Chapter 1.1

Question 1

Discuss how to classify all pythagorean triangles

Answer 1

x^2 + y^2 = z^2 in integers same as x^2 + y^2 = 1 in rational numbers. Comes down to finding rational points on circle. Find birational equivalence between circle and y-axis.
B2

Question 2

Discuss difference between vector space and affine space

Answer 2

Affine space is the set of tuples (a1, … , an) where ai in k.

Different automorphism groups. Vector space: GL_n.
Affine space: GL_n and translations

Vector space has origin. Affine space forgets origin

Question 3

Discuss relationship between affine space and coordinate ring

Noetherian properties?

Answer 3

First, the coordinate ring is just all polynomials on A^n. There is a bijection: affine space homomorphisms to k maximal ideal of k[x1, … , xn].

B17

Question 4

Define: algebraic set

Answer 4

An algebraic set is the set of zeros of a set of polynomials in k[x1, … , xn]. Clear that the ideal generated by set of of polynomials has same set of zeros.

Question 5

Discuss Zariski topology. Prove a topology

Answer 5

Define Zariski topology on A^n by taking open sets to be the complements of the algebraic sets.

pg2

Question 6

Define: Noetherian ring, prove equivalence

Answer 6

1. Every ideal is fg
2. Every nonempty set of ideals has maximal element
3. Every chain of ideals I_0 < I_1 < … is eventually constant acc

Question 7

Prove: Hilbert basis theorem

Answer 7

B22

Question 8

Define: Noetherian topological space

Answer 8

A topological space is called Noetherian if:

1. Closed sets satisfy descending chain condition: any decreasing sequence of closed sets stabilizes
2. Any nonempty collection of closed sets has a minimal element

Question 9

Discuss relationship between algebraic set and its coordinate ring in terms of Noetherian property

Answer 9

Use correspondence ideals - closed sets

Question 10

Define: Irreducible space

Answer 10

A set/topological space is called irreducible if it is nonempty and not a union of two proper nonempty closed subsets

Question 11

Prove: Any Noetherian space is a finite union of irreducible subspaces

Answer 11

This thm allows us to reduce the study of Noetherian spaces to irreducible Noetherian spaces.

By Noetherian induction. If not every closed subset is a finite union of irreducibles, pick a minimal counter example C. Then C = C1 U C2 where C1 and C2 are smaller proper nonempty subsets. By induction, C1 and C2 are finite unions of irreducibles. Contradiction.

pg 5

Question 12

Define: Irreducible space, affine algebraic variety, quasi-affine variety

Answer 12

A set/topological space is called irreducible if it is nonempty and not a union of two proper nonempty closed subsets

An affine algebraic variety (affine variety) is an irreducible closed subset of A^n (with the induced topology). A quasi-affine variety is an open subset of an affine variety.

Question 13

Discuss the maps I and Z and 5 properties

Answer 13

I takes a subset X of affine space to the ideal of polynomials vanishing on X. Z takes an ideal J to the subset of affine space on which every element of J vanishes.

These maps are certainly not inverses. What is the relationship?

1. If T1 < T2, then Z(T1) > Z(T2)
2. If Y1 < Y2, then I(Y1) > I(Y2)
3. I(Y1 U Y2) = I(Y1) intersect I(Y2)
4. I(Z(a)) = rad a
5. Z(I(Y)) = Y closure in Zariski topology

J =  (x^2>) < k[x] then Z(J) = {0} and I(Z(J)) = (x) != (x^2)
J = (x^2+1), Z(J) = empty, so I(Z(J)) = R[x]

Question 14

Discuss and prove Weak Nullstellensatz

Answer 14

Max ideals <=> points

if working over algebraically closed field B30

Question 15

Discuss and prove Strong Nullstellensatz

Answer 15

I(Z(J)) = rad J

More formally, let k be an algebraically closed field field, let a be an ideal in A = k[x1, … xn], and let f in A be a polynomial which vanishes at all points of Z(a). Then f^r in a for some integer r > 0.

B32

Question 16

Discuss dictionary between algebraic sets and coordinate rings

Answer 16

There is a one-to-one inclusion reversing correspondence between algebraic sets in A^n and radical ideals in A given by I and Z.

Also prove below correspondences or state theorem
points  max ideals (Weak N) 
varieties  prime ideals (prove)
alg sets  radical ideals (Strong N)
closed subschemes  ideals

Question 17

Examples of algebraic sets. Irreducible or not?

