Commutative Algebra - academic year 2026/27

• 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 (and, eventually + 3 credits course on some extra topic). For the academic year 2026/27 the standard 6 credits course are 42 our lectures (and no additional 3 credits). There is an official Ariel Portal homepage Commutative Algebra 2025/26. See also the Algant local page for schedules, etc.


• Course (6 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. Specifically, in the course, we afford primary decompositions, integral extensions, regular rings, homological dimension & the key 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 localisations, 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 & homework are available: you can get it using the Ariel Portal homepage Commutative Algebra


• 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: dimension zero and one
• derivations & Zariski tangent space
• primary decomposition of modules, support & associated primes
• filtered/graded modules & Artin-Rees
• Hilbert-Samuel polynomial & the dimension theorem
• homological dimension, Ext & Tor
• A homological characterisation of regular local rings


• Exams The exam consists of written homework on given subjects and a discussion on the matters treated in the lectures, explaining in your own words the dimension theory and/or some specific technical issues in the proof of the main theorems. Eventually, a seminar on some additional related matter.
• Commutative Algebra (6 credits) Some homework will be assigned during the lectures. After the final assignment you can send the solutions to me as a pdf file by email. For the final exam there will be a final oral interview: the homework 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, usually, for acquiring the next 3 credits, you are asked to give a 45 minutes lecture on a subject related to the additional part of the program.


• 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 basic part of the course. For the dimension theorems we follow the Chapters 3 & 4 of the lecture notes by S. Raghavan, Balwant Singh & R. Sridharan Homological Methods in Commutative Algebra pdf version, Oxford Univ. Press/TIFR, 1975
• Commutative Algebra (6 + 3 credits) Eventually, for acquiring the next 3 credits, some additional references can be useful depending on the matter.


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