jacobson_of_surjective | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ideal",
"ideal.ker_le_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_quotient [is_jacobson R] : is_jacobson (R ⧸ I) | is_jacobson_of_surjective ⟨quotient.mk I, (by rintro ⟨x⟩; use x; refl)⟩ | instance | ideal.is_jacobson_quotient | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_iso (e : R ≃+* S) : is_jacobson R ↔ is_jacobson S | ⟨λ h, @is_jacobson_of_surjective _ _ _ _ h ⟨(e : R →+* S), e.surjective⟩,
λ h, @is_jacobson_of_surjective _ _ _ _ h ⟨(e.symm : S →+* R), e.symm.surjective⟩⟩ | lemma | ideal.is_jacobson_iso | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_of_is_integral [algebra R S] (hRS : algebra.is_integral R S)
(hR : is_jacobson R) : is_jacobson S | begin
rw is_jacobson_iff_prime_eq,
introsI P hP,
by_cases hP_top : comap (algebra_map R S) P = ⊤,
{ simp [comap_eq_top_iff.1 hP_top] },
{ haveI : nontrivial (R ⧸ comap (algebra_map R S) P) := quotient.nontrivial hP_top,
rw jacobson_eq_iff_jacobson_quotient_eq_bot,
refine eq_bot_of_comap_eq_bot (is_int... | lemma | ideal.is_jacobson_of_is_integral | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"Inf_le_Inf",
"algebra",
"algebra.is_integral",
"algebra_map",
"bot_le",
"eq_bot_iff",
"is_integral_quotient_of_is_integral",
"nontrivial",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_of_is_integral' (f : R →+* S) (hf : f.is_integral)
(hR : is_jacobson R) : is_jacobson S | @is_jacobson_of_is_integral _ _ _ _ f.to_algebra hf hR | lemma | ideal.is_jacobson_of_is_integral' | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_powers_iff_not_mem (hI : I.is_radical) :
disjoint ((submonoid.powers y) : set R) ↑I ↔ y ∉ I.1 | begin
refine ⟨λ h, set.disjoint_left.1 h (mem_powers _), λ h, disjoint_iff.mpr (eq_bot_iff.mpr _)⟩,
rintros x ⟨⟨n, rfl⟩, hx'⟩,
exact h (hI $ mem_radical_of_pow_mem $ le_radical hx')
end | lemma | ideal.disjoint_powers_iff_not_mem | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"disjoint",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_iff_is_maximal_disjoint [H : is_jacobson R] (J : ideal S) :
J.is_maximal ↔ (comap (algebra_map R S) J).is_maximal ∧ y ∉ ideal.comap (algebra_map R S) J | begin
split,
{ refine λ h, ⟨_, λ hy, h.ne_top (ideal.eq_top_of_is_unit_mem _ hy
(map_units _ ⟨y, submonoid.mem_powers _⟩))⟩,
have hJ : J.is_prime := is_maximal.is_prime h,
rw is_prime_iff_is_prime_disjoint (submonoid.powers y) at hJ,
have : y ∉ (comap (algebra_map R S) J).1 :=
set.disjoint_l... | lemma | ideal.is_maximal_iff_is_maximal_disjoint | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"algebra_map",
"eq_top_iff",
"ideal",
"ideal.comap",
"ideal.eq_top_of_is_unit_mem",
"is_localization.order_embedding",
"powers",
"submonoid.mem_powers",
"submonoid.powers"
] | If `R` is a Jacobson ring, then maximal ideals in the localization at `y`
correspond to maximal ideals in the original ring `R` that don't contain `y`.
This lemma gives the correspondence in the particular case of an ideal and its comap.
See `le_rel_iso_of_maximal` for the more general relation isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_maximal_of_is_maximal_disjoint [is_jacobson R] (I : ideal R) (hI : I.is_maximal)
(hy : y ∉ I) : (map (algebra_map R S) I).is_maximal | begin
rw [is_maximal_iff_is_maximal_disjoint S y,
comap_map_of_is_prime_disjoint (powers y) S I (is_maximal.is_prime hI)
((disjoint_powers_iff_not_mem y hI.is_prime.is_radical).2 hy)],
exact ⟨hI, hy⟩
end | lemma | ideal.is_maximal_of_is_maximal_disjoint | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"algebra_map",
"ideal",
"powers"
] | If `R` is a Jacobson ring, then maximal ideals in the localization at `y`
correspond to maximal ideals in the original ring `R` that don't contain `y`.
