Theoretical / proof-based
Commutative algebra · Graduate · Math
Learning objectives from the Commutative algebra syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Commutative Algebra — scope drawn from open commutative algebra notes and typical US graduate commutative algebra syllabi.
Rings and modules
- Noetherian and Artinian rings and modules — Noetherian and Artinian rings and modules
- Primary decomposition in Noetherian rings — Primary decomposition in Noetherian rings
- Integral extensions — Integral extensions
- Going-up/going-down theorems — Going-up/going-down theorems
- Dedekind domains and DVRs — Dedekind domains and DVRs
- Tensor products — Tensor products
- Flatness (introduction) — Flatness (introduction)
Localization and dimension
- Localization of rings and modules — Localization of rings and modules
- Spec(R) and Zariski topology (commutative algebra view) — Spec(R) and Zariski topology (commutative algebra view)
- Krull dimension and height of primes — Krull dimension and height of primes
- Hilbert's Nullstellensatz — Hilbert's Nullstellensatz
- Connections to geometry — Connections to geometry
- Regular sequences — Regular sequences
- Cohen–Macaulay rings (introduction) — Cohen–Macaulay rings (introduction)
Homological algebra tools
- Projective — Projective
- Injective modules — Injective modules
- Ext and Tor functors (definitions and basic computations) — Ext and Tor functors (definitions and basic computations)
- Depth and Auslander–Buchsbaum formula (introduction) — Depth and Auslander–Buchsbaum formula (introduction)
- Completion and Hensel's lemma — Completion and Hensel's lemma
- Smooth and étale morphisms (preview) — Smooth and étale morphisms (preview)
Learning objectives
Click an objective for study materials.
Rings and modules
- Noetherian and Artinian rings and modules — Noetherian and Artinian rings and modules
- Primary decomposition in Noetherian rings — Primary decomposition in Noetherian rings
- Integral extensions — Integral extensions
- Going-up/going-down theorems — Going-up/going-down theorems
- Dedekind domains and DVRs — Dedekind domains and DVRs
- Tensor products — Tensor products
- Flatness (introduction) — Flatness (introduction)
Localization and dimension
- Localization of rings and modules — Localization of rings and modules
- Spec(R) and Zariski topology (commutative algebra view) — Spec(R) and Zariski topology (commutative algebra view)
- Krull dimension and height of primes — Krull dimension and height of primes
- Hilbert's Nullstellensatz — Hilbert's Nullstellensatz
- Connections to geometry — Connections to geometry
- Regular sequences — Regular sequences
- Cohen–Macaulay rings (introduction) — Cohen–Macaulay rings (introduction)
Homological algebra tools
- Projective — Projective
- Injective modules — Injective modules
- Ext and Tor functors (definitions and basic computations) — Ext and Tor functors (definitions and basic computations)
- Depth and Auslander–Buchsbaum formula (introduction) — Depth and Auslander–Buchsbaum formula (introduction)
- Completion and Hensel's lemma — Completion and Hensel's lemma
- Smooth and étale morphisms (preview) — Smooth and étale morphisms (preview)
Definitions and structure
- Noetherian and Artinian rings and modules — Noetherian and Artinian rings and modules
- Primary decomposition in Noetherian rings — Primary decomposition in Noetherian rings
- Integral extensions — Integral extensions
- Going-up/going-down theorems — Going-up/going-down theorems
- Dedekind domains and DVRs — Dedekind domains and DVRs
Proofs and reasoning
- Tensor products — Tensor products
- Flatness (introduction) — Flatness (introduction)
- Localization of rings and modules — Localization of rings and modules
- Spec(R) and Zariski topology (commutative algebra view) — Spec(R) and Zariski topology (commutative algebra view)
- Krull dimension and height of primes — Krull dimension and height of primes
Abstraction and generalization
- Hilbert's Nullstellensatz — Hilbert's Nullstellensatz
- Connections to geometry — Connections to geometry
- Regular sequences — Regular sequences
- Cohen–Macaulay rings (introduction) — Cohen–Macaulay rings (introduction)
- Projective — Projective
Modeling and computation
- Apply noetherian and artinian rings and modules in engineering contexts — Apply noetherian and artinian rings and modules in engineering contexts
- Apply primary decomposition in noetherian rings in engineering contexts — Apply primary decomposition in noetherian rings in engineering contexts
- Apply integral extensions in engineering contexts — Apply integral extensions in engineering contexts
- Apply going-up/going-down theorems in engineering contexts — Apply going-up/going-down theorems in engineering contexts
- Apply dedekind domains and dvrs in engineering contexts — Apply dedekind domains and dvrs in engineering contexts
Data and technology
- Use software to explore commutative algebra problems numerically — Use software to explore commutative algebra problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Tensor products — Tensor products
- Flatness (introduction) — Flatness (introduction)
- Localization of rings and modules — Localization of rings and modules
- Spec(R) and Zariski topology (commutative algebra view) — Spec(R) and Zariski topology (commutative algebra view)
- Krull dimension and height of primes — Krull dimension and height of primes
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Rings and modules
Noetherian and Artinian rings and modules
Coming soon - Localization and dimension
Localization of rings and modules
Coming soon - Homological algebra tools
Projective
Coming soon
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