The Stacks project

9 Miscellany

  • Chapter 107: Examples
    • Section 107.1: Introduction
    • Section 107.2: An empty limit
    • Section 107.3: A zero limit
    • Section 107.4: Non-quasi-compact inverse limit of quasi-compact spaces
    • Section 107.5: A nonintegral connected scheme whose local rings are domains
    • Section 107.6: Noncomplete completion
    • Section 107.7: Noncomplete quotient
    • Section 107.8: Completion is not exact
    • Section 107.9: The category of complete modules is not abelian
    • Section 107.10: The category of derived complete modules
    • Section 107.11: Nonflat completions
    • Section 107.12: Nonabelian category of quasi-coherent modules
    • Section 107.13: Regular sequences and base change
    • Section 107.14: A Noetherian ring of infinite dimension
    • Section 107.15: Local rings with nonreduced completion
    • Section 107.16: A non catenary Noetherian local ring
    • Section 107.17: Existence of bad local Noetherian rings
    • Section 107.18: Dimension in Noetherian Jacobson rings
    • Section 107.19: Non-quasi-affine variety with quasi-affine normalization
    • Section 107.20: A locally closed subscheme which is not open in closed
    • Section 107.21: Nonexistence of suitable opens
    • Section 107.22: Nonexistence of quasi-compact dense open subscheme
    • Section 107.23: Affines over algebraic spaces
    • Section 107.24: Pushforward of quasi-coherent modules
    • Section 107.25: A nonfinite module with finite free rank 1 stalks
    • Section 107.26: A noninvertible ideal invertible in stalks
    • Section 107.27: A finite flat module which is not projective
    • Section 107.28: A projective module which is not locally free
    • Section 107.29: Zero dimensional local ring with nonzero flat ideal
    • Section 107.30: An epimorphism of zero-dimensional rings which is not surjective
    • Section 107.31: Finite type, not finitely presented, flat at prime
    • Section 107.32: Finite type, flat and not of finite presentation
    • Section 107.33: Topology of a finite type ring map
    • Section 107.34: Pure not universally pure
    • Section 107.35: A formally smooth non-flat ring map
    • Section 107.36: A formally étale non-flat ring map
    • Section 107.37: A formally étale ring map with nontrivial cotangent complex
    • Section 107.38: Ideals generated by sets of idempotents and localization
    • Section 107.39: A ring map which identifies local rings which is not ind-étale
    • Section 107.40: Non flasque quasi-coherent sheaf associated to injective module
    • Section 107.41: A non-separated flat group scheme
    • Section 107.42: A non-flat group scheme with flat identity component
    • Section 107.43: A non-separated group algebraic space over a field
    • Section 107.44: Specializations between points in fibre étale morphism
    • Section 107.45: A torsor which is not an fppf torsor
    • Section 107.46: Stack with quasi-compact flat covering which is not algebraic
    • Section 107.47: Limit preserving on objects, not limit preserving
    • Section 107.48: A non-algebraic classifying stack
    • Section 107.49: Sheaf with quasi-compact flat covering which is not algebraic
    • Section 107.50: Sheaves and specializations
    • Section 107.51: Sheaves and constructible functions
    • Section 107.52: The lisse-étale site is not functorial
    • Section 107.53: Derived pushforward of quasi-coherent modules
    • Section 107.54: A big abelian category
    • Section 107.55: Weakly associated points and scheme theoretic density
    • Section 107.56: Example of non-additivity of traces
    • Section 107.57: Being projective is not local on the base
    • Section 107.58: Non-effective descent data for projective schemes
    • Section 107.59: A family of curves whose total space is not a scheme
    • Section 107.60: Derived base change
    • Section 107.61: An interesting compact object
    • Section 107.62: Two differential graded categories
    • Section 107.63: The stack of proper algebraic spaces is not algebraic
    • Section 107.64: An example of a non-algebraic Hom-stack
    • Section 107.65: An algebraic stack not satisfying strong formal effectiveness
    • Section 107.66: A counter example to Grothendieck's existence theorem
    • Section 107.67: Affine formal algebraic spaces
    • Section 107.68: Flat maps are not directed limits of finitely presented flat maps
    • Section 107.69: The category of modules modulo torsion modules
    • Section 107.70: Different colimit topologies
    • Section 107.71: Universally submersive but not V covering
    • Section 107.72: The spectrum of the integers is not quasi-compact
  • Chapter 108: Exercises
    • Section 108.1: Algebra
    • Section 108.2: Colimits
    • Section 108.3: Additive and abelian categories
    • Section 108.4: Tensor product
    • Section 108.5: Flat ring maps
    • Section 108.6: The Spectrum of a ring
    • Section 108.7: Localization
    • Section 108.8: Nakayama's Lemma
    • Section 108.9: Length
    • Section 108.10: Associated primes
    • Section 108.11: Ext groups
    • Section 108.12: Depth
    • Section 108.13: Cohen-Macaulay modules and rings
    • Section 108.14: Singularities
