9 Miscellany

• Chapter 110: Examples
  • Section 110.1: Introduction
  • Section 110.2: An empty limit
  • Section 110.3: A zero limit
  • Section 110.4: Non-quasi-compact inverse limit of quasi-compact spaces
  • Section 110.5: The structure sheaf on the fibre product
  • Section 110.6: A nonintegral connected scheme whose local rings are domains
  • Section 110.7: Noncomplete completion
  • Section 110.8: Noncomplete quotient
  • Section 110.9: Completion is not exact
  • Section 110.10: The category of complete modules is not abelian
  • Section 110.11: The category of derived complete modules
  • Section 110.12: Nonflat completions
  • Section 110.13: Nonabelian category of quasi-coherent modules
  • Section 110.14: Nonsplit locally split sequence
  • Section 110.15: Regular sequences and base change
  • Section 110.16: A Noetherian ring of infinite dimension
  • Section 110.17: Local rings with nonreduced completion
  • Section 110.18: Another local ring with nonreduced completion
  • Section 110.19: A non catenary Noetherian local ring
  • Section 110.20: Existence of bad local Noetherian rings
  • Section 110.21: Dimension in Noetherian Jacobson rings
  • Section 110.22: Underlying space Noetherian not Noetherian
  • Section 110.23: Non-quasi-affine variety with quasi-affine normalization
  • Section 110.24: Taking scheme theoretic images
  • Section 110.25: Images of locally closed subsets
  • Section 110.26: A locally closed subscheme which is not open in closed
  • Section 110.27: Nonexistence of suitable opens
  • Section 110.28: Nonexistence of quasi-compact dense open subscheme
  • Section 110.29: Affines over algebraic spaces
  • Section 110.30: Pushforward of quasi-coherent modules
  • Section 110.31: A nonfinite module with finite free rank 1 stalks
  • Section 110.32: A noninvertible ideal invertible in stalks
  • Section 110.33: A finite flat module which is not projective
  • Section 110.34: A projective module which is not locally free
  • Section 110.35: Zero dimensional local ring with nonzero flat ideal
  • Section 110.36: An epimorphism of zero-dimensional rings which is not surjective
  • Section 110.37: Finite type, not finitely presented, flat at prime
  • Section 110.38: Finite type, flat and not of finite presentation
  • Section 110.39: Topology of a finite type ring map
  • Section 110.40: Pure not universally pure
  • Section 110.41: A formally smooth non-flat ring map
  • Section 110.42: A formally étale non-flat ring map
  • Section 110.43: A formally étale ring map with nontrivial cotangent complex
  • Section 110.44: Flat and formally unramified is not formally étale
  • Section 110.45: Ideals generated by sets of idempotents and localization
  • Section 110.46: A ring map which identifies local rings which is not ind-étale
  • Section 110.47: Non flasque quasi-coherent sheaf associated to injective module
  • Section 110.48: A non-separated flat group scheme
  • Section 110.49: A non-flat group scheme with flat identity component
  • Section 110.50: A non-separated group algebraic space over a field
  • Section 110.51: Specializations between points in fibre étale morphism
  • Section 110.52: A torsor which is not an fppf torsor
  • Section 110.53: Stack with quasi-compact flat covering which is not algebraic
  • Section 110.54: Limit preserving on objects, not limit preserving
  • Section 110.55: A non-algebraic classifying stack
  • Section 110.56: Sheaf with quasi-compact flat covering which is not algebraic
  • Section 110.57: Sheaves and specializations
  • Section 110.58: Sheaves and constructible functions
  • Section 110.59: The lisse-étale site is not functorial
