111 Exercises

• Section 111.1: Algebra
  • Exercise 111.1.1
  • Exercise 111.1.2
  • Exercise 111.1.3
  • Exercise 111.1.4
  • Remark 111.1.5
  • Exercise 111.1.6
  • Exercise 111.1.7
  • Exercise 111.1.8
  • Exercise 111.1.9
  • Exercise 111.1.10
• Section 111.2: Colimits
  • Definition 111.2.1
  • Exercise 111.2.2
  • Definition 111.2.3
  • Exercise 111.2.4
  • Exercise 111.2.5
  • Exercise 111.2.6
  • Exercise 111.2.7
  • Definition 111.2.8
  • Exercise 111.2.9
• Section 111.3: Additive and abelian categories
  • Exercise 111.3.1
  • Exercise 111.3.2
  • Exercise 111.3.3
• Section 111.4: Tensor product
  • Exercise 111.4.1
• Section 111.5: Flat ring maps
  • Exercise 111.5.1
  • Exercise 111.5.2
  • Exercise 111.5.3
  • Remark 111.5.4
  • Exercise 111.5.5
  • Exercise 111.5.6
  • Exercise 111.5.7
  • Remark 111.5.8
• Section 111.6: The Spectrum of a ring
  • Exercise 111.6.1
  • Exercise 111.6.2
  • Exercise 111.6.3
  • Definition 111.6.4
  • Exercise 111.6.5
  • Definition 111.6.6
  • Exercise 111.6.7
  • Remark 111.6.8
  • Definition 111.6.9
  • Exercise 111.6.10
  • Exercise 111.6.11
  • Definition 111.6.12
  • Exercise 111.6.13
  • Exercise 111.6.14
  • Exercise 111.6.15
  • Definition 111.6.16
  • Exercise 111.6.17
  • Definition 111.6.18
  • Exercise 111.6.19
  • Exercise 111.6.20
  • Exercise 111.6.21
  • Definition 111.6.22
  • Exercise 111.6.23
  • Exercise 111.6.24
  • Exercise 111.6.25
  • Definition 111.6.26
  • Exercise 111.6.27
  • Exercise 111.6.28
  • Exercise 111.6.29
  • Exercise 111.6.30
  • Exercise 111.6.31
• Section 111.7: Localization
  • Exercise 111.7.1
  • Exercise 111.7.2
  • Exercise 111.7.3
  • Exercise 111.7.4
  • Exercise 111.7.5
• Section 111.8: Nakayama's Lemma
  • Exercise 111.8.1
• Section 111.9: Length
  • Definition 111.9.1
  • Exercise 111.9.2
  • Exercise 111.9.3
  • Exercise 111.9.4
• Section 111.10: Associated primes
  • Exercise 111.10.1
  • Exercise 111.10.2
  • Exercise 111.10.3
• Section 111.11: Ext groups
  • Exercise 111.11.1
  • Exercise 111.11.2
  • Exercise 111.11.3
  • Exercise 111.11.4
• Section 111.12: Depth
  • Exercise 111.12.1
  • Exercise 111.12.2
  • Exercise 111.12.3
  • Exercise 111.12.4
  • Exercise 111.12.5
  • Exercise 111.12.6
  • Exercise 111.12.7
  • Exercise 111.12.8
• Section 111.13: Cohen-Macaulay modules and rings
  • Exercise 111.13.1
• Section 111.14: Singularities
  • Exercise 111.14.1
  • Remark 111.14.2
  • Exercise 111.14.3
• Section 111.15: Constructible sets
  • Exercise 111.15.1
  • Exercise 111.15.2
  • Exercise 111.15.3
• Section 111.16: Hilbert Nullstellensatz
  • Exercise 111.16.1
  • Remark 111.16.2
  • Exercise 111.16.3
  • Remark 111.16.4
  • Exercise 111.16.5
• Section 111.17: Dimension
  • Exercise 111.17.1
  • Exercise 111.17.2
  • Exercise 111.17.3
• Section 111.18: Catenary rings
  • Definition 111.18.1
  • Exercise 111.18.2
  • Exercise 111.18.3
  • Exercise 111.18.4
  • Exercise 111.18.5
• Section 111.19: Fraction fields
  • Exercise 111.19.1
• Section 111.20: Transcendence degree
  • Exercise 111.20.1
  • Exercise 111.20.2
• Section 111.21: Dimension of fibres
  • Exercise 111.21.1
  • Exercise 111.21.2
• Section 111.22: Finite locally free modules
