Commutative algebra
Graduate · Math
Syllabus focus
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)
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
Notes
Graduate Commutative Algebra — scope drawn from open commutative algebra notes and typical US graduate commutative algebra syllabi. Topic outline: `content/topics/graduate/commutative_algebra.json`.