Commutative Algebra
A.Y. 2026/2027
Learning objectives
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.
Expected learning outcomes
Theory and computations of primary decompositions, integral extensions, regular rings & a first step in dimension theory.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
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. Homological characterisation of regular local rings.
Prerequisites for admission
We assume known the basic language of categories and functors up to the Yoneda Lemma and other basic algebra like ideals, polynomial rings, modules, UFD, PID and their completions, as explained in "What is a number? An introduction to algebra" by L. Barbieri Viale Raffaello Cortina Editore (New Edition, 2026). We also assume the standard notions of multiplicatively closed subsets and localisations, tensor products of modules and Noetherian rings. 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.
Teaching methods
Lectures.
Teaching Resources
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
Extra Ref:
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)
Extra Ref:
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)
Assessment methods and Criteria
Some written homework will be assigned during class. These homework, which count as a written exam, must be completed and returned before the oral exam. For the oral exam, a seminar will be assigned on a preferred subject, based on the topics discussed in class, explaining in your own words the dimension theory and/or some specific technical issues in the proof of the main theorems. During the seminar presentation, questions will be asked on course topics relevant to the seminar. The final grade will take into account the written exam (insufficient, sufficient, good, or excellent) and the oral exam, i.e., the quality of the presentation and answers to the questions.
MATH-02/A - Algebra - University credits: 6
Lessons: 42 hours
Professor:
Barbieri Viale Luca
Shifts:
Turno
Professor:
Barbieri Viale LucaProfessor(s)
Reception:
Email contact (usually for Tuesday h. 2-4 p.m.)
Office 2092 - Math Department