First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
MATHEMATICS
Course unit
COMMUTATIVE ALGEBRA
SCP3050935, A.A. 2015/16

Information concerning the students who enrolled in A.Y. 2015/16

Information on the course unit
Degree course Second cycle degree in
MATHEMATICS
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2015/16
N0
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Degree course track ALGANT [001PD]
Number of ECTS credits allocated 8.0
Type of assessment Mark
Course unit English denomination COMMUTATIVE ALGEBRA
Website of the academic structure http://matematica.scienze.unipd.it/2015/laurea_magistrale
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Branch PADOVA
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Lecturers
Teacher in charge MARCO-ANDREA GARUTI MAT/03

Mutuating
Course unit code Course unit name Teacher in charge Degree course code
SCP3050935 COMMUTATIVE ALGEBRA MARCO-ANDREA GARUTI SC1172

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/03 Geometry 8.0

Course unit organization
Period First semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
hours
Hours of
Individual study
Shifts
Practice 4.0 32 68.0 No turn
Lecture 4.0 32 68.0 No turn

Calendar
Start of activities 01/10/2015
End of activities 28/01/2016
Show course schedule 2019/20 Reg.2011 course timetable

Examination board
Board From To Members of the board
7 Algebra Commutativa - a.a. 2019/2020 01/10/2019 30/09/2020 KLOOSTERMAN REMKE NANNE (Presidente)
GARUTI MARCO-ANDREA (Membro Effettivo)
BALDASSARRI FRANCESCO (Supplente)
CAILOTTO MAURIZIO (Supplente)
CHIARELLOTTO BRUNO (Supplente)
LONGO MATTEO (Supplente)
6 Algebra Commutativa - a.a. 2018/2019 01/10/2018 30/09/2019 KLOOSTERMAN REMKE NANNE (Presidente)
GARUTI MARCO-ANDREA (Membro Effettivo)
BALDASSARRI FRANCESCO (Supplente)
CAILOTTO MAURIZIO (Supplente)
CHIARELLOTTO BRUNO (Supplente)
LONGO MATTEO (Supplente)
5 Algebra Commutativa - 2017/2018 01/10/2017 30/09/2018 KLOOSTERMAN REMKE NANNE (Presidente)
GARUTI MARCO-ANDREA (Membro Effettivo)
BALDASSARRI FRANCESCO (Supplente)
CAILOTTO MAURIZIO (Supplente)
CHIARELLOTTO BRUNO (Supplente)
LONGO MATTEO (Supplente)
4 Algebra Commutativa - 2016/2017 01/10/2016 30/11/2017 GARUTI MARCO-ANDREA (Presidente)
BERTAPELLE ALESSANDRA (Membro Effettivo)
BALDASSARRI FRANCESCO (Supplente)
CAILOTTO MAURIZIO (Supplente)
CHIARELLOTTO BRUNO (Supplente)
KLOOSTERMAN REMKE NANNE (Supplente)
LONGO MATTEO (Supplente)
3 Algebra Commutativa - a.a. 2015/2016 01/10/2015 30/09/2016 GARUTI MARCO-ANDREA (Presidente)
BERTAPELLE ALESSANDRA (Membro Effettivo)
BALDASSARRI FRANCESCO (Supplente)
CAILOTTO MAURIZIO (Supplente)
CHIARELLOTTO BRUNO (Supplente)
LONGO MATTEO (Supplente)
MISTRETTA ERNESTO CARLO (Supplente)

Syllabus
Prerequisites: Basic notions of algebra (groups, rings, ideals, fields, quotients, etc.), as acquired in the "Algebra 1" course.
Target skills and knowledge: A good knowledge of the algeraic objects used in Algebraic Geometry and Number Theory:
- Modules;
- Tensor products;
- Prime spectrum;
- Localization;
- Integral Extensions;
- Noetherian rings;
- Dedekind domains and dicrete valuation rings;
- Basics on dimension theory.
Examination methods: - A compulsory written test for everyone.
- An optional oral test, according to the results of the homeworks and of the written tests.
Assessment criteria: The student will be evaluated on his/her understanding of the topics, on the acquisition of concepts and methodologies proposed and on the ability to apply them in full independence and awareness.
Course unit contents: Commutative rings with unit, ideals, homomorphisms, quotient rings. Fields, integral domains, zero divisors, nilpotent elements. Prime ideals and maximal ideals. Local rings and their characterization. Operations on ideals (sum, intersection, product). Extension and contraction of ideals w.r.t. homomorphisms. Annihilator, radical ideal, nilradical and Jacobson radical of a ring. The Zariski topology on the prime spectrum Spec(R). Spec(R/I) as closed subset of Spec(A). Direct product of rings.

Modules, submodules and their operations (sums, intersection). Annihilator of a module. Faithful modules. Direct sums and direct products of modules. Exact sequences of modules, snake lemma. Projective and injective modules. Finitely generated and finitely presented modules, free modules. Cayley-Hamilton theorem and Nakayama's lemma.

Tensor product and its properties. Extension of scalars for modules. Algebras over a ring and their tensor product. Adjunction and exactness of the Hom and tensor product functors. Flat modules. Kahler differentials

Rings of fractions and localisation. Exactness of localisation. of rings and modules. Localisation and open subsets of Spec(R). Local properties. faithfully flat modules and descent theory. Projective and locally free modules.

Integral elements, integral extension of rings and integral closure. Going Up, Going Down and geometric translation. Norm, trace, discriminant. Valuation rings. Overview of completions.

Chain conditions, Artinian and Noetherian rings and modules. Hilbert's basis theorem. Normalization Lemma and Nullstellensatz.

Discrete valuation rings. Fractional ideals and invertible modules. Cartier and Weil divisors, Picard group, cycle map. Dedekind domains and their extensions. Decomposition of ideals, inertia, ramification.

Krull dimension, height of a prime ideal. Principal ideal theorem. Characterisation of factorial domains. Regular local rings. Finiteness of dimension for local noetherian rings.
Planned learning activities and teaching methods: Lectures, exercises. Recommended exercises.
Additional notes about suggested reading: Lecture notes available at http://mgaruti.weebly.com/ca.html
Other material (exercises, past exam subjects) available at the same homepage.
Textbooks (and optional supplementary readings)
  • Garuti, M.A., Commutative Algebra Lecture notes. Padova: --, 2015. Disponibile gratuitamente alla pagina web del corso.
  • Atiyah, Michael Francis; Mac Donald, Ian Grant, Introduction to commutative algebra. Reading [etc.]: Addison-Wesley, --. Versione italiana edita da Feltrinelli, 1981. Cerca nel catalogo
  • Eisenbud, David, Commutative algebra with a view toward algebraic geometry. New York [etc.]: Springer, --. Graduate Texts in Mathematics, No. 150 Cerca nel catalogo
  • Ramero, Lorenzo, Grimoire d'alg├Ębre commutative. Lille: Les Presses Insoumises, 2015. Link dalla pagina web del corso