## MAT 536 ALGEBRA III, FALL 2003

## Commutative Ring Theory

####
The main goal is to overview the theory of commutative rings.
Some attention will be paid
to how this theory applies to the language of Algebraic Geometry,
but it will be first and foremost
an Algebra course.

**Textbook: **
Commutative ring theory, by H. Matsumura, Cambridge University Press.

**Office hours:** TU 12:45-2:45pm, Mat Tower P-148 and
TH 3-4pm, Mat Tower 3-115.
**The grade** will be based on three homework assignments
and on one in-class presentation
on a topic chosen by the student. Students should contact me as
soon as possible to discuss a choice of topic.

**Topics:**
Schemes (basic properties), quasi-coherent sheaves on schemes,
flatness, normality...

**HMK 1, due on Tu Oct 7: **
Only for people registered in the class.
Give complete solutions, but keep them as short as possible
and readable. Staple the hmk.
1.2, 1.4, 1.5,
Prove NAK by induction on the number of generators,
2.1, 2.3, 2.5,
3.3, 3.4, 3.5.

**HMK 2, due on Th Nov 6: **
Only for people registered in the class.
Give complete solutions, but keep them as short as possible
and readable. Staple the hmk.
Matsumura: 4.3, 4.4, 4.5, 4.7, 4.9, 4.12;
Hartshorne: Ex II.3.10.
Study the map induced by Z --> Z[x,y]/(x^2+y^2-1)
(i.e. show it is a map of intgral schemes, compute the fibers,
which fibers are integral schemes which are not and why).
Let n be an integer and f(x,y) be in k[x,y]. Study the map
induced by k[x,y] --> k[x,y,z]/(z^n - f(x,y)) (first concentrate
on the case k=C with f irreducible).
What is the Galois group of the field extension obtained at the generic points
(when the surface in question is irreducible)?

**HMK 3, due on Th Dec 11: **
Only for people registered in the class.
Give complete solutions, but keep them as short as possible
and readable. Staple the hmk.
Matsumura: 5.1, 5.2, 6.1, 6.3, 6.6, 6.7.
Also: A: construct the scheme P^1_R (projective line over any
ring); use a glueing with two patches; describe
the locally free sheaves of rank one on this space
via modules (on the rings that give the two affine patches) that glue.
B. Consider A^2xP^1 (over a field k); describe the surface
S inside of it defined by the equation: xu=yv (x,y coordinates in A^2,
u,v homogeneous coordinates for P^1); find a covering of S by two affine
schemes (less is not possible); if k = complex numbers,
show S ``is" a complex manifold of dimension 2;
study the map b: S -> A^2 (the ring-theoretic expression, fibers,...);
show that b is not flat; compute b^* m_o (pull-back of the
sheaf of ideals of the origin of A^2) (there is a surprise!).
For fun: study Spec K, K = the direct product of countably many
copies of the same field.