The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

9 Miscellany

  • Chapter 102: Examples
    • Section 102.1: Introduction
    • Section 102.2: An empty limit
    • Section 102.3: A zero limit
    • Section 102.4: Non-quasi-compact inverse limit of quasi-compact spaces
    • Section 102.5: A nonintegral connected scheme whose local rings are domains
    • Section 102.6: Noncomplete completion
    • Section 102.7: Noncomplete quotient
    • Section 102.8: Completion is not exact
    • Section 102.9: The category of complete modules is not abelian
    • Section 102.10: The category of derived complete modules
    • Section 102.11: Nonflat completions
    • Section 102.12: Nonabelian category of quasi-coherent modules
    • Section 102.13: Regular sequences and base change
    • Section 102.14: A Noetherian ring of infinite dimension
    • Section 102.15: Local rings with nonreduced completion
    • Section 102.16: A non catenary Noetherian local ring
    • Section 102.17: Existence of bad local Noetherian rings
    • Section 102.18: Dimension in Noetherian Jacobson rings
    • Section 102.19: Non-quasi-affine variety with quasi-affine normalization
    • Section 102.20: A locally closed subscheme which is not open in closed
    • Section 102.21: Nonexistence of suitable opens
    • Section 102.22: Nonexistence of quasi-compact dense open subscheme
    • Section 102.23: Affines over algebraic spaces
    • Section 102.24: Pushforward of quasi-coherent modules
    • Section 102.25: A nonfinite module with finite free rank 1 stalks
    • Section 102.26: A noninvertible ideal invertible in stalks
    • Section 102.27: A finite flat module which is not projective
    • Section 102.28: A projective module which is not locally free
    • Section 102.29: Zero dimensional local ring with nonzero flat ideal
    • Section 102.30: An epimorphism of zero-dimensional rings which is not surjective
    • Section 102.31: Finite type, not finitely presented, flat at prime
    • Section 102.32: Finite type, flat and not of finite presentation
    • Section 102.33: Topology of a finite type ring map
    • Section 102.34: Pure not universally pure
    • Section 102.35: A formally smooth non-flat ring map
    • Section 102.36: A formally étale non-flat ring map
    • Section 102.37: A formally étale ring map with nontrivial cotangent complex
    • Section 102.38: Ideals generated by sets of idempotents and localization
    • Section 102.39: A ring map which identifies local rings which is not ind-étale
    • Section 102.40: Non flasque quasi-coherent sheaf associated to injective module
    • Section 102.41: A non-separated flat group scheme
    • Section 102.42: A non-flat group scheme with flat identity component
    • Section 102.43: A non-separated group algebraic space over a field
    • Section 102.44: Specializations between points in fibre étale morphism
    • Section 102.45: A torsor which is not an fppf torsor
    • Section 102.46: Stack with quasi-compact flat covering which is not algebraic
    • Section 102.47: Limit preserving on objects, not limit preserving
    • Section 102.48: A non-algebraic classifying stack
    • Section 102.49: Sheaf with quasi-compact flat covering which is not algebraic
    • Section 102.50: Sheaves and specializations
    • Section 102.51: Sheaves and constructible functions
    • Section 102.52: The lisse-étale site is not functorial
    • Section 102.53: Derived pushforward of quasi-coherent modules
    • Section 102.54: A big abelian category
    • Section 102.55: Weakly associated points and scheme theoretic density
    • Section 102.56: Example of non-additivity of traces
    • Section 102.57: Being projective is not local on the base
    • Section 102.58: Descent data for schemes need not be effective, even for a projective morphism
    • Section 102.59: A family of curves whose total space is not a scheme
    • Section 102.60: Derived base change
    • Section 102.61: An interesting compact object
    • Section 102.62: Two differential graded categories
    • Section 102.63: The stack of proper algebraic spaces is not algebraic
    • Section 102.64: An example of a non-algebraic Hom-stack
    • Section 102.65: An algebraic stack not satisfying strong formal effectiveness
    • Section 102.66: A counter example to Grothendieck's existence theorem
    • Section 102.67: Affine formal algebraic spaces
    • Section 102.68: Flat maps are not directed limits of finitely presented flat maps
    • Section 102.69: The category of modules modulo torsion modules
    • Section 102.70: Different colimit topologies
    • Section 102.71: Universally submersive but not V covering
    • Section 102.72: The spectrum of the integers is not quasi-compact
  • Chapter 103: Exercises
    • Section 103.1: Algebra
    • Section 103.2: Colimits
    • Section 103.3: Additive and abelian categories
    • Section 103.4: Tensor product
    • Section 103.5: Flat ring maps
    • Section 103.6: The Spectrum of a ring
    • Section 103.7: Localization
    • Section 103.8: Nakayama's Lemma
    • Section 103.9: Length
    • Section 103.10: Associated primes
    • Section 103.11: Ext groups
