chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
5,004 | lean_file_chunk | ed31c5f13ec7a62b795be15dce249ce740cd1738fcf66adc20d78d1a3870e9ee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Init.Data.ULift
public import Init.Data.Fin.Fold
public import Init.Data.List.Nat.Pairwise
public import Init.Data.List.Nat.Range
public import Ma... |
23,977 | lean_file_chunk | 836e694cace323fcc3dda22ff2bb5ef0d22626a5c07ca3c08985ade22f6fe059 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone
public import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord
p... |
15,103 | lean_file_chunk | f224e3b52eebdffa261453c5d18ef7b4dd0d33c8699337a65dd43425250a97aa | lean_source_file | pkg_mathlib_source_file_chunks | enter [2, 1, 2, w, 2, i]
rw [if_neg i.prop]
simp_rw [sum_apply, ← sum_fn, map_sum, Pi.smul_apply, ← Pi.smul_def, map_smul,
completeBasis_apply_of_ne, expMap_symm_apply, normAtAllPlaces_mixedEmbedding,
← logEmbedding_component, logEmbedding_fundSystem, Finsupp.coe_finset_sum, Finsupp.coe_smul,
sum_appl... |
23,974 | lean_file_chunk | 0068b0a6507aa0bb826645de4d2b09507f2d425142835a17160094db5e57e090 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.Algebra.Module.ZLattice.Basic
public import Mathlib.Analysis.InnerProductSpace.ProdL2
public import Mathlib.MeasureTheory.Measure.Haar.Uni... |
23,950 | lean_file_chunk | d273cc8ed9404acc1cfeb626aa2909911148497c2d0907771a35dea72b236a8d | lean_source_file | pkg_mathlib_source_file_chunks | · exact ⟨Sum.inr ⟨InfinitePlace.mkComplex ⟨φ, hφ⟩, 0⟩, by simp [hw]⟩
· exact ⟨Sum.inr ⟨InfinitePlace.mkComplex ⟨φ, hφ⟩, 1⟩,
by simp [(embedding_mk_eq φ).resolve_left hw]⟩
· rw [Embeddings.card, ← mixedEmbedding.finrank K,
← Module.finrank_eq_card_basis (stdBasis K)]
variable {K}
@[simp]
theore... |
5,852 | lean_file_chunk | a6fe1186598f1dd58af94e1237b9319072f235a8db4644de7f58db5d79d20add | lean_source_file | pkg_mathlib_source_file_chunks | volume_negAt_plusPart hm, tsum_fintype, sum_const, card_univ, Fintype.card_set, nsmul_eq_mul,
Nat.cast_pow, Nat.cast_ofNat, nrRealPlaces]
end plusPart
noncomputable section realSpace
open MeasureTheory
/--
The `realSpace` associated to a number field `K` is the real vector space indexed by the
infinite places o... |
22,739 | lean_file_chunk | 7ead896741bdccb84d4fe2d011ebc1d9c9fe8ce6c2552dd4e97299774d871804 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.Analysis.SpecialFunctions.PolarCoord
public import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
public import Mathlib.Topolog... |
23,956 | lean_file_chunk | 45eb2f71609b588b1fec7f0f9bae817ba871760fa851fd13d9b1c6e4045dd0ea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.MeasureTheory.Group.GeometryOfNumbers
public import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
public import Mathlib.NumberTheor... |
7,638 | lean_file_chunk | c3db7ee550903af109a94c43712d8715fda90e4ba894ca44d4eb7044130bd641 | lean_source_file | pkg_mathlib_source_file_chunks | h_fund (convexBodyLT_neg_mem K f) (convexBodyLT_convex K f) h
rw [mem_toAddSubgroup, mem_span_fractionalIdealLatticeBasis] at hx
obtain ⟨a, ha, rfl⟩ := hx
exact ⟨a, ha, by simpa using h_nz, (convexBodyLT_mem K f).mp h_mem⟩
open scoped Classical in
/-- A version of `exists_ne_zero_mem_ideal_lt` where the absolute... |
23,983 | lean_file_chunk | 8afc27dfef62db88a84ab01c4017dfa7be8c9f8c925127f6101409c038d19983 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.RingTheory.Ideal.IsPrincipal
public import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
public import Mathlib.RingTheory.ClassG... |
4,966 | lean_file_chunk | 7f747df9786a98d7e5c108c95e7a79e07213612f16063634f3732fe6f81f2f36 | lean_source_file | pkg_mathlib_source_file_chunks | ((idealSetEquiv K J).symm a : mixedSpace K) = a := by
rw [← (idealSetEquiv_apply ((idealSetEquiv K J).symm a)), Equiv.apply_symm_apply]
theorem intNorm_idealSetEquiv_apply (a : idealSet K J) :
intNorm (idealSetEquiv K J a).val = mixedEmbedding.norm (a : mixedSpace K) := by
rw [intNorm_coe, idealSetEquiv_apply]... |
7,704 | lean_file_chunk | 59e32f715d16e1136cd2c97638e9c9af1036057799238cc1652652b95ebe93d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Daniel Weber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Weber
-/
module
public import Mathlib.RingTheory.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
/-!
