9 Miscellany

Chapter 106: Examples

Section 106.1: Introduction

Section 106.2: An empty limit

Section 106.3: A zero limit

Section 106.4: Nonquasicompact inverse limit of quasicompact spaces

Section 106.5: A nonintegral connected scheme whose local rings are domains

Section 106.6: Noncomplete completion

Section 106.7: Noncomplete quotient

Section 106.8: Completion is not exact

Section 106.9: The category of complete modules is not abelian

Section 106.10: The category of derived complete modules

Section 106.11: Nonflat completions

Section 106.12: Nonabelian category of quasicoherent modules

Section 106.13: Regular sequences and base change

Section 106.14: A Noetherian ring of infinite dimension

Section 106.15: Local rings with nonreduced completion

Section 106.16: A non catenary Noetherian local ring

Section 106.17: Existence of bad local Noetherian rings

Section 106.18: Dimension in Noetherian Jacobson rings

Section 106.19: Nonquasiaffine variety with quasiaffine normalization

Section 106.20: A locally closed subscheme which is not open in closed

Section 106.21: Nonexistence of suitable opens

Section 106.22: Nonexistence of quasicompact dense open subscheme

Section 106.23: Affines over algebraic spaces

Section 106.24: Pushforward of quasicoherent modules

Section 106.25: A nonfinite module with finite free rank 1 stalks

Section 106.26: A noninvertible ideal invertible in stalks

Section 106.27: A finite flat module which is not projective

Section 106.28: A projective module which is not locally free

Section 106.29: Zero dimensional local ring with nonzero flat ideal

Section 106.30: An epimorphism of zerodimensional rings which is not surjective

Section 106.31: Finite type, not finitely presented, flat at prime

Section 106.32: Finite type, flat and not of finite presentation

Section 106.33: Topology of a finite type ring map

Section 106.34: Pure not universally pure

Section 106.35: A formally smooth nonflat ring map

Section 106.36: A formally étale nonflat ring map

Section 106.37: A formally étale ring map with nontrivial cotangent complex

Section 106.38: Ideals generated by sets of idempotents and localization

Section 106.39: A ring map which identifies local rings which is not indétale

Section 106.40: Non flasque quasicoherent sheaf associated to injective module

Section 106.41: A nonseparated flat group scheme

Section 106.42: A nonflat group scheme with flat identity component

Section 106.43: A nonseparated group algebraic space over a field

Section 106.44: Specializations between points in fibre étale morphism

Section 106.45: A torsor which is not an fppf torsor

Section 106.46: Stack with quasicompact flat covering which is not algebraic

Section 106.47: Limit preserving on objects, not limit preserving

Section 106.48: A nonalgebraic classifying stack

Section 106.49: Sheaf with quasicompact flat covering which is not algebraic

Section 106.50: Sheaves and specializations

Section 106.51: Sheaves and constructible functions

Section 106.52: The lisseétale site is not functorial

Section 106.53: Derived pushforward of quasicoherent modules

Section 106.54: A big abelian category

Section 106.55: Weakly associated points and scheme theoretic density

Section 106.56: Example of nonadditivity of traces

Section 106.57: Being projective is not local on the base

Section 106.58: Noneffective descent data for projective schemes

Section 106.59: A family of curves whose total space is not a scheme

Section 106.60: Derived base change

Section 106.61: An interesting compact object

Section 106.62: Two differential graded categories

Section 106.63: The stack of proper algebraic spaces is not algebraic

Section 106.64: An example of a nonalgebraic Homstack

Section 106.65: An algebraic stack not satisfying strong formal effectiveness

Section 106.66: A counter example to Grothendieck's existence theorem

Section 106.67: Affine formal algebraic spaces

Section 106.68: Flat maps are not directed limits of finitely presented flat maps

