Large part of this teaching in commutative algebra will be centered on the issue of the lack of bases for modules over a ring: most modules do not admit a basis, but one can "approximate" them through free modules (i.e., those modules who do admit a basis). The better is the approximation, the better is the ring, in a sense that agrees with the geometric concept of singularities.
Provide students with the basics of homological algebra and notions such as free resolution and depth of a module; introduce/explore regular rings, Cohen-Macaulay rings and UFDs.
The detailed aims of the teaching are:
1) To present basic concepts of homological algebra in order to define projective and injective resolutions, derived functors and their properties.
2) To generalize the concept of non-zero divisor to that of regular sequence, in order to study the notion of grade.
3) To state and prove Auslander-Buchsbaum-Serre's Theorem, which allows to characterize regular rings. To introduce some singularities and to study their good properties.
The expected learning outcomes are:
1) At the end of Algebra Commutativa 2 a student will know the theory of resolutions of a module, and how to compite them in certain cases of ideals inside polynomial rings or inside power series rings over a field. The student will also know how to compute derived functors such as Ext and Tor and will know their main properties.
2) At the end of Algebra Commutativa 2 a student will know the theory of regular sequences and depth, also in relation to the vanishing of functors such as Ext and Tor, or Koszul homology.
3) At the end of Algebra Commutativa 2 a student will know how to characterize regular rings, and the main properties of certain notable singularities such as Cohen-Macaulay rings.
Algebra Commutativa 1 and Algebra 3. Istituzioni di Geometria Superiore and Introduzione alla Geometria Algebrica could also be very useful for this course.
Lessons will be in presence. Most of the available hours will be devoted to the development of the theoretical part of the course; exercises will be proposed during the semester and will be discussed collectively in the remaining hours.
Homological algebra: projective and injective modules, resolutions, derived functors. Regular sequences, grade and depth, Koszul complex. Regular rings, Cohen-Macaulay rings and UFD from a higher point of view.
Bruns, Herzog, "Cohen-Macaulay rings", Cambridge studies in advances mathematica 39, 1994.
Eisenbud "Commutative algebra with a view toward algebraic geometry", Springer GTM 150, 1996
Matsumura "Commutative ring theory", Cambridge University Press, 1980
Ricevimento: The office hours will be
Lessons will start on February 23, 2026, see here for the schedule: https://corsi.unige.it/corsi/9011/studenti-orario
The examination is oral.
The student will be evaluated on the theoretical aspects developed during the lectures and on the capacity to analyze and tackle problems related to the contents of the teaching.
The assessment will be based both on the knowledge of the topics and on the ability to present them in a formal, coincise and correct way.
Teacher's webpage: https://rubrica.unige.it/personale/UkNGX1Jq
Attendance in person is highly recommended.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the Settore Servizi di supporto alla disabilità e agli studenti con DSA of UNIGE, and to agree with the teacher at the beginning of the course the methods of examination which, in compliance with the teaching objectives, will take into account individual learning arrangements and will provide appropriate compensatory tools.