# Differential Fields
This file defines the logarit... |
8,861 | lean_file_chunk | 891ac2f4efb7b1ac808d88cd30d62ea2b85f094f3ce2b5a4b9a4b0ddbceb1e95 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Daniel Weber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Weber
-/
module
public import Mathlib.Algebra.Algebra.Field
public import Mathlib.Algebra.BigOperators.Field
public import Mathlib.FieldTheory.Differential.Basic
public import Ma... |
16,175 | lean_file_chunk | e05e55ff448fa50216982dc01bd2772ea95d1520fd8f8e0a1d6aead5a7aebae1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.FieldTheory.LinearDisjoint
public import Mathlib.FieldTheory.PurelyInseparable.PerfectClosure
/-!
# Tower law for purely inseparable extensions
This ... |
23,961 | lean_file_chunk | 968336f38994c4a2744b0cd6bbd6fbc6e8873420309716bab8488800b13e9469 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.Algebra.CharP.IntermediateField
public import Mathlib.FieldTheory.SeparableClosure
/-!
# Basic results about purely inseparable extensions
This file ... |
10,094 | lean_file_chunk | dc007facbe35336b0948816fb5dbdf469b29bf44625615d33ae7101c350c835c | lean_source_file | pkg_mathlib_source_file_chunks | mk_prod, mk_eq_one (Emb _ E), lift_one, mul_one, lift_id]
lemma finInsepDegree_eq_pow (q : ℕ) [ExpChar F q] [FiniteDimensional F E] :
∃ n, finInsepDegree F E = q ^ n :=
IsPurelyInseparable.finrank_eq_pow ..
/-- An intermediate field of `E / F` contains the separable closure of `F` in `E`
if `E` is purely insepa... |
14,142 | lean_file_chunk | 63df0d47864e0690a2286347c31b4df5430789435fefc8ff91c8ad10794cdc12 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michal Staromiejski. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michal Staromiejski
-/
module
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathlib.LinearAlgebra.Charpoly.Basic
/-!
# The exponent of purely insepara... |
21,957 | lean_file_chunk | ecb138740c0a8e4e54c95cae0c6adfc49537d877c369af236c27da0cfe158a2a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.Algebra.CharP.Lemmas
public import Mathlib.Algebra.CharP.IntermediateField
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathl... |
1,186 | lean_file_chunk | dfd3e8ffdee08e1550a6e69a3dc635d0c7d726bf17b9bee3e7cbcb9e7ff7e711 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 María Inés de Frutos-Fernández, Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Xavier Généreux
-/
module
public import Mathlib.FieldTheory.Finite.Basic
public import Mathlib.RingTheory.Valuation.Ba... |
14,470 | lean_file_chunk | d6a8af1481f089be93aacba933af86de70867687c5c3e6e548c28a3622c6cfb4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
-/
module
public import Mathlib.Algebra.Algebra.ZMod
public import Mathlib.FieldTheory.Finite.Basic
pu... |
6,685 | lean_file_chunk | 56ff8b88a905e318f276a8fbaabe6d802a920bac932aa99211260900f04a5f4b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Kevin Buzzard
-/
module
public import Mathlib.FieldTheory.Finite.GaloisField
/-!