Section 106.69: The category of modules modulo torsion modules

Section 106.70: Different colimit topologies

Section 106.71: Universally submersive but not V covering

Section 106.72: The spectrum of the integers is not quasicompact

Chapter 107: Exercises

Section 107.1: Algebra

Section 107.2: Colimits

Section 107.3: Additive and abelian categories

Section 107.4: Tensor product

Section 107.5: Flat ring maps

Section 107.6: The Spectrum of a ring

Section 107.7: Localization

Section 107.8: Nakayama's Lemma

Section 107.9: Length

Section 107.10: Associated primes

Section 107.11: Ext groups

Section 107.12: Depth

Section 107.13: CohenMacaulay modules and rings

Section 107.14: Singularities

Section 107.15: Constructible sets

Section 107.16: Hilbert Nullstellensatz

Section 107.17: Dimension

Section 107.18: Catenary rings

Section 107.19: Fraction fields

Section 107.20: Transcendence degree

Section 107.21: Dimension of fibres

Section 107.22: Finite locally free modules

Section 107.23: Glueing

Section 107.24: Going up and going down

Section 107.25: Fitting ideals

Section 107.26: Hilbert functions

Section 107.27: Proj of a ring

Section 107.28: CohenMacaulay rings of dimension 1

Section 107.29: Infinitely many primes

Section 107.30: Filtered derived category

Section 107.31: Regular functions

Section 107.32: Sheaves

Section 107.33: Schemes

Section 107.34: Morphisms

Section 107.35: Tangent Spaces

Section 107.36: Quasicoherent Sheaves

Section 107.37: Proj and projective schemes

Section 107.38: Morphisms from the projective line

Section 107.39: Morphisms from surfaces to curves

Section 107.40: Invertible sheaves

Section 107.41: Čech Cohomology

Section 107.42: Cohomology

Section 107.43: More cohomology

Section 107.44: Cohomology revisited

Section 107.45: Cohomology and Hilbert polynomials

Section 107.46: Curves

Section 107.47: Moduli

Section 107.48: Global Exts

Section 107.49: Divisors

Section 107.50: Differentials

Section 107.51: Schemes, Final Exam, Fall 2007

Section 107.52: Schemes, Final Exam, Spring 2009

Section 107.53: Schemes, Final Exam, Fall 2010

Section 107.54: Schemes, Final Exam, Spring 2011

Section 107.55: Schemes, Final Exam, Fall 2011

Section 107.56: Schemes, Final Exam, Fall 2013

Section 107.57: Schemes, Final Exam, Spring 2014

Section 107.58: Commutative Algebra, Final Exam, Fall 2016

Section 107.59: Schemes, Final Exam, Spring 2017

Section 107.60: Commutative Algebra, Final Exam, Fall 2017

Section 107.61: Schemes, Final Exam, Spring 2018

Chapter 108: A Guide to the Literature

Section 108.1: Short introductory articles

Section 108.2: Classic references

Section 108.3: Books and online notes

Section 108.4: Related references on foundations of stacks

Section 108.5: Papers in the literature

Section 108.6: Stacks in other fields

Section 108.7: Higher stacks

Chapter 109: Desirables

Section 109.1: Introduction

Section 109.2: Conventions

Section 109.3: Sites and Topoi

Section 109.4: Stacks

Section 109.5: Simplicial methods

Section 109.6: Cohomology of schemes

Section 109.7: Deformation theory à la Schlessinger

Section 109.8: Definition of algebraic stacks

Section 109.9: Examples of schemes, algebraic spaces, algebraic stacks

Section 109.10: Properties of algebraic stacks

Section 109.11: Lisse étale site of an algebraic stack

Section 109.12: Things you always wanted to know but were afraid to ask

Section 109.13: Quasicoherent sheaves on stacks

Section 109.14: Flat and smooth

Section 109.15: Artin's representability theorem

Section 109.16: DM stacks are finitely covered by schemes

Section 109.17: Martin Olsson's paper on properness

Section 109.18: Proper pushforward of coherent sheaves

Section 109.19: Keel and Mori

Section 109.20: Add more here

Chapter 110: Coding Style

Section 110.1: List of style comments

Chapter 111: Obsolete

Section 111.1: Introduction

Section 111.2: Homological algebra

Section 111.3: Obsolete algebra lemmas

Section 111.4: Lemmas related to ZMT

Section 111.5: Formally smooth ring maps

Section 111.6: Sites and sheaves

Section 111.7: Cohomology

Section 111.8: Simplicial methods

Section 111.9: Obsolete lemmas on schemes

Section 111.10: Functor of quotients

Section 111.11: Spaces and fpqc coverings

Section 111.12: Very reasonable algebraic spaces

Section 111.13: Obsolete lemma on algebraic spaces

Section 111.14: Variants of cotangent complexes for schemes

Section 111.15: Deformations and obstructions of flat modules

Section 111.16: The stack of coherent sheaves in the nonflat case

Section 111.17: Modifications

Section 111.18: Intersection theory

Section 111.19: Commutativity of intersecting divisors

Section 111.20: Dualizing modules on regular proper models

Section 111.21: Duplicate and split out references

Chapter 112: GNU Free Documentation License

Section 112.1: APPLICABILITY AND DEFINITIONS

Section 112.2: VERBATIM COPYING

Section 112.3: COPYING IN QUANTITY

Section 112.4: MODIFICATIONS

Section 112.5: COMBINING DOCUMENTS

Section 112.6: COLLECTIONS OF DOCUMENTS

Section 112.7: AGGREGATION WITH INDEPENDENT WORKS

Section 112.8: TRANSLATION

Section 112.9: TERMINATION

Section 112.10: FUTURE REVISIONS OF THIS LICENSE

Section 112.11: ADDENDUM: How to use this License for your documents