This is a first course in commutative algebraic geometry, for
who have some background in basic algebra, and in particular in the
theory of rings and fields. The first part covers some basic
commutative algebra, building towards
the basic theory of affine and projective algebraic varieties.
completion of this module, the student
1. be familiar with the basic
concepts, methods and results of elementary Algebraic Geometry and
elementary Commutative Algebra.
2. be able to recognise affine
and projective algebraic varieties, describe them them in the language
of Commutative Algebra
and analyse their basic
properties using the methods and tools of the later.
3. be able to describe, construct
and analyze affine and projective algebraic varieties, including their
3. be able to apply the
techniques of ideal theory to basic problems in the theory of affine
and projective algebraic curves and surfaces.
4. be able to apply the
techniques of basic elimination theory to problems in elementary
5. be familiar with the basic
concepts of classical intersection theory.
I. Dolgacev, Introduction
D. Mumford, The
Red Book of Varieties and Schemes (we will only cover parts of this
somewhat advanced text, given that it requires more commutative algebra
than I can assume known).
For basic algebra and commutative
Atiyah & McDonald: "An introduction to commutative algebra".
Serge Lang: "Algebra".
Bourbaki: "Algebra I", "Algebra
II" and "Commutative Algebra"
H. Matsumura: "Commutative Ring Theory".
Topics covered in class:
1. Rings and ideals; definition of prime and
multiplicative sets; quotient of a ring by an ideal; integral domains
and principal ideal domains (P.I.D.s).
2. Ideal generated by a set; ideal intersections and ideal sums.
3. Characterizations of prime and maximal ideals. The prime and maximal
spectra of a ring. Krull's theorem; the Krull dimension of a ring.
Noetherian rings. Ideal-theoretic characterization of fields and
4. Basic theory of semilattices and order lattices. The lattice
($J(R),+,\cap)$ of ideals of a ring. The
product of ideals. The semiring $(J(R),+,\cdot)$.
5. Radical of an ideal; roots of an
nilpotent elements; the
nilradical of a ring. Reduced rings. Radical ideals. The quotient
criterion for radical ideals. Prime ideals are radical. The reduction
of a ring. I-order of elements and order of nilpotency. The
radicalization map as a closure operator. The semilattice of radical
6. Closure and kernel operators.
7. Modules and algebras over a commutative ring; morphism and
isomorphism of algebras; monoid algebras and polynomial algebras;
basic terminology and conventions for polynomials. Z-modules are
the same as Abelian groups; Z-algebras are the same as commutative
rings. First isomorphism theorem for commutative R-algebras;
application to commutative rings.
8. Finitely generated algebras. Presentation of a finitely-generated
algebra through generators and relations; polynomial algebras are the
free commutative algebras on finite sets of generators; the
universality property of polynomial algebras.
9. Galois correspondences; examples.
10. Hilbert's basis theorem.
The case of coefficients in a field.
10. Hilbert's weal and hard nullstellensatz. The (affine) ideal
11. Affine varieties; the Zarsiski tangent space; singular points.
12. Projective space; homogeneous polynomials; projective varieties and
Tutorial 1: The ideals of the ring of integers
-The ring of integers is a P.I.D.
-Divisibility as a partial order relation;
connection with the inclusion relation on ideals
-Ideal intersections and sums correspond to the lcm
Tutorial 2: The ideals of the ring of integers -- part 2:
-The lattice J(Z) and the spectrum Spec(Z). Meaning
of Noetherianness of Z.
-Irreducibility versus primeness. Primality.
Primality implies that all irreducibles are prime.
-The Euclidean algorithm and the unique
factorization theorem (the fundamental theorem of arithmetic).
-Basics on generalizations: Arithmetic in Euclidean
Rings, P.I.Ds, Bezout domains, UFD's, GCDs, (pre-)Schreier domains etc.
Tutorial 3: Square-free content of a natural number; radicalization map
and radical ideals of Z; the nilradical of the rings Z_n;
description of the nilpotent elements of Z_n.
Tutorial 4: Viete's relations.
Tutorial 5: Resultants and discriminants.
Revision Tutorial: Affine curves; computing intersection points
through resultants; some examples of intersection multiplicity;
ordinary double points and cusps for affine real and complex curves.
Some lecture notes (with
apologies for any typos):
Notes on radical ideals
Notes on closure and contraction
Notes on Galois Correspondences
Notes on projective spaces
Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
The exam will consist of 4 questions (each worth 25 points), of which
you have to answer 3 questions correctly in order to get a maximum
score (a score of 75 points will be renormalized to 100%). For those
questions which have sub-questions, each sub-question will be numbered
and contain an indication of how many points it is worth. For example:
Question 1 [25 p] ....
a. [10 p]
b. [8 p]
Question 2 [25 p] ...