9 Miscellany

Chapter 109: Examples

Section 109.1: Introduction

Section 109.2: An empty limit

Section 109.3: A zero limit

Section 109.4: Nonquasicompact inverse limit of quasicompact spaces

Section 109.5: The structure sheaf on the fibre product

Section 109.6: A nonintegral connected scheme whose local rings are domains

Section 109.7: Noncomplete completion

Section 109.8: Noncomplete quotient

Section 109.9: Completion is not exact

Section 109.10: The category of complete modules is not abelian

Section 109.11: The category of derived complete modules

Section 109.12: Nonflat completions

Section 109.13: Nonabelian category of quasicoherent modules

Section 109.14: Regular sequences and base change

Section 109.15: A Noetherian ring of infinite dimension

Section 109.16: Local rings with nonreduced completion

Section 109.17: Another local ring with nonreduced completion

Section 109.18: A non catenary Noetherian local ring

Section 109.19: Existence of bad local Noetherian rings

Section 109.20: Dimension in Noetherian Jacobson rings

Section 109.21: Underlying space Noetherian not Noetherian

Section 109.22: Nonquasiaffine variety with quasiaffine normalization

Section 109.23: Taking scheme theoretic images

Section 109.24: Images of locally closed subsets

Section 109.25: A locally closed subscheme which is not open in closed

Section 109.26: Nonexistence of suitable opens

Section 109.27: Nonexistence of quasicompact dense open subscheme

Section 109.28: Affines over algebraic spaces

Section 109.29: Pushforward of quasicoherent modules

Section 109.30: A nonfinite module with finite free rank 1 stalks

Section 109.31: A noninvertible ideal invertible in stalks

Section 109.32: A finite flat module which is not projective

Section 109.33: A projective module which is not locally free

Section 109.34: Zero dimensional local ring with nonzero flat ideal

Section 109.35: An epimorphism of zerodimensional rings which is not surjective

Section 109.36: Finite type, not finitely presented, flat at prime

Section 109.37: Finite type, flat and not of finite presentation

Section 109.38: Topology of a finite type ring map

Section 109.39: Pure not universally pure

Section 109.40: A formally smooth nonflat ring map

Section 109.41: A formally étale nonflat ring map

Section 109.42: A formally étale ring map with nontrivial cotangent complex

Section 109.43: Flat and formally unramified is not formally étale

Section 109.44: Ideals generated by sets of idempotents and localization

Section 109.45: A ring map which identifies local rings which is not indétale

Section 109.46: Non flasque quasicoherent sheaf associated to injective module

Section 109.47: A nonseparated flat group scheme

Section 109.48: A nonflat group scheme with flat identity component

Section 109.49: A nonseparated group algebraic space over a field

Section 109.50: Specializations between points in fibre étale morphism

Section 109.51: A torsor which is not an fppf torsor

Section 109.52: Stack with quasicompact flat covering which is not algebraic

Section 109.53: Limit preserving on objects, not limit preserving

Section 109.54: A nonalgebraic classifying stack

Section 109.55: Sheaf with quasicompact flat covering which is not algebraic

Section 109.56: Sheaves and specializations

Section 109.57: Sheaves and constructible functions

Section 109.58: The lisseétale site is not functorial

Section 109.59: Sheaves on the category of Noetherian schemes

Section 109.60: Derived pushforward of quasicoherent modules

Section 109.61: A big abelian category

Section 109.62: Weakly associated points and scheme theoretic density

Section 109.63: Example of nonadditivity of traces

Section 109.64: Being projective is not local on the base

Section 109.65: Noneffective descent data for projective schemes

Section 109.66: A family of curves whose total space is not a scheme

Section 109.67: Derived base change

Section 109.68: An interesting compact object

Section 109.69: Two differential graded categories

Section 109.70: The stack of proper algebraic spaces is not algebraic

Section 109.71: An example of a nonalgebraic Homstack

Section 109.72: An algebraic stack not satisfying strong formal effectiveness

Section 109.73: A counter example to Grothendieck's existence theorem

Section 109.74: Affine formal algebraic spaces

Section 109.75: Flat maps are not directed limits of finitely presented flat maps

