Learning objectives
Knowledge and understanding
The student will be introduced to the basic concepts and techniques of commutaive algebra and algebraic geometry.
Applying knowledge and understanding
The student will be able to: i) solve simple exercise of algebraic geometry, ii) simple
exercises of commutaive algebra.
Prerequisites
Theory of groups.
Course unit content
The course is an introduction to the basic notions of commutative algebra and basic notions of algebraic geometry.
The first part studies commutative rings with unit ideals, Nullstellensatz Theorem, Zarisky Topology while the second part is devoted to the study of modules, operations on modules, Hamilton-Cayley Theorem, Nakayama Lemma and flat modules.
In the third part of the course we study localization of rings and modules, primary decomposition, Noetherian and Artin rings (modules) , Hilbert basis Theorem .
The course ends with a study of integral dependence and valutations, Krull dimension and basic notions on algebraic varieties.
Full programme
Commutative rings with unit, prime ideal, radical, Nilradical, Nullestellensatz Theorem, Zarisky Topologyt.
Modules, operations on modules, Hamilton-Cayley Theorem and Lemma di Nakayama.
Localizations of rings and modules, primary decomposition, Noetherian and Artin rings, Hilbert basis Theorem.
Integral dependence and valutations, Krull dimension and basic notions on algebraic varieties.
Bibliography
Atyah e Mc Donald,
Algebra commutativa
Teaching methods
During lectures, the material of the course is presented using formal
definitions and proofs; abstract concepts are illustrated through
significant examples, applications, and exercises. The discussion of
examples and exercises is of fundamental importance for grasping the
meaning of the abstract mathematical concepts.
Assessment methods and criteria
Course grades will be based on an oral interview.
In the colloquium, students should establish that students have learned the course
materials to a sufficient level and should be able
to repeat definitions, theorems and proofs given in the lectures using a
proper mathematical language and formalism. In the colloquium, students should be able
to repeat definitions, theorems and proofs given in the lectures using a
proper mathematical language and formalism.
Other information
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