  • Definition 111.22.1
  • Exercise 111.22.2
  • Definition 111.22.3
  • Exercise 111.22.4
  • Exercise 111.22.5
  • Exercise 111.22.6
  • Exercise 111.22.7
• Section 111.23: Glueing
  • Exercise 111.23.1
  • Remark 111.23.2
  • Exercise 111.23.3
• Section 111.24: Going up and going down
  • Definition 111.24.1
  • Exercise 111.24.2
  • Exercise 111.24.3
  • Exercise 111.24.4
  • Remark 111.24.5
• Section 111.25: Fitting ideals
  • Exercise 111.25.1
• Section 111.26: Hilbert functions
  • Definition 111.26.1
  • Definition 111.26.2
  • Definition 111.26.3
  • Exercise 111.26.4
  • Exercise 111.26.5
  • Exercise 111.26.6
  • Exercise 111.26.7
  • Exercise 111.26.8
  • Exercise 111.26.9
  • Exercise 111.26.10
• Section 111.27: Proj of a ring
  • Definition 111.27.1
  • Definition 111.27.2
  • Definition 111.27.3
  • Exercise 111.27.4
  • Remark 111.27.5
  • Exercise 111.27.6
  • Exercise 111.27.7
  • Exercise 111.27.8
  • Exercise 111.27.9
  • Exercise 111.27.10
• Section 111.28: Cohen-Macaulay rings of dimension 1
  • Definition 111.28.1
  • Exercise 111.28.2
  • Exercise 111.28.3
  • Exercise 111.28.4
  • Exercise 111.28.5
  • Exercise 111.28.6
  • Exercise 111.28.7
  • Exercise 111.28.8
  • Remark 111.28.9
• Section 111.29: Infinitely many primes
  • Exercise 111.29.1
  • Exercise 111.29.2
  • Exercise 111.29.3
  • Remark 111.29.4
  • Exercise 111.29.5
  • Exercise 111.29.6
• Section 111.30: Filtered derived category
  • Exercise 111.30.1
  • Exercise 111.30.2
  • Definition 111.30.3
  • Definition 111.30.4
  • Exercise 111.30.5
  • Definition 111.30.6
  • Definition 111.30.7
  • Exercise 111.30.8
  • Exercise 111.30.9
  • Exercise 111.30.10
  • Exercise 111.30.11
  • Exercise 111.30.12
  • Exercise 111.30.13
• Section 111.31: Regular functions
  • Exercise 111.31.1
  • Exercise 111.31.2
  • Exercise 111.31.3
  • Exercise 111.31.4
  • Exercise 111.31.5
• Section 111.32: Sheaves
  • Exercise 111.32.1
  • Exercise 111.32.2
  • Exercise 111.32.3
  • Exercise 111.32.4
  • Exercise 111.32.5
  • Exercise 111.32.6
  • Exercise 111.32.7
  • Remark 111.32.8
  • Exercise 111.32.9
  • Exercise 111.32.10
  • Exercise 111.32.11
• Section 111.33: Schemes
  • Exercise 111.33.1
  • Exercise 111.33.2
  • Exercise 111.33.3
  • Exercise 111.33.4
  • Exercise 111.33.5
  • Remark 111.33.6
  • Exercise 111.33.7
  • Exercise 111.33.8
  • Exercise 111.33.9
  • Exercise 111.33.10
  • Exercise 111.33.11
  • Definition 111.33.12
  • Exercise 111.33.13
  • Exercise 111.33.14
  • Remark 111.33.15
  • Exercise 111.33.16
  • Exercise 111.33.17
  • Remark 111.33.18
• Section 111.34: Morphisms
  • Exercise 111.34.1
  • Exercise 111.34.2
  • Exercise 111.34.3
  • Remark 111.34.4
  • Exercise 111.34.5
  • Exercise 111.34.6
  • Exercise 111.34.7
  • Exercise 111.34.8
  • Remark 111.34.9
  • Exercise 111.34.10
  • Exercise 111.34.11
  • Exercise 111.34.12
• Section 111.35: Tangent Spaces
  • Definition 111.35.1
  • Exercise 111.35.2
  • Definition 111.35.3
  • Exercise 111.35.4
  • Exercise 111.35.5
  • Exercise 111.35.6
  • Remark 111.35.7
  • Exercise 111.35.8
  • Exercise 111.35.9
  • Exercise 111.35.10
  • Exercise 111.35.11
• Section 111.36: Quasi-coherent Sheaves
  • Definition 111.36.1
  • Definition 111.36.2
  • Exercise 111.36.3
  • Exercise 111.36.4
  • Definition 111.36.5
  • Definition 111.36.6
  • Exercise 111.36.7
  • Remark 111.36.8
• Section 111.37: Proj and projective schemes
  • Exercise 111.37.1
  • Exercise 111.37.2
  • Exercise 111.37.3
  • Exercise 111.37.4
  • Exercise 111.37.5
  • Exercise 111.37.6
  • Exercise 111.37.7
  • Exercise 111.37.8
  • Exercise 111.37.9
• Section 111.38: Morphisms from the projective line
  • Exercise 111.38.1
  • Exercise 111.38.2
  • Exercise 111.38.3
  • Exercise 111.38.4
• Section 111.39: Morphisms from surfaces to curves
  • Exercise 111.39.1
  • Exercise 111.39.2
  • Exercise 111.39.3
  • Exercise 111.39.4
  • Exercise 111.39.5
• Section 111.40: Invertible sheaves
  • Definition 111.40.1
  • Exercise 111.40.2
  • Exercise 111.40.3
  • Definition 111.40.4
  • Exercise 111.40.5
  • Exercise 111.40.6
  • Definition 111.40.7
  • Exercise 111.40.8
• Section 111.41: Čech Cohomology
  • Exercise 111.41.1
  • Definition 111.41.2
• Section 111.42: Cohomology
  • Exercise 111.42.1
  • Exercise 111.42.2
  • Exercise 111.42.3
  • Exercise 111.42.4
  • Exercise 111.42.5
  • Exercise 111.42.6
  • Exercise 111.42.7
  • Exercise 111.42.8
  • Exercise 111.42.9
• Section 111.43: More cohomology
  • Exercise 111.43.1
  • Exercise 111.43.2
  • Exercise 111.43.3
  • Exercise 111.43.4
  • Exercise 111.43.5
  • Exercise 111.43.6
  • Exercise 111.43.7
  • Exercise 111.43.8
  • Exercise 111.43.9
  • Exercise 111.43.10
• Section 111.44: Cohomology revisited
  • Exercise 111.44.1
  • Exercise 111.44.2
  • Exercise 111.44.3
  • Exercise 111.44.4
  • Exercise 111.44.5
  • Exercise 111.44.6
  • Exercise 111.44.7
• Section 111.45: Cohomology and Hilbert polynomials
  • Situation 111.45.1
  • Exercise 111.45.2
  • Exercise 111.45.3
  • Exercise 111.45.4
  • Exercise 111.45.5
  • Exercise 111.45.6
• Section 111.46: Curves
  • Exercise 111.46.1
  • Exercise 111.46.2
  • Exercise 111.46.3
  • Exercise 111.46.4
  • Exercise 111.46.5
  • Exercise 111.46.6
  • Exercise 111.46.7
• Section 111.47: Moduli
  • Exercise 111.47.1
  • Exercise 111.47.2
• Section 111.48: Global Exts
  • Exercise 111.48.1
  • Exercise 111.48.2
  • Exercise 111.48.3
  • Exercise 111.48.4
  • Exercise 111.48.5
  • Exercise 111.48.6
• Section 111.49: Divisors
  • Definition 111.49.1
  • Remarks 111.49.2
  • Exercise 111.49.3
  • Exercise 111.49.4
  • Exercise 111.49.5
  • Exercise 111.49.6
  • Exercise 111.49.7
  • Exercise 111.49.8
  • Exercise 111.49.9
• Section 111.50: Differentials
  • Exercise 111.50.1
  • Remark 111.50.2
  • Exercise 111.50.3
  • Exercise 111.50.4
  • Exercise 111.50.5
• Section 111.51: Schemes, Final Exam, Fall 2007
  • Exercise 111.51.1: Definitions
  • Exercise 111.51.2
  • Exercise 111.51.3
  • Exercise 111.51.4
  • Exercise 111.51.5
• Section 111.52: Schemes, Final Exam, Spring 2009
  • Exercise 111.52.1
  • Remark 111.52.2
  • Exercise 111.52.3
  • Remark 111.52.4
  • Exercise 111.52.5
  • Remark 111.52.6
  • Exercise 111.52.7
  • Exercise 111.52.8
• Section 111.53: Schemes, Final Exam, Fall 2010
  • Exercise 111.53.1: Definitions
  • Exercise 111.53.2
  • Exercise 111.53.3