    • Section 103.12: Depth
    • Section 103.13: Cohen-Macaulay modules and rings
    • Section 103.14: Singularities
    • Section 103.15: Hilbert Nullstellensatz
    • Section 103.16: Dimension
    • Section 103.17: Catenary rings
    • Section 103.18: Fraction fields
    • Section 103.19: Transcendence degree
    • Section 103.20: Dimension of fibres
    • Section 103.21: Finite locally free modules
    • Section 103.22: Glueing
    • Section 103.23: Going up and going down
    • Section 103.24: Fitting ideals
    • Section 103.25: Hilbert functions
    • Section 103.26: Proj of a ring
    • Section 103.27: Cohen-Macaulay rings of dimension 1
    • Section 103.28: Infinitely many primes
    • Section 103.29: Filtered derived category
    • Section 103.30: Regular functions
    • Section 103.31: Sheaves
    • Section 103.32: Schemes
    • Section 103.33: Morphisms
    • Section 103.34: Tangent Spaces
    • Section 103.35: Quasi-coherent Sheaves
    • Section 103.36: Proj and projective schemes
    • Section 103.37: Morphisms from the projective line
    • Section 103.38: Morphisms from surfaces to curves
    • Section 103.39: Invertible sheaves
    • Section 103.40: Čech Cohomology
    • Section 103.41: Cohomology
    • Section 103.42: More cohomology
    • Section 103.43: Cohomology revisited
    • Section 103.44: Cohomology and Hilbert polynomials
    • Section 103.45: Curves
    • Section 103.46: Moduli
    • Section 103.47: Global Exts
    • Section 103.48: Divisors
    • Section 103.49: Differentials
    • Section 103.50: Schemes, Final Exam, Fall 2007
    • Section 103.51: Schemes, Final Exam, Spring 2009
    • Section 103.52: Schemes, Final Exam, Fall 2010
    • Section 103.53: Schemes, Final Exam, Spring 2011
    • Section 103.54: Schemes, Final Exam, Fall 2011
    • Section 103.55: Schemes, Final Exam, Fall 2013
    • Section 103.56: Schemes, Final Exam, Spring 2014
    • Section 103.57: Commutative Algebra, Final Exam, Fall 2016
    • Section 103.58: Schemes, Final Exam, Spring 2017
    • Section 103.59: Commutative Algebra, Final Exam, Fall 2017
    • Section 103.60: Schemes, Final Exam, Spring 2018
  • Chapter 104: A Guide to the Literature
    • Section 104.1: Short introductory articles
    • Section 104.2: Classic references
    • Section 104.3: Books and online notes
    • Section 104.4: Related references on foundations of stacks
    • Section 104.5: Papers in the literature
    • Section 104.6: Stacks in other fields
    • Section 104.7: Higher stacks
  • Chapter 105: Desirables
    • Section 105.1: Introduction
    • Section 105.2: Conventions
    • Section 105.3: Sites and Topoi
    • Section 105.4: Stacks
    • Section 105.5: Simplicial methods
    • Section 105.6: Cohomology of schemes
    • Section 105.7: Deformation theory à la Schlessinger
    • Section 105.8: Definition of algebraic stacks
    • Section 105.9: Examples of schemes, algebraic spaces, algebraic stacks
    • Section 105.10: Properties of algebraic stacks
    • Section 105.11: Lisse étale site of an algebraic stack
    • Section 105.12: Things you always wanted to know but were afraid to ask
    • Section 105.13: Quasi-coherent sheaves on stacks
    • Section 105.14: Flat and smooth
    • Section 105.15: Artin's representability theorem
    • Section 105.16: DM stacks are finitely covered by schemes
    • Section 105.17: Martin Olsson's paper on properness
    • Section 105.18: Proper pushforward of coherent sheaves
    • Section 105.19: Keel and Mori
    • Section 105.20: Add more here
  • Chapter 106: Coding Style
    • Section 106.1: List of style comments
  • Chapter 107: Obsolete
    • Section 107.1: Introduction
    • Section 107.2: Homological algebra
    • Section 107.3: Obsolete algebra lemmas
    • Section 107.4: Lemmas related to ZMT
    • Section 107.5: Formally smooth ring maps
    • Section 107.6: Sites and sheaves
    • Section 107.7: Cohomology
    • Section 107.8: Simplicial methods
    • Section 107.9: Obsolete lemmas on schemes
    • Section 107.10: Functor of quotients
    • Section 107.11: Spaces and fpqc coverings
    • Section 107.12: Very reasonable algebraic spaces
    • Section 107.13: Obsolete lemma on algebraic spaces
    • Section 107.14: Variants of cotangent complexes for schemes
    • Section 107.15: Deformations and obstructions of flat modules
    • Section 107.16: The stack of coherent sheaves in the non-flat case
    • Section 107.17: Modifications
    • Section 107.18: Intersection theory
    • Section 107.19: Dualizing modules on regular proper models
    • Section 107.20: Duplicate and split out references
  • Chapter 108: GNU Free Documentation License
    • Section 108.1: APPLICABILITY AND DEFINITIONS
    • Section 108.2: VERBATIM COPYING
    • Section 108.3: COPYING IN QUANTITY
    • Section 108.4: MODIFICATIONS
    • Section 108.5: COMBINING DOCUMENTS
    • Section 108.6: COLLECTIONS OF DOCUMENTS
    • Section 108.7: AGGREGATION WITH INDEPENDENT WORKS
    • Section 108.8: TRANSLATION
    • Section 108.9: TERMINATION
    • Section 108.10: FUTURE REVISIONS OF THIS LICENSE
    • Section 108.11: ADDENDUM: How to use this License for your documents