Commutative algebra for geometry

Selected full lessons supplying algebraic prerequisites for flat, smooth and étale morphisms.

  1. Spectra of rings
  2. Localization, local properties and support
  3. Noetherian and Artinian rings
  4. Associated primes and primary decomposition
  5. Integral extensions: lying over, going up and going down
  6. The Nullstellensatz and Jacobson rings
  7. Resolutions, Tor and Ext
  8. Tor and flat modules
  9. Faithful flatness and the local criterion for flatness
  10. Krull dimension and Noether normalization
  11. Graded modules and Hilbert–Samuel functions
  12. Dimension theory of Noetherian local rings
  13. Regular sequences, depth and Cohen–Macaulay modules
  14. Projective dimension and the Auslander–Buchsbaum formula
  15. Regular local rings
  16. Discrete valuation rings, normal rings and Serre's criterion
  17. Kähler differentials
  18. Formally smooth, unramified and étale ring maps
  19. Smooth algebras over a field and the Jacobian criterion
  20. Completion
  21. Coefficient rings and the Cohen structure theorem
  22. Henselian local rings and henselization

Exact prerequisites · Attribution and licences

Further lessons