    • Section 108.15: Constructible sets
    • Section 108.16: Hilbert Nullstellensatz
    • Section 108.17: Dimension
    • Section 108.18: Catenary rings
    • Section 108.19: Fraction fields
    • Section 108.20: Transcendence degree
    • Section 108.21: Dimension of fibres
    • Section 108.22: Finite locally free modules
    • Section 108.23: Glueing
    • Section 108.24: Going up and going down
    • Section 108.25: Fitting ideals
    • Section 108.26: Hilbert functions
    • Section 108.27: Proj of a ring
    • Section 108.28: Cohen-Macaulay rings of dimension 1
    • Section 108.29: Infinitely many primes
    • Section 108.30: Filtered derived category
    • Section 108.31: Regular functions
    • Section 108.32: Sheaves
    • Section 108.33: Schemes
    • Section 108.34: Morphisms
    • Section 108.35: Tangent Spaces
    • Section 108.36: Quasi-coherent Sheaves
    • Section 108.37: Proj and projective schemes
    • Section 108.38: Morphisms from the projective line
    • Section 108.39: Morphisms from surfaces to curves
    • Section 108.40: Invertible sheaves
    • Section 108.41: Čech Cohomology
    • Section 108.42: Cohomology
    • Section 108.43: More cohomology
    • Section 108.44: Cohomology revisited
    • Section 108.45: Cohomology and Hilbert polynomials
    • Section 108.46: Curves
    • Section 108.47: Moduli
    • Section 108.48: Global Exts
    • Section 108.49: Divisors
    • Section 108.50: Differentials
    • Section 108.51: Schemes, Final Exam, Fall 2007
    • Section 108.52: Schemes, Final Exam, Spring 2009
    • Section 108.53: Schemes, Final Exam, Fall 2010
    • Section 108.54: Schemes, Final Exam, Spring 2011
    • Section 108.55: Schemes, Final Exam, Fall 2011
    • Section 108.56: Schemes, Final Exam, Fall 2013
    • Section 108.57: Schemes, Final Exam, Spring 2014
    • Section 108.58: Commutative Algebra, Final Exam, Fall 2016
    • Section 108.59: Schemes, Final Exam, Spring 2017
    • Section 108.60: Commutative Algebra, Final Exam, Fall 2017
    • Section 108.61: Schemes, Final Exam, Spring 2018
  • Chapter 109: A Guide to the Literature
    • Section 109.1: Short introductory articles
    • Section 109.2: Classic references
    • Section 109.3: Books and online notes
    • Section 109.4: Related references on foundations of stacks
    • Section 109.5: Papers in the literature
    • Section 109.6: Stacks in other fields
    • Section 109.7: Higher stacks
  • Chapter 110: Desirables
    • Section 110.1: Introduction
    • Section 110.2: Conventions
    • Section 110.3: Sites and Topoi
    • Section 110.4: Stacks
    • Section 110.5: Simplicial methods
    • Section 110.6: Cohomology of schemes
    • Section 110.7: Deformation theory à la Schlessinger
    • Section 110.8: Definition of algebraic stacks
    • Section 110.9: Examples of schemes, algebraic spaces, algebraic stacks
    • Section 110.10: Properties of algebraic stacks
    • Section 110.11: Lisse étale site of an algebraic stack
    • Section 110.12: Things you always wanted to know but were afraid to ask
    • Section 110.13: Quasi-coherent sheaves on stacks
    • Section 110.14: Flat and smooth
    • Section 110.15: Artin's representability theorem
    • Section 110.16: DM stacks are finitely covered by schemes
    • Section 110.17: Martin Olsson's paper on properness
    • Section 110.18: Proper pushforward of coherent sheaves
    • Section 110.19: Keel and Mori
    • Section 110.20: Add more here
  • Chapter 111: Coding Style
    • Section 111.1: List of style comments
  • Chapter 112: Obsolete
    • Section 112.1: Introduction
    • Section 112.2: Homological algebra
    • Section 112.3: Obsolete algebra lemmas
    • Section 112.4: Lemmas related to ZMT
    • Section 112.5: Formally smooth ring maps
    • Section 112.6: Sites and sheaves
    • Section 112.7: Cohomology
    • Section 112.8: Differential graded algebra
    • Section 112.9: Simplicial methods
    • Section 112.10: Obsolete lemmas on schemes
    • Section 112.11: Functor of quotients
    • Section 112.12: Spaces and fpqc coverings
    • Section 112.13: Very reasonable algebraic spaces
    • Section 112.14: Obsolete lemma on algebraic spaces
    • Section 112.15: Variants of cotangent complexes for schemes
    • Section 112.16: Deformations and obstructions of flat modules
    • Section 112.17: The stack of coherent sheaves in the non-flat case
    • Section 112.18: Modifications
    • Section 112.19: Intersection theory
    • Section 112.20: Commutativity of intersecting divisors
    • Section 112.21: Dualizing modules on regular proper models
    • Section 112.22: Duplicate and split out references
  • Chapter 113: GNU Free Documentation License
    • Section 113.1: APPLICABILITY AND DEFINITIONS
    • Section 113.2: VERBATIM COPYING
    • Section 113.3: COPYING IN QUANTITY
    • Section 113.4: MODIFICATIONS
    • Section 113.5: COMBINING DOCUMENTS
    • Section 113.6: COLLECTIONS OF DOCUMENTS
    • Section 113.7: AGGREGATION WITH INDEPENDENT WORKS
    • Section 113.8: TRANSLATION
    • Section 113.9: TERMINATION
    • Section 113.10: FUTURE REVISIONS OF THIS LICENSE
    • Section 113.11: ADDENDUM: How to use this License for your documents