Alphabeta MathRegular Local Rings and Homological Dimension
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1 · Prerequisites
- Abelian Categories
- Adjunctions Units and Counits
- Algebraic Extensions, Extension Degree, and Finite Fields
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Chain Homotopy and the Homotopy Category
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Dedekind Domains and Ideal Classes
- Delta Functors and Universality
- Depth and Cohen Macaulay Modules
- Derived Functors
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Finite Counting, Factorials and Binomial Coefficients
- Flatness and Faithful Flatness
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Integral Extensions and Going Up
- Inverse Limits and Noetherian Completion
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Limits and Colimits
- Linear Independence, Bases and Dimension
- Localisation of Modules and Support
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Polynomial Rings, the Division Algorithm and Roots
- Preadditive and Additive Categories and Biproducts
- Prime Spectra and Radicals
- Projective and Injective Resolutions
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Subobject Lattices Generators and the Grothendieck Axioms
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Diagram Lemmas in an Abelian Category
- The Field of Fractions and Localisation
- The ZFC Axioms and the Basic Set Constructions
- Tor Flatness and Global Dimension
- Universal Properties, Representables and the Yoneda Lemma
- Valuation Rings and Discrete Valuation Rings
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Yoneda Extensions and Homological Dimension
2 · Logical flowchart
3 · Items
Definition
embedding dimension and regular local ring
Lemma
embedding dimension is minimal maximal ideal generator number
Theorem
dimension at most embedding dimension
Definition
regular system of parameters
Lemma
regular system of parameters equivalent basis
Lemma
associated graded polynomial surjection
Lemma
regular local graded surjection has zero kernel
Theorem
associated graded ring of a regular local ring
Corollary
regular local hilbert samuel multiplicity one
Lemma
regular local domain induction
Lemma
regular local parameter is nonzerodivisor
Lemma
regular local quotient by parameter is regular
Theorem
quotient and lifting regularity across a regular element
Theorem
regular local rings are domains and cohen macaulay
Theorem
one dimensional regular local rings are dvrs
Lemma
regular local regular quotient ideal is parameter generated
Lemma
finite local modules admit minimal free resolutions
Lemma
minimal free resolution differentials land in maximal ideal
Lemma
minimal free resolution reduces to zero differential
Definition
betti numbers of a finite local module
Corollary
betti number is rank in minimal resolution
Lemma
minimal free resolutions unique up to chain isomorphism
Lemma
projective dimension from last nonzero betti number
Lemma
auslander buchsbaum syzygy projective dimension
Lemma
auslander buchsbaum base case free module
Lemma
minimal free matrix induces zero on residue ext
Lemma
auslander buchsbaum projective dimension one
Lemma
auslander buchsbaum first syzygy depth
Theorem
auslander buchsbaum formula
Lemma
global dimension is detected on cyclic modules
Lemma
local global dimension equals residue field projective dimension
Lemma
positive depth ring has regular minimal generator
Lemma
regular element reduction preserves minimal resolution
Lemma
residue field splits off reduced maximal ideal
Lemma
finite residue field projective dimension forces depth equals dimension
Lemma
regular local residue field koszul resolution
Corollary
regular local residue field projective dimension dimension
Theorem
auslander buchsbaum serre regularity criterion
Corollary
localisations of regular local rings are regular
Definition
regular noetherian ring
Lemma
flat local ascent of regularity
Lemma
polynomial local regularity fibre step
Theorem
localisation and polynomial extension of regular rings
Lemma
completion preserves embedding dimension
Theorem
completion preserves regular local rings
Definition
normal noetherian ring
Definition
serre r k and s k conditions
Corollary
regular local ring satisfies r one
Corollary
regular local ring satisfies s two
Lemma
normal domain implies r one
Lemma
normal domain implies s two
Lemma
r one s two intersection of height one localisations
Lemma
r one s two integral element membership
Corollary
serre normality criterion two directions
Lemma
serre r zero s one characterises reducedness
Definition
total ring of fractions
Lemma
reduced noetherian total fractions and normal components
Lemma
depth two excludes finite punctured extension
Theorem
serre normality criterion
Theorem
regular local rings are normal
Sources
- 12.3–12.5, p.115
- 12.3, p.115
- Remark 12.4, p.115
- 12.5, p.115
- 10.106.1, first proof paragraph
- 10.106.1 proof; Li Proposition 25.6
- Proposition 25.6, p.67
- 10.106.1, monomial-count consequence
- 10.106.2
- 10.106.2–10.106.3
- Proposition 12.7, p.115
- Proposition 12.8 and Exercise 12.16, pp.116–117
- Example 12.10, p.116
- Proposition 12.13, pp.116–117
- Construction before Proposition 12.27, p.120
- Definition 1.49 and discussion after Lemma 1.50, pp.23–24
- Proposition 12.27 proof, p.121
- Unnumbered Betti-number definition after Lemma 1.50, p.24
- Mircea Mustață, Introduction to Commutative Algebra, Proposition 12.27 and Remark 12.28, pp.120–121
- Proposition 12.27, pp.120–121
- Remark 12.28, p.121
- Corollary 12.29, p.121
- Theorem 12.31 proof, Case 3, p.122
- Theorem 1.53, pd=0 case, p.24
- Theorem 1.53 proof, Claim, p.25
- Theorem 1.53, pp.24–25
- Theorem 1.53, final induction step, p.25
- Theorem 1.53, pp.24–25; Mustata 12.31
- Propositions 12.24–12.25, p.120
- Corollary 12.30, p.121
- Theorem 12.33 proof, p.123; Lemma 5.1 variant
- Lemma 12.32 and proof of 12.31 Case 2, p.122
- Theorem 12.33 proof, p.123; Jeffries 1.60
- Theorem 12.33, p.123
- Theorem 12.33 forward proof, p.123
- 12.27 and 12.33, pp.121–123
- Theorem 12.33 and Corollary 12.30, pp.121–123
- Corollary 12.34, p.123
- Definition 12.14 and Corollary 12.34
- Exercise 12.40(i), p.124
- Lemma 10.110.9 full proof (minimal-resolution version of its syzygy argument)
- Proposition 12.36 proof, p.124
- Corollary 12.34 and Proposition 12.36, pp.123–124
- Corollary 12.35, pp.123–124
- Lecture 25, completion properties (1), (5), (6), pp.68–69
- Lecture 25, property (6), p.69
- Definition 8.34 and Remarks 8.35–8.36, p.56
- Definition 10.157.1
- 10.157.5, R1 implication
- 10.157.5, S2 implication
- 10.157.4 forward implication
- Lemma 8.40 proof, pp.56–57 (normal hypothesis required); 8.41
- Proposition 8.41 proof, pp.57–58; Stacks 10.157.6(1)–(2)
- Proposition 8.41 final proof paragraph, p.58
- Proposition 8.41 and Lemma 8.40, pp.56–58
- 10.157.2–10.157.3
- 10.37.16 proof, total fraction ring used there
- 10.37.16, full proof
- Lemma 10.25.4, full proof
- 10.119.2 last proof paragraph, restricted finite-extension rigidity
- 10.157.4 complete proof, with 030C and 0BHZ
- 10.157.5