9 Miscellany

• Chapter 109: Examples
  • Section 109.1: Introduction
  • Section 109.2: An empty limit
  • Section 109.3: A zero limit
  • Section 109.4: Non-quasi-compact inverse limit of quasi-compact spaces
  • Section 109.5: The structure sheaf on the fibre product
  • Section 109.6: A nonintegral connected scheme whose local rings are domains
  • Section 109.7: Noncomplete completion
  • Section 109.8: Noncomplete quotient
  • Section 109.9: Completion is not exact
  • Section 109.10: The category of complete modules is not abelian
  • Section 109.11: The category of derived complete modules
  • Section 109.12: Nonflat completions
  • Section 109.13: Nonabelian category of quasi-coherent modules
  • Section 109.14: Regular sequences and base change
  • Section 109.15: A Noetherian ring of infinite dimension
  • Section 109.16: Local rings with nonreduced completion
  • Section 109.17: Another local ring with nonreduced completion
  • Section 109.18: A non catenary Noetherian local ring
  • Section 109.19: Existence of bad local Noetherian rings
  • Section 109.20: Dimension in Noetherian Jacobson rings
  • Section 109.21: Underlying space Noetherian not Noetherian
  • Section 109.22: Non-quasi-affine variety with quasi-affine normalization
  • Section 109.23: Taking scheme theoretic images
  • Section 109.24: Images of locally closed subsets
  • Section 109.25: A locally closed subscheme which is not open in closed
  • Section 109.26: Nonexistence of suitable opens
  • Section 109.27: Nonexistence of quasi-compact dense open subscheme
  • Section 109.28: Affines over algebraic spaces
  • Section 109.29: Pushforward of quasi-coherent modules
  • Section 109.30: A nonfinite module with finite free rank 1 stalks
  • Section 109.31: A noninvertible ideal invertible in stalks
  • Section 109.32: A finite flat module which is not projective
  • Section 109.33: A projective module which is not locally free
  • Section 109.34: Zero dimensional local ring with nonzero flat ideal
  • Section 109.35: An epimorphism of zero-dimensional rings which is not surjective
  • Section 109.36: Finite type, not finitely presented, flat at prime
  • Section 109.37: Finite type, flat and not of finite presentation
  • Section 109.38: Topology of a finite type ring map
  • Section 109.39: Pure not universally pure
  • Section 109.40: A formally smooth non-flat ring map
  • Section 109.41: A formally étale non-flat ring map
  • Section 109.42: A formally étale ring map with nontrivial cotangent complex
  • Section 109.43: Flat and formally unramified is not formally étale
  • Section 109.44: Ideals generated by sets of idempotents and localization
  • Section 109.45: A ring map which identifies local rings which is not ind-étale
  • Section 109.46: Non flasque quasi-coherent sheaf associated to injective module
  • Section 109.47: A non-separated flat group scheme
  • Section 109.48: A non-flat group scheme with flat identity component
  • Section 109.49: A non-separated group algebraic space over a field
  • Section 109.50: Specializations between points in fibre étale morphism
  • Section 109.51: A torsor which is not an fppf torsor
  • Section 109.52: Stack with quasi-compact flat covering which is not algebraic
  • Section 109.53: Limit preserving on objects, not limit preserving
  • Section 109.54: A non-algebraic classifying stack
  • Section 109.55: Sheaf with quasi-compact flat covering which is not algebraic
  • Section 109.56: Sheaves and specializations
  • Section 109.57: Sheaves and constructible functions
  • Section 109.58: The lisse-étale site is not functorial
  • Section 109.59: Sheaves on the category of Noetherian schemes
  • Section 109.60: Derived pushforward of quasi-coherent modules
  • Section 109.61: A big abelian category
  • Section 109.62: Weakly associated points and scheme theoretic density
  • Section 109.63: Example of non-additivity of traces
  • Section 109.64: Being projective is not local on the base
  • Section 109.65: Non-effective descent data for projective schemes
  • Section 109.66: A family of curves whose total space is not a scheme
  • Section 109.67: Derived base change
  • Section 109.68: An interesting compact object
  • Section 109.69: Two differential graded categories
  • Section 109.70: The stack of proper algebraic spaces is not algebraic
  • Section 109.71: An example of a non-algebraic Hom-stack
  • Section 109.72: An algebraic stack not satisfying strong formal effectiveness
  • Section 109.73: A counter example to Grothendieck's existence theorem
  • Section 109.74: Affine formal algebraic spaces
  • Section 109.75: Flat maps are not directed limits of finitely presented flat maps
  • Section 109.76: The category of modules modulo torsion modules
  • Section 109.77: Different colimit topologies
  • Section 109.78: Universally submersive but not V covering
  • Section 109.79: The spectrum of the integers is not quasi-compact
• Chapter 110: Exercises
  • Section 110.1: Algebra
  • Section 110.2: Colimits
  • Section 110.3: Additive and abelian categories
  • Section 110.4: Tensor product
  • Section 110.5: Flat ring maps
  • Section 110.6: The Spectrum of a ring
  • Section 110.7: Localization
  • Section 110.8: Nakayama's Lemma
  • Section 110.9: Length
  • Section 110.10: Associated primes
  • Section 110.11: Ext groups
  • Section 110.12: Depth
  • Section 110.13: Cohen-Macaulay modules and rings
