Commutative algebra
Graduate · Math
Syllabus focus
Theoretical / proof-based
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Topics typically covered
Theoretical / proof-based
Rings and modules
- Noetherian and Artinian rings and modules
- Primary decomposition in Noetherian rings
- Integral extensions and going-up/going-down theorems
- Dedekind domains and DVRs
- Tensor products and flatness (introduction)
Localization and dimension
- Localization of rings and modules
- Spec(R) and Zariski topology (commutative algebra view)
- Krull dimension and height of primes
- Hilbert's Nullstellensatz and connections to geometry
- Regular sequences and Cohen–Macaulay rings (introduction)
Homological algebra tools
- Projective and injective modules; resolutions
- Ext and Tor functors (definitions and basic computations)
- Depth and Auslander–Buchsbaum formula (introduction)
- Completion and Hensel's lemma
- Smooth and étale morphisms (preview)
Notes
Topics reflect common graduate commutative algebra syllabi at US universities. This course underpins algebraic geometry and is often taken concurrently or sequentially with scheme theory.