Answer 17

1. A^1 finite complement (irreducible since only proper closed subsets are finite, yet it is infinite)
2. A^2 complements of points and curves
3. A^n irreducible (corresponds to 0 ideal - a prime ideal)
4. Determinantal Varieties
5. Nonempty open subset of irreducible space is irreducible and dense
6. If Y is irreducible subset of X, then Y closure in X is also irreducible

B19-20

Question 18

Define: affine curve, surface, hypersurface

Answer 18

Let f be an irreducible polynomial in k[x,y] = A. Then (f) is a prime ideal in A since A is a UFD. So the zero set Z(f) is irreducible - call it the AFFINE CURVE defined by f(x,y) = 0.

More generally if f is irreducible in k[x1, … , x^n] we obtain an affine variety called a surface (n=2) or hypersurface (n>2).

Question 19

Define: affine coordinate ring

Discuss correspondence between coordinate rings and finitely generated k-algebras which are domains.

Answer 19

If Y < A^n is an affine algebraic set, we define the affine coordinate ring A(Y) of Y to be A / I(Y)

If Y is an affine variety, then I(Y) is prime, so A(Y) is an integral domain. Further, being a quotient of a polynomial ring, A(Y) is a fg k-algebra.

Conversely, a fg k-algebra B which is a domain is the affine coordinate ring of some affine variety. Write B as quotient of polynomial ring by ideal a. a must be prime. So Y = Z(a) is affine variety with B as coordinate ring.

Question 20

Define: dimension of topological space, height of prime ideal, Krull dimension

Answer 20

dim X is defined to be the supremum of all integers n such that there exists a chain Z0 < Z1 < … < Zn of distinct irreducible closed subsets of X. We define the dimension of an affine or quasi-affine variety to be its dimension as a topological space

In a ring A, the height of a prime ideal p is the supremum of all integers n such that there exists a chain p0 < p1 < … < pn = p of distinct prime ideals.

We define the Krull dimension of A to be the supremum of the heights of all prime ideals.

Question 21

Prove: If Y is an affine algebraic set, then the dimension of Y is equal to the dimension of its affine coordinate ring A(Y)

Answer 21

If Y is an affine irreducible set in A^n, then the closed irreducible subsets of Y correspond to prime ideals of A = k[x1, … , xn] containing I(Y). These in tern correspond to prime ideals of A(Y)…pg 6

Question 22

Discuss how to use dimension theory of Noetherian rings to get results about dimension of affine sets

Answer 22

We saw above that dim Y = Krull dim coordinate ring.

Thm. Let k be a field and B be an integral domain which is fg k-algebra. Then:

(a) dim B = trancendence degree of quotient field K(B) of B over k
(b) For any prime ideal p in B, height p + dim B/p = dim B.

(a) implies dim A^n = n
(b) implies if Y a quasi-affine variety, then dim Y = dim Y closure.

A variety Y in A^n has dim n-1 <=> Y = Z(f) and f irreducible.

pg 6-7

Question 23

Define: primary ideal, coprimary module

examples?

Answer 23

Two definitions P PRIMARY
1. ab in P, then a in P or b^n in P for some n

EXAMPLE
k[x] = R. (x) prime, (x^m) primary

Notice if P is primary, then viewing R/P as a module over R ab = 0 with a in R and b in R/P implies either b = 0 or a^n = 0 for some n i.e. we are either multiplying by 0 or a is nilpotent. Any module with this property is called COPRIMARY.

2. An R-module M is COPRIMARY if it has exactly one ASSOCIATED PRIME P  (A prime P s.t. R/P is isomorphic to a submodule of M). 
If M > N, then N is PRIMARY <=> M/N COPRIMARY.

Question 24

What is Lasker-Noether Thm? Proof

Answer 24

We know from decomposition of alg set into finite union of irreducibles that radical ideal is finite intersection of prime ideals.

Lasker: Idea a of k[x1, … , xn] is finite intersection of PRIMARY ideals.

Noether: Also true for Noetherian rings.

Better to think of this as a thm about modules.

LN for f.g. modules over Noetherian rings: 0 is finite intersection of primary submodules of M. Alternatively, M < finite sum coprimary modules.

Question 25

Discuss quotients of varieties by groups. Examples?

