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