  • Section 110.14: Singularities
  • Section 110.15: Constructible sets
  • Section 110.16: Hilbert Nullstellensatz
  • Section 110.17: Dimension
  • Section 110.18: Catenary rings
  • Section 110.19: Fraction fields
  • Section 110.20: Transcendence degree
  • Section 110.21: Dimension of fibres
  • Section 110.22: Finite locally free modules
  • Section 110.23: Glueing
  • Section 110.24: Going up and going down
  • Section 110.25: Fitting ideals
  • Section 110.26: Hilbert functions
  • Section 110.27: Proj of a ring
  • Section 110.28: Cohen-Macaulay rings of dimension 1
  • Section 110.29: Infinitely many primes
  • Section 110.30: Filtered derived category
  • Section 110.31: Regular functions
  • Section 110.32: Sheaves
  • Section 110.33: Schemes
  • Section 110.34: Morphisms
  • Section 110.35: Tangent Spaces
  • Section 110.36: Quasi-coherent Sheaves
  • Section 110.37: Proj and projective schemes
  • Section 110.38: Morphisms from the projective line
  • Section 110.39: Morphisms from surfaces to curves
  • Section 110.40: Invertible sheaves
  • Section 110.41: Čech Cohomology
  • Section 110.42: Cohomology
  • Section 110.43: More cohomology
  • Section 110.44: Cohomology revisited
  • Section 110.45: Cohomology and Hilbert polynomials
  • Section 110.46: Curves
  • Section 110.47: Moduli
  • Section 110.48: Global Exts
  • Section 110.49: Divisors
  • Section 110.50: Differentials
  • Section 110.51: Schemes, Final Exam, Fall 2007
  • Section 110.52: Schemes, Final Exam, Spring 2009
  • Section 110.53: Schemes, Final Exam, Fall 2010
  • Section 110.54: Schemes, Final Exam, Spring 2011
  • Section 110.55: Schemes, Final Exam, Fall 2011
  • Section 110.56: Schemes, Final Exam, Fall 2013
  • Section 110.57: Schemes, Final Exam, Spring 2014
  • Section 110.58: Commutative Algebra, Final Exam, Fall 2016
  • Section 110.59: Schemes, Final Exam, Spring 2017
  • Section 110.60: Commutative Algebra, Final Exam, Fall 2017
  • Section 110.61: Schemes, Final Exam, Spring 2018
  • Section 110.62: Commutative Algebra, Final Exam, Fall 2019
  • Section 110.63: Algebraic Geometry, Final Exam, Spring 2020
  • Section 110.64: Commutative Algebra, Final Exam, Fall 2021
  • Section 110.65: Algebraic Geometry, Final Exam, Spring 2022
• Chapter 111: A Guide to the Literature
  • Section 111.1: Short introductory articles
  • Section 111.2: Classic references
  • Section 111.3: Books and online notes
  • Section 111.4: Related references on foundations of stacks
  • Section 111.5: Papers in the literature
  • Section 111.6: Stacks in other fields
  • Section 111.7: Higher stacks
• Chapter 112: Desirables
  • Section 112.1: Introduction
  • Section 112.2: Conventions
  • Section 112.3: Sites and Topoi
  • Section 112.4: Stacks
  • Section 112.5: Simplicial methods
  • Section 112.6: Cohomology of schemes
  • Section 112.7: Deformation theory à la Schlessinger
  • Section 112.8: Definition of algebraic stacks
  • Section 112.9: Examples of schemes, algebraic spaces, algebraic stacks
  • Section 112.10: Properties of algebraic stacks
  • Section 112.11: Lisse étale site of an algebraic stack
  • Section 112.12: Things you always wanted to know but were afraid to ask
  • Section 112.13: Quasi-coherent sheaves on stacks
  • Section 112.14: Flat and smooth
  • Section 112.15: Artin's representability theorem
  • Section 112.16: DM stacks are finitely covered by schemes
  • Section 112.17: Martin Olsson's paper on properness
  • Section 112.18: Proper pushforward of coherent sheaves
  • Section 112.19: Keel and Mori
  • Section 112.20: Add more here
• Chapter 113: Coding Style
  • Section 113.1: List of style comments
• Chapter 114: Obsolete
  • Section 114.1: Introduction
  • Section 114.2: Preliminaries
  • Section 114.3: Homological algebra
  • Section 114.4: Obsolete algebra lemmas
  • Section 114.5: Lemmas related to ZMT
  • Section 114.6: Formally smooth ring maps
  • Section 114.7: Sites and sheaves
  • Section 114.8: Cohomology
  • Section 114.9: Differential graded algebra
  • Section 114.10: Simplicial methods
  • Section 114.11: Results on schemes
  • Section 114.12: Derived categories of varieties
  • Section 114.13: Functor of quotients
  • Section 114.14: Spaces and fpqc coverings
  • Section 114.15: Very reasonable algebraic spaces
  • Section 114.16: Obsolete lemmas on algebraic spaces
  • Section 114.17: Obsolete lemmas on algebraic stacks
  • Section 114.18: Variants of cotangent complexes for schemes
  • Section 114.19: Deformations and obstructions of flat modules
  • Section 114.20: The stack of coherent sheaves in the non-flat case
  • Section 114.21: Modifications
  • Section 114.22: Intersection theory
  • Section 114.23: Commutativity of intersecting divisors
  • Section 114.24: Dualizing modules on regular proper models
  • Section 114.25: Duplicate and split out references
• Chapter 115: GNU Free Documentation License
  • Section 115.1: APPLICABILITY AND DEFINITIONS
  • Section 115.2: VERBATIM COPYING
  • Section 115.3: COPYING IN QUANTITY
  • Section 115.4: MODIFICATIONS
  • Section 115.5: COMBINING DOCUMENTS
  • Section 115.6: COLLECTIONS OF DOCUMENTS
  • Section 115.7: AGGREGATION WITH INDEPENDENT WORKS
  • Section 115.8: TRANSLATION
  • Section 115.9: TERMINATION
  • Section 115.10: FUTURE REVISIONS OF THIS LICENSE
  • Section 115.11: ADDENDUM: How to use this License for your documents