| {"id":"MatrixRing.isCentralSimple","label":"MatrixRing.isCentralSimple","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_6.html","dependencies":[],"nlp_text":"If \\(n\\geq 1\\) then the \\(n\\times n\\) matrices \\(M_n(K)\\) are a central simple algebra over \\(K\\).","doc_url":null,"latex_link":"sect0003.html#MatrixRing.isCentralSimple","lean_link":null,"lean_decl_name":"MatrixRing.isCentralSimple","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"IsCentralSimple.baseChange","label":"IsCentralSimple.baseChange","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_6.html","dependencies":[],"nlp_text":"If \\(D\\) is a central simple algebra over \\(K\\) and \\(L/K\\) is a field extension, then \\(L\\otimes _KD\\) is a central simple algebra over \\(L\\).","doc_url":null,"latex_link":"sect0003.html#IsCentralSimple.baseChange","lean_link":null,"lean_decl_name":"IsCentralSimple.baseChange","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch6","source_path":"./FLT/Deformations/RepresentationTheory/GaloisRep.lean","lean4_code":"def GaloisRep.baseChange [IsTopologicalRing B] [Algebra A B] [ContinuousSMul A B]\n [Module.Finite A M] [Module.Free A M]\n (ρ : GaloisRep K A M) : GaloisRep K B (B ⊗[A] M) :=\n letI := moduleTopology A (Module.End A M)\n letI := moduleTopology B (Module.End B (B ⊗[A] M))\n letI : ContinuousMul _ := ⟨IsModuleTopology.continuous_mul_of_finite B (Module.End B (B ⊗[A] M))⟩\n letI := IsModuleTopology.toContinuousAdd B (Module.End B (B ⊗[A] M))\n let F : Module.End A M →+* Module.End B (B ⊗[A] M) := Module.End.baseChangeHom A B M\n have : Continuous F := by\n have : IsTopologicalSemiring (Module.End B (B ⊗[A] M)) := ⟨⟩\n have : Continuous (algebraMap A (Module.End B (B ⊗[A] M))) := by\n rw [IsScalarTower.algebraMap_eq A B, RingHom.coe_comp]\n exact (continuous_algebraMap _ _).comp (continuous_algebraMap _ _)\n exact IsModuleTopology.continuous_of_ringHom (R := A) F (by simpa [F])\n .comp ⟨F, this⟩ ρ\n\nomit [IsTopologicalRing A] [NumberField K] in\nopen TensorProduct in\n","line":191,"end_line":208,"proof_status":"statement_formalized"} |
