9 Miscellany

Chapter 108: Examples

Section 108.1: Introduction

Section 108.2: An empty limit

Section 108.3: A zero limit

Section 108.4: Nonquasicompact inverse limit of quasicompact spaces

Section 108.5: The structure sheaf on the fibre product

Section 108.6: A nonintegral connected scheme whose local rings are domains

Section 108.7: Noncomplete completion

Section 108.8: Noncomplete quotient

Section 108.9: Completion is not exact

Section 108.10: The category of complete modules is not abelian

Section 108.11: The category of derived complete modules

Section 108.12: Nonflat completions

Section 108.13: Nonabelian category of quasicoherent modules

Section 108.14: Regular sequences and base change

Section 108.15: A Noetherian ring of infinite dimension

Section 108.16: Local rings with nonreduced completion

Section 108.17: Another local ring with nonreduced completion

Section 108.18: A non catenary Noetherian local ring

Section 108.19: Existence of bad local Noetherian rings

Section 108.20: Dimension in Noetherian Jacobson rings

Section 108.21: Underlying space Noetherian not Noetherian

Section 108.22: Nonquasiaffine variety with quasiaffine normalization

Section 108.23: Taking scheme theoretic images

Section 108.24: A locally closed subscheme which is not open in closed

Section 108.25: Nonexistence of suitable opens

Section 108.26: Nonexistence of quasicompact dense open subscheme

Section 108.27: Affines over algebraic spaces

Section 108.28: Pushforward of quasicoherent modules

Section 108.29: A nonfinite module with finite free rank 1 stalks

Section 108.30: A noninvertible ideal invertible in stalks

Section 108.31: A finite flat module which is not projective

Section 108.32: A projective module which is not locally free

Section 108.33: Zero dimensional local ring with nonzero flat ideal

Section 108.34: An epimorphism of zerodimensional rings which is not surjective

Section 108.35: Finite type, not finitely presented, flat at prime

Section 108.36: Finite type, flat and not of finite presentation

Section 108.37: Topology of a finite type ring map

Section 108.38: Pure not universally pure

Section 108.39: A formally smooth nonflat ring map

Section 108.40: A formally étale nonflat ring map

Section 108.41: A formally étale ring map with nontrivial cotangent complex

Section 108.42: Flat and formally unramified is not formally étale

Section 108.43: Ideals generated by sets of idempotents and localization

Section 108.44: A ring map which identifies local rings which is not indétale

Section 108.45: Non flasque quasicoherent sheaf associated to injective module

Section 108.46: A nonseparated flat group scheme

Section 108.47: A nonflat group scheme with flat identity component

Section 108.48: A nonseparated group algebraic space over a field

Section 108.49: Specializations between points in fibre étale morphism

Section 108.50: A torsor which is not an fppf torsor

Section 108.51: Stack with quasicompact flat covering which is not algebraic

Section 108.52: Limit preserving on objects, not limit preserving

Section 108.53: A nonalgebraic classifying stack

Section 108.54: Sheaf with quasicompact flat covering which is not algebraic

Section 108.55: Sheaves and specializations

Section 108.56: Sheaves and constructible functions

Section 108.57: The lisseétale site is not functorial

Section 108.58: Sheaves on the category of Noetherian schemes

Section 108.59: Derived pushforward of quasicoherent modules

Section 108.60: A big abelian category

Section 108.61: Weakly associated points and scheme theoretic density

Section 108.62: Example of nonadditivity of traces

Section 108.63: Being projective is not local on the base

Section 108.64: Noneffective descent data for projective schemes

Section 108.65: A family of curves whose total space is not a scheme

Section 108.66: Derived base change

Section 108.67: An interesting compact object

Section 108.68: Two differential graded categories

Section 108.69: The stack of proper algebraic spaces is not algebraic

Section 108.70: An example of a nonalgebraic Homstack

Section 108.71: An algebraic stack not satisfying strong formal effectiveness

Section 108.72: A counter example to Grothendieck's existence theorem

Section 108.73: Affine formal algebraic spaces

Section 108.74: Flat maps are not directed limits of finitely presented flat maps

