
Course unit
COMMUTATIVE ALGEBRA
SCP3050935, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2017/18
Mutuated
Course unit code 
Course unit name 
Teacher in charge 
Degree course code 
SCP3050935 
COMMUTATIVE ALGEBRA 
REMKE NANNE KLOOSTERMAN 
SC1172 
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/03 
Geometry 
8.0 
Mode of delivery (when and how)
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Organisation of didactics
Type of hours 
Credits 
Hours of teaching 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Start of activities 
02/10/2017 
End of activities 
19/01/2018 
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 discrete valuation rings;
 Basics on dimension theory. 
Examination methods:

Written exam 
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. CayleyHamilton 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. Recommended exercises. 
Additional notes about suggested reading:

Lecture notes available at http://mgaruti.weebly.com/ca.html 
Textbooks (and optional supplementary readings) 

Garuti, M.A., Commutative Algebra Lecture notes. Padova: , 2015. Disponibile gratuitamente alla pagina web dell'autore.

Atiyah, Michael Francis; Mac_Donald, Ian Grant, Introduction to commutative algebraM. F. Atiyah, I. G. Macdonald. Reading [etc.]: AddisonWesley, .

Atiyah, Michael Francis; Mac_Donald, Ian Grant; Maroscia, Paolo, Introduzione all'algebra commutativaM. F. Atiyah e I. G. Macdonaldappendice all'edizione italiana di Paolo Maroscia. Milano: Feltrinelli, 1981.

Eisenbud, David, Commutative algebra with a view toward algebraic geometry. New York [etc.]: Springer, .

Gathmann, A., Commutative Algebra. Kaiserslautern: , 2013. Disponibile gratuitamente alla pagina web dell'autore.


