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 |
|---|---|---|---|---|---|
13,265 | lean_file_chunk | 7cf0e1c171186da6982acd70c6c00149256a9883c00f01b38425acfa0a764455 | 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, Robert Y. Lewis
-/
module
public import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vect... |
18,294 | lean_file_chunk | 741763e66bc72b6846e5d3431d013380e45da566ab9f0f5a459d4418e3c7c63d | 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, Robert Y. Lewis
-/
module
public import Mathlib.Algebra.MvPolynomial.Funext
public import Mathlib.Algebra.Ring.ULift
public import Mathlib.RingTheory.WittVector.Basic
... |
15,120 | lean_file_chunk | d962b620fd3adfc9f9174d6050a10916b936072ef75dec2876cfed482b9a12d0 | 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, Robert Y. Lewis
-/
module
public import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a ... |
10,686 | lean_file_chunk | 4ff586f956dd332274db7ca6ef08ca3bd1105cbd369739ce97b1065573709b8d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
module
public import Mathlib.Data.Nat.Cast.WithTop
public import Mathlib.FieldTheory.IsAlgClosed.Basic
public import Mathlib.RingTheory.WittVector... |
2,552 | lean_file_chunk | b971509884cb8c1d338d6296113abe4f9d0b49304d2df0c2550dd443cb9951db | 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.RingTheory.WittVector.IsPoly
/-!
## Multiplication by `n` in the ring of Witt vectors
In this file we show that multiplication by `n` ... |
8,572 | lean_file_chunk | e77fd51fb6ca68fa14b3a6578b5a8a01af89b26af0ee81c394daa3c5efaae11f | 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.RingTheory.WittVector.Frobenius
public import Mathlib.RingTheory.WittVector.Verschiebung
public import Mathlib.RingTheory.WittVector.Mul... |
8,426 | lean_file_chunk | 16fde0abb826fd2da852cec907c654195c1914dc7c904552b3f6f5108201b377 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
module
public import Mathlib.RingTheory.WittVector.FrobeniusFractionField
/-!
## F-isocrystals over a perfect field
When `k` is an integral domain, so is `𝕎 k`... |
4,488 | lean_file_chunk | b5d73afc72aa8bf8cbd0f6225b72e006797944de8df9d291574f4db7d92ad496 | 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.RingTheory.WittVector.Basic
/-!
# Teichmüller lifts
This file defines `WittVector.teichmuller`, a monoid hom `R →* 𝕎 R`, which embeds... |
5,511 | lean_file_chunk | f63b9bd3568246a0fb9c04e90b7d55c2b984467443287dddd53f8ef6b4a0a060 | 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
-/
module
public import Mathlib.LinearAlgebra.TensorProduct.Decomposition
public import Mathlib.RingTheory.GradedAlgebra.AlgHom
public import Mathlib.RingTheory.TensorProduct.Bas... |
14,727 | lean_file_chunk | bba987067ab3890f413a4efc3cf40720bc6fd0d3a9b54ad5b3d70bbe330d9b6f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
module
public import Mathlib.Algebra.DirectSum.Algebra
public import Mathlib.Algebra.DirectSum.Decomposition
public import Mathli... |
9,551 | lean_file_chunk | fe1d2b3e8e14f365229b7315a9b9edea5c0bb6a69ff282fe83abeb58998ae622 | 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
-/
module
public import Mathlib.Data.FunLike.Graded
public import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Homomorphisms of graded (semi)rings
This file defines bundled hom... |
23,983 | lean_file_chunk | 9b86776cb0acdb4e824bf3b28b1545f59981784deb51b809ff76e60ac06e6d27 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
module
public import Mathlib.Algebra.Group.Submonoid.Finsupp
public import Mathlib.Order.Filter.AtTopBot.Defs
public import Mathlib.RingTheory.Adjoin.Basic
... |
19,862 | lean_file_chunk | 3b03c5067399dd4bac58553a55f96978b7eec26266440e3c3685cf508d6dd658 | lean_source_file | pkg_mathlib_source_file_chunks | apply_fun IsLocalization.map (Localization Q) g.toRingHom comap_le at e
simp_rw [HomogeneousLocalization.NumDenSameDeg.embedding, Localization.mk_eq_mk',
IsLocalization.map_mk', ← Localization.mk_eq_mk'] at e
exact e
map_add' := Quotient.ind₂' fun x y ↦ by
simp only [← mk_add, Quotient.map'_mk... |
1,961 | lean_file_chunk | 1e400746049b6a2cc12a8df10e233f9b61a737bc3e88eec6b7f1ca919a1a1946 | 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: Kevin Buzzard, Harald Carlens, Andrew Yang, Jujian Zhang
-/
module
public import Mathlib.RingTheory.FiniteType
public import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Graded rings o... |
9,215 | lean_file_chunk | c9ba368fc78ed336ed83914a648d759d42148aa33ab74f8333bcde163bfd7efe | 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
-/
module
public import Mathlib.RingTheory.GradedAlgebra.RingHom
/-!
