9 Miscellany

Chapter 104: Examples

Section 104.1: Introduction

Section 104.2: An empty limit

Section 104.3: A zero limit

Section 104.4: Nonquasicompact inverse limit of quasicompact spaces

Section 104.5: A nonintegral connected scheme whose local rings are domains

Section 104.6: Noncomplete completion

Section 104.7: Noncomplete quotient

Section 104.8: Completion is not exact

Section 104.9: The category of complete modules is not abelian

Section 104.10: The category of derived complete modules

Section 104.11: Nonflat completions

Section 104.12: Nonabelian category of quasicoherent modules

Section 104.13: Regular sequences and base change

Section 104.14: A Noetherian ring of infinite dimension

Section 104.15: Local rings with nonreduced completion

Section 104.16: A non catenary Noetherian local ring

Section 104.17: Existence of bad local Noetherian rings

Section 104.18: Dimension in Noetherian Jacobson rings

Section 104.19: Nonquasiaffine variety with quasiaffine normalization

Section 104.20: A locally closed subscheme which is not open in closed

Section 104.21: Nonexistence of suitable opens

Section 104.22: Nonexistence of quasicompact dense open subscheme

Section 104.23: Affines over algebraic spaces

Section 104.24: Pushforward of quasicoherent modules

Section 104.25: A nonfinite module with finite free rank 1 stalks

Section 104.26: A noninvertible ideal invertible in stalks

Section 104.27: A finite flat module which is not projective

Section 104.28: A projective module which is not locally free

Section 104.29: Zero dimensional local ring with nonzero flat ideal

Section 104.30: An epimorphism of zerodimensional rings which is not surjective

Section 104.31: Finite type, not finitely presented, flat at prime

Section 104.32: Finite type, flat and not of finite presentation

Section 104.33: Topology of a finite type ring map

Section 104.34: Pure not universally pure

Section 104.35: A formally smooth nonflat ring map

Section 104.36: A formally étale nonflat ring map

Section 104.37: A formally étale ring map with nontrivial cotangent complex

Section 104.38: Ideals generated by sets of idempotents and localization

Section 104.39: A ring map which identifies local rings which is not indétale

Section 104.40: Non flasque quasicoherent sheaf associated to injective module

Section 104.41: A nonseparated flat group scheme

Section 104.42: A nonflat group scheme with flat identity component

Section 104.43: A nonseparated group algebraic space over a field

Section 104.44: Specializations between points in fibre étale morphism

Section 104.45: A torsor which is not an fppf torsor

Section 104.46: Stack with quasicompact flat covering which is not algebraic

Section 104.47: Limit preserving on objects, not limit preserving

Section 104.48: A nonalgebraic classifying stack

Section 104.49: Sheaf with quasicompact flat covering which is not algebraic

Section 104.50: Sheaves and specializations

Section 104.51: Sheaves and constructible functions

Section 104.52: The lisseétale site is not functorial

Section 104.53: Derived pushforward of quasicoherent modules

Section 104.54: A big abelian category

Section 104.55: Weakly associated points and scheme theoretic density

Section 104.56: Example of nonadditivity of traces

Section 104.57: Being projective is not local on the base

Section 104.58: Descent data for schemes need not be effective, even for a projective morphism

Section 104.59: A family of curves whose total space is not a scheme

Section 104.60: Derived base change

Section 104.61: An interesting compact object

Section 104.62: Two differential graded categories

Section 104.63: The stack of proper algebraic spaces is not algebraic

Section 104.64: An example of a nonalgebraic Homstack

Section 104.65: An algebraic stack not satisfying strong formal effectiveness

Section 104.66: A counter example to Grothendieck's existence theorem

Section 104.67: Affine formal algebraic spaces

Section 104.68: Flat maps are not directed limits of finitely presented flat maps

