Unit COMMUTATIVE AND COMPUTATIONAL ALGEBRA
- Course
- Mathematics
- Study-unit Code
- 55A00076
- Curriculum
- Matematica per l'economia e la finanza
- Teacher
- Giuliana Fatabbi
- Teachers
-
- Giuliana Fatabbi
- Daniele Bartoli (Codocenza)
- Hours
- 42 ore - Giuliana Fatabbi
- 21 ore (Codocenza) - Daniele Bartoli
- CFU
- 6
- Course Regulation
- Coorte 2020
- Offered
- 2021/22
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/02
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Introduction to commutative algebra: module and primary decomposition.
Groebner bases. Introduction to
Introduction to algebraic varieties. Introduction to algebraic curves. Zeroes and poles, divisors, Riemann-Roch spaces. Elliptic curves. - Reference texts
- William Fulton, Algebraic Curves - An Introduction to Algebraic Geometry , 2008 (available online)
Atiyah-Macdonald ,Intoduction to Commutative algebra, Addison-Wesley, 1969
Cox-Little-O'Shea, Ideal s, Varieties, and Algorithms, Springer , 1997 - Educational objectives
- Knowledge of concepts proposed. Capability of the usage of a symbolic programme.
- Prerequisites
- Basic concepts of rings and ideals, in particular ring of polynomials in one indeterminate over a field.
- Teaching methods
- face to face lessons
- Other information
- According with the students attending the course, the course can be given partially or entirely in English. also the exam can be given in English, upon request of the student.
- Learning verification modality
- Oral examination, lasting 45-60 minutes, which tends to evaluate the level of understanding of the topics treated and of critic study and personal rethinking.
- Extended program
- Modules and their properties.
Noetherian ring and modules. Primary decomposition in noetherian rings.
Polynomial in several indetermibnates. Monomial ideals. Dickson's Lemma. Monomial orderings. Division algorithm. Groebner bases. Holbert basis Theorem. Buchberger's criterion and algorithm. Membership algorithm. Radical membership criterion and algorithm. Eliminationa nd ins\ntersection algorithm. Affine and projective varieties. Hilbert zeroes Theorems. Introduction to dimension theoery.
Plane algebraic curves. Affine and Projective algebraic varieties. Irreducible components. Tangent spaces and dimension. Singular points. Rational maps and morphisms. Smooth curves. Differentials and canonical divisors. The genus of an algebraic curve. Riemann-Roch spaces. Elliptic curves, Isogenies. Torsion points. Weil pairing.