# `R`-linear homomorphisms of graded algebras
This file defines bundled `R`-linear homomorphisms of graded ... |
969 | lean_file_chunk | 8c5a603ccc849ea7b82677300aa3cb588f58c485a57ad217183f80c66bf161e5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Fangming Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fangming Li
-/
module
public import Mathlib.RingTheory.GradedAlgebra.Basic
public import Mathlib.RingTheory.Noetherian.Basic
/-!
# The properties of a graded Noetherian ring.
This file... |
8,239 | lean_file_chunk | 8cc0b19db72bc4f04f407c41ee7300448012ed44b0c273f6e8d18bcbc1c92d7c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
module
public import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
/-!
This file contains a proof that the radical of any homogeneous ideal is a homo... |
2,107 | lean_file_chunk | c9e9e390f38286fbdbd9edf81c45bf72b27bdc4707b1005aa4e852f05d8aa29b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors
/-!
# Finiteness of divisors
## Main results
* ... |
3,236 | lean_file_chunk | ad0f711b529abcf365237da2a69fa0df90aa9b5e1c8cf6e05deebac55b26287d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
public import Mathlib.RingTheory.UniqueFactorizationDomai... |
8,122 | lean_file_chunk | 8171814bd6b11d56ac36e4adbd02583228e76cc2bad3a695e5a1872d1aedad86 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors
/-!
# Multiplicative maps on unique factorizatio... |
2,095 | lean_file_chunk | 8a5d3c3e6ab674c5e707e249796066f2decb094cad0f471e6e84536376b7f5fd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.Data.ENat.Basic
public import Mathlib.Data.Nat.Factors
public import Mathlib.RingTheory.UniqueFactorizat... |
6,353 | lean_file_chunk | 87659cd797a67218ea3a70a54fd1de223ef04b1890f7dbc0019180da2f366b35 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.RingThe... |
2,755 | lean_file_chunk | a0f78a89c6f44f3c39e9c9cc2f7200727ec6b69b65108efcedf1064e82e3ca3e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
module
public import Mathlib.NumberTheory.ArithmeticFunction.Moebius
public import Mathlib.RingTheory.UniqueFactorizationDomain.Basic
/-!
# The Moebius function o... |
5,379 | lean_file_chunk | 4c92c05ed11397ff6f9657e70ccb312458a63ba9a6c30a5196546875c0924491 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Boris Bilich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Boris Bilich, Alexei Piskunov, Jonathan Shneyer
-/
module
public import Mathlib.RingTheory.ClassGroup
/-!
# The class group of a Unique Factorization Domain is trivial
This file proves... |
20,083 | lean_file_chunk | 7713525aa3de8272b1c59ca6d45867d9a329ffd5bd3f247e8c6bed17936b37ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.Algebra.BigOperators.Associated
public import Mathlib.Data.ENat.Basic
public import Mathlib.RingTheory.U... |
17,288 | lean_file_chunk | 5cb266415ce9988ab8f442e6cca191c259a3c3a0a4e9a0a9a1edef0662a303d8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.Algebra.GCDMonoid.Basic
public import Mathlib.Data.Multiset.OrderedMonoid
public import Mathlib.RingTheo... |
2,487 | lean_file_chunk | 5ee065b6a0255993e55281f6954c7e58095a088120f8071cc6c0a544f7516c30 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.RingTheory.Ideal.Operations
public import Mathlib.RingTheory.UniqueFactorizationDomain.Defs
/-!