Section 104.69: The category of modules modulo torsion modules

Section 104.70: Different colimit topologies

Section 104.71: Universally submersive but not V covering

Section 104.72: The spectrum of the integers is not quasicompact

Chapter 105: Exercises

Section 105.1: Algebra

Section 105.2: Colimits

Section 105.3: Additive and abelian categories

Section 105.4: Tensor product

Section 105.5: Flat ring maps

Section 105.6: The Spectrum of a ring

Section 105.7: Localization

Section 105.8: Nakayama's Lemma

Section 105.9: Length

Section 105.10: Associated primes

Section 105.11: Ext groups

Section 105.12: Depth

Section 105.13: CohenMacaulay modules and rings

Section 105.14: Singularities

Section 105.15: Hilbert Nullstellensatz

Section 105.16: Dimension

Section 105.17: Catenary rings

Section 105.18: Fraction fields

Section 105.19: Transcendence degree

Section 105.20: Dimension of fibres

Section 105.21: Finite locally free modules

Section 105.22: Glueing

Section 105.23: Going up and going down

Section 105.24: Fitting ideals

Section 105.25: Hilbert functions

Section 105.26: Proj of a ring

Section 105.27: CohenMacaulay rings of dimension 1

Section 105.28: Infinitely many primes

Section 105.29: Filtered derived category

Section 105.30: Regular functions

Section 105.31: Sheaves

Section 105.32: Schemes

Section 105.33: Morphisms

Section 105.34: Tangent Spaces

Section 105.35: Quasicoherent Sheaves

Section 105.36: Proj and projective schemes

Section 105.37: Morphisms from the projective line

Section 105.38: Morphisms from surfaces to curves

Section 105.39: Invertible sheaves

Section 105.40: Čech Cohomology

Section 105.41: Cohomology

Section 105.42: More cohomology

Section 105.43: Cohomology revisited

Section 105.44: Cohomology and Hilbert polynomials

Section 105.45: Curves

Section 105.46: Moduli

Section 105.47: Global Exts

Section 105.48: Divisors

Section 105.49: Differentials

Section 105.50: Schemes, Final Exam, Fall 2007

Section 105.51: Schemes, Final Exam, Spring 2009

Section 105.52: Schemes, Final Exam, Fall 2010

Section 105.53: Schemes, Final Exam, Spring 2011

Section 105.54: Schemes, Final Exam, Fall 2011

Section 105.55: Schemes, Final Exam, Fall 2013

Section 105.56: Schemes, Final Exam, Spring 2014

Section 105.57: Commutative Algebra, Final Exam, Fall 2016

Section 105.58: Schemes, Final Exam, Spring 2017

Section 105.59: Commutative Algebra, Final Exam, Fall 2017

Section 105.60: Schemes, Final Exam, Spring 2018

Chapter 106: A Guide to the Literature

Section 106.1: Short introductory articles

Section 106.2: Classic references

Section 106.3: Books and online notes

Section 106.4: Related references on foundations of stacks

Section 106.5: Papers in the literature

Section 106.6: Stacks in other fields

Section 106.7: Higher stacks

Chapter 107: Desirables

Section 107.1: Introduction

Section 107.2: Conventions

Section 107.3: Sites and Topoi

Section 107.4: Stacks

Section 107.5: Simplicial methods

Section 107.6: Cohomology of schemes

Section 107.7: Deformation theory à la Schlessinger

Section 107.8: Definition of algebraic stacks

Section 107.9: Examples of schemes, algebraic spaces, algebraic stacks

Section 107.10: Properties of algebraic stacks

Section 107.11: Lisse étale site of an algebraic stack

Section 107.12: Things you always wanted to know but were afraid to ask

Section 107.13: Quasicoherent sheaves on stacks

Section 107.14: Flat and smooth

Section 107.15: Artin's representability theorem

Section 107.16: DM stacks are finitely covered by schemes

Section 107.17: Martin Olsson's paper on properness

Section 107.18: Proper pushforward of coherent sheaves

Section 107.19: Keel and Mori

Section 107.20: Add more here

Chapter 108: Coding Style

Section 108.1: List of style comments

Chapter 109: Obsolete

Section 109.1: Introduction

Section 109.2: Homological algebra

Section 109.3: Obsolete algebra lemmas

Section 109.4: Lemmas related to ZMT

Section 109.5: Formally smooth ring maps

Section 109.6: Sites and sheaves

Section 109.7: Cohomology

Section 109.8: Simplicial methods

Section 109.9: Obsolete lemmas on schemes

Section 109.10: Functor of quotients

Section 109.11: Spaces and fpqc coverings

Section 109.12: Very reasonable algebraic spaces

Section 109.13: Obsolete lemma on algebraic spaces

Section 109.14: Variants of cotangent complexes for schemes

Section 109.15: Deformations and obstructions of flat modules

Section 109.16: The stack of coherent sheaves in the nonflat case

Section 109.17: Modifications

Section 109.18: Intersection theory

Section 109.19: Dualizing modules on regular proper models

Section 109.20: Duplicate and split out references

Chapter 110: GNU Free Documentation License

Section 110.1: APPLICABILITY AND DEFINITIONS

Section 110.2: VERBATIM COPYING

Section 110.3: COPYING IN QUANTITY

Section 110.4: MODIFICATIONS

Section 110.5: COMBINING DOCUMENTS

Section 110.6: COLLECTIONS OF DOCUMENTS

Section 110.7: AGGREGATION WITH INDEPENDENT WORKS

Section 110.8: TRANSLATION

Section 110.9: TERMINATION

Section 110.10: FUTURE REVISIONS OF THIS LICENSE

Section 110.11: ADDENDUM: How to use this License for your documents