Derived Category Methods in Commutative Algebra II

In our report on the first “Derived Category Methods in Commutative Algebra” workshop (2008) we wrote: “Derived category methods have proved to be very successful in ring theory, in particular in commutative algebra. Evidence is provided by [1, 8, 4, 5, 6, 7, 11, 12, 15, 16, 19, 20, 23, 24, 27], to list some work of considerable importance. Surprisingly, there is no accessible introduction or reference to the applications of derived category methods in commutative algebra, or in general ring theory for that matter. To be an effective practitioner of these methods, one must be well-versed in a series of research articles and lecture notes, including unpublished ones: [10, 14, 17, 16, 22, 25, 28, 13, 3, 2, 9, 18, 29]. To get an overview of their applications in commutative algebra, the list grows further. The purpose of the BIRS workshop was to make progress on a book manuscript—authored by L.W. Christensen, H.-B. Foxby, and H. Holm—that will remedy this deficiency. As implied in the discussion above, the book has no direct competition. Many books cover applications of classical homological algebra in (commutative) ring theory, but only a few books address derived category methods and their applications in this field: In Homological Algebra [9] by Cartan and Eilenberg, resolutions of complexes and derived functors are briefly discussed in the final chapter; no applications are given. In Weibel’s An introduction to homological algebra [30], derived categories are introduced in the final chapter; a few applications to ring theory are included as exercises. Derived categories are also covered in Methods of Homological Algebra [21] by Gelfand and Manin, but applications to ring theory are not. A very thorough construction of derived categories is given in Categories and Sheaves [26] by Kashiwara and Schapira. However, the aim of [26] is sheaf theory, so beyond the construction of derived categories, there is barely any overlap with this book. Finally, Christensen’s Gorenstein Dimensions [10] has an appendix on derived category methods. It provides a rudimentary and incomplete survey of technical results without proofs. The fact that it has, nevertheless, become a frequently cited reference betrays a significant gap in the existing literature.”