9 Miscellany

Chapter 107: Examples

Section 107.1: Introduction

Section 107.2: An empty limit

Section 107.3: A zero limit

Section 107.4: Nonquasicompact inverse limit of quasicompact 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 quasicoherent 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: Nonquasiaffine variety with quasiaffine 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 quasicompact dense open subscheme

Section 107.23: Affines over algebraic spaces

Section 107.24: Pushforward of quasicoherent 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 zerodimensional 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 nonflat ring map

Section 107.36: A formally étale nonflat 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 indétale

Section 107.40: Non flasque quasicoherent sheaf associated to injective module

Section 107.41: A nonseparated flat group scheme

Section 107.42: A nonflat group scheme with flat identity component

Section 107.43: A nonseparated 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 quasicompact flat covering which is not algebraic

Section 107.47: Limit preserving on objects, not limit preserving

Section 107.48: A nonalgebraic classifying stack

Section 107.49: Sheaf with quasicompact flat covering which is not algebraic

Section 107.50: Sheaves and specializations

Section 107.51: Sheaves and constructible functions

Section 107.52: The lisseétale site is not functorial

Section 107.53: Derived pushforward of quasicoherent modules

Section 107.54: A big abelian category

Section 107.55: Weakly associated points and scheme theoretic density

Section 107.56: Example of nonadditivity of traces

Section 107.57: Being projective is not local on the base

Section 107.58: Noneffective 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 nonalgebraic Homstack

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 quasicompact

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: CohenMacaulay 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: CohenMacaulay 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: Quasicoherent 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: Quasicoherent 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 nonflat 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