This lemma gives the correspondence in the particular case of an ideal and its map.
See `le_rel_iso_of_maximal` for the more general statement, and the reverse of this ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_of_maximal [is_jacobson R] :
{p : ideal S // p.is_maximal} ≃o {p : ideal R // p.is_maximal ∧ y ∉ p} | { to_fun := λ p,
⟨ideal.comap (algebra_map R S) p.1, (is_maximal_iff_is_maximal_disjoint S y p.1).1 p.2⟩,
inv_fun := λ p,
⟨ideal.map (algebra_map R S) p.1, is_maximal_of_is_maximal_disjoint y p.1 p.2.1 p.2.2⟩,
left_inv := λ J, subtype.eq (map_comap (powers y) S J),
right_inv := λ I, subtype.eq (comap_map_... | def | ideal.order_iso_of_maximal | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"algebra_map",
"ideal",
"ideal.map_mono",
"inv_fun",
"powers"
] | If `R` is a Jacobson ring, then maximal ideals in the localization at `y`
correspond to maximal ideals in the original ring `R` that don't contain `y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_localization [H : is_jacobson R] : is_jacobson S | begin
rw is_jacobson_iff_prime_eq,
refine λ P' hP', le_antisymm _ le_jacobson,
obtain ⟨hP', hPM⟩ := (is_localization.is_prime_iff_is_prime_disjoint (powers y) S P').mp hP',
have hP := H.out hP'.is_radical,
refine (is_localization.map_comap (powers y) S P'.jacobson).ge.trans
((map_mono _).trans (is_localiz... | lemma | ideal.is_jacobson_localization | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"Inf_eq_infi",
"algebra_map",
"ideal",
"ideal.jacobson",
"infi_le_infi_of_subset",
"is_localization.comap_map_of_is_prime_disjoint",
"is_localization.is_prime_iff_is_prime_disjoint",
"is_localization.map_comap",
"powers"
] | If `S` is the localization of the Jacobson ring `R` at the submonoid generated by `y : R`, then
`S` is Jacobson. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_is_localization_polynomial_quotient
(P : ideal R[X]) (pX : R[X]) (hpX : pX ∈ P)
[algebra (R ⧸ P.comap (C : R →+* _)) Rₘ]
[is_localization.away (pX.map (quotient.mk (P.comap (C : R →+* R[X])))).leading_coeff Rₘ]
[algebra (R[X] ⧸ P) Sₘ]
[is_localization ((submonoid.powers (pX.map
(quotient.mk (P... | begin
let P' : ideal R := P.comap C,
let M : submonoid (R ⧸ P') :=
submonoid.powers (pX.map (quotient.mk (P.comap (C : R →+* R[X])))).leading_coeff,
let M' : submonoid (R[X] ⧸ P) :=
(submonoid.powers (pX.map (quotient.mk (P.comap (C : R →+* R[X])))).leading_coeff).map
(quotient_map P C le_rfl),
let φ : ... | lemma | ideal.polynomial.is_integral_is_localization_polynomial_quotient | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"algebra",
"algebra_map",
"ideal",
"is_integral",
"is_integral_of_mem_closure''",
"is_localization",
"is_localization.away",
"is_localization.map",
"is_localization.map_comp",
"le_rfl",
"mul_assoc",
"mul_comm",
"polynomial.induction_on",
"pow_one",
"pow_succ",
"quotient_map",
"ring_h... | If `I` is a prime ideal of `R[X]` and `pX ∈ I` is a non-constant polynomial,
then the map `R →+* R[x]/I` descends to an integral map when localizing at `pX.leading_coeff`.
