Commutative algebra
Algebraic geometry is a part of mathematics
dealing with commutative rings, ideals and
modules over commutative rings. The methods
of commutative algera are central in algebraic
geometry, and the algebraic-geometric
reasoning plays an important role underlying
the intuition of commutative altebra.
The methods of commutative algebra are
used to treat the algebraic number theory
and the geometry of complex manifolds, all
included under the bigger umbrella of
"algebraic geometry".
I will give an introduction to
commutative algebra, treating it
geometrically.
Syllabus
1. Hilbert Nulstellensatz. Algebraic sets and algebraic
varieties. Categories and functors.
2. Localization. Ideals. Maximal ideals, prime ideals,
nilradical. Strong Nulstellensatz. Equivalence of the
categories of affine varieties and of finitely generated rings.
3. Irreducible components. Noetherian rings.
Hilbert basis theorem. Modules over a ring.
Noether theorem on rings of invariants.
4. Tensor product of rings and tensor products
of modules over a ring.
5. Zariski topology, dominant morphisms, integrally
closed domains. Integral closure and integral dependency.
Quotient of a variety by a group action.
6. Noether normalization lemma and its applications.
7. Dimension and its properties.
8. Smooth varieties and regular local rings.
Basic knowledge of algebra (rings, fields, groups,
vector spaces) is necessary. I would explain more advanced
algebraic notions, such as ideals and modules over a ring,
very briefly. Some knowledge of geometry (implicit function
theorem, definition of a manifold) would also be helpful.
The last two sections will be given only if time permits.
Useful books:
Atiyah, MacDonald, "Introduction To Commutative Algebra"
Hartshorne, "Algebraic geometry"
Mumford, "Algebraic Geometry I: Complex Projective Varieties"
Miles Reid, "Undergraduate Commutative Algebra"
Van der Waerden, "Modern Algebra"