# Extensions of finite fields
In this file we develop the theory of extensions of finite field... |
23,991 | lean_file_chunk | a6fdb31f650d887152f09a667c141766600df4dd7eca54a55574d976ba10f74e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
module
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.Algebra.CharP.Reduced
public import Mathlib.Algebra.Field.ZMod
pub... |
11,537 | lean_file_chunk | 26627dc682127aa51187806fe0207c9225699cd0dd51f58ff48a2f6820157f72 | lean_source_file | pkg_mathlib_source_file_chunks | have := zsmul_mem (one_mem (⊥ : Subfield (ZMod p))) n.val
by rwa [natCast_zsmul, Nat.smul_one_eq_cast, ZMod.natCast_zmod_val] at this
theorem fieldRange_castHom_eq_bot (p : ℕ) [Fact p.Prime] [DivisionRing K] [CharP K p] :
(ZMod.castHom (m := p) dvd_rfl K).fieldRange = (⊥ : Subfield K) := by
rw [RingHom.fieldRa... |
2,900 | lean_file_chunk | 961d3b168ba6db2254f771b20d0d9c271e0d26be674cb345dd74eb0617832a23 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
module
public import Mathlib.RingTheory.Trace.Basic
public import Mathlib.FieldTheory.Finite.GaloisField
/-!
# The trace and norm maps for finite fields
We state sev... |
7,834 | lean_file_chunk | f4d7ca9e0f5bad2ea9a368a13d191f6264df8efcb9e2446abbfb559291ff0ce9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
module
public import Mathlib.Algebra.MvPolynomial.Expand
public import Mathlib.FieldTheory.Finite.Basic
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import... |
12,862 | lean_file_chunk | 20bee71b823a8a85a64d725233eb0b9311b79f276f471a9a5fc64416b88cb0a9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.FieldTheory.SplittingField.IsSplittingField
public import Mathlib.RingTheory.Algebraic.Basic
/-... |
8,590 | lean_file_chunk | 144ff563c11779b3fae349374129591283ad9c9eaee6ff14aacd661247aa9995 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra
public import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
public import Mathlib.... |
1,440 | lean_file_chunk | 3cbdd7e9531949022fe87d195df97420b24aa392d29caa4016d45fc5f67ea273 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
module
public import Mathlib.Algebra.MvPolynomial.Cardinal
public import Mathlib.RingTheory.Algebraic.LinearIndependent
public import Mathlib.RingTheory.Algebraic.MvPolynomial
publi... |
4,521 | lean_file_chunk | 1c3717e6d9233937e2c45e4a6dad65b73eed695f1658e5366b9b2fa7ab08c78f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Artie Khovanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Artie Khovanov
-/
module
public import Mathlib.Algebra.Polynomial.Degree.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
pub... |
8,716 | lean_file_chunk | 53f2736617be13308312e1724dcb970165be60d8eff7132f9cd76aa15cf5279b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.Algebra.Algebra.ZMod
public import Mathlib.Algebra.Field.ZMod
public import Mathlib.Algebra.MvPolynomial.Cardinal
public import Mathlib.Fiel... |
9,542 | lean_file_chunk | 56c5e499af989edc954f15d3fb060c7748f3144f9022a198032713ef3af22386 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.Data.Multiset.Fintype
public import Mathlib.FieldTheory.IsAlgClosed.Basic
public import Mathlib.FieldT... |
23,965 | lean_file_chunk | 610e6227c30f6e670c8832caaaff508d1ca2d8bc6203fca809ad904be7b0067e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.FieldTheory.Extension
public import Mathlib.FieldTheory.Normal.Defs
public import Mathlib.FieldTheory.Perfect
public import Mathlib.RingTheory.Loc... |
2,658 | lean_file_chunk | 7703b2a377dbbdcada5a94ce068b747a39d863484d9d6d9dbcfef1da5cc22ca8 | lean_source_file | pkg_mathlib_source_file_chunks | (hL : ∀ x : K, ((minpoly F x).map (algebraMap F L)).Splits) (f : K →ₐ[F] L) (x : K) :
algHomEquivAlgHomOfSplits A L hL f x = algebraMap L A (f x) := rfl
/-- All `F`-embeddings of a field `K` into another field `A` factor through any subextension
of `A/F` in which the minimal polynomial of elements of `K` splits. -... |
8,739 | lean_file_chunk | 8920f948710cd06b9a195e467d690b927fe94d5e459248b1e2169c260e745f5d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
public import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Spectrum mapping theorem
This file deve... |
9,151 | lean_file_chunk | 945232bb67b235e16e3622e6991e8bc4aedfcf93d890b917d1b73dae30ff5b0a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
public import Mathlib.FieldTheory.PrimitiveElement
public import Mathlib.FieldTheory.IsAlgClosed.Basic
... |
13,726 | lean_file_chunk | dda77787d09c02c4cc6063026c149f1d91d4206e79d2a26ca2a578f9618eceef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
module
public import Mathlib.RingTheory.AdjoinRoot
public import Mathlib.FieldTheory.Minpoly.Field
public import Mathlib.RingTheory.Polynom... |
10,959 | lean_file_chunk | c00f9ed6bc50e88afa272b32c906490de580f8002e4d304ff68b49dc13416c23 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
module
public import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an eleme... |
6,364 | lean_file_chunk | f843a19cb4b1b00d3f9b2a2cf22452916f068ace18204e14aa5039c5942930c6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
module
public import Mathlib.FieldTheory.Minpoly.IsConjRoot
/-!