Answer 25

We have an affine variety Y acted on by some group G. Want to form quotient Y/G. As Y is topological space, so is Y/G. However, Y/G is hard to view as affine algebraic set. How do we embed in affine space? Much easier to look at coordinate rings. Functions on Y/G Functions on Y invariant under G. Given an action of G on Y, we have an action of G on coordinate algebra A(Y) = k[x1, ... , xn] / I(Y). Look at ring of invariants ( k[x1, ... , xn] / I(Y) ) ^G If this ring satisfies: 1. Algebra over k (easy - subalgebra of A(Y) ) 2. finitely generated (HARD) 3. No nilpotent elements (easy - subalgebra of A(Y) which is reduced) Then it is the coordinate ring of some affine algebraic set EXAMPLES 1. G = S^n acting on A^n. Ring of invariants = symmetric functions. k[e1, ... , en] no relations isomorphuc to polynomial ring so A^n / S^n = A^n 2. GL_n(k) acting on k^n. The oribits are 0 and everything else. So forming the topological quotient, we have a two point space. However, using the method above, the only invariant polynomials are constants - invariant ring = k = the coordinate ring of a point. 3. Binary quantics pg 45-50

Question 26

Discuss when the ring of invariants is finitely generated. Proofs?

Answer 26

We need G to have Reynolds operator p: an A^G-module hom from A to A^G which retracts A onto A^G leaving A^G fixed. This is most easily obtained by averaging over group. We can do this easily for finite and compact groups over algebraically closed fields. Extend to noncompact classical groups via Weyl's Unitarian trick. Proof. A GRADED by degree. Look at irrelvent ideal I. I is f.g. since A Noetherian. Let i1, ... , in be generators for ideal. Show by induction on degree that if x in A^G, then x in algebra generated by i1, ... , in. Degree 0, done. If degree > 0, write x = a1i1 + ... anin for some ai in A. Now apply Renolds operator.

Question 27

Discuss cyclic quotient singularity

Answer 27

Take A^2, k[x,y]. G = Z/nZ generated by z of order n. G acts on x and y by zeta*x and zeta * y where zeta is an nth root of unity. The quotient in the sense of alg geometry is obtained by looking at invariants of coordinate ring under G. x^iy^j invariant <=> i+j = 0 mod n Generated by x^3, x^2y, xy^2, y^3

Question 28

Discuss Parameter spaces and Moduli Spaces Examples?

Answer 28

Parameter spaces: Here we consider a space where points = configurations. Nice example of molecule cyclohexane. Nearly identical to our bar and truss stuff, tensegrity frameworks. Moduli Spaces: Points = isomorphism classes of things. Not a whole lot different than Parameter space. Traditional to use parameter space for things embedded in something else, Moduli space for things not embedded Example: Moduli space of elliptic curves Over C an elliptic curve is a non-singular curve topologically isomorphic to the torus. Can always be put into this form: y^2 = x^3 + ax^2 +bx + c y^2 = (x-alpha)(x-beta)(x-gamma) 1/lambda) and (lambda --> 1 - lambda). These two transformations generate a group isomorphic to S_3. What we want to look at is the affine quotient variety (A^1 - {0,1})/S_3. Look at action of S_3 on coordinate algebra of A^1 - {0,1} --> k[lambda, 1/lambda, 1/(1-lambda)] invariant j = 2^8(lambda^2- lmbda + 1)^3/lamda^2(lamda-1)^2 generates the invariant ring k[j].

Question 29

Survey different topological definitions of dimension

Answer 29

1. Number of parameters needed to define a point: fails space filling curves 2. Lebesgue covering dimension: Set has dim <= n if every open cover has refinement so that each point in <= n+1 sets 0. No use in alg geometry 6. Transfinite dimensions

Question 30

Survey different algebraic definitions of dimension

Answer 30

Idea: A set of high dimension has MANY functions on it 1. Suppose B a variety over k. Look at quotient field of coordinate ring of B. dim(B) = transcendence dimension of quotient field (largest number of algebraically independent elements of field). e.g. A^2 --> k(x,y) dim 2 Doesn't work well for sets other than varieties in algebraic geometry 2. Hilbert Polynomial. Works for local rings A, max ideal m. dim(A/m^k) is a polynomial in k (large k) of degree d. Define dim of local ring = d. For algebraic set, define dim at a point to be dimension of local ring at that point 3. Gelfand-Kirillov Dimension: nope 4. dim tangent space at a point: works for nonsingular varieties or schemes. Fails with singularities -- still useful because you can use it to define singular point: dim tangent space != dim of scheme 5. Min # of elements of system of parameters: nope 6. Homological dimension: define by asking when do various homology groups vanish

Question 31

Exercise 1.2

Question 32

Exercise 1.3

Question 33

Exercise 1.5

Question 34

Exercise 1.6

Question 35

Exercise 1.7

Question 36

Exercise 1.1

Question 37

Exercise 1.8

Question 38

Exercise 1.9

Question 39

Exercise 1.10

Question 40

Exercise 1.11

Question 41

Exercise 1.12