Infinite free resolutions. (English) Zbl 0934.13008

Elias, J. (ed.) et al., Six lectures on commutative algebra. Lectures presented at the summer school, Bellaterra, Spain, July 16-26, 1996. Basel: Birkhäuser. Prog. Math. 166, 1-118 (1998).
This paper is based on the notes for a series of six lectures to the Barcelona Summer School in Commutative Algebra in 1996. It describes techniques for finding and using free resolutions of finite modules over commutative noetherian rings. The prerequisites are a basic preparation in commutative ring theory, including a few homological routines. Modulo that, complete proofs are given for all but a few results in the text. There are also a number of helpful and amusing footnotes. Here is an outline of the contents:
1. Complexes: 1.1 Basic constructions; 1.2 Syzygies; 1.3 Differential graded algebra.
2. Multiplicative structures on resolutions: 2.1 Differential graded algebra resolutions; 2.2 Differential graded module resolutions; 2.3 Products versus minimality.
3. Change of rings: 3.1 Universal resolutions; 3.2 Spectral sequences.
4. Growth of resolutions: 4.1 Regular presentations; 4.2 Complexity and curvature; 4.3 Growth problems.
5. Modules over Golod rings: 5.1 Hypersurfaces; 5.2 Golod rings; 5.3 Golod modules.
6. Tate resolutions: 6.1 Construction; 6.2 Derivations; 6.3 Acyclic closures.
7. Deviations of a local ring: 7.1 Deviations and Betti numbers; 7.2 Minimal modules; 7.3 Complete intersections; 7.4 Localization.
8. Test modules: 8.1 Residue field; 8.2 Residue domains; 8.3 Conormal modules.
9. Modules over complete intersections: 9.1 Cohomology operators; 9.2 Betti numbers; 9.3 Complexity and Tor.
10. Homotopy Lie algebras of a local ring: 10.1 Products in cohomology; 10.2 Homotopy Lie algebras; 10.3 Applications.
For the entire collection see [Zbl 0892.00031].


13D02 Syzygies, resolutions, complexes and commutative rings
13D25 Complexes (MSC2000)
13E05 Commutative Noetherian rings and modules