# Conjugate root classes
In this file, we define the `ConjRootClass` of a field extension `L / K` as the ... |
16,181 | lean_file_chunk | 507ddef091404315132e61c4e5612c672e9cb8a9d712d2f8fc621172a67406c4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
module
public import Mathlib.Algebra.Polynomial.FieldDivision
public import Mathlib.Algebra.Polynomial.Lifts
public import Mathlib.FieldTheory.Minpo... |
15,278 | lean_file_chunk | 381a3f1db57a72293651e36e10dea20c2332fbe4897280db1763887f017ab0eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang
-/
module
public import Mathlib.FieldTheory.Extension
public import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
public import Mathlib.FieldTheory.Minpoly.Basic
pub... |
14,683 | lean_file_chunk | 8bcaca01e615f09652b240edffea90a5f6fc31c7d58b3d138a581f142f75e9bd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.RingTheory.SimpleRing.Basic
public import Mathlib.FieldTheory.Normal.Basic
public import Mathlib.Order.Closure
public import Mathlib.L... |
13,431 | lean_file_chunk | 5a41e0e6f21158850d841f2be1840a7b8b487b978551606ede3b8f15b99a2a29 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.FieldTheory.Extension
public import Mathlib.FieldTheory.Normal.Defs
public import Mathlib.GroupTheory.Solvable
publ... |
10,563 | lean_file_chunk | c5963e382b3a90647915d054d3283fce4a15375a5571cde842fcc0fdca9294a0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.Algebra.Polynomial.Splits
public import Mathlib.FieldTheory.Galois.Notation
public import Mathlib.FieldTheory.Inter... |
2,473 | lean_file_chunk | f8b14fc08bf1ac5a91e14d5551607de37b41486e01cb7e2cb666a52d578c0727 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.FieldTheory.Galois.Infinite
/-!
# Abelian extensions
In this file, we define the typeclass of abelian extensions and provide some basic API... |
23,990 | lean_file_chunk | ff1e14a6a4e4cfc46f11362ef7cd487e59e7a5dabdd2a6e4346b8644fb28c2b5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz, Yongle Hu, Jingting Wang
-/
module
public import Mathlib.FieldTheory.Fixed
public import Mathlib.FieldTheory.Normal.Closure
public import... |
5,817 | lean_file_chunk | fdf9dec0aa42e3468ef1587588688580a32a509c276be72fffb55b84b35bfe8f | lean_source_file | pkg_mathlib_source_file_chunks | /-- Equivalent characterizations of a Galois extension of finite degree. -/
theorem tfae [FiniteDimensional F E] : List.TFAE [
IsGalois F E,
IntermediateField.fixedField (⊤ : Subgroup Gal(E/F)) = ⊥,
Nat.card Gal(E/F) = finrank F E,
∃ p : F[X], p.Separable ∧ p.IsSplittingField F E] := by
tfae_have 1 → ... |
2,108 | lean_file_chunk | 0b9ea685ca7b8c47ac2fcdb04771437c2c84d5199d916538167fc23918d90ecb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Algebra.Equiv
public import Lean.PrettyPrinter.Delaborator.Basic
/-!
