\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: Nonquasicompact inverse limit of quasicompact 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 quasicoherent 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: Nonquasiaffine variety with quasiaffine 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 quasicompact dense open subscheme

Section 102.23: Affines over algebraic spaces

Section 102.24: Pushforward of quasicoherent 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 zerodimensional 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 nonflat ring map

Section 102.36: A formally étale nonflat 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 indétale

Section 102.40: Non flasque quasicoherent sheaf associated to injective module

Section 102.41: A nonseparated flat group scheme

Section 102.42: A nonflat group scheme with flat identity component

Section 102.43: A nonseparated 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 quasicompact flat covering which is not algebraic

Section 102.47: Limit preserving on objects, not limit preserving

Section 102.48: A nonalgebraic classifying stack

Section 102.49: Sheaf with quasicompact flat covering which is not algebraic

Section 102.50: Sheaves and specializations

Section 102.51: Sheaves and constructible functions

Section 102.52: The lisseétale site is not functorial

Section 102.53: Derived pushforward of quasicoherent modules

Section 102.54: A big abelian category

Section 102.55: Weakly associated points and scheme theoretic density

Section 102.56: Example of nonadditivity 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 nonalgebraic Homstack

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 quasicompact

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: CohenMacaulay 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: CohenMacaulay 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: Quasicoherent 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: Quasicoherent 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 nonflat 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