Alphabeta MathKoszul Euler Characteristics and Hilbert–Samuel Multiplicity
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1 · Prerequisites
Abstract Algebra
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- 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
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
Category Theory
- Abelian Categories
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
Homological Algebra
Linear Algebra
Real Analysis
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Lemma
The module-relative Hilbert–Samuel polynomial exists without a dimension theorem
Definition
koszul euler characteristic and degree indexed multiplicity
Lemma
koszul homology finite length for an ideal of definition
Lemma
bounded finite length complex euler identities
Lemma
shifted adic koszul filtration euler comparison
Theorem
degree-r Hilbert–Samuel coefficient as Koszul Euler characteristic
Lemma
koszul euler characteristic first element reduction
Theorem
Multiplicity under parameter reduction with annihilator correction
Sources
- Stacks Project, Proposition 10.58.7 and its finite-difference induction
- Stacks Project, Proposition 10.59.5 (associated graded reduction)
- Stacks Project, 43.15.4–6; local proof with stated module-relative and coefficient conventions
- Hochster, Math 615 Winter 2012, pp.104–108: Euler characteristics and the multiplicity theorem
- Hochster, Math 615 Winter 2012, printed p.165, Proposition and following Corollary (last-element Euler reduction); first-element form via signed wedge order
- Stacks Project, Lemmas 10.60.13 and 10.60.14
- Stacks Project, Theorem 43.15.5: multiplicity as Koszul Euler characteristic
- Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §21