109 Examples

• Section 109.1: Introduction
• Section 109.2: An empty limit
• Section 109.3: A zero limit
• Section 109.4: Non-quasi-compact inverse limit of quasi-compact spaces
  • Lemma 109.4.1
• 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
  • Equation 109.7.0.1
  • Lemma 109.7.1
• Section 109.8: Noncomplete quotient
  • Lemma 109.8.1
• Section 109.9: Completion is not exact
  • Lemma 109.9.1 slogan
• Section 109.10: The category of complete modules is not abelian
  • Lemma 109.10.1
• Section 109.11: The category of derived complete modules
  • Lemma 109.11.1
• Section 109.12: Nonflat completions
  • Lemma 109.12.1
  • Lemma 109.12.2
  • Lemma 109.12.3
  • Lemma 109.12.4
  • Lemma 109.12.5
  • Lemma 109.12.6
    • Equation 109.12.6.1
  • Lemma 109.12.7
• Section 109.13: Nonabelian category of quasi-coherent modules
  • Lemma 109.13.1
• Section 109.14: Regular sequences and base change
  • Lemma 109.14.1
  • Lemma 109.14.2
• 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
  • Lemma 109.17.1
• 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
  • Lemma 109.20.1
• Section 109.21: Underlying space Noetherian not Noetherian
  • Example 109.21.1
  • Example 109.21.2
• Section 109.22: Non-quasi-affine variety with quasi-affine normalization
  • Lemma 109.22.1
• Section 109.23: Taking scheme theoretic images
• Section 109.24: Images of locally closed subsets
  • Lemma 109.24.1
• Section 109.25: A locally closed subscheme which is not open in closed
• Section 109.26: Nonexistence of suitable opens
  • Lemma 109.26.1
  • Lemma 109.26.2
• Section 109.27: Nonexistence of quasi-compact dense open subscheme
  • Lemma 109.27.1
• Section 109.28: Affines over algebraic spaces
  • Lemma 109.28.1
• Section 109.29: Pushforward of quasi-coherent modules
  • Lemma 109.29.1
• Section 109.30: A nonfinite module with finite free rank 1 stalks
• Section 109.31: A noninvertible ideal invertible in stalks
  • Lemma 109.31.1
• Section 109.32: A finite flat module which is not projective
  • Lemma 109.32.1
• Section 109.33: A projective module which is not locally free
  • Lemma 109.33.1
  • Lemma 109.33.2
  • Lemma 109.33.3
  • Lemma 109.33.4
• Section 109.34: Zero dimensional local ring with nonzero flat ideal
  • Lemma 109.34.1 slogan
• Section 109.35: An epimorphism of zero-dimensional rings which is not surjective
  • Lemma 109.35.1
• Section 109.36: Finite type, not finitely presented, flat at prime
  • Lemma 109.36.1
• Section 109.37: Finite type, flat and not of finite presentation
  • Lemma 109.37.1
• Section 109.38: Topology of a finite type ring map
  • Lemma 109.38.1
• Section 109.39: Pure not universally pure
  • Lemma 109.39.1
• Section 109.40: A formally smooth non-flat ring map
  • Lemma 109.40.1
• Section 109.41: A formally étale non-flat ring map
  • Lemma 109.41.1
• Section 109.42: A formally étale ring map with nontrivial cotangent complex
  • Lemma 109.42.1
• Section 109.43: Flat and formally unramified is not formally étale
  • Lemma 109.43.1
  • Lemma 109.43.2
  • Remark 109.43.3
  • Lemma 109.43.4
• Section 109.44: Ideals generated by sets of idempotents and localization
  • Lemma 109.44.1
• Section 109.45: A ring map which identifies local rings which is not ind-étale
  • Lemma 109.45.1
• Section 109.46: Non flasque quasi-coherent sheaf associated to injective module
  • Lemma 109.46.1
  • Lemma 109.46.2
• Section 109.47: A non-separated flat group scheme
  • Lemma 109.47.1
  • Lemma 109.47.2
• Section 109.48: A non-flat group scheme with flat identity component
  • Lemma 109.48.1
• Section 109.49: A non-separated group algebraic space over a field
  • Lemma 109.49.1
• Section 109.50: Specializations between points in fibre étale morphism
  • Lemma 109.50.1
• Section 109.51: A torsor which is not an fppf torsor
  • Lemma 109.51.1
• Section 109.52: Stack with quasi-compact flat covering which is not algebraic
  • Lemma 109.52.1
• Section 109.53: Limit preserving on objects, not limit preserving
  • Lemma 109.53.1
• Section 109.54: A non-algebraic classifying stack
• Section 109.55: Sheaf with quasi-compact flat covering which is not algebraic
  • Lemma 109.55.1
• Section 109.56: Sheaves and specializations
  • Lemma 109.56.1
  • Remark 109.56.2
• Section 109.57: Sheaves and constructible functions
  • Lemma 109.57.1
• Section 109.58: The lisse-étale site is not functorial
  • Lemma 109.58.1
• Section 109.59: Sheaves on the category of Noetherian schemes
  • Lemma 109.59.1
• Section 109.60: Derived pushforward of quasi-coherent modules
  • Lemma 109.60.1
  • Lemma 109.60.2
  • Lemma 109.60.3
• Section 109.61: A big abelian category
  • Lemma 109.61.1
• Section 109.62: Weakly associated points and scheme theoretic density
  • Lemma 109.62.1
• Section 109.63: Example of non-additivity of traces
  • Equation 109.63.0.1
  • Lemma 109.63.1
• Section 109.64: Being projective is not local on the base
  • Lemma 109.64.1
• Section 109.65: Non-effective descent data for projective schemes
  • Lemma 109.65.1
• Section 109.66: A family of curves whose total space is not a scheme
  • Equation 109.66.0.1
  • Lemma 109.66.1
• Section 109.67: Derived base change
  • Lemma 109.67.1
• Section 109.68: An interesting compact object
  • Lemma 109.68.1
• Section 109.69: Two differential graded categories
• Section 109.70: The stack of proper algebraic spaces is not algebraic
  • Lemma 109.70.1
• Section 109.71: An example of a non-algebraic Hom-stack
  • Lemma 109.71.1
  • Lemma 109.71.2
  • Lemma 109.71.3
  • Proposition 109.71.4
  • Remark 109.71.5
• Section 109.72: An algebraic stack not satisfying strong formal effectiveness
  • Lemma 109.72.1
• Section 109.73: A counter example to Grothendieck's existence theorem
  • Lemma 109.73.1
• Section 109.74: Affine formal algebraic spaces
  • Lemma 109.74.1
  • Lemma 109.74.2
  • Lemma 109.74.3
• Section 109.75: Flat maps are not directed limits of finitely presented flat maps
  • Lemma 109.75.1
• Section 109.76: The category of modules modulo torsion modules
  • Proposition 109.76.1
  • Proposition 109.76.2
  • Proposition 109.76.3
  • Proposition 109.76.4
• Section 109.77: Different colimit topologies
  • Lemma 109.77.1
• Section 109.78: Universally submersive but not V covering
  • Lemma 109.78.1
• Section 109.79: The spectrum of the integers is not quasi-compact
  • Lemma 109.79.1
  • Lemma 109.79.2