# Uniqu... |
8,470 | lean_file_chunk | caf1f0b50f48a64b514abc0796f46c437f6c1050c9590acad8a38f5600ade207 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
public import Mathlib.Algebra.Group.Submonoid.BigOperators
pub... |
4,600 | lean_file_chunk | cbf072113057a2a8460ee76a23044c3111f0f05a6b829c98d84455b2607f33a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Emilie Uthaiwat
-/
module
public import Mathlib.RingTheory.UniqueFactorizationDomain.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.Ideal
/-!
# K... |
2,218 | lean_file_chunk | 10719205024016446ef06c54c463b6ab0795c99887b639c045880340000b509e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.Data.Finsupp.Multiset
public import Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors
/-!
... |
23,999 | lean_file_chunk | 788fe7014ba6c908d800a36ede419b3491e7df1e16ffbc055d0d9ae233f754e2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
module
public import Mathlib.RingTheory.UniqueFactorizationDomain.Basic
public import Mathlib.Tactic.Ring
/-!
# Set of factors
## M... |
1,299 | lean_file_chunk | 0520fef8dc4d529707f09e1e4aa4067f75c6a8666202ae30de642dcaf111b08d | lean_source_file | pkg_mathlib_source_file_chunks | exact eq_pow_count_factors_of_dvd_pow hp h
· have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
rcases subsingleton_or_nontrivial α with hα | hα
· simp [eq_iff_true_of_subsingleton] at hph
convert count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)
rw [... |
997 | lean_file_chunk | 23ab7db2c1fbf522f681f3f77eb38cffbb82e94c99a34f73f1617d3481d4dec5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
public import Mathlib.FieldTheory.Finite.Basic
/-!
# Torsion group of `ZMod p` for prime `... |
13,996 | lean_file_chunk | 8f1e4111fa6c82f188609241ac102c9f568f63e7cad2be3541daa5b99b8f9f58 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Junyan Xu
-/
module
public import Mathlib.Algebra.Order.Star.Basic
public import Mathlib.Analysis.Normed.Ring.Lemmas
public import Mathlib.Data.Nat.Choos... |
3,248 | lean_file_chunk | b6fbfd2c31471a9877be5c0678cbb2c163546392dd9b83e10c5a586e447a4047 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.MvPolynomial.PDeriv
public import Mathlib.RingTheory.MvPolynomial.Homogeneous
/-!
# Euler's homogeneous identity
## Main results
* `IsH... |
1,425 | lean_file_chunk | 85056d3fc803a5caabcbc5b93c3018c25e4ab07b9bb2ed25ff9f76cce63c4969 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wenrong Zou
-/
module
public import Mathlib.Algebra.MvPolynomial.Expand
public import Mathlib.RingTheory.MvPolynomial.Basic
public import Mathlib.Algebra.CharP.Frobenius
/-!
# Result... |
2,630 | lean_file_chunk | 07ff15e8b8d720a25293be7d134009ab31e92e2d4a47487d5d2aaf5108ca8d0d | 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.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Algebra towers for multivariate polynomial
This file ... |
8,557 | lean_file_chunk | e8d4efe4975294b93857e18267f91c41ef4c999025d3c0430274ff061856be2a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Algebra.CharP.Defs
public import Mathlib.Algebra.MvPolynomial.Degrees
public import Mathlib.Data.DFinsupp.Small
public import Mathlib.Da... |
7,939 | lean_file_chunk | 090c2da09bb33c4c33244653e8f3191a4b67557febe808fa2279cc98e9809a9a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.MonoidAlgebra.Ideal
public import Mathlib.Algebra.MvPolynomial.Division
public import Mathlib.RingTheory.Ideal.Quotient.Operations
pub... |
10,042 | lean_file_chunk | cf3bb65ad868ad0c9897d982295c64a011241089bbf2c84540657d18058af7d4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Data.Finsupp.MonomialOrder
public import Mathlib.Data.Finsupp.WellFounded
public im... |
3,936 | lean_file_chunk | 2cff5ea5cf4dce33dace3677020603481e2a07ad18282e6e73b32f5d9ad2dc39 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
module
public import Mathlib.RingTheory.FreeCommRing
/-!