# Notation for Galois group
The Galois group `Gal(L/K)` is impl... |
15,962 | lean_file_chunk | 90b1f79956c67a07c2332eb33262bf706ff5e8831b44e6d12c6f9f628b90f5b8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan, Yuyang Zhao, Jujian Zhang
-/
module
public import Mathlib.FieldTheory.KrullTopology
public import Mathlib.FieldTheory.Galois.GaloisClosure
public import Mathlib.Topology.Alg... |
23,082 | lean_file_chunk | 14751ab6fb6301cb65482f3ea68f070d3c7f9487be87c681ffb3703d98b91c1c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.FieldTheory.Galois.Infinite
public import Mathlib.NumberTheory.NumberField.Basic
public import Mathlib.RingTheory.Invariant.Basic
/-!... |
13,399 | lean_file_chunk | 1a134e9bc09be1fdd87804d7b966952c89232d574d1d9f1dacaeda3c77903b32 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.FieldTheory.KrullTopology
public import Mathlib.FieldTheory.Galois.GaloisClosure
public import Mathlib.Topology.Algebra.Group.ClosedSubgroup
/... |
6,522 | lean_file_chunk | 0d8e4e01d6dd62af1b681b697289b2a590e631761f03c3a73306a2f77d0481f9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Madison Crim, Aaron Liu, Justus Springer, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Madison Crim, Aaron Liu, Justus Springer, Junyan Xu
-/
module
public import Mathlib.Algebra.Module.PID
public import Mathlib.Algebra.MvPolynomial.... |
5,971 | lean_file_chunk | 9e34b95d8dad288b080c500967055afdf88a55f7dcfcd234ff217990111a2a8c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan, Yuyang Zhao
-/
module
public import Mathlib.FieldTheory.Normal.Closure
public import Mathlib.FieldTheory.SeparableClosure
/-!
# Main definitions and results
In a field ex... |
5,949 | lean_file_chunk | 24d5f52ed2791dfbb23693f7834bdc8442e21f693dd1253f09554c4b1eb7617a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Ashvni Narayanan
-/
module
public import Mathlib.FieldTheory.RatFunc.Degree
/-!
# Valuations on F(t)
This file defines the valuation at infinity on the field of rational f... |
8,589 | lean_file_chunk | 214cf1b4cf3c8465db79fa8568db8adf818e07be5c6ad7447340ba3ecc6b8262 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Miriam Philipp, Justus Springer and Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miriam Philipp, Justus Springer, Junyan Xu
-/
module
public import Mathlib.Algebra.Polynomial.Bivariate
public import Mathlib.FieldTheory.RatFunc.AsPoly... |
13,646 | lean_file_chunk | 4ef7426ae0432aa89af922abed08a9420f4dd1f5e53b13d362bd5ecaa50c813b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
module
public import Mathlib.FieldTheory.RatFunc.Basic
public import Mathlib.RingTheory.EuclideanDomain
public import... |
23,997 | lean_file_chunk | 7d0415df077327dfdf259a43a17f15c02da76c4475dfc6cb0d8307e3342dbf34 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.FieldTheory.RatFunc.Defs
public import Mathlib.RingTheory.Polynomial.Content
public import Mathlib... |
20,289 | lean_file_chunk | ac13d9cad3935dadef75aa63fe02909664a46f84aeaf749dcf3f5ef0ba1be6aa | lean_source_file | pkg_mathlib_source_file_chunks | { mapRingHom φ hφ with
commutes' := fun r => by
simp_rw [RingHom.toFun_eq_coe, coe_mapRingHom_eq_coe_map, algebraMap_apply r, map_apply_div,
map_one, AlgHom.commutes] }
theorem coe_mapAlgHom_eq_coe_map (φ : K[X] →ₐ[S] R[X]) (hφ : K[X]⁰ ≤ R[X]⁰.comap φ) :
(mapAlgHom φ hφ : K⟮X⟯ → R⟮X⟯) = map φ hφ ... |
22,663 | lean_file_chunk | d5a7487da8a347cd8c68d085e2c7163cd807f7c6a7907004d0e89ed96d2f49da | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Miriam Philipp, Justus Springer and Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miriam Philipp, Justus Springer, Junyan Xu
-/
module
public import Mathlib.Algebra.Polynomial.Basis
public import Mathlib.FieldTheory.RatFunc.Intermedia... |
9,183 | lean_file_chunk | 1f51e8c7d6acee06e41c0c7eb52d90e06e38c576838d2d71b6cdaf1f9a669a48 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.Algebra.Polynomial.Basic
public import Mathlib.Algebra.Ring.NonZeroDivisors
public import Mathlib.RingTheory.Localization.FractionRing
/-!
