Acquire a good knowledge of some methods and fundamental results in the study of the commutative rings and their modules, with particular reference to the study of ring classes of interest for the algebraic theory of numbers and for algebraic geometry.
Curriculum
scheda docente
materiale didattico
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.
Programma
Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.Testi Adottati
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.
Bibliografia Di Riferimento
As aboveModalità Erogazione
Frontal lectures, suggested exercises.Modalità Frequenza
Attending lessons is strongly suggested.Modalità Valutazione
written exam and seminar
scheda docente
materiale didattico
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 LELLI CHIESA MARGHERITA
Programma
Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.Testi Adottati
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.
Bibliografia Di Riferimento
As aboveModalità Erogazione
Frontal lectures, suggested exercises.Modalità Frequenza
Attending lessons is strongly suggested.Modalità Valutazione
written exam and seminar