Commutative Algebra - academic year 2019/20

• General The main task is to give an introduction to modern commutative algebra with a special regard to commutative ring theory, arithmetic, homological methods and algebraic geometry. This is a 6 + 3 credits course titled Commutative Algebra. The first 6 credits are 28 hour lectures & 20 hour exercises. The additional 3 credits are 21 hour lectures. The 20 hours of exercises & 7 hours of lectures are provided by F. Binda See also the Algant local page for schedules, etc.


• Course (6 + 3 credits)
• Description Commutative Algebra studies commutative rings (with identity), their ideals, and modules based on such rings. Both algebraic geometry and algebraic number theory are based on commutative algebra. Algebraic geometry combines commutative algebra with geometry. For example, solutions of systems of polynomial equations, the so called algebraic sets, are combined with related algebraic structures which are ideals in the polynomial ring. For the 6 credits course we afford primary decompositions, integral extensions, regular rings & a first step in dimension theory. The additional 3 credits course is providing the next step in dimension theory.


• Prerequisites We assume known the basic language of categories and functors up to the Yoneda Lemma. We also assume the standard notions: ideals, polynomial rings, multiplicatively closed subsets and localizations, tensor products of modules, Noetherian rings & modules. For example, with respect to M. Reid Undergraduate Commutative Algebra LMS student text series C.U.P. 1995 we will be dealing fast with Chapters 4, 5 & 7, 8 in the commutative algebra course assuming the other Chapters.


• Notes Course notes & homeworks are available: click here or you can get it using the Ariel Portal


• Program outline (6 credits)
• substitution principle, prime spectrum & points
• Hilbert's Nullstellensatz
• primary decomposition & regular rings
• integral ring extensions & valuations
• Noether's normalization
• a first step in dimension theory
• derivations & Zariski tangent space


• Extras outline (3 credits)
• primary decomposition of modules, support & associated primes
• filtered/graded modules & Artin-Rees
• Hilbert-Samuel polynomial & the dimension theorem


• Exams The exam consists of a written exam, homeworks on given subjects and a discussion on the matters treated in the lectures: eventually, a seminar.
• Commutative Algebra (6 credits) Some homeworks will be assigned during the lectures. After the final assignment you can send the solutions to F. Binda as a pdf file by email. For the final written exam a few exercises will be given. There will be a final oral interview: the homeworks and the written exam will be discussed, with all desired documentation available. Please check here the dates for the current academic year. Those who for some reason wish to postpone the examination procedure can make a motivated agreement (by email) with me for a later date if there are good reasons for that. You need to fill the SIFA form online here to get your name listed for the exam (but the page is in Italian so you might need help).
• Commutative Algebra (6 + 3 credits) In addition, for acquiring the next 3 credits, you are asked to give a 45 minutes lecture on the additional part of the program, explaining in your own words the dimension theorem and/or some specific technical issues in its proof.


• References You find below a list of recent and classical books which are a good reference for both rings and schemes.
• Commutative Algebra (6 credits) Course notes are available so that a textbook is not really necessary for the first 6 credits course. Here you find the link: Commutative Algebra pdf version (2014), 93 p.
• Commutative Algebra (6 + 3 credits) For the additional 3 credits we follow the Chapters 1 & 3 of the lecture notes by S. Raghavan, Balwant Singh & R. Sridharan Homological Methods in Commutative Algebra pdf version, Oxford Univ. Press/TIFR, 1975


Extras
Rings
• A. Altman & S. Kleiman A Term of Commutative Algebra Available in Digital Full Color pdf (for free) or Print. (Course notes of the MIT Course Commutative Algebra)
• J.S. Milne A Primer of Commutative Algebra pdf version (2014) available at Milne's homepage
• M. Artin Commutative Rings MIT Course Notes, 1966.
• M.F. Atiyah & I.G. MacDonald Introduction to Commutative Algebra Addison-Wesley 1969 (ed. Feltrinelli, 1981)
• H. Matsumura Commutative Ring Theory Cambridge University Press, 1986
• D. Eisenbud Commutative Algebra with a view toward Algebraic Geometry Graduate Texts in Math., Springer-Verlag, 1994.
• Jean-Pierre Serre Local Algebra Springer Monographs in Math, 2000 (an english translation of Algèbre Locale - Multiplicités Springer LNM 11, 1965)
Schemes
• S. Bosch Algebraic Geometry and Commutative Algebra Universitext, Springer, 2013, 504 p.
• Qing Liu Algebraic Geometry and Arithmetic Curves Oxford University Press, 2002
• R. Hartshorne Algebraic Geometry Springer-Verlag, 1977
• D. Mumford Red Book of Varieties and Schemes, Springer LNM 1358, second ed. 1999