In particular `X` is integral because it satisfies `pX`, and constants are trivially integral,
so integrality of the entire extension follows ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
jacobson_bot_of_integral_localization
{R : Type*} [comm_ring R] [is_domain R] [is_jacobson R]
(Rₘ Sₘ : Type*) [comm_ring Rₘ] [comm_ring Sₘ]
(φ : R →+* S) (hφ : function.injective φ) (x : R) (hx : x ≠ 0)
[algebra R Rₘ] [is_localization.away x Rₘ]
[algebra S Sₘ] [is_localization ((submonoid.powers x).map φ : su... | begin
have hM : ((submonoid.powers x).map φ : submonoid S) ≤ non_zero_divisors S :=
map_le_non_zero_divisors_of_injective φ hφ (powers_le_non_zero_divisors_of_no_zero_divisors hx),
letI : is_domain Sₘ := is_localization.is_domain_of_le_non_zero_divisors _ hM,
let φ' : Rₘ →+* Sₘ := is_localization.map _ φ (sub... | lemma | ideal.polynomial.jacobson_bot_of_integral_localization | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"Inf_le_Inf",
"algebra",
"algebra_map",
"comm_ring",
"ideal",
"is_domain",
"is_localization",
"is_localization.away",
"is_localization.injective",
"is_localization.is_domain_of_le_non_zero_divisors",
"is_localization.map",
"is_localization.map_comp",
"is_localization.surjective_quotient_map_... | If `f : R → S` descends to an integral map in the localization at `x`,
and `R` is a Jacobson ring, then the intersection of all maximal ideals in `S` is trivial | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_polynomial_of_domain
(R : Type*) [comm_ring R] [is_domain R] [hR : is_jacobson R]
(P : ideal R[X]) [is_prime P] (hP : ∀ (x : R), C x ∈ P → x = 0) :
P.jacobson = P | begin
by_cases Pb : P = ⊥,
{ exact Pb.symm ▸ jacobson_bot_polynomial_of_jacobson_bot
(hR.out is_radical_bot_of_no_zero_divisors) },
{ rw jacobson_eq_iff_jacobson_quotient_eq_bot,
haveI : (P.comap (C : R →+* R[X])).is_prime := comap_is_prime C P,
obtain ⟨p, pP, p0⟩ := exists_nonzero_mem_of_ne_bot Pb ... | lemma | ideal.polynomial.is_jacobson_polynomial_of_domain | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"ideal",
"is_domain",
"le_rfl",
"localization",
"localization.away",
"polynomial.map",
"quotient_map",
"submonoid",
"submonoid.powers"
] | Used to bootstrap the proof of `is_jacobson_polynomial_iff_is_jacobson`.
That theorem is more general and should be used instead of this one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_jacobson_polynomial_of_is_jacobson (hR : is_jacobson R) :
is_jacobson R[X] | begin
refine is_jacobson_iff_prime_eq.mpr (λ I, _),
introI hI,
let R' : subring (R[X] ⧸ I) := ((quotient.mk I).comp C).range,
let i : R →+* R' := ((quotient.mk I).comp C).range_restrict,
have hi : function.surjective (i : R → R') := ((quotient.mk I).comp C).range_restrict_surjective,
have hi' : (polynomial.... | lemma | ideal.polynomial.is_jacobson_polynomial_of_is_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"le_rfl",
"le_sup_of_le_left",
"polynomial",
"polynomial.map",
"polynomial.map_ring_hom",
"polynomial.map_surjective",
"ring_hom.comp_apply",
"subring",
"subtype.ext_iff",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_polynomial_iff_is_jacobson :
is_jacobson R[X] ↔ is_jacobson R | begin
refine ⟨_, is_jacobson_polynomial_of_is_jacobson⟩,
introI H,
exact is_jacobson_of_surjective ⟨eval₂_ring_hom (ring_hom.id _) 1, λ x,
⟨C x, by simp only [coe_eval₂_ring_hom, ring_hom.id_apply, eval₂_C]⟩⟩,
end | theorem | ideal.polynomial.is_jacobson_polynomial_iff_is_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ring_hom.id",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_comap_C_of_is_maximal [nontrivial R] (hP' : ∀ (x : R), C x ∈ P → x = 0) :
is_maximal (comap (C : R →+* R[X]) P : ideal R) | begin
haveI hp'_prime : (P.comap (C : R →+* R[X]) : ideal R).is_prime := comap_is_prime C P,
obtain ⟨m, hm⟩ := submodule.nonzero_mem_of_bot_lt (bot_lt_of_maximal P polynomial_not_is_field),
have : (m : R[X]) ≠ 0, rwa [ne.def, submodule.coe_eq_zero],
let φ : R ⧸ P.comap (C : R →+* R[X]) →+* R[X] ⧸ P := quotient... | lemma | ideal.polynomial.is_maximal_comap_C_of_is_maximal | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"algebra_map",
"bot_le",
"eq_bot_iff",
"ideal",
"is_domain",
