The aim of the course is for the student to know how to use the main techniques of commutative algebra with full knowledge. In particular, the student should be able to solve basic exercises on extension and contraction of ideals, to compute the primary decomposition of ideals. Moreover he should havethoroughly understood the nature of Noetherian and Artinian rings, and know how to use localization. He is also expected to have understood the geometric meaning of these concepts, in the basic transition from rings to spectra of rings.
Course Prerequisites
Contents of the courses of Geometria 1, Algebra 1 and Algebra 2
Teaching Methods
Lessons and exercises classes. Two hours per week will be devoted to solving exercises together. The students that will solve the weekly exercises (also working in groups) will be exempt from the written part of the exam.
Assessment Methods
The exam will be both written and oral. The writing evaluates the skills that the student has achieved in calculating and solving problems regarding the topics of the course. The exercises will be divided into questions of varying difficulty aimed at establishing the degree of depth in the acquisition of these skills. In the oral part we will focus above all on verifying the level of knowledge of the notions presented during the course, the clarity with which they are presented and the student's ability to apply them. The formulation of the grade will be obtained by considering the overall breadth and depth of learning, as well as the clarity of the presentation and the skills demonstrated in problem solving. The grade will be obtained from the comparison, not necessarily reduced to an arithmetic mean, of the evaluation of the written part and the oral part.
Texts
Principal reference: - M.F. Atiyah, I.G. MacDonald: "Introduzione all'algebra commutativa", Feltrinelli, (1981) (This is also available in english) Other useful references: - S. Bosch, “Algebraic Geometry and Commutative Algebra”, Springer, (2013) - D. Eisenbud, “Commutative Algebra with a View Toward Algebraic Geometry,” SpringerVerlag, (1999) -A Ferretti, “Commutative algebra”, Open Math Notes of the AMS (2020) - S. Lang, “Algebra”, Addison-Wisley Publishing Company (1965) - H. Matsumura: "Commutative Ring Theory", Cambridge University Press, (1989) - M. Reid, “Undergraduate Commutative Algebra,” Cambridge University Press, (1995)
Contents
The course program covers the fundamental aspects of commutative algebra and the Zariski topology associated with a commutative ring. In the last part of the course, complementary topics will be selected based on the interests of the students attending the course. More in detail, these are the main topics covered in the course: - Recalls on rings and modules - Mentions on sequences of modules - Tensor product of modules - Localization of rings and modules - Primary decomposition of ideals - Artinian and Noetherian rings and modules - Dimension theory - Integral closures and normalization - Noether normalization lemma - Hilbert's Nullstellensatz - Spectrum of a commutative ring - Zariski topology - Affine algebraic sets
Course Language
Italian
More information
The teachers are available for student reception by appointment via mail message. Students belonging to the categories involved with the innovative teaching will have the possibility of having online reception (by appointment) or will have the possibility of seeing the drafts of the lectures.