  • Exercise 111.53.4: Examples
  • Exercise 111.53.5
  • Exercise 111.53.6
  • Exercise 111.53.7
  • Exercise 111.53.8
• Section 111.54: Schemes, Final Exam, Spring 2011
  • Exercise 111.54.1: Definitions
  • Exercise 111.54.2: Results
  • Exercise 111.54.3
  • Exercise 111.54.4
  • Exercise 111.54.5
• Section 111.55: Schemes, Final Exam, Fall 2011
  • Exercise 111.55.1: Definitions
  • Exercise 111.55.2: Results
  • Exercise 111.55.3
  • Exercise 111.55.4
  • Exercise 111.55.5
  • Exercise 111.55.6
  • Exercise 111.55.7
  • Exercise 111.55.8
  • Exercise 111.55.9
  • Exercise 111.55.10
• Section 111.56: Schemes, Final Exam, Fall 2013
  • Exercise 111.56.1: Definitions
  • Exercise 111.56.2: Results
  • Exercise 111.56.3
  • Exercise 111.56.4
  • Situation 111.56.5: Notation plane curve
  • Exercise 111.56.6
  • Exercise 111.56.7
  • Exercise 111.56.8
  • Exercise 111.56.9
  • Exercise 111.56.10
• Section 111.57: Schemes, Final Exam, Spring 2014
  • Exercise 111.57.1: Definitions
  • Exercise 111.57.2: Theorems
  • Exercise 111.57.3
  • Exercise 111.57.4
  • Exercise 111.57.5
  • Exercise 111.57.6
• Section 111.58: Commutative Algebra, Final Exam, Fall 2016
  • Exercise 111.58.1: Definitions
  • Exercise 111.58.2: Theorems
  • Exercise 111.58.3
  • Exercise 111.58.4
  • Exercise 111.58.5
  • Exercise 111.58.6
  • Exercise 111.58.7
  • Exercise 111.58.8
• Section 111.59: Schemes, Final Exam, Spring 2017
  • Exercise 111.59.1: Definitions
  • Exercise 111.59.2: Theorems
  • Exercise 111.59.3
  • Exercise 111.59.4
  • Exercise 111.59.5
  • Exercise 111.59.6
  • Exercise 111.59.7
• Section 111.60: Commutative Algebra, Final Exam, Fall 2017
  • Exercise 111.60.1: Definitions
  • Exercise 111.60.2: Theorems
  • Exercise 111.60.3
  • Exercise 111.60.4
  • Exercise 111.60.5
  • Exercise 111.60.6
  • Exercise 111.60.7
• Section 111.61: Schemes, Final Exam, Spring 2018
  • Exercise 111.61.1: Definitions
  • Exercise 111.61.2: Theorems
  • Exercise 111.61.3
  • Exercise 111.61.4
  • Exercise 111.61.5
  • Exercise 111.61.6
  • Exercise 111.61.7
• Section 111.62: Commutative Algebra, Final Exam, Fall 2019
  • Exercise 111.62.1: Definitions
  • Exercise 111.62.2: Theorems
  • Exercise 111.62.3: Making primes
  • Exercise 111.62.4: No prime
  • Exercise 111.62.5: Frobenius
  • Exercise 111.62.6: Dimension
  • Exercise 111.62.7: Tor computation
  • Exercise 111.62.8
• Section 111.63: Algebraic Geometry, Final Exam, Spring 2020
  • Exercise 111.63.1: Definitions
  • Exercise 111.63.2: Theorems
  • Exercise 111.63.3
  • Exercise 111.63.4
  • Exercise 111.63.5
  • Exercise 111.63.6
  • Exercise 111.63.7
• Section 111.64: Commutative Algebra, Final Exam, Fall 2021
  • Exercise 111.64.1: Definitions
  • Exercise 111.64.2: Theorems
  • Exercise 111.64.3: Units
  • Exercise 111.64.4: Ideals
  • Exercise 111.64.5: Tor and Ext
  • Exercise 111.64.6: Two vectors
  • Exercise 111.64.7: Injectives
• Section 111.65: Algebraic Geometry, Final Exam, Spring 2022
  • Exercise 111.65.1: Definitions
  • Exercise 111.65.2: Theorems
  • Exercise 111.65.3: Cubic hypersurfaces
  • Exercise 111.65.4: Heights
  • Exercise 111.65.5: Monomorphisms
  • Exercise 111.65.6: Fixed points