# Constructing Ring terms from MvPolynomial
This file provides tools for constructing ring terms that can be ... |
4,888 | lean_file_chunk | 22671019e21967008baea0a6a49cac83597d08c41158128f67d68a31d812965b | 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.Algebra.Module.LocalizedModule.IsLocalization
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.RingTheory.I... |
9,663 | lean_file_chunk | f57708284c830c38d642eeaa5e4b55fbf3108be3e46c6c3cbdbdf46b8b98c813 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Johan Commelin, Andrew Yang
-/
module
public import Mathlib.Algebra.MvPolynomial.Division
public import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
impor... |
23,956 | lean_file_chunk | cb714d2d30a670a2fa31d71410f88b3451a5f4504056938e9f4c5da442ce8e87 | 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, Eric Wieser
-/
module
public import Mathlib.Algebra.MvPolynomial.CommRing
public import Mathlib.Algebra.MvPolynomial.Equiv
public import Mathlib.Algebra.Polynomial.Roo... |
2,731 | lean_file_chunk | 1fa07426655f31bac43bcd0c7d609ae228cdace08b31b6e4596aa3ec92a609f2 | lean_source_file | pkg_mathlib_source_file_chunks | fast_instance% weightedDecomposition R (1 : σ → ℕ)
/-- `MvPolynomial σ R` as a graded algebra, graded by the degree.
We do not make this a global instance because one may want to consider a different
graded algebra structure on `MvPolynomial σ R`, induced by another weight function.
To make it a local instance, you ma... |
23,994 | lean_file_chunk | 828d8c1b03eaf7ec648d33d43fdd15836bbeda8258b95e4a0b33c63a06784f63 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
module
public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.Algebra.DirectSum.Decompositio... |
3,913 | lean_file_chunk | 7634e694ba7ee453fe0ae5e16a65f61e758b314e4d9899cff2e3153bef05026c | lean_source_file | pkg_mathlib_source_file_chunks | degree zero if and only if for every `m ∈ p.support` and `x : σ`, `m x = 0`. -/
theorem weightedTotalDegree_eq_zero_iff (hw : NonTorsionWeight w) (p : MvPolynomial σ R) :
p.weightedTotalDegree w = 0 ↔ ∀ (m : σ →₀ ℕ) (_ : m ∈ p.support) (x : σ), m x = 0 := by
rw [← isWeightedHomogeneous_zero_iff_weightedTotalDegre... |
23,971 | lean_file_chunk | 8cd710795b51fc719096687c6bc96e862ac4b429ff54ef851ee91f7b30f8330e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
module
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Data.Finsupp.MonomialOrder
public import Mathlib.Data.Finsupp.WellFounded
public im... |
20,839 | lean_file_chunk | 5c4db13d1aad771cf29c375f18414f106f6fb6ed6c334ed5ee497541de58fd0a | lean_source_file | pkg_mathlib_source_file_chunks | exact congr_arg₂ _ rfl hrec
theorem degree_prod_of_mem_nonZeroDivisors {ι : Type*}
{P : ι → MvPolynomial σ R} {s : Finset ι}
(H : ∀ i ∈ s, m.leadingCoeff (P i) ∈ nonZeroDivisors _) :
m.degree (∏ i ∈ s, P i) = ∑ i ∈ s, m.degree (P i) := by
cases subsingleton_or_nontrivial R with
| inl _ => simp [Subsing... |
4,168 | lean_file_chunk | d59a623c514b8b5de03f3902e1a46f727f8619dfee84a1cb17118aa5dbaa9992 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
module
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.RingTheory.Derivation.Basic
public import Mathlib.Algebra.Lie.Prod
/-!... |
3,794 | lean_file_chunk | 44a67547cfc5931e9b735aebe596a0ddd99242f40b31de73bf5c54181a0cec14 | 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.Basic
/-!
# Differential and Algebras
This file defines derivations from a commutative ring to itself as a typeclass... |
18,320 | lean_file_chunk | 68682f996e5b3e3a006e9f15ae109595e91c6fc438ca24d35787336aa84d4868 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
module
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
/-!
# Derivations
This file defines... |
7,720 | lean_file_chunk | b43bfb4964c66cb03ccc1c5018293c28013b4fd6f224e2e48fb9c3c97f2c3aad | 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.DifferentialRing
public import Mathlib.Algebra.Polynomial.Module.Basic
public import Mathlib.Algebra.Polynomial.Deriva... |
5,482 | lean_file_chunk | 576522361a90c9467ad31d4d21b24161dd0fdc506ca98c7b2b576c510848f05e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
module
public import Mathlib.RingTheory.Derivation.Basic
public import Mathlib.RingTheory.Ideal.Quotient.Operations
/-!
