Jan 29

Standard idea: the inclusion I -> Y of a subset.

Easy related theorem: given any function f: X -> Y, we can uniquely factor f as a composite X -> I -> Y where the map X -> I is onto, and I -> Y is an inclusion. The subset I is exactly the image of f.

That separates f into a part involving X and (part of) Y, and a part wholly about Y. Can we have a part wholly about X?

Define a quotient p : X -> Q to be a function that's 1:1, where each p(x) is a subset of X and contains x, and if y is in p(x) then p(y) = p(x). So the collection Q = {p(x)} is a bunch of subsets exactly covering X. You can look up a version of this definition here. They are hard to count (check out the asymptotics!).

Now, given any function f:X -> Y, we can factor it uniquely as X ->> Q -> I -> Y, a quotient followed by a perfect correspondence Q -> I followed by the inclusion of I as a subset of Y.

What do quotients of vector spaces look like? Now p: V -> Q, and we could ask that Q be a vector space, and p be linear. Then (theorem) the elements of Q are the translates of ker(p).

Definition: if W is a subspace of V, let V/W be the set of all translates of W in V, and p : V -> V/W be the quotient map taking v to v+W := {v+w : w in W}, an element of V/W.

It's not hard to prove that dim(V/W) = dim V - dim W, if dim V is finite.

A ring is a set with a +,-,x,0,1 satisfying some fairly obvious conditions, like 0+r = r = 1r, and multiplication distributes over addition. Examples: Z, R, Z[x], {evens,odds}, Z/nZ. We only care about "commutative" rings, where multiplication should be commutative. (So not, e.g., NxN matrices which is a perfectly good noncommutative ring. Nor the quaternions.)

If R is a ring, and p : R -> Q is a quotient map, we could ask that Q be a ring and p take +,-,1,times on R to the corresponding operation on Q, i.e. p(rs) = p(r) p(s). Theorem: the elements of Q must be the R-translates of an ideal I in R.

(Next time: given an ideal, we can form the quotient R/I, and we should!)

Jan 24

Definition of a radical ideal, and the radical of an ideal.

Proof of the easy half of the Nullstellensatz, that I(V(I)) contains the radical of I.
(The hard half is the opposite containment; we'll do that later.)

An algebraic subset is one of the form V(I). The Nullstellensatz says that they correspond 1:1 to radical ideals.

Proof that V(I+J) = V(I) intersect V(J).

Some examples of V(I intersect J), i.e., why it's interesting to write an ideal as an intersection.

We're not ready to approach the book. First, I'm going to motivate using algebraic geometry.

---

Class 1/22: Definition: the ideal generated by a set of polynomials.
Definition: an ideal. (1) should contain zero (2) should be closed under addition (3) should be closed under multiplying by any polynomial.
Definition: the vanishing set V(I) of an ideal, those points in C^n (we'll be working with the complex numbers hereafter) where all the polynomials vanish.
Definition: the ideal I(X) of all polynomials vanishing on a set X in C^n.

---

Homework #1, due Thursday Jan 31. If you're wondering whether "show" means "prove", the answer is... this is a real math class. Of course it does. 1. Let X be a set in C^n. Show V(I(X)) contains X.
2. Give an example of X where they're equal, and an example where they're not.
3. Let I be an ideal in C[x_1,...,x_n]. Show I(V(I)) contains I.
4. If I,J are ideals, let I+J := {i+j : i in I, j in J}. Show I+J is an ideal.
5. Show I+I = I.
6. Let I = < g_1, ..., g_m >. Show that V(I) = the set of x in C^n where every g_i vanishes. (Make sure you understand why that's different from the definition!)
[To show two sets are equal, show both containments -- if x is in the left-hand-side, then x is in the RHS, and separately, if x is in the RHS, then x is in the LHS.]
7. Assume g_1, ..., g_m are homogeneous linear polynomials, and let I be the ideal generated by them. Let p be another homogeneous linear polynomial. How would you test whether p is in I? (Describe an algorithm, perhaps, that correctly answers "yes" or "no" after finite time.)
8. What if the (g_i) and p in #7 are all homogeneous of the same degree, but that degree isn't necessarily 1?

This is where I'm going to put daily summaries of what we did in class, and the homework assignments. If you use an RSS reader, you can have it tell you when this is updated.