Misha Verbitsky

Instituto Nacional de Matemática Pura e Aplicada

Commutative Algebra

15:00-17:00, Mondays, Fridays - Sala 232, Thursdays - Aud. 2, January-February 2022

Slides:

• Lecture 1: Algebraic sets and Hilbert Nullstellensatz (January 3).
• Lecture 2: Categories and Hilbert Nullstellensatz (January 6).
• Lecture 3: Strong Nullstellensatz (January 7).
• Lecture 4: Noetherian rings (January 10).
• Lecture 5: Primary decomposition (January 13).
• Lecture 6: The ring of $G$-invariants is finitely generated (January 14).
• Lecture 7: Tensor product of $R$-modules (January 17).
• Lecture 8: Tensor product of rings (January 21).
• Lecture 9: Yoneda lemma and the fibered product (January 24).
• Lecture 10: Tensor product of field extensions (January 27).
• Lecture 11: Primitive element theorem (January 28).
• Lecture 12: Normal varieties (January 31).
• Lecture 13: Gauss Lemma and normalization (February 3).
• Lecture 14: Nakayama lemma (February 4).
• Lecture 15: Finite quotient and branched covers (February 7).
• Lecture 16: Noether normalization lemma (February 10).
• Lecture 17: Krull dimension and algebraic dimension (February 11).
• Lecture 18: Discrete valuation rings (February 14).
• Lecture 19: Discrete valuation rings (2) (February 17).
• Lecture 20: Dedekind rings (February 18).
• Lecture 21: Smooth points in complex varieties (February 21).
• Lecture 22: Hilbert polynomials (February 24).
• Lecture 23: Regular rings and smooth varieties (February 25).

Other materials

• Assignment 1: Zorn lemma (January 6)
• Assignment 2: Applications of Axiom of Choice (January 6)
• Assignment 3: Noetherian rings (January 10)
• Assignment 4: Tensor product (January 17)
• Assignment 5: Factorial rings (January 27)
• Assignment 6: Limits and colimits (January 28)
• Assignment 7: Yoneda lemma (February 3)
• Assignment 8: Exercises in divisibility (February 7)

• Exam problems.
• Personal problem sets, obtained randomly with this program.
• Final results (pending).

Misha Verbitsky IMPA