# Derivations into Square-Z... |
23,914 | lean_file_chunk | 2f18356e90417ed51c6ea40ea4a0d6c9e253e22970cb043a652c152574d77f07 | 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.RingTheory.Polynomial.UniversalFactorizationRing
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public import Mathlib.RingTheor... |
2,075 | lean_file_chunk | 0f48a19e5e183a88f9e896d78d15ada10217983d9d6bedb28d30c5cd0b00ee10 | lean_source_file | pkg_mathlib_source_file_chunks | have : .powers (f ⊗ₜ[R] 1) = Algebra.algebraMapSubmonoid (R' ⊗[R] S) (.powers f) := by simp
rw [IsLocalization.Away, this]
refine IsLocalization.tensorProduct_tensorProduct _ _ _ _ ?_
ext; simp [RingHom.algebraMap_toAlgebra, φ]
let P'f := P'.map (algebraMap _ (Localization.Away f ⊗[R] S))
have hP'f : Di... |
18,582 | lean_file_chunk | fcdef4a9561a2655eed7a04d9688aa4aa2f87b1edf0917b6c76b1bd5710029ef | 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.RingTheory.Etale.Basic
public import Mathlib.RingTheory.Kaehler.JacobiZariski
public import Mathlib.RingTheory.Localization.BaseChange
public ... |
1,205 | lean_file_chunk | f2e228972b69dd8abb4bd3d906f99abb7ace3b77d5240ed85e5325b25c17d5f6 | 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.RingTheory.Smooth.Pi
public import Mathlib.RingTheory.Unramified.Pi
public import Mathlib.RingTheory.Etale.Basic
public import Mathlib.RingThe... |
9,691 | lean_file_chunk | bdba22014adc10de438ba91416bf69e29259d9d2427a87935a575d419e0c9302 | 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.RingTheory.Ideal.Quotient.Nilpotent
public import Mathlib.RingTheory.Smooth.Basic
public import Mathlib.RingTheory.Unramified.Basic
/-!
# Ét... |
4,665 | lean_file_chunk | 011b5aa5cd4b19c3d070aba6e5cd82e9b4db2d67de042ca68bdc174169a97943 | 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.RingTheory.Smooth.Locus
public import Mathlib.RingTheory.Unramified.Locus
/-!
# Etale locus of an algebra
## Main results
Let `A` be a `R`-a... |
9,996 | lean_file_chunk | 8cb03043528c5bffd6b9d3ac436ca5b1fecff144d8ecfa855b4e6fe36780df50 | 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.RingTheory.Etale.Pi
public import Mathlib.RingTheory.Unramified.Field
/-!
# Étale algebras over fields
## Main results
Let `K` be a field, ... |
21,272 | lean_file_chunk | 9979f4646cff2f32d64d781af39536e2a6397b7240b50fc642b4ef75d3f67fe8 | 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.Polynomial.Bivariate
public import Mathlib.Algebra.Polynomial.Taylor
public import Mathlib.RingTheory.Etale.Basic
public import Mathli... |
12,829 | lean_file_chunk | 6770ee5181258916d236369a8f2d336d887fa46bd8ccf558c33dd901ba02ee68 | 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.RingTheory.Etale.Field
public import Mathlib.RingTheory.Flat.Equalizer
public import Mathlib.RingTheory.Kaehler.TensorProduct
public import Ma... |
2,703 | lean_file_chunk | 15b87afd34289bcde74dbd72e40352febc9fe52583c900767b5138005048ce8c | 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: Judith Ludwig, Christian Merten
-/
module
public import Mathlib.RingTheory.AdicCompletion.AsTensorProduct
public import Mathlib.RingTheory.Flat.Stability
public import Mathlib.RingT... |
9,011 | lean_file_chunk | 2d875ef46c437502d272c490971b3a36db58f1e892464dc49645b5840ef99de8 | 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.RingTheory.Extension.Cotangent.Basis
public import Mathlib.RingTheory.Extension.Cotangent.Free
public import Mathlib.RingTheory.Smoo... |
7,533 | lean_file_chunk | 5297342af7cac01ca90537b86c9e5119c530ecea79ccc85ee7348e864c38f0d6 | 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: Jung Tao Cheng, Christian Merten, Andrew Yang
-/
module
public import Mathlib.RingTheory.Extension.Presentation.Submersive
/-!
