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
Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars.
Rings and modules of fractions, localization. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.
A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 R TURCHET AMOS
Programma
Rings and ideals, maximal ideals and prime ideals, nilradical and Jacobson radical, 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. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.
Testi Adottati
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021
Modalità Valutazione
Written exam on exercises and an oral exam consisting of a seminar plus a standard part on theorems and proofs.
scheda docente
materiale didattico
Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars.
Rings and modules of fractions, localization. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.
A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 R TURCHET AMOS
Programma
Rings and ideals, maximal ideals and prime ideals, nilradical and Jacobson radical, 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. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.
Testi Adottati
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021
Modalità Valutazione
Written exam on exercises and an oral exam consisting of a seminar plus a standard part on theorems and proofs.
scheda docente
materiale didattico
Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars.
Rings and modules of fractions, localization. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.
A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021
Mutuazione: 20410445 AL410 - ALGEBRA COMMUTATIVA in Matematica LM-40 R TURCHET AMOS
Programma
Rings and ideals, maximal ideals and prime ideals, nilradical and Jacobson radical, 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. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.
Testi Adottati
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021
Modalità Valutazione
Written exam on exercises and an oral exam consisting of a seminar plus a standard part on theorems and proofs.