# Math 6670, Fall 2017

## Allen Knutson

## Tues/Thurs 11:40-12:55

Algebra is the offer made by the devil to the mathematician. The devil
says: I will give you this powerful machine, it will answer any
question you like. All you need to do is give me your soul: give up
geometry and you will have this marvelous machine.

â€”Sir Michael Atiyah, 2002

The book we'll use for reference is at the bottom
here.
Now that we're getting into scheme theory, we'll use
Eisenbud and Harris'
*The Geometry of Schemes*.
Topics:

Initial notes here.
Next notes.
If you're getting a grade in this class, turn in HW. Due 8/31:

Ex 1.1, 1.2 from those notes
HW due 9/7:
Give the analogue of Taylor's theorem when expanding a
function NN->ZZ as f(d) = \sum_n c_n (d+n choose n).
In particular, show that a polynomial f is integer-valued iff
these c_n are all integer.
What does your analogue give for the non-polynomial function f(d)=2^d?
Prove the Hilbert syzygy theorem for the case of monomial ideals
in x,y.
Let I = < xy-z > and J = < z-t > be ideals in C[x,y,z], where t is
a number. What's the prime decomposition of I+J?
Let I be a homogeneous radical ideal, the intersection of some
minimal prime ideals {P}. Find a formula for the degree of I
(the leading coefficient of the Hilbert polynomial) in terms of
the {P}.
HW due 9/21. Personally, I find the easiest
way to use Macaulay2 to be on my machine from within emacs, but
it *is* possible to
use it online.
HW due 9/28

HW due 10/5

HW due 10/26 **(or the following Tuesday, for undergrads who'd rather
be studying for the GRE)**:
exercises I-9, 10, 15, 17, 20 from Eisenbud and Harris.

HW due 11/9:

1. Let X be a scheme and R = Gamma(X; O_X) the global functions.
Show in excruciating detail that there is a map of schemes from
X to Spec R.
2. Let R be a commutative O_X-algebra, i.e. a sheaf of algebras
over X with a compatible O_X-module structure. Show in detail
how to define the scheme Spec R in terms of the Specs of the stalks,
and its map to X.
In particular if R = O_X you'd better get the identity X -> X.
3. Exercise I-43 of [EH]
4. Exercise I-46 of [EH]
5a. Compute the fibers of Spec of the map RR[y] -> RR[x,y] / < y-x^2 >
over the points < y-r > for r in RR (here RR = the reals).
5b. Compute the fibers of Spec of the map ZZ -> ZZ[x] / < x^2+1 >
over the points < p > for p in Spec ZZ (here ZZ = the integers).