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COMMUTATIVE ALGEBRA 2

COURSE DETAILS

• CODE: 42911
• ACADEMIC YEAR: 2025/2026
• CREDITS:
  • 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA
  • 6 cfu anno 1 MATEMATICA 11907 (LM-40 R) - GENOVA
• SCIENTIFIC DISCIPLINARY SECTOR: MAT/02
• LANGUAGE: Italian (English on demand)
• TEACHING LOCATION: GENOVA
• SEMESTER: 2° Semester
• TEACHING MATERIALS: AULAWEB

OVERVIEW

The course focuses on commutative algebra, particularly on the issue of the lack of bases for modules over a ring. It explores how modules can be "approximated" through free modules and how the quality of this approximation reflects on the ring, aligning with geometric concepts of singularities.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide students with the basics of homological algebra, including notions such as free resolution and depth of a module. It introduces regular rings, Cohen-Macaulay rings, and UFDs.

AIMS AND LEARNING OUTCOMES

Detailed aims include:

1. Presenting basic concepts of homological algebra to define projective and injective resolutions, derived functors, and their properties.
2. Generalizing the concept of non-zero divisor to regular sequence, to study the notion of grade.
3. Stating and proving Auslander-Buchsbaum-Serre's Theorem to characterize regular rings, introducing singularities and their good properties.

Expected learning outcomes are:

1. Knowledge of the theory of resolutions of a module and how to compute them in certain cases.
2. Understanding of the theory of regular sequences and depth, in relation to the vanishing of functors like Ext and Tor, or Koszul homology.
3. Ability to characterize regular rings and understand properties of notable singularities like Cohen-Macaulay rings.

PREREQUISITES

• Algebra Commutativa 1
• Algebra 3
• Istituzioni di Geometria Superiore and Introduzione alla Geometria Algebrica are also recommended.

TEACHING METHODS

Lessons will be in-person, with most hours devoted to theoretical development and some to exercises and collective discussion.

SYLLABUS/CONTENT

• Homological algebra: projective and injective modules, resolutions, derived functors.
• Regular sequences, grade, and depth, Koszul complex.
• Regular rings, Cohen-Macaulay rings, and UFDs from a higher perspective.

RECOMMENDED READING/BIBLIOGRAPHY

• Bruns, Herzog, "Cohen-Macaulay rings", Cambridge studies in advances mathematics 39, 1994.
• Eisenbud "Commutative algebra with a view toward algebraic geometry", Springer GTM 150, 1996.
• Matsumura "Commutative ring theory", Cambridge University Press, 1980.

TEACHERS AND EXAM BOARD

• Teacher: MATTEO VARBARO
• Exam Board:
  • President: MATTEO VARBARO
  • Members: EMANUELA DE NEGRI, ALDO CONCA (President Substitute), ALESSANDRO DE STEFANI (President Substitute)

LESSONS

LESSONS START

Lessons start on February 23, 2026. The schedule is available on the Portale EasyAcademy.

EXAMS

EXAM DESCRIPTION

The examination is oral.

ASSESSMENT METHODS

Students are evaluated on theoretical aspects and the ability to analyze and tackle problems related to the course content. Assessment is based on knowledge and the ability to present topics formally, concisely, and correctly.

EXAM SCHEDULE

• Data appello: 20/02/2026, Orario: 09:00, Luogo: GENOVA, Degree type: Esame su appuntamento
• Data appello: 18/09/2026, Orario: 09:00, Luogo: GENOVA, Degree type: Esame su appuntamento

FURTHER INFORMATION

• Attendance in person is highly recommended.
• Students with DSA certification, disability, or other special educational needs should contact the Settore Servizi di supporto alla disabilitŕ e agli studenti con DSA of UNIGE and agree with the teacher on examination methods and compensatory tools.

AGENDA 2030 - SUSTAINABLE DEVELOPMENT GOALS

• Quality education
• Gender equality