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COMMUTATIVE ALGEBRA Contents - Columbia University

COMMUTATIVE ALGEBRA . 00AO. Contents 1. Introduction 4. 2. Conventions 4. 3. Basic notions 5. 4. Snake lemma 7. 5. Finite modules and finitely presented modules 7. 6. Ring maps of finite type and of finite presentation 9. 7. Finite ring maps 10. 8. Colimits 11. 9. Localization 14. 10. Internal Hom 19. 11. Characterizing finite and finitely presented modules 20. 12. Tensor products 21. 13. Tensor ALGEBRA 26. 14. Base change 28. 15. Miscellany 29. 16. Cayley-Hamilton 31. 17. The spectrum of a ring 33. 18. Local rings 37. 19. The Jacobson radical of a ring 38. 20. Nakayama's lemma 39. 21. Open and closed subsets of spectra 40. 22. Connected components of spectra 42. 23. Glueing properties 43. 24. Glueing functions 46. 25. Zerodivisors and total rings of fractions 48. 26. Irreducible components of spectra 49. 27. Examples of spectra of rings 50. 28. A meta-observation about prime ideals 53.

00AP Basic commutative algebra will be explained in this document. A reference is [Mat70]. 2. Conventions 00AQ A ring is commutative with 1. The zero ring is a ring. In fact it is the only ring thatdoesnothaveaprimeideal. TheKroneckersymbolδ ijwillbeused. IfR→S isaringmapandq aprimeofS,thenweusethenotation“p = R∩q”toindicate

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