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The tensor calculation uses the stated finite Galois splitting field itself.", "readings": [ { "label": "Profinite groups and infinite Galois theory, Section 0", "reader": "../NT-CFT/profinite-groups-and-infinite-galois-theory.html#0-the-finite-algebra-behind-the-limits" }, { "label": "Section 3", "reader": "../NT-CFT/profinite-groups-and-infinite-galois-theory.html#3-the-correspondence-for-an-infinite-extension" }, { "label": "Extensions of complete valued fields, Proposition 0.1", "reader": "../NT-LOC/NT-LOC-04.html#0-algebraic-generators" }, { "label": "Hensel's lemma, squares and roots of unity, Lemma 4.0", "reader": "../NT-LOC/NT-LOC-03.html#4-residues-lift-multiplicatively" }, { "label": "Normal integral bases, Lemma 5.23", "reader": "../NOE-ABS/NOE-ABS-05.html#5-the-group-determinant-and-the-global-obstruction" }, { "label": "Integrality of characters and Burnside's theorem, Lemma 4.2", "reader": "../RT-FIN/integrality-of-characters-and-burnsides-theorem.html#4-a-class-of-prime-power-size-prevents-simplicity" }, { "label": "Profinite groups and infinite Galois theory, Section 0", "reader": "../NT-CFT/profinite-groups-and-infinite-galois-theory.html#0-the-finite-algebra-behind-the-limits" }, { "label": "Faithful flatness, Theorem 1.1 and Theorem 2.1", "reader": "../AG-CA/AG-CA-08.html#1-detecting-nonzero-modules" }, { "label": "same Section 0", "reader": "../NT-CFT/profinite-groups-and-infinite-galois-theory.html#0-the-finite-algebra-behind-the-limits" } ] }, { "title": "Finite extensions of the complex Laurent-series field", "scope": "Finite separable normalization over a Noetherian normal domain, simple-root lifting in complete local rings, local factors of a finite algebra over a henselian ring, dimension under integral inclusion, the one-dimensional normal local DVR criterion, and completeness of finite modules. These apply to the displayed integral closure over C[[t]].", "readings": [ { "label": "Normalization, Theorem 4.1", "reader": "../AG-MO/normalization.html#4-a-trace-lattice-proves-separable-finiteness" }, { "label": "Completion, Theorem 5.1", "reader": "../AG-CA/AG-CA-19.html#5-hensel-lifting-in-every-complete-local-ring" }, { "label": "Henselian local rings and henselization, Theorem 2.2", "reader": "../AG-CA/AG-CA-20.html#2-factorization-and-the-finite-algebra-criterion" }, { "label": "Integral extensions, Theorem 5.1", "reader": "../AG-CA/AG-CA-05.html#5-chains-and-dimension" }, { "label": "Discrete valuation rings, Theorem 1.2", "reader": "../AG-CA/AG-CA-15.html#1-a-single-parameter-and-a-discrete-valuation" }, { "label": "Completion, Theorem 3.1", "reader": "../AG-CA/AG-CA-19.html#3-noetherian-exactness-tensoring-and-flatness" } ] }, { "title": "Geometry and coherent finiteness for torsion on curves", "scope": "Finite normalization, the separated quasi-finite factorization and coherent cohomology under proper morphisms. Projective finiteness and vanishing supply the stated preceding coherent input; the curve cohomology arguments themselves remain in the course.", "readings": [ { "label": "Normalization, Theorem 5.1", "reader": "../AG-MO/normalization.html#5-finiteness-over-fields-including-inseparability" }, { "label": "Zariski's Main Theorem, Theorem 4.2", "reader": "../AG-MO/AG-MO-12.html#4-the-separated-scheme-theorem" }, { "label": "Coherence of higher direct images, Theorem 4.1", "reader": "../AG-QC/proper-morphisms-and-coherent-direct-images.html#4-a-projective-modification-supplies-the-witness" }, { "label": "Corollary 5.1", "reader": "../AG-QC/proper-morphisms-and-coherent-direct-images.html#5-finite-cohomology-and-global-functions" }, { "label": "Serre's projective finiteness and vanishing, Theorem 2.1 and Corollary 2.3", "reader": "../AG-QC/serres-theorems-on-projective-schemes.html#2-descending-induction-proves-serre-s-theorems" } ] }, { "title": "Proper