Alphabeta MathCohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
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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
- Cyclic Groups and Direct Products
- 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 over a Principal Ideal Domain and the Canonical Forms
- 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
- The Fundamental Theorem of Finite Abelian Groups
Algebraic Geometry
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- 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
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
- Algebraic Zariski Main for Quasi-Finite Morphisms
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Homogeneous Resultants and Projective Intersection Length
- 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
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Functional Analysis
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
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
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
- Topology of ℝ
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
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
Enough injective sheaves of modules
Definition
Higher direct image of a sheaf
Lemma
Local-section formula for derived direct image
Lemma
Exact principal-open Cech resolution
Theorem
Affine acyclicity of quasi-coherent sheaves
Lemma
Acyclicity on intersections of a standard affine cover
Theorem
Cech cohomology computes quasi-coherent cohomology on a separated scheme
Theorem
Higher direct images of quasi-coherent modules vanish along affine morphisms
Lemma
Affine pushforward is compatible with sheaf cohomology
Lemma
Closed immersion preserves cohomology and coherent pushforward
Definition
Twists of a quasi-coherent sheaf
Lemma
Laurent-monomial decomposition of the projective Cech complex
Theorem
Cohomology of O(d) on projective space
Corollary
Global sections of projective twists
Corollary
Intermediate cohomology of projective twists vanishes
Corollary
Top cohomology of projective twists
Theorem
Projective n-space has quasi-coherent cohomological dimension at most n
Theorem
Dimension bound for quasi-coherent cohomology on a Noetherian scheme
Lemma
Eventual generation of coherent projective twists
Lemma
Projective coherent finiteness and large twist vanishing
Theorem
Serre vanishing for coherent sheaves and ample twists
Lemma
Higher direct images localize over an affine base
Lemma
Affine neighbourhood containing component generic points
Lemma
Schematic closure and agreement on a dense open
Lemma
Projective-space projection is universally closed by finite graded pieces
Lemma
Affine finite-type source immerses into relative projective space
Lemma
Chow lemma for proper Noetherian schemes
Lemma
Noetherian devissage for coherent proper pushforward
Theorem
Coherent higher direct images under proper morphisms
Theorem
Finite coherent cohomology for proper schemes
Corollary
Finite-dimensional coherent cohomology over a field
Definition
Euler characteristic of a coherent sheaf
Lemma
Euler characteristic is additive in short exact sequences
Definition
Hilbert function and Euler characteristic on a projective scheme
Lemma
High-degree section module is finite graded
Lemma
Flat field extension commutes with coherent cohomology
Lemma
Support dimension under field extension
Lemma
Regular hyperplane step for coherent support induction
Theorem
Euler characteristic is a Hilbert polynomial
Theorem
Degree of the coherent Hilbert polynomial
Lemma
Sections of a sheaf flat over the base are flat over affine opens
Lemma
Finite projective complex for proper flat coherent cohomology
Lemma
Finite-stage descent of finitely presented schemes and their morphisms
Lemma
Finite-stage descent of finitely presented quasi-coherent sheaves
Lemma
Finite-stage descent of relative flatness for a finitely presented sheaf
Lemma
Finite-stage descent of properness for finitely presented schemes
Lemma
Noetherian approximation of proper flat finitely presented sheaf data
Lemma
Universal finite projective cohomology complex over any base
Definition
Cohomology and base-change map
Lemma
Finite-free local criterion for cohomology and base change
Theorem
Cohomology and base change for proper flat coherent families
Corollary
Upper semicontinuity of fibre cohomology dimensions
Corollary
Euler characteristic in a proper flat family is locally constant
Lemma
Hypersurface cohomology sequence
Corollary
Global functions on geometrically connected and geometrically reduced proper schemes
Remark
Coherence is essential for proper finiteness
Sources
- The Stacks Project, Tag 07D6 (examples of Grothendieck abelian categories) and Tag 05AB (injectives in Grothendieck categories)
- The Stacks Project, Cohomology of Schemes, Chapter 30, §§30.2–30.22
- Ravi Vakil, The Rising Sea (29 August 2022), §§19.1, 19.6, 19.9, 28.1–28.2
- The Stacks Project, Cohomology of Schemes, Chapter 30, Sections 30.2-30.22
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9, 28.1-28.2
- Ravi Vakil, The Rising Sea (29 August 2022), Section 18.2
- The Stacks Project, Cohomology of Schemes, Section 30.8 (tag 01XS), Lemma 30.8.1 (tag 01XT)
- The Stacks Project, Cohomology of Schemes, Section 30.2 (tag 01X9)
