Alphabeta MathCommutative Algebra
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
1 page- Noetherian Rings and Hilbert Basis43 results
This page works in commutative rings with identity, using the regular-module definition of a Noetherian ring and the module theorem identifying finite generation, the ascending chain condition, and the maximal condition.
3 definitions, 18 lemmas, 9 theorems, 9 corollaries, 2 examples, 2 remarksExamples & counterexamples →
Level 1
2 pages- Localisation of Modules and Support35 results
Ring localisation, exact sequences, tensor products, module homomorphisms, finitely generated modules, and finitely presented modules supply the background for extending scalars from R to S⁻¹R.
3 definitions, 13 lemmas, 14 theorems, 5 corollariesExamples & counterexamples → - Prime Spectra and Radicals45 results
Radicals turn power-membership into an ideal-theoretic closure operation, and prime ideals detect that closure.
6 definitions, 24 lemmas, 6 theorems, 9 corollariesExamples & counterexamples →
Level 2
5 pages- Artinian Rings and Length12 results
Artinian rings are the commutative rings whose ideals cannot descend forever.
1 lemma, 9 theorems, 2 corollariesExamples & counterexamples → This page develops the associated-prime language from annihilators of elements and cyclic submodules, then proves its behavior in short exact sequences, localization, prime…
3 definitions, 19 lemmas, 13 theorems, 4 corollariesExamples & counterexamples →- Flatness and Faithful Flatness22 results
This page records the flatness tools needed downstream in commutative algebra and algebraic geometry without opening the Tor and derived-functor machinery yet.
11 theorems, 4 corollaries, 7 examplesExamples & counterexamples → - Integral Extensions and Going Up18 results
This page develops integral extensions from the one-element finite-module criterion through the Cohen-Seidenberg theorems.
2 definitions, 4 lemmas, 8 theorems, 4 corollariesExamples & counterexamples → - Koszul Complexes and Regular Sequences51 results
A finite, ordered treatment of exterior constructions, Koszul homology, regular sequences, and their local finite consequences.
6 definitions, 25 lemmas, 7 theorems, 13 corollariesExamples & counterexamples →
Level 3
2 pagesThis page assembles the classical affine-algebra spine in the order the later geometry pages use it.
19 lemmas, 9 corollariesExamples & counterexamples →This page fixes the ordered-group conventions behind valuations, then develops valuation rings from their field-theoretic comparability condition to the value-group construction that recovers a valuation from the ring itself.
7 definitions, 1 lemma, 9 theorems, 2 corollariesExamples & counterexamples →
Level 4
4 pages- Dedekind Domains and Ideal Classes33 results
This draft page packages the abstract Dedekind-domain toolkit around one-dimensional normal domains: the local DVR characterization, localization closure, finite separable…
7 definitions, 10 lemmas, 12 theorems, 3 corollaries, 1 remarkExamples & counterexamples → - Krull Dimension and Height Theorems37 results
This page turns the earlier spectrum-and-height definitions into working dimension theory.
1 definition, 16 lemmas, 8 theorems, 11 corollaries, 1 remarkExamples & counterexamples → This page proves the finiteness of normalisation for affine domains: the integral closure of a finitely generated domain over a field in its fraction field is again a finite…
7 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →- Zariski Topology on Prime Spectra32 results
This page upgrades the set-theoretic prime-spectrum material to the Zariski topology.
4 definitions, 15 lemmas, 4 theorems, 9 corollariesExamples & counterexamples →
Level 5
2 pagesThis page develops the algebraic form of Zariski's main theorem for quasi-finite morphisms of affine spectra, in the version proved in the Stacks Project's Section 10.123 and…
3 definitions, 8 lemmas, 2 theorems, 2 corollariesExamples & counterexamples →This page builds the standard graded and Rees constructions used to move from ideal-adic filtrations to graded algebra.
6 definitions, 1 lemma, 7 theorems, 2 corollaries, 7 examplesExamples & counterexamples →
Level 6
3 pagesThe page begins with the elimination-theoretic resultant of two binary forms of nominated positive degrees, defined as the determinant of the Sylvester multiplication map (A,B)↦ AF+BG.
3 definitions, 10 lemmas, 2 theorems, 2 corollariesExamples & counterexamples →- Inverse Limits and Noetherian Completion21 results
This page builds I-adic completion entirely as an inverse limit of quotient modules.
6 definitions, 1 proposition, 13 theorems, 1 corollaryExamples & counterexamples → This page derives the equality between Koszul Euler characteristic and the Hilbert–Samuel coefficient indexed by the number of generators.
1 definition, 5 lemmas, 2 theoremsExamples & counterexamples →
Level 7
2 pages- Depth and Cohen Macaulay Modules57 results
This collection develops depth through regular sequences, Ext, and Koszul cohomology, proves the Depth Lemma and localization bounds, and applies them to Cohen--Macaulay…
3 definitions, 25 lemmas, 13 theorems, 16 corollariesExamples & counterexamples → This page organizes three linked themes. Henselian pairs are introduced through coprime factor lifting, simple-root lifting, and idempotent lifting; complete, nilpotent…
2 definitions, 23 lemmas, 1 proposition, 1 theorem, 11 corollariesExamples & counterexamples →
Level 8
1 pageEmbedding dimension and associated graded rings lead to regular parameters, minimal resolutions, the Auslander–Buchsbaum formula, and the homological criterion for regularity.
7 definitions, 35 lemmas, 11 theorems, 7 corollariesExamples & counterexamples →