# Standard smooth algebras
A standard smooth algebra... |
14,036 | lean_file_chunk | f4eaba9b1aff05ee536994983f6b010c6db46fe602d89cdaefaaa0dbf9cc7bfd | 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: Judith Ludwig, Christian Merten
-/
module
public import Mathlib.RingTheory.Extension.Presentation.Core
public import Mathlib.RingTheory.MvPolynomial.Homogeneous
public import Mathli... |
18,413 | lean_file_chunk | 5add7bae4b9086c78e12d37699efb0fa360e4d2c31681fc68b4289b0dc8408cd | 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.RingTheory.Extension.Cotangent.Basic
/-!
# Relation of smoothness and `Ω[S⁄R]`
## Main results
- `retractionKerToTensorEquivSection`:
Giv... |
5,721 | lean_file_chunk | 940f1869dfcf44251ae0cb4a054d8c9f7156802b5e9d0ef40f6db2d0de372a1a | 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.RingTheory.Idempotents
public import Mathlib.RingTheory.Smooth.Basic
/-!
# Formal-smoothness of finite products of rings
## Main result
- ... |
4,230 | lean_file_chunk | 4d8dcfcecdd71fa70ddcb60681042ee9ff0dec72daae3012796cab2c88bccad8 | 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: Judith Ludwig, Christian Merten
-/
module
public import Mathlib.RingTheory.Smooth.Basic
public import Mathlib.RingTheory.AdicCompletion.Algebra
/-!
# Formally smooth algebras and a... |
23,965 | lean_file_chunk | 6004861edde9a6d82075473578490252a137c2f4f61faa89361b2cdb25249e5a | 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.RingTheory.FiniteStability
public import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
public import Mathlib.RingTheory.Localization.Away.Adjoin... |
1,998 | lean_file_chunk | 4bd09004955f21b4dde8eeb37371fc28a7c92da0dbaa65f03c26986e1e52d76a | lean_source_file | pkg_mathlib_source_file_chunks | theorem localization_map [FormallySmooth R A] : FormallySmooth Rₘ Sₘ := by
haveI : FormallySmooth A Sₘ := FormallySmooth.of_isLocalization (M.map (algebraMap R A))
haveI : FormallySmooth R Sₘ := FormallySmooth.comp R A Sₘ
exact FormallySmooth.localization_base M
end Localization
end FormallySmooth
section
var... |
4,602 | lean_file_chunk | 3ac533f2f9a1ff043c0e53690549756f8c627ba7ae67ad6366be5bd4c12f97ea | 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.RingTheory.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.LocalRing.Module
public import Mathlib.RingTheory.Smooth.Basic
public im... |
12,444 | lean_file_chunk | 90dd47bec6cc1c782cc7e4d68543361d6f8cef288aa068469d37cd3d4ddbc86d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
module
public import Mathlib.RingTheory.Kaehler.TensorProduct
public import Mathlib.RingTheory.Regular.RegularSequence
public import Mathlib.RingTheory.RingHom.Flat
public... |
7,329 | lean_file_chunk | ceb1307b14dddd69c109a704ae31db8d6cf70d8215a7b8e40705e8f98c587c01 | 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.RingTheory.Etale.Kaehler
public import Mathlib.RingTheory.Spectrum.Prime.FreeLocus
public import Mathlib.RingTheory.Support
/-!
# Smooth locu... |
14,936 | lean_file_chunk | ce68c3efb2b80b039493074fd09d2e7108b3e9701106573134c7a2b4bb8112f1 | 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.LinearAlgebra.Basis.Exact
public import Mathlib.RingTheory.Extension.Cotangent.Basic
public import Mathlib.RingTheory.Smooth.Standar... |
21,988 | lean_file_chunk | 9a1754438d22ecf6479f6d38ec31391296ebae0879c2195290829f25a1b6cd4c | 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.RingTheory.Smooth.Flat
public import Mathlib.RingTheory.Unramified.LocalStructure
/-!
# Smooth base change commutes with integral closure
In... |
2,372 | lean_file_chunk | ddd33cd4e0410823ecaccbbe6fb023298ff771b49ff4f51d92e8d3d00a59c796 | 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.RingTheory.Etale.Field
public import Mathlib.FieldTheory.SeparablyGenerated
/-!