geometry and henselian comparison", "scope": "General Stein factorization, proper approximation over an arbitrary base, coherent formal existence and the full finite-type Chow construction support the geometric reductions in proper base change.", "readings": [ { "label": "General Stein factorization, Theorem 6.1", "reader": "../AG-QC/zariski-connectedness-and-stein-factorization.html#6-examples-and-the-general-base-extension" }, { "label": "Proper approximation, Appendix A.4.1 and A.4.2", "reader": "../AG-QC/zariski-connectedness-and-stein-factorization.html#a-4-from-a-proper-scheme-to-proper-finitely-presented-thickenings" }, { "label": "Coherent formal existence, Theorems 3.2 and 6.1", "reader": "../AG-QC/grothendiecks-existence-theorem.html#6-proper-schemes-and-proper-supports" }, { "label": "Chow's lemma over a qcqs base, Theorem 5.1", "reader": "../AG-MO/projective-morphisms-and-chows-lemma.html#5-the-forms-over-a-qcqs-base" }, { "label": "Finite Zariski Main, Theorem 4.2", "reader": "../AG-MO/AG-MO-12.html#4-the-separated-scheme-theorem" }, { "label": "Finite-presentation descent, Sections 3–4", "reader": "../AG-MO/AG-MO-03.html#4-descending-objects-maps-and-properties" }, { "label": "Henselian-pair idempotents and étale lifting, Lemmas 5.3 and 5.4", "reader": "../AG-FSE/AG-FSE-07.html#henselian-pairs-and-finite-étale-algebras" } ] } ], "reader_aliases": [ { "reader": "../AG-CA/noetherian-and-artinian-rings.html", "same_reading_as": "../AG-CA/AG-CA-03.html", "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d", "title": "Noetherian and Artinian rings", "course_lesson": "AG-CA-03" }, { "reader": "../AG-CA/integral-extensions-lying-over-going-up-and-going-down.html", "same_reading_as": "../AG-CA/AG-CA-05.html", "reader_sha256": "f4e1cea75e1cb7980a0a96248b0ee8288b89b443dcff5a9b3801ff5d0c7ed8b9", "title": "Integral extensions: lying over, going up and going down", "course_lesson": "AG-CA-05" }, { "reader": "../AG-CA/AG-CA-09.html", "same_reading_as": "../AG-CA/krull-dimension-and-noether-normalization.html", "reader_sha256": "a7db947490ce1eed7862660b870a5e0037f3e5e2c3eae4895b0ee56bbdce8298", "title": "Krull dimension and Noether normalization", "course_lesson": "AG-CA-09" }, { "reader": "../AG-CA/dimension-theory-of-noetherian-local-rings.html", "same_reading_as": "../AG-CA/AG-CA-11.html", "reader_sha256": "760e9c283547986d656b0b346ff1b93854966219b1b7ed0ff1e43be81ffc24ec", "title": "Dimension theory of Noetherian local rings", "course_lesson": "AG-CA-11" }, { "reader": "../AG-CA/regular-sequences-depth-and-cohen-macaulay-modules.html", "same_reading_as": "../AG-CA/AG-CA-12.html", "reader_sha256": "300459db5236d2b07a2c17f6fde00f5f7f48f38146722d938e6c7052a3f6a6a2", "title": "Regular sequences, depth and Cohen–Macaulay modules", "course_lesson": "AG-CA-12" }, { "reader": "../AG-CA/regular-local-rings.html", "same_reading_as": "../AG-CA/AG-CA-14.html", "reader_sha256": "99644c59866594525e218805d04cb5b7f0b2f28171f160a1521025f05acc62f8", "title": "Regular local rings", "course_lesson": "AG-CA-14" }, { "reader": "../AG-CA/completion.html", "same_reading_as": "../AG-CA/AG-CA-19.html", "reader_sha256": "ad6bc0253616c5e04a038ebe76caa5916403edfb09b416d2b4e51c08db1ed4a0", "title": "Completion", "course_lesson": "AG-CA-19" }, { "reader": "../AG-CA/graded-modules-and-hilbert-samuel-functions.html", "same_reading_as": "../AG-CA/AG-CA-10.html", "reader_sha256": "2d5974ca8eb5fe427161695ef08ff490a018e49577d7646f0c1287c03b5e7329", "title": "Graded modules and Hilbert–Samuel functions", "course_lesson": "AG-CA-10" }, { "reader": "../AG-CA/associated-primes-and-primary-decomposition.html", "same_reading_as": "../AG-CA/AG-CA-04.html", "reader_sha256": "64de66dba4c915f86ab71ae99bcfcc2dde04b1dc2551e243afdd1ec028dc1727", "title": "Associated primes and primary decomposition", "course_lesson": "AG-CA-04" } ] }