"is_field.localization_map_bijective",
"is_localization.comap_map_of_is_prime_disjoint",
"is_localization.is_domain_localization",
"is_localization.map_injective_of_injective",
"le_non_zero_divisors_of_no_zero_divisors",
"le_rfl",
"l... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_mk_comp_C_is_integral_of_jacobson' [nontrivial R] (hR : is_jacobson R)
(hP' : ∀ (x : R), C x ∈ P → x = 0) :
((quotient.mk P).comp C : R →+* R[X] ⧸ P).is_integral | begin
refine (is_integral_quotient_map_iff _).mp _,
let P' : ideal R := P.comap C,
obtain ⟨pX, hpX, hp0⟩ :=
exists_nonzero_mem_of_ne_bot (ne_of_lt (bot_lt_of_maximal P polynomial_not_is_field)).symm hP',
let M : submonoid (R ⧸ P') := submonoid.powers (pX.map (quotient.mk P')).leading_coeff,
let φ : R ⧸ P'... | lemma | ideal.polynomial.quotient_mk_comp_C_is_integral_of_jacobson' | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"algebra_map",
"ideal",
"is_field.localization_map_bijective",
"is_integral",
"is_integral_of_surjective",
"is_integral_quotient_map_iff",
"is_integral_tower_bot_of_is_integral",
"is_localization.injective",
"is_localization.map",
"is_localization.map_comp",
"le_non_zero_divisors_of_no_zero_divi... | Used to bootstrap the more general `quotient_mk_comp_C_is_integral_of_jacobson` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_mk_comp_C_is_integral_of_jacobson :
((quotient.mk P).comp C : R →+* R[X] ⧸ P).is_integral | begin
let P' : ideal R := P.comap C,
haveI : P'.is_prime := comap_is_prime C P,
let f : R[X] →+* polynomial (R ⧸ P') := polynomial.map_ring_hom (quotient.mk P'),
have hf : function.surjective f := map_surjective (quotient.mk P') quotient.mk_surjective,
have hPJ : P = (P.map f).comap f,
{ rw comap_map_of_sur... | lemma | ideal.polynomial.quotient_mk_comp_C_is_integral_of_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ideal",
"ideal.is_jacobson_quotient",
"is_integral",
"is_integral_of_surjective",
"le_rfl",
"le_sup_of_le_left",
"polynomial",
"polynomial.map_ring_hom",
"ring_hom.is_integral_tower_bot_of_is_integral",
"ring_hom.is_integral_trans",
"sup_le"
] | If `R` is a Jacobson ring, and `P` is a maximal ideal of `R[X]`,
then `R → R[X]/P` is an integral map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_maximal_comap_C_of_is_jacobson :
(P.comap (C : R →+* R[X])).is_maximal | begin
rw [← @mk_ker _ _ P, ring_hom.ker_eq_comap_bot, comap_comap],
exact is_maximal_comap_of_is_integral_of_is_maximal' _
(quotient_mk_comp_C_is_integral_of_jacobson P) ⊥ ((bot_quotient_is_maximal_iff _).mpr hP),
end | lemma | ideal.polynomial.is_maximal_comap_C_of_is_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"ring_hom.ker_eq_comap_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_C_integral_of_surjective_of_jacobson
{S : Type*} [field S] (f : R[X] →+* S) (hf : function.surjective f) :
(f.comp C).is_integral | begin
haveI : (f.ker).is_maximal := ring_hom.ker_is_maximal_of_surjective f hf,
let g : R[X] ⧸ f.ker →+* S := ideal.quotient.lift f.ker f (λ _ h, h),
have hfg : (g.comp (quotient.mk f.ker)) = f := ring_hom_ext' rfl rfl,
rw [← hfg, ring_hom.comp_assoc],
refine ring_hom.is_integral_trans _ g (quotient_mk_comp_C... | lemma | ideal.polynomial.comp_C_integral_of_surjective_of_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"field",
"function.surjective.of_comp",
"ideal.quotient.lift",
"is_integral",
"ring_hom.comp_assoc",
"ring_hom.is_integral_trans",
"ring_hom.ker_is_maximal_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_jacobson_mv_polynomial_fin {R : Type*} [comm_ring R] [H : is_jacobson R] :
∀ (n : ℕ), is_jacobson (mv_polynomial (fin n) R) | | 0 := ((is_jacobson_iso ((rename_equiv R
(equiv.equiv_pempty (fin 0))).to_ring_equiv.trans (is_empty_ring_equiv R pempty))).mpr H)
| (n+1) := (is_jacobson_iso (fin_succ_equiv R n).to_ring_equiv).2
(polynomial.is_jacobson_polynomial_iff_is_jacobson.2 (is_jacobson_mv_polynomial_fin n)) | lemma | ideal.mv_polynomial.is_jacobson_mv_polynomial_fin | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"equiv.equiv_pempty",
"fin_succ_equiv",
"mv_polynomial",
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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