Alphabeta MathHomogeneous Resultants and Projective Intersection Length
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- 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
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- 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
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
Algebraic Geometry
Combinatorics
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Zariski Topology on Prime Spectra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
Scheme Theory
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
Definition
Sylvester resultant of two positive-degree binary forms
Lemma
Scaling, specialization, and the affine and infinite charts of a binary resultant
Theorem
The binary Sylvester resultant detects a common geometric projective root
Lemma
Coprime positive-degree plane forms form a regular sequence
Lemma
Hilbert series and eventual Hilbert value of a two-form plane complete intersection
Definition
Projective scheme of a homogeneous quotient and its standard affine charts
Lemma
Prime and local-ring correspondence on standard projective charts
Lemma
The standard open D_+(f) of a projective quotient is the affine chart Spec((S_f)₀)
Corollary
A plane intersection with no common component is nonempty and zero-dimensional
Lemma
A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings
Definition
Total length of a zero-dimensional projective scheme
Lemma
Localisation at a homogeneous element is graded, with graded kernels and dehomogenised degree-zero parts
Lemma
The spectrum of a finite product ring is the disjoint union of the factor spectra
Lemma
Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient
Lemma
The eventual Hilbert function of a zero-dimensional projective quotient equals its total length
Theorem
Two coprime projective plane forms meet in total length equal to their degree product
Corollary
Algebraic Bezout formula as a sum of local scheme lengths
Sources
- J. S. Milne, Algebraic Geometry v6.10, elimination theory, p. 166
- J. S. Milne, Algebraic Geometry v6.10, Proposition 7.27, p. 166
- J. S. Milne, Algebraic Geometry v6.10, Proposition 7.28, p. 167
- J. S. Milne, Algebraic Geometry v6.10, Proposition 1.24 and the discussion of height-one primes, p. 22
- Andreas Gathmann, Algebraic Geometry class notes (2002), Lemma 6.1.4 and Remark 6.1.6, pp. 92-93
- J. S. Milne, Algebraic Geometry v6.10, discussion of Hilbert functions, pp. 152-155
- J. S. Milne, Algebraic Geometry v6.10, section on projective schemes and Proj
- The Stacks Project, Sections 26.5 and 27.8: affine schemes and Proj
- J. S. Milne, Algebraic Geometry v6.10, Chapter 6 (Proj and its standard affine charts), pp. 152-166
- The Stacks Project, Section 27.8: Projective schemes (tag 01M3)
- The Stacks Project, Section 27.8: Projective schemes (tag 01M3), Lemma 27.8.4
- J. S. Milne, Algebraic Geometry v6.10, Chapter 6 (Proj and its standard affine charts)
- J. S. Milne, Algebraic Geometry v6.10, Chapter 6 (projective schemes; dimension of projective intersections)
- Andreas Gathmann, Algebraic Geometry class notes (2002), Section 6.1 (Bezout and complete intersections), pp. 92-95
- J. S. Milne, Algebraic Geometry v6.10, Chapter 6 (Proj and dimension), and Michael Artin, Algebraic Geometry notes
- Andreas Gathmann, Algebraic Geometry class notes (2002), Section 6.1, pp. 92-94
- A. Gathmann, Algebraic Geometry class notes (2002), Lemma 6.1.4 and Example 6.1.8(i), pp. 93-95
- J. S. Milne, Algebraic Geometry v6.10, Remark 6.38, p. 153
- The Stacks Project, Section 10.56 (tag 00JL): graded rings and localisation at a homogeneous element
- J. S. Milne, Algebraic Geometry v6.10, Chapter 6 (graded rings and Proj)
- The Stacks Project, Lemma 10.21.2 (tag 00ED): the spectrum of a product of rings
- The Stacks Project, Lemma 10.17.6 (tag 00DY) and Section 26.6 (tag 01HX)
- J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring
- The Stacks Project, Section 27.8 (tag 01M3) and Lemma 33.20.2 (tag 06LH)
- A. Gathmann, Algebraic Geometry class notes (2002), Lemma 6.1.4 and Remark 6.1.6, pp. 93-94
- The Stacks Project, Lemma 33.20.2 (tag 06LH) and Section 27.8 (tag 01M3)
- A. Gathmann, Algebraic Geometry class notes (2002), Lemma 6.1.4 and Theorem 6.2.1, pp. 93-96
- J. S. Milne, Algebraic Geometry v6.10, Theorem 6.37 and Remark 6.38, pp. 152-153
- A. Gathmann, Algebraic Geometry class notes (2002), Example 6.2.2, p. 96