# ... |
4,641 | lean_file_chunk | 43a7b129aeacb723b884637d7b242bf808995fa1863d787946dddf4594f49b74 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.FieldTheory.RatFunc.AsPolynomial
public import Mathlib.RingTheory.EuclideanDomain
public import Mathlib.RingTheory.Localization.FractionRing
p... |
7,029 | lean_file_chunk | 04310392c67079f1fd958797f72159e129a4fae4132e0752d08e87c6b4ee0245 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.FieldTheory.IntermediateField.Basic
public import Mathlib.FieldTheory.Minpoly.Basic
public import Mathlib.FieldTheory.Tower
public import Math... |
23,932 | lean_file_chunk | 647bc2fc4466e83c8fc8fd55e7be71cb4d051b7980c4b48b2df669a0d213d1b9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.Field.IsField
public import Mathlib.Algebra.Field.Subfield.Basic
public import ... |
7,089 | lean_file_chunk | df2501ef60055028c0ee22c5019622060f3d10305ac062105037ebcd24689942 | lean_source_file | pkg_mathlib_source_file_chunks | theorem restrictScalars_toSubalgebra {E : IntermediateField L' L} :
(E.restrictScalars K).toSubalgebra = E.toSubalgebra.restrictScalars K :=
SetLike.coe_injective rfl
@[simp]
theorem restrictScalars_toSubfield {E : IntermediateField L' L} :
(E.restrictScalars K).toSubfield = E.toSubfield :=
SetLike.coe_inj... |
5,572 | lean_file_chunk | 98f996a9dd3fb0d8fa71573f2bbccb8f81449526ef809f821aca9912320667eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
module
public import Mathlib.FieldTheory.IntermediateField.Basic
public import Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
/-!
# Extending intermediate... |
23,993 | lean_file_chunk | dbdec36a94d99cdea9a9074fe7dbc0fef3460cf5a7dd4ce45db20244777ab49e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Directed
public import Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder
public ... |
11,242 | lean_file_chunk | 177fbdb93130af6e2b0f9bf8fb479731984835b99b4668b6e843238d7ebb11e1 | lean_source_file | pkg_mathlib_source_file_chunks | rw [dvd_iff_exists_eq_mul_left, ← IntermediateField.adjoin.finrank hx]
use finrank K⟮x⟯ L
rw [mul_comm, finrank_mul_finrank]
theorem _root_.Polynomial.Irreducible.natDegree_dvd_finrank {f : K[X]} (hi : Irreducible f)
(hs : (f.map (algebraMap K L)).Splits) : f.natDegree ∣ finrank K L := by
have := hi.degree_p... |
23,913 | lean_file_chunk | add31da32fd1e8114c47fba0e3b535697379ba439d16241f01ce16f40879ddf0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.FieldTheory.IntermediateField.Basic
/-!
# Adjoining Elements to Fields
In this file we introduce the not... |
4,063 | lean_file_chunk | 68afc75b9f2deb925f14e2484814b0b219edf1cdf6d42ae9efe19ee59da01d61 | lean_source_file | pkg_mathlib_source_file_chunks | theorem fg_def {S : IntermediateField F E} : S.FG ↔ ∃ t : Set E, Set.Finite t ∧ adjoin F t = S :=
Iff.symm Set.exists_finite_iff_finset
theorem fg_adjoin_of_finite {t : Set E} (h : Set.Finite t) : (adjoin F t).FG :=
fg_def.mpr ⟨t, h, rfl⟩
theorem fg_bot : (⊥ : IntermediateField F E).FG :=
⟨∅, by simp only [Fins... |
15,553 | lean_file_chunk | 4f46a2121fb6f4425a8aad3b772a4377650725a29335a04359b0c4a1cdb2e232 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
module
public import Mathlib.FieldTheory.Finiteness
public import Mathlib.FieldTheory.IntermediateField.Adjoin.Defs
public import Mathl... |
9,427 | lean_file_chunk | 2b98a7e306fa9312e84d784587c1633078778a85ce310a7d2fc3164bd4c54545 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Integral
public import Mathlib.Algebra.Category.Ring.Epi
public import Mathlib.RingTheory.F... |
9,358 | lean_file_chunk | 4d7081e0b7d3a6762bc140cdd35554815a16d0c8207f9510f33dcc47f9f640cb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
public import Mathlib.RingTheory.RingHom.Surjective
public import Mathlib.RingTheory.Spectrum.Pr... |
23,917 | lean_file_chunk | 93e5211b6acbe2ff991a15916927e5bc0fd6706de7d6d309183004eddec62181 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Andrew Yang
-/
module
public import Mathlib.Algebra.Category.Ring.Under.Limits
public import Mathlib.AlgebraicGeometry.Morphisms.Affine
