Cover -- Half-Title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Commutative Algebra in the Cohomology of Groups -- Contents -- 1. Introduction -- 2. Some Examples -- 3. Group Cohomology -- 4. Finite Generation -- 5. Krull Dimension -- 6. Depth -- 7. Associated Primes and Steenrod Operations -- 8. Associated Primes and Transfer -- 9. Idempotent Modules and Varieties -- 10. Modules with Injective Cohomology -- 11. Duality Theorems -- 12. More Duality Theorems -- 13. Dual Localization -- 14. Quasiregular Sequences -- Appendix: Two-Groups of Order 64 and Their mod 2 Cohomology -- Acknowledgements -- References -- Modules and Cohomology over Group Algebras: One Commutative Algebraist's Perspective -- Contents -- Introduction -- 1. The Group Algebra -- 2. Modules over Group Algebras -- 3. Projective Modules -- 4. Structure of Projectives -- 5. Cohomology of Supplemented Algebras -- 6. Group Cohomology -- 7. Finite Generation of the Cohomology Algebra -- References -- An Informal Introduction to Multiplier Ideals -- Contents -- 1. Introduction -- 2. Definition and Examples -- 3. The Multiplier Ideal of Monomial Ideals -- 4. Invariants Arising from Multiplier Ideals and Applications -- 5. Further Local Properties of Multiplier Ideals -- 6. Asymptotic Constructions -- References -- Lectures on the Geometry of Syzygies -- Contents -- 1. Hilbert Functions and Syzygies -- 2. Points in the Plane and an Introduction to Castelnuovo-Mumford Regularity -- 3. The Size of Free Resolutions -- 4. Linear Complexes and the Strands of Resolutions -- References -- Commutative Algebra of n Points in the Plane -- Contents -- Introduction -- Lecture 1: A Subspace Arrangement -- Lecture 2: A Ring of Invariants -- Lecture 3: A Remarkable Grobner Basis -- Appendix: Hilbert Schemes of Points in the Plane -- References.

Tight Closure Theory and Characteristic p Methods -- Contents -- Introduction -- 1. Reasons for Thinking About Tight Closure -- 2. The Definition of Tight Closure in Positive Characteristic -- 3. Basic Properties of Tight Closure and the Briancon-Skoda Theorem -- 4. Direct Summands of Regular Rings are Cohen-Macaulay -- 5. The Ein-Lazarsfeld-Smith Comparison Theorem -- 6. Extending the Theory to Affine Algebras in Characteristic 0 -- 7. Test Elements -- 8. Test Elements Using the Lipman-Sathaye Theorem -- 9. Tight Closure for Submodules -- 10. Further Thoughts and Questions -- Appendix: Some Examples in Tight Closure -- References -- Monomial Ideals, Binomial Ideals, Polynomial Ideals -- Contents -- 1. Introduction -- 2. Strong Principalization of Monomial Ideals by Toric Maps -- 3. The Integral Closure of Ideals -- 4. The Monomial Briancon-Skoda Theorem -- 5. Polynomial Ideals and Nondegeneracy -- 6. Resolution of Binomial Varieties -- 7. Resolution of Singularities of Branches -- Appendix: Multiplicities, Volumes and Nondegeneracy -- References -- Some Facts About Canonical Subalgebra Bases -- Contents -- 1. Introduction -- 2. SABGI Bases Versus Grobner Bases -- 3. When Are SAGBI Bases Finite? -- 4. Finite SAGBI Bases -- 5. An Algorithm to Compute SAGBI bases -- 6. Geometric Interpretation -- Acknowledgments -- References.