Section 109.76: The category of modules modulo torsion modules

Section 109.77: Different colimit topologies

Section 109.78: Universally submersive but not V covering

Section 109.79: The spectrum of the integers is not quasicompact

Chapter 110: Exercises

Section 110.1: Algebra

Section 110.2: Colimits

Section 110.3: Additive and abelian categories

Section 110.4: Tensor product

Section 110.5: Flat ring maps

Section 110.6: The Spectrum of a ring

Section 110.7: Localization

Section 110.8: Nakayama's Lemma

Section 110.9: Length

Section 110.10: Associated primes

Section 110.11: Ext groups

Section 110.12: Depth

Section 110.13: CohenMacaulay modules and rings

Section 110.14: Singularities

Section 110.15: Constructible sets

Section 110.16: Hilbert Nullstellensatz

Section 110.17: Dimension

Section 110.18: Catenary rings

Section 110.19: Fraction fields

Section 110.20: Transcendence degree

Section 110.21: Dimension of fibres

Section 110.22: Finite locally free modules

Section 110.23: Glueing

Section 110.24: Going up and going down

Section 110.25: Fitting ideals

Section 110.26: Hilbert functions

Section 110.27: Proj of a ring

Section 110.28: CohenMacaulay rings of dimension 1

Section 110.29: Infinitely many primes

Section 110.30: Filtered derived category

Section 110.31: Regular functions

Section 110.32: Sheaves

Section 110.33: Schemes

Section 110.34: Morphisms

Section 110.35: Tangent Spaces

Section 110.36: Quasicoherent Sheaves

Section 110.37: Proj and projective schemes

Section 110.38: Morphisms from the projective line

Section 110.39: Morphisms from surfaces to curves

Section 110.40: Invertible sheaves

Section 110.41: Čech Cohomology

Section 110.42: Cohomology

Section 110.43: More cohomology

Section 110.44: Cohomology revisited

Section 110.45: Cohomology and Hilbert polynomials

Section 110.46: Curves

Section 110.47: Moduli

Section 110.48: Global Exts

Section 110.49: Divisors

Section 110.50: Differentials

Section 110.51: Schemes, Final Exam, Fall 2007

Section 110.52: Schemes, Final Exam, Spring 2009

Section 110.53: Schemes, Final Exam, Fall 2010

Section 110.54: Schemes, Final Exam, Spring 2011

Section 110.55: Schemes, Final Exam, Fall 2011

Section 110.56: Schemes, Final Exam, Fall 2013

Section 110.57: Schemes, Final Exam, Spring 2014

Section 110.58: Commutative Algebra, Final Exam, Fall 2016

Section 110.59: Schemes, Final Exam, Spring 2017

Section 110.60: Commutative Algebra, Final Exam, Fall 2017

Section 110.61: Schemes, Final Exam, Spring 2018

Section 110.62: Commutative Algebra, Final Exam, Fall 2019

Section 110.63: Algebraic Geometry, Final Exam, Spring 2020

Section 110.64: Commutative Algebra, Final Exam, Fall 2021

Section 110.65: Algebraic Geometry, Final Exam, Spring 2022

Chapter 111: A Guide to the Literature

Section 111.1: Short introductory articles

Section 111.2: Classic references

Section 111.3: Books and online notes

Section 111.4: Related references on foundations of stacks

Section 111.5: Papers in the literature

Section 111.6: Stacks in other fields

Section 111.7: Higher stacks

Chapter 112: Desirables

Section 112.1: Introduction

Section 112.2: Conventions

Section 112.3: Sites and Topoi

Section 112.4: Stacks

Section 112.5: Simplicial methods

Section 112.6: Cohomology of schemes

Section 112.7: Deformation theory à la Schlessinger

Section 112.8: Definition of algebraic stacks

Section 112.9: Examples of schemes, algebraic spaces, algebraic stacks

Section 112.10: Properties of algebraic stacks

Section 112.11: Lisse étale site of an algebraic stack

Section 112.12: Things you always wanted to know but were afraid to ask

Section 112.13: Quasicoherent sheaves on stacks

Section 112.14: Flat and smooth

Section 112.15: Artin's representability theorem

Section 112.16: DM stacks are finitely covered by schemes

Section 112.17: Martin Olsson's paper on properness

Section 112.18: Proper pushforward of coherent sheaves

Section 112.19: Keel and Mori

Section 112.20: Add more here

Chapter 113: Coding Style

Section 113.1: List of style comments

Chapter 114: Obsolete

Section 114.1: Introduction

Section 114.2: Preliminaries

Section 114.3: Homological algebra

Section 114.4: Obsolete algebra lemmas

Section 114.5: Lemmas related to ZMT

Section 114.6: Formally smooth ring maps

Section 114.7: Sites and sheaves

Section 114.8: Cohomology

Section 114.9: Differential graded algebra

Section 114.10: Simplicial methods

Section 114.11: Results on schemes

Section 114.12: Derived categories of varieties

Section 114.13: Functor of quotients

Section 114.14: Spaces and fpqc coverings

Section 114.15: Very reasonable algebraic spaces

Section 114.16: Obsolete lemmas on algebraic spaces

Section 114.17: Obsolete lemmas on algebraic stacks

Section 114.18: Variants of cotangent complexes for schemes

Section 114.19: Deformations and obstructions of flat modules

Section 114.20: The stack of coherent sheaves in the nonflat case

Section 114.21: Modifications

Section 114.22: Intersection theory

Section 114.23: Commutativity of intersecting divisors

Section 114.24: Dualizing modules on regular proper models

Section 114.25: Duplicate and split out references

Chapter 115: GNU Free Documentation License

Section 115.1: APPLICABILITY AND DEFINITIONS

Section 115.2: VERBATIM COPYING

Section 115.3: COPYING IN QUANTITY

Section 115.4: MODIFICATIONS

Section 115.5: COMBINING DOCUMENTS

Section 115.6: COLLECTIONS OF DOCUMENTS

Section 115.7: AGGREGATION WITH INDEPENDENT WORKS

Section 115.8: TRANSLATION

Section 115.9: TERMINATION

Section 115.10: FUTURE REVISIONS OF THIS LICENSE

Section 115.11: ADDENDUM: How to use this License for your documents