# Smooth algebras over fields
We show that separably genera... |
3,482 | lean_file_chunk | dc01c70c699bd524cc9259ee5251c29c7df4796955cdb6bf6fb1fd6f55bca916 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
module
public import Mathlib.RingTheory.Ideal.Maximal
/-!
# The set of non-invertible elements of a monoid
## Main definitions
* `nonunits` is... |
2,997 | lean_file_chunk | 7d157d0fc4e6ce3d11913b432fc1d905e78dd1d6f4eab109a90a27147fa6189f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Anthony Fernandes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anthony Fernandes, Marc Robin
-/
module
public import Mathlib.RingTheory.Ideal.Colon
/-!
# Oka predicates
This file introduces the notion of Oka predicates and standard results a... |
4,845 | lean_file_chunk | 17229d7fce097f7099c7735d352edae8552fc6509c1aa68d043579e14eb8fa3b | 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.Algebra.Field.ZMod
public import Mathlib.RingTheory.Ideal.Norm.AbsNorm
/-!
# Ideal of `ℤ`
We prove results about ideals of `ℤ` or ideals... |
7,257 | lean_file_chunk | 1fc6d39ea18bae49d6ed3e94881fe444a22f30bcf8a4adb6d9942a6cecdf60dd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten, Yi Song, Sihan Su. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten, Yi Song, Sihan Su
-/
module
public import Mathlib.RingTheory.Ideal.GoingUp
public import Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra
public i... |
5,514 | lean_file_chunk | 25f64313cece0d777984bd5693561578e4653b80db1e47389985b4aec5e4b960 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.Algebra.EuclideanDomain.Int
public import Mathlib.Algebra.Order.Ring.Int
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.RingTheory... |
6,251 | lean_file_chunk | fe8410e45f43639fb2c419be573d1bd263e3279f90db13f57d1ddf98b29c24a2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 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.Ring.Action.Pointwise.Set
public import Mathlib.LinearAlgebra.Quotient.Defs
public import Mathlib.RingTheory.Ideal.Maps
/-!
# The colon i... |
23,928 | lean_file_chunk | cf26a7e9044957d04eaaf38fdef699cf565faaba4eaaa3025ffd9717cbc29288 | 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: Wanyi He, Jiedong Jiang, Christian Merten, Jingting Wang, Andrew Yang, Shouxin Zhang
-/
module
public import Mathlib.RingTheory.HopkinsLevitzki
public import Mathlib.RingTheory.Ideal.Goi... |
1,439 | lean_file_chunk | b980085ec085adbd3056603b86fd287ed8aefa1051e6d3b4def1f6e59aedb21e | lean_source_file | pkg_mathlib_source_file_chunks | (P.map (Quotient.mk (p.map (algebraMap R S)))).exists_ltSeries_length_eq_height
let l' : LTSeries (PrimeSpectrum S) :=
lq.map (PrimeSpectrum.comap (Quotient.mk (p.map (algebraMap R S))))
(RingHom.strictMono_comap_of_surjective Quotient.mk_surjective)
have : l'.head.asIdeal.LiesOver lp.last.asIdeal := by
... |
23,966 | lean_file_chunk | dcdee35d4f2f14b29ade5c718d0e3411ce9dc9b7e569d22d137a988d5145d0f6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
module
public import Mathlib.Data.DFinsupp.Module
public import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `I... |
23,955 | lean_file_chunk | 99340831de96ee8bf81a8d57a3a6a0889b518681a2efc1896bfa31fe07a2d4e2 | lean_source_file | pkg_mathlib_source_file_chunks | variable {F : Type*} [CommSemiring R] [CommSemiring S]
variable [FunLike F R S] [rc : RingHomClass F R S]
variable (f : F)
variable (I J : Ideal R) (K L : Ideal S)
protected theorem map_mul {R} [Semiring R] [FunLike F R S] [RingHomClass F R S]
(f : F) (I J : Ideal R) :
map f (I * J) = map f I * map f J :=
le... |
2,907 | lean_file_chunk | cca16c59913eac806e2e0abab4bd46fbec54d4fef92530845797b3e4fe530bb2 | lean_source_file | pkg_mathlib_source_file_chunks | e.idealComapOrderIso.symm I = I.map e :=
rfl
namespace AlgHom
variable {R A B : Type*} [CommSemiring R] [Semiring A] [Semiring B]
[Algebra R A] [Algebra R B] (f : A →ₐ[R] B)
lemma ker_coe : RingHom.ker f = RingHom.ker (f : A →+* B) := rfl
lemma coe_ideal_map (I : Ideal A) :
Ideal.map f I = Ideal.map (f : ... |
7,622 | lean_file_chunk | 57508a3e435856fa65701d40b5e0d339ba37bee45b4b3459c52be7e292cd64d5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.RingTheory.Ideal.Maps
/-!
