Lecture Notes on Local Rings

1. Dimension of a local ring. 1.1. Nakayama's lemma. 1.2. Prime ideals. 1.3. Noetherian modules. 1.4. Modules of finite length. 1.5. Hilbert's basis theorem. 1.6. Graded rings. 1.7. Filtered rings. 1.8. Local rings. 1.9. Regular local rings -- 2. Modules over a local ring. 2.1. Support of a module. 2.2. Associated prime ideals. 2.3. Dimension of a module. 2.4. Depth of a module. 2.5. Cohen-Macaulay modules. 2.6. Modules of finite projective dimension. 2.7. The Koszul complex. 2.8. Regular local rings. 2.9. Projective dimension and depth -- 2.10. [symbol]-depth. 2.11. The acyclicity theorem. 2.12. An example -- 3. Divisor theory. 3.1. Discrete valuation rings. 3.2. Normal domains. 3.3. Divisors. 3.4. Unique factorization. 3.5. Torsion modules. 3.6. The first Chern class. 3.7. Regular local rings. 3.8. Picard groups. 3.9. Dedekind domains -- 4. Completion. 4.1. Exactness of the completion functor. 4.2. Separation of the [symbol]-adic topology. 4.3. Complete filtered rings. 4.4. Completion of local rings. 4.5. Structure of complete local rings -- 5. Injective modules. 5.1. Injective modules. 5.2. Injective envelopes. 5.3. Decomposition of injective modules. 5.4. Matlis duality. 5.5. Minimal injective resolutions. 5.6. Modules of finite injective dimension. 5.7. Gorenstein rings -- 6. Local cohomology. 6.1. Basic properties. 6.2. Local cohomology and dimension. 6.3. Local cohomology and depth. 6.4. Support in the maximal ideal. 6.5. Local duality for Gorenstein rings -- 7. Dualizing complexes. 7.1. Complexes of injective modules. 7.2. Complexes with finitely generated cohomology. 7.3. The evaluation map. 7.4. Existence of dualizing complexes. 7.5. The codimension function. 7.6. Complexes of flat modules. 7.7. Generalized evaluation maps. 7.8. Uniqueness of dualizing complexes -- 8. Local duality. 8.1. Poincare series. 8.2. Grothendieck's local duality theorem. 8.3. Duality for Cohen-Macaulay modules. 8.4. Dualizing modules. 8.5. Locally factorial domains. 8.6. Conductors. 8.7. Formal fibers -- 9. Amplitude and dimension. 9.1. Depth of a complex. 9.2. The dual of a module. 9.3. The amplitude formula. 9.4. Dimension of a complex. 9.5. The tensor product formula. 9.6. Depth inequalities. 9.7. Condition Sr of Serre. 9.8. Factorial rings and condition Sr. 9.9. Condition S[symbol]. 9.10. Specialization of Poincare series -- 10. Intersection multiplicities. 10.1. Introduction to Serre's conjectures. 10.2. Filtration of the Koszul complex. 10.3. Euler characteristic of the Koszul complex. 10.4. A projection formula. 10.5. Power series over a field. 10.6. Power series over a discrete valuation ring. 10.7. Application of Cohen's structure theorem. 10.8. The amplitude inequality. 10.9. Translation invariant operators. 10.10. Todd operators. 10.11. Serre's conjecture in the graded case -- 11. Complexes of free modules. 11.1. McCoy's theorem. 11.2. The rank of a linear map. 11.3. The Eisenbud-Buchsbaum criterion. 11.4. Fitting's ideals. 11.5. The Euler characteristic. 11.6. McRae's invariant. 11.7. The integral character of McRae's invariant