Homological Methods in Local Algebra
Termine und Zeiten:
Vorlesungen: Montag, 16-18 ct, Raum B 039.Ãœbungen: Dienstag, 12-14 ct, Raum B 039.
Zusammenfassung:
One of the first famous applications of methods of homological algebra was the proof by Auslander–Buchsbaum and Serre of the conjectures on the factoriality of regular local rings and on the regularity of their localizations, which have now become classical results [1–4]. Since then, methods of homological algebra have become an integral part of commutative algebra and algebraic geometry.In the course we will start with the very basics of homological algebra, then discuss global dimension, Koszul complex, and homological characterization of regular rings following [4–7]. Then we will prove mentioned results about localizations and factoriality. If time permits, we will also discuss Cohen–Macaulay modules and depth, or K-theory of regular rings.
Voraussetzung:
This course will be of interest to Master students majoring in Algebra. I will assume that the students are familiar with Algebra I & II courses. Knowledge of Algebraic Geometry is not strictly required, but will be helpful. The course will be taught in English.Literatur:
[1] M. Auslander, D. Buchsbaum, "Homological dimension in local rings", Trans. Amer. Math. Soc. 85 (1957) 390–405[2] M. Auslander, D. Buchsbaum, "Unique factorization in local rings", Proc. Nat. Acad. Sci. USA 45 (1959) 733–734
[3] J.-P. Serre, "Sur la dimension homologique des anneaux et des modules noetheriens", Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko (1955) 175–189
[4] J.-P. Serre, "Local algebra", Springer Monographs in Mathematics, Springer-Verlag (2000)
[5] H. Cartan, S. Eilenberg, "Homological algebra", Princeton Univ. Press (1956)
[6] D. Eisenbud, "Commutative Algebra with a View Toward Algebraic Geometry", Graduate Texts in Math. 150, Springer-Verlag (1995)
[7] M. Atiyah, I. Macdonald, "Introduction to Commutative algebra", Addison-Wesley (1969)
Dr. Andrei Lavrenov