# Ideals in product rings
For commutative rings `R` and `S` and ideals `I ≤ R`, `J ≤ S`, we define `Ideal.prod... |
8,278 | lean_file_chunk | b1870c34872cc79386cb77200108993c3d75af211201755d4a1d2df07d7e0b3b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
module
public import Mathlib.Algebra.Ring.Action.End
public import Mathlib.RingTheory.Ideal.Maps
/-! # Pointwise instances on `Ideal`s
This file provides the action `Ide... |
13,384 | lean_file_chunk | e46a5050f538d22bbab42423cb3608e85b3e792cc6eddf34cb15fa4f9d122797 | 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, Yongle Hu
-/
module
public import Mathlib.Algebra.Algebra.Tower
public import Mathlib.Algebra.Group.Subgroup.Actions
public import Mathlib.RingTheory.Ideal.Pointwise
public ... |
5,463 | lean_file_chunk | a480a670914f6cfc24703bcd13a7834c598166f20f99b892928adff70922118e | 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.LinearAlgebra.TensorProduct.Quotient
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.RingTheory.Ideal.Cotangent
pu... |
2,894 | lean_file_chunk | 87b98247f1daf8d28ab875d05f3e4cf1291bb817722db426bb28140fcff5e4c2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yakov Pechersky
-/
module
public import Mathlib.RingTheory.IsPrimary
public import Mathlib.RingTheory.Ideal.Maps
public import Mathlib.RingTheory.Ideal.Operations
/-!
# Primary... |
1,620 | lean_file_chunk | 0e2a44817e73faf4a8ae8704e23bec93e544806a829cb62fbcd54d736890c702 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
module
public import Mathlib.Algebra.Ring.Idempotent
public import Mathlib.Order.Basic
public import Mathlib.RingTheory.Finiteness.Naka... |
20,506 | lean_file_chunk | 3f7656161c853bf24d0480c2b76f3cfca939cfad8d9e8246529c5085a32fc548 | 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, Yongle Hu
-/
module
public import Mathlib.RingTheory.Ideal.Over
public import Mathlib.RingTheory.Localization.AtPrime.Basic
public import Mathlib.RingTheory.Localization.Int... |
11,218 | lean_file_chunk | e8a81f2f18864e95caeb93850cf40e01343e1c4d38907bfb9f392eae9ad6fa04 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
module
public import Mathlib.Algebra.Field.IsField
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Data.Nat.Choose.Sum
public im... |
23,958 | lean_file_chunk | 2cf0e2c2a8cb8dbcc6a359cfb2260c11a57791b4a4454ad4ea3a866e633a242c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 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.Algebra.Operations
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.Data.Fintype.Lattice
public import Mathlib.Alge... |
23,969 | lean_file_chunk | b40853b480d9a1800a5fe4119c7d10d2069b762d65fffda41d4cc94511abe105 | lean_source_file | pkg_mathlib_source_file_chunks | (hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i ∈ s, I i} := by
ext x
simp only [Submodule.mem_finsetInf, Ideal.mem_span_singleton]
exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩
theorem iInf_span_singleton ... |
11,775 | lean_file_chunk | f4dd9f09aeeb4f22abc3c1184e2d85742eff58c9781e27c317c5e61c4448d7b6 | lean_source_file | pkg_mathlib_source_file_chunks | · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s :=
⟨s.erase b, Finset.notMem_erase b s, Finset.insert_erase hbs⟩
have hp' : ∀ j ∈ t, IsPrime (f j) := by
intro j hj
refine hp j (Finset.mem_insert_of_mem hj) ?_ ?_ <;> rintro rfl <;>
solve_by_elim only [Finset.mem_ins... |
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