  • Section 110.60: Sheaves on the category of Noetherian schemes
  • Section 110.61: Derived pushforward of quasi-coherent modules
  • Section 110.62: A big abelian category
  • Section 110.63: Weakly associated points and scheme theoretic density
  • Section 110.64: Example of non-additivity of traces
  • Section 110.65: Being projective is not local on the base
  • Section 110.66: Non-effective descent data for projective schemes
  • Section 110.67: A family of curves whose total space is not a scheme
  • Section 110.68: Derived base change
  • Section 110.69: An interesting compact object
  • Section 110.70: Two differential graded categories
  • Section 110.71: The stack of proper algebraic spaces is not algebraic
  • Section 110.72: An example of a non-algebraic Hom-stack
  • Section 110.73: An algebraic stack not satisfying strong formal effectiveness
  • Section 110.74: A counter example to Grothendieck's existence theorem
  • Section 110.75: Affine formal algebraic spaces
  • Section 110.76: Flat maps are not directed limits of finitely presented flat maps
  • Section 110.77: The category of modules modulo torsion modules
  • Section 110.78: Different colimit topologies
  • Section 110.79: Universally submersive but not V covering
  • Section 110.80: The spectrum of the integers is not quasi-compact
• Chapter 111: Exercises
  • Section 111.1: Algebra
  • Section 111.2: Colimits
  • Section 111.3: Additive and abelian categories
  • Section 111.4: Tensor product
  • Section 111.5: Flat ring maps
  • Section 111.6: The Spectrum of a ring
  • Section 111.7: Localization
  • Section 111.8: Nakayama's Lemma
  • Section 111.9: Length
  • Section 111.10: Associated primes
  • Section 111.11: Ext groups
  • Section 111.12: Depth
  • Section 111.13: Cohen-Macaulay modules and rings
  • Section 111.14: Singularities
  • Section 111.15: Constructible sets
  • Section 111.16: Hilbert Nullstellensatz
  • Section 111.17: Dimension
  • Section 111.18: Catenary rings
  • Section 111.19: Fraction fields
  • Section 111.20: Transcendence degree
  • Section 111.21: Dimension of fibres
  • Section 111.22: Finite locally free modules
  • Section 111.23: Glueing
  • Section 111.24: Going up and going down
  • Section 111.25: Fitting ideals
  • Section 111.26: Hilbert functions
  • Section 111.27: Proj of a ring
  • Section 111.28: Cohen-Macaulay rings of dimension 1
  • Section 111.29: Infinitely many primes
  • Section 111.30: Filtered derived category
  • Section 111.31: Regular functions
  • Section 111.32: Sheaves
  • Section 111.33: Schemes
  • Section 111.34: Morphisms
  • Section 111.35: Tangent Spaces
  • Section 111.36: Quasi-coherent Sheaves
  • Section 111.37: Proj and projective schemes
  • Section 111.38: Morphisms from the projective line
  • Section 111.39: Morphisms from surfaces to curves
  • Section 111.40: Invertible sheaves
  • Section 111.41: Čech Cohomology
  • Section 111.42: Cohomology
  • Section 111.43: More cohomology
  • Section 111.44: Cohomology revisited
  • Section 111.45: Cohomology and Hilbert polynomials
  • Section 111.46: Curves
  • Section 111.47: Moduli
  • Section 111.48: Global Exts
  • Section 111.49: Divisors
  • Section 111.50: Differentials
  • Section 111.51: Schemes, Final Exam, Fall 2007
  • Section 111.52: Schemes, Final Exam, Spring 2009
  • Section 111.53: Schemes, Final Exam, Fall 2010
  • Section 111.54: Schemes, Final Exam, Spring 2011
  • Section 111.55: Schemes, Final Exam, Fall 2011
  • Section 111.56: Schemes, Final Exam, Fall 2013
  • Section 111.57: Schemes, Final Exam, Spring 2014
  • Section 111.58: Commutative Algebra, Final Exam, Fall 2016