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.9, 28.1-28.2
- The Stacks Project, Cohomology of Schemes, Lemma 30.8.2 (Tag 01XV) and Section 30.8
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1-19.2
- The Stacks Project, Cohomology of Schemes, Lemma 30.8.2 (Tag 01XV)
- Ravi Vakil, The Rising Sea (29 August 2022), Section 19.2
- The Stacks Project, Properties of Schemes, Proposition 28.27.13 (Tag 01Q3)
- The Stacks Project, Morphisms of Schemes, Lemma 29.44.5 (Tag 01WC)
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 17.4, 17.6
- The Stacks Project, Cohomology of Schemes, Sections 30.15-30.16 (Tags 0B5S, 01XS)
- Robin Hartshorne, Algebraic Geometry, Theorem III.5.2
- Ravi Vakil, The Rising Sea (29 August 2022), Section 28.2
- The Stacks Project, Cohomology of Schemes, Section 30.17 (Tag 02Y0)
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 18.6 and 19.2
- The Stacks Project, Cohomology of Schemes, Section 30.4 (tags 01XJ, 01XK)
- The Stacks Project, Cohomology of Schemes, Section 30.5 (tag 02KH)
- The Stacks Project, Properties of Schemes, Lemma 28.30.4 (tag 01ZX) and Lemma 28.30.1 (tag 01ZV)
- Ravi Vakil, The Rising Sea (29 August 2022), §§8.4, 11.5, 19.1, 19.6, 19.8–19.9, 28.1–28.2
- The Stacks Project, Morphisms of Schemes, Section 29.6 (tag 01R5), Lemmas 29.6.1-29.6.3
- The Stacks Project, Schemes, Lemma 26.10.1 (tag 01IN)
- The Stacks Project, Morphisms of Schemes, Lemma 29.40.3 (Tag 01VS), affine-source case
- The Stacks Project, Coherent Cohomology, Lemma 30.18.1 and Section 29.7
- Ravi Vakil, The Rising Sea (29 August 2022), Section 28.1
- The Stacks Project, Cohomology of Schemes, Section 30.12 (Tags 01YD, 01YE, 01YF, 01YG, 01YH)
- The Stacks Project, Cohomology of Schemes, Proposition 30.19.1 (Tag 02O3)
- The Stacks Project, Cohomology of Schemes, Lemma 30.18.1 (Tag 0200)
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 18.6, 19.2, 28.1-28.2
- The Stacks Project, Cohomology of Schemes, Section 30.16 (Tag 01YS) and Section 30.15
- The Stacks Project, Cohomology of Schemes, Section 30.5, Lemma 30.5.2 (tag 02KH, flat base change) and Lemma 30.5.1 (tag 02KG, affine case)
- Ravi Vakil, The Rising Sea (29 August 2022), Theorem 25.2.9 with Exercise 25.2.M, and Exercise 19.8.B(b)
- Robin Hartshorne, Algebraic Geometry, Chapter III, Section 5 (hyperplane induction in the proofs of Serre's vanishing and finiteness theorems)
- The Stacks Project, Morphisms of Schemes, Lemma 29.26.2
- The Stacks Project, Algebra, Lemma 10.39.18
- The Stacks Project, Cohomology of Schemes, Lemma 30.22.1
- The Stacks Project, More on Algebra, Lemma 15.60.7 (bounded above flat complexes are K-flat)
- The Stacks Project, More on Algebra, Lemma 15.66.5 (finite free replacement of a bounded above complex)
- The Stacks Project, More on Algebra, Lemma 15.68.2 (flatness of the cokernel at the truncation degree)
- The Stacks Project, More on Algebra, Lemma 15.76.2 (perfect complexes)
- The Stacks Project, Morphisms of Schemes, Lemma 29.26.2 (sections of a flat sheaf over affine opens)
- Ravi Vakil, The Rising Sea, 29 August 2022, Sections 19.1, 19.6, 19.9, 28.1-28.2
- The Stacks Project, Limits of Schemes, Lemma 32.10.1 (Tag 01ZM)
- The Stacks Project, Morphisms of Schemes, Definition 29.22.1 and Lemma 29.22.2 (Tags 01TP and 01TQ)
- The Stacks Project, Limits of Schemes, Lemma 32.10.2 (Tag 01ZR)
- The Stacks Project, Algebra, Lemma 10.127.6, finite-presentation module maps under filtered colimits
- The Stacks Project, Limits of Schemes, Lemma 32.10.4 (Tag 05LY)
- The Stacks Project, Algebra, Lemma 10.168.1(3) (Tag 02JO)
- The Stacks Project, Limits of Schemes, Lemma 32.13.1 (Tag 081F)
- The Stacks Project, Coherent Cohomology, Lemma 30.18.1 (Tag 0200)
- The Stacks Project, Limits of Schemes, Sections 32.8-32.13, especially Lemma 32.13.1
- The Stacks Project, Algebra, Lemma 10.168.1 (Tag 02JO)
- The Stacks Project, Cohomology of Schemes, Section 30.19
- The Stacks Project, Derived Categories of Schemes, Section 36.30 (perfect complexes and base change)
- The Stacks Project, Cohomology of Schemes, Lemma 30.22.1 (Tag 07VK)
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 28.1-28.2
- Ravi Vakil, The Rising Sea (29 August 2022), §28.2, in particular 28.2.10–28.2.11
- The Stacks Project, Derived Categories of Schemes, §§36.26–36.32
- The Stacks Project, Derived Categories of Schemes, Lemma 36.32.1 (Tag 0BDN)
- The Stacks Project, Derived Categories of Schemes, Lemma 36.31.1 (Tag 0BDI)
- The Stacks Project, Derived Categories of Schemes, Sections 36.30-36.32
- The Stacks Project, Derived Categories of Schemes, Lemma 36.32.2 (Tag 0B9T)
- The Stacks Project, Derived Categories of Schemes, Section 36.30 (Remark 30.2, Tag 0A1I, and Lemma 30.4, Tag 0A1K)
- Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9
- The Stacks Project, Cohomology of Schemes, Chapter 30, Sections 30.2-30.22 (proper finiteness and flat base change, tag 02KH)
- The Stacks Project, Algebra, Section 10.53 (Artinian rings), tag 00J4, and Section 10.166 (conjugation and base change), tags 0381, 0382
- The Stacks Project, Topology, Section 5.7 and Algebra Section 10.22 (connected components of spectra), tag 04PP
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Theorem 16.7 and Lemma 14.2