public import Mathlib.Algeb... |
1,237 | lean_file_chunk | 30571afe364720737cb6b1ebf1cf2502cfd26f5d6f5fe8f55a36ddde2c826461 | lean_source_file | pkg_mathlib_source_file_chunks | exact mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlat _ _ _ _ hUS hUX hUT hiX
set_option backward.isDefEq.respectTransparency false in
include H in
lemma isIso_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat [Flat f]
(hUS : IsAffineOpen US) (hUT : IsCompact (X := T) UT) (hUT' : IsQuasiSeparat... |
6,739 | lean_file_chunk | 614ab7a16aa8f9551f7c653550c7f96143e720762534473d21c6034d558f1c94 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation
public import Mathlib.AlgebraicGeometry.Morphisms.Flat
public import Mathlib.Algebrai... |
6,777 | lean_file_chunk | b7170e3a6094a44e7aec63681f1c0c96aebe693901f609dd8ba314ca3925991f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Separated
public import Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed
public import Mathlib.RingTheory.Rin... |
2,201 | lean_file_chunk | ef43d299e845ca82fece8f071d616443a1cf5048d338cc9e33b57c5ffc2f2c3d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.FlatDescent
/-!
# Flat monomorphisms of finite presentation are open immersions
We show the titular re... |
21,414 | lean_file_chunk | bf997e7bc574f735ddc2bba4a68fe33bcb6710f04d25f5d9f7b890c7227456e6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Artinian
public import Mathlib.AlgebraicGeometry.Fiber
public import Mathlib.AlgebraicGeometry.Morphisms.Finite
public impor... |
4,789 | lean_file_chunk | 30b576672ce56a2acc409df35b38a33f598982c3f935c4ad0e770bb58c8a2850 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
/-!
# Open immersions
A morphism is an open immersion if the underlying map of spaces is an open ... |
16,319 | lean_file_chunk | 7148788ff2a096344ab50fabb95a59317df06e0d237dd4368424d04f17283d6c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mathl... |
14,757 | lean_file_chunk | 0dccc6a3de3727120cc0a24f8a8e21e003ae2581b3d5fc35553518cc7ce1a1c8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
public import Mathlib.Topology.Spectral.Hom
public import Mathlib.AlgebraicGeometry.Limits
/-!
# Qu... |
5,490 | lean_file_chunk | 33e5b03865baa170cb3f0a0ee3bbc15f0e01dff158620ba60567a846d73bf8a9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Algebra.Category.Ring.Adjunctions
public import Mathlib.AlgebraicGeometry.IdealSheaf.Functorial
public import Mathlib.AlgebraicGeometry.Morphi... |
4,112 | lean_file_chunk | e77e10f4bfa4f2d8119eca9d2e03f238cae60dbf557b3a549356b174a13149ed | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Basic
/-!
# Local closure of morphism properties
We define the source local closure of a property `P` ... |
2,256 | lean_file_chunk | 215f91e7397ea4f0032023f6481f64f2cc455ec96b88c57ffd57cd075f458975 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.LocalClosure
/-!
# Local isomorphisms
A local isomorphism of schemes is a morphism that is source-loca... |
11,933 | lean_file_chunk | 5819d1e36dbdff21edebabeb72c2fc9c2129e1abb64170044d4d060a4b197e02 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Topology.LocalAtTarget
public import Mathlib.AlgebraicGeometry.Morphisms.Constructors
/-!
# Properties on the underlying functions of morphis... |
9,625 | lean_file_chunk | 01b8d637ce52269c6fb880e2b3843b37ef6f044fade3ba42448bedc2e6561dca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.AffineAnd
public import Mathlib.AlgebraicGeometry.Morphisms.LocalIso
public import Mathlib.CategoryTheor... |
2,246 | lean_file_chunk | 0cd6d4286153d64fe6eb377d1d8215daba46a1654b6a7f19ad931961fc39c4d8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Fiber
public import Mathlib.AlgebraicGeometry.Morphisms.Smooth
/-!
# Smooth morphisms and their fibers
## Main r... |
23,999 | lean_file_chunk | e4365bb83c8cedd29c833830a70c964c8e7ec9de33b77c9a3a304cbf4d0b58e9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Limits
public import Mathlib.CategoryTheory.MorphismProperty.Local
public import Mathlib.Data.List.TFAE
/-!