Section 108.75: The category of modules modulo torsion modules

Section 108.76: Different colimit topologies

Section 108.77: Universally submersive but not V covering

Section 108.78: The spectrum of the integers is not quasicompact

Chapter 109: Exercises

Section 109.1: Algebra

Section 109.2: Colimits

Section 109.3: Additive and abelian categories

Section 109.4: Tensor product

Section 109.5: Flat ring maps

Section 109.6: The Spectrum of a ring

Section 109.7: Localization

Section 109.8: Nakayama's Lemma

Section 109.9: Length

Section 109.10: Associated primes

Section 109.11: Ext groups

Section 109.12: Depth

Section 109.13: CohenMacaulay modules and rings

Section 109.14: Singularities

Section 109.15: Constructible sets

Section 109.16: Hilbert Nullstellensatz

Section 109.17: Dimension

Section 109.18: Catenary rings

Section 109.19: Fraction fields

Section 109.20: Transcendence degree

Section 109.21: Dimension of fibres

Section 109.22: Finite locally free modules

Section 109.23: Glueing

Section 109.24: Going up and going down

Section 109.25: Fitting ideals

Section 109.26: Hilbert functions

Section 109.27: Proj of a ring

Section 109.28: CohenMacaulay rings of dimension 1

Section 109.29: Infinitely many primes

Section 109.30: Filtered derived category

Section 109.31: Regular functions

Section 109.32: Sheaves

Section 109.33: Schemes

Section 109.34: Morphisms

Section 109.35: Tangent Spaces

Section 109.36: Quasicoherent Sheaves

Section 109.37: Proj and projective schemes

Section 109.38: Morphisms from the projective line

Section 109.39: Morphisms from surfaces to curves

Section 109.40: Invertible sheaves

Section 109.41: Čech Cohomology

Section 109.42: Cohomology

Section 109.43: More cohomology

Section 109.44: Cohomology revisited

Section 109.45: Cohomology and Hilbert polynomials

Section 109.46: Curves

Section 109.47: Moduli

Section 109.48: Global Exts

Section 109.49: Divisors

Section 109.50: Differentials

Section 109.51: Schemes, Final Exam, Fall 2007

Section 109.52: Schemes, Final Exam, Spring 2009

Section 109.53: Schemes, Final Exam, Fall 2010

Section 109.54: Schemes, Final Exam, Spring 2011

Section 109.55: Schemes, Final Exam, Fall 2011

Section 109.56: Schemes, Final Exam, Fall 2013

Section 109.57: Schemes, Final Exam, Spring 2014

Section 109.58: Commutative Algebra, Final Exam, Fall 2016

Section 109.59: Schemes, Final Exam, Spring 2017

Section 109.60: Commutative Algebra, Final Exam, Fall 2017

Section 109.61: Schemes, Final Exam, Spring 2018

Section 109.62: Commutative Algebra, Final Exam, Fall 2019

Section 109.63: Algebraic Geometry, Final Exam, Spring 2020

Chapter 110: A Guide to the Literature

Section 110.1: Short introductory articles

Section 110.2: Classic references

Section 110.3: Books and online notes

Section 110.4: Related references on foundations of stacks

Section 110.5: Papers in the literature

Section 110.6: Stacks in other fields

Section 110.7: Higher stacks

Chapter 111: Desirables

Section 111.1: Introduction

Section 111.2: Conventions

Section 111.3: Sites and Topoi

Section 111.4: Stacks

Section 111.5: Simplicial methods

Section 111.6: Cohomology of schemes

Section 111.7: Deformation theory à la Schlessinger

Section 111.8: Definition of algebraic stacks

Section 111.9: Examples of schemes, algebraic spaces, algebraic stacks

Section 111.10: Properties of algebraic stacks

Section 111.11: Lisse étale site of an algebraic stack

Section 111.12: Things you always wanted to know but were afraid to ask

Section 111.13: Quasicoherent sheaves on stacks

Section 111.14: Flat and smooth

Section 111.15: Artin's representability theorem

Section 111.16: DM stacks are finitely covered by schemes

Section 111.17: Martin Olsson's paper on properness

Section 111.18: Proper pushforward of coherent sheaves

Section 111.19: Keel and Mori

Section 111.20: Add more here

Chapter 112: Coding Style

Section 112.1: List of style comments

Chapter 113: Obsolete

Section 113.1: Introduction

Section 113.2: Preliminaries

Section 113.3: Homological algebra

Section 113.4: Obsolete algebra lemmas

Section 113.5: Lemmas related to ZMT

Section 113.6: Formally smooth ring maps

Section 113.7: Sites and sheaves

Section 113.8: Cohomology

Section 113.9: Differential graded algebra

Section 113.10: Simplicial methods

Section 113.11: Obsolete lemmas on schemes

Section 113.12: Functor of quotients

Section 113.13: Spaces and fpqc coverings

Section 113.14: Very reasonable algebraic spaces

Section 113.15: Obsolete lemmas on algebraic spaces

Section 113.16: Obsolete lemmas on algebraic stacks

Section 113.17: Variants of cotangent complexes for schemes

Section 113.18: Deformations and obstructions of flat modules

Section 113.19: The stack of coherent sheaves in the nonflat case

Section 113.20: Modifications

Section 113.21: Intersection theory

Section 113.22: Commutativity of intersecting divisors

Section 113.23: Dualizing modules on regular proper models

Section 113.24: Duplicate and split out references

Chapter 114: GNU Free Documentation License

Section 114.1: APPLICABILITY AND DEFINITIONS

Section 114.2: VERBATIM COPYING

Section 114.3: COPYING IN QUANTITY

Section 114.4: MODIFICATIONS

Section 114.5: COMBINING DOCUMENTS

Section 114.6: COLLECTIONS OF DOCUMENTS

Section 114.7: AGGREGATION WITH INDEPENDENT WORKS

Section 114.8: TRANSLATION

Section 114.9: TERMINATION

Section 114.10: FUTURE REVISIONS OF THIS LICENSE

Section 114.11: ADDENDUM: How to use this License for your documents