  • Section 111.59: Schemes, Final Exam, Spring 2017
  • Section 111.60: Commutative Algebra, Final Exam, Fall 2017
  • Section 111.61: Schemes, Final Exam, Spring 2018
  • Section 111.62: Commutative Algebra, Final Exam, Fall 2019
  • Section 111.63: Algebraic Geometry, Final Exam, Spring 2020
  • Section 111.64: Commutative Algebra, Final Exam, Fall 2021
  • Section 111.65: Algebraic Geometry, Final Exam, Spring 2022
• Chapter 112: A Guide to the Literature
  • Section 112.1: Short introductory articles
  • Section 112.2: Classic references
  • Section 112.3: Books and online notes
  • Section 112.4: Related references on foundations of stacks
  • Section 112.5: Papers in the literature
  • Section 112.6: Stacks in other fields
  • Section 112.7: Higher stacks
• Chapter 113: Desirables
  • Section 113.1: Introduction
  • Section 113.2: Conventions
  • Section 113.3: Sites and Topoi
  • Section 113.4: Stacks
  • Section 113.5: Simplicial methods
  • Section 113.6: Cohomology of schemes
  • Section 113.7: Deformation theory à la Schlessinger
  • Section 113.8: Definition of algebraic stacks
  • Section 113.9: Examples of schemes, algebraic spaces, algebraic stacks
  • Section 113.10: Properties of algebraic stacks
  • Section 113.11: Lisse étale site of an algebraic stack
  • Section 113.12: Things you always wanted to know but were afraid to ask
  • Section 113.13: Quasi-coherent sheaves on stacks
  • Section 113.14: Flat and smooth
  • Section 113.15: Artin's representability theorem
  • Section 113.16: DM stacks are finitely covered by schemes
  • Section 113.17: Martin Olsson's paper on properness
  • Section 113.18: Proper pushforward of coherent sheaves
  • Section 113.19: Keel and Mori
  • Section 113.20: Add more here
• Chapter 114: Coding Style
  • Section 114.1: List of style comments
• Chapter 115: Obsolete
  • Section 115.1: Introduction
  • Section 115.2: Preliminaries
  • Section 115.3: Homological algebra
  • Section 115.4: Obsolete algebra lemmas
  • Section 115.5: Lemmas related to ZMT
  • Section 115.6: Formally smooth ring maps
  • Section 115.7: Sites and sheaves
  • Section 115.8: Cohomology
  • Section 115.9: Differential graded algebra
  • Section 115.10: Simplicial methods
  • Section 115.11: Results on schemes
  • Section 115.12: Derived categories of varieties
  • Section 115.13: Representability in the regular proper case
  • Section 115.14: Functor of quotients
  • Section 115.15: Spaces and fpqc coverings
  • Section 115.16: Very reasonable algebraic spaces
  • Section 115.17: Obsolete lemmas on algebraic spaces
  • Section 115.18: Obsolete lemmas on algebraic stacks
  • Section 115.19: Variants of cotangent complexes for schemes
  • Section 115.20: Deformations and obstructions of flat modules
  • Section 115.21: The stack of coherent sheaves in the non-flat case
  • Section 115.22: Modifications
  • Section 115.23: Intersection theory
  • Section 115.24: Commutativity of intersecting divisors
  • Section 115.25: Dualizing modules on regular proper models
  • Section 115.26: Duplicate and split out references
• Chapter 116: GNU Free Documentation License
  • Section 116.1: APPLICABILITY AND DEFINITIONS
  • Section 116.2: VERBATIM COPYING
  • Section 116.3: COPYING IN QUANTITY
  • Section 116.4: MODIFICATIONS
  • Section 116.5: COMBINING DOCUMENTS
  • Section 116.6: COLLECTIONS OF DOCUMENTS
  • Section 116.7: AGGREGATION WITH INDEPENDENT WORKS
  • Section 116.8: TRANSLATION
  • Section 116.9: TERMINATION
  • Section 116.10: FUTURE REVISIONS OF THIS LICENSE
  • Section 116.11: ADDENDUM: How to use this License for your documents