# Properties of... |
6,120 | lean_file_chunk | b411d0530c0e47a9c4c9e713fb33233bb2b1ac472eaadcdbb5254de4130d4793 | lean_source_file | pkg_mathlib_source_file_chunks | exact (Q.arrow_mk_iso_iff
(morphismRestrictRestrictBasicOpen f _ r)).mp this
| openCover U s hs H =>
apply AffineTargetMorphismProperty.IsLocal.of_basicOpenCover _
(s.image (Scheme.Opens.topIso _).inv) (by simp [← Ideal.map_span, hs, Ideal.map_top])
intro ⟨r, hr⟩
obtain ⟨r, hr', rfl⟩ := Finset... |
21,688 | lean_file_chunk | 39c7c879f611e199f7ed10bc9aca51d3caa9060a0d714e416d173dc28232bf1f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Jonas van der Schaaf. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Christian Merten, Jonas van der Schaaf
-/
module
public import Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme
public import Mathlib.AlgebraicGeometry.Morphism... |
5,986 | lean_file_chunk | 4e492ec7184b1322f8f839f9675bdc010d2fe4e16861f85beacaa7cc0c3acf93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Smooth
public import Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified
public import Mathlib.Catego... |
2,423 | lean_file_chunk | f9d41c22ab0d7f1d158e8a36dad6476245d0660e731d3699039f08e34c1a3ad6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion
/-!
# Being an isomorphism is local at the target
-/
universe u
@[expose] public section
open ... |
6,316 | lean_file_chunk | 0a2d5b12da59c08d618c616656ef93255bd979755ffcdc539766832a052fc4f1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten, Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Separated
public import Mathlib.AlgebraicGeometry.Morphisms.Finite
/-!
# Pro... |
5,909 | lean_file_chunk | cb34c3f7d1c6ebd5c93226852baa6d730be6c7094f879ea0605ac531a1b23f87 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
public import Mathlib.RingTheory.RingHom.EssFiniteType
public import Mathlib.RingTheory.RingHom.... |
3,465 | lean_file_chunk | 0e701b8043990b4421d42ad1cc05bb6b093403cd3f6c1c233a85e0edc845226b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mathlib.Topology.LocalAtTarget
/-!
# Universally injective morphism
A morphism of schemes `f... |
7,991 | lean_file_chunk | bf427eb1499783714f093f5f85fe015bb551d0513ccdb3afa657ed4a79eb1dbd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mathlib.Topology.LocalAtTarg... |
19,024 | lean_file_chunk | 699ac05eb8593ca7c0afd7f8105b9a346ba98ed0584a8a74db2fc03d3397bd44 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
public import Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation
public import Mat... |
11,806 | lean_file_chunk | c8a66e94224a4cc995791736a0ce58cb406ceec6fb18de5ba6c621e12adf7f31 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Preimmersion
public import Mathlib.AlgebraicGeometry.Morphisms.Separated
public import Mathlib.AlgebraicGeometry.I... |
3,654 | lean_file_chunk | 1acfd6009736539f4256cfa0ce8ce8ec99f66e834322d20830408bf5c107407f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Jiedong Jiang, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang, Christian Merten
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Etale
/-!
# Weakly étale morphisms
A morphism of schemes is weakly étal... |
23,980 | lean_file_chunk | 6b7d7dc93259da52806d7bb5426319b8e5ab5243d09edd295a4634854ab23665 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.AlgebraicGeometry.Morphisms.Constructors
public import Mathlib.RingTheory.LocalProperties.Basic
public import Mathlib.RingTheory.RingHom.Local... |
12,863 | lean_file_chunk | fc3803fd49e3f89b7e1967bd6793ed821b8ad392ee2d86a5b5773d0e0e611b12 | lean_source_file | pkg_mathlib_source_file_chunks | {X Y Z : Scheme.{u}} {f : X ⟶ Y} {g : Y ⟶ Z} (h : P (f ≫ g)) : P f := by
wlog hZ : IsAffine Z generalizing X Y Z
· rw [IsZariskiLocalAtTarget.iff_of_iSup_eq_top (P := P) _
(g.iSup_preimage_eq_top (iSup_affineOpens_eq_top Z))]
intro U
have H := IsZariskiLocalAtTarget.restrict h U.1
rw [morphismRest... |
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