Chapter 1.1 Flashcards
Discuss how to classify all pythagorean triangles
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
Discuss difference between vector space and affine space
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
Discuss relationship between affine space and coordinate ring
Noetherian properties?
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
Define: algebraic set
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.
Discuss Zariski topology. Prove a topology
Define Zariski topology on A^n by taking open sets to be the complements of the algebraic sets.
pg2
Define: Noetherian ring, prove equivalence
- Every ideal is fg
- Every nonempty set of ideals has maximal element
- Every chain of ideals I_0 < I_1 < … is eventually constant acc
Prove: Hilbert basis theorem
B22
Define: Noetherian topological space
A topological space is called Noetherian if:
- Closed sets satisfy descending chain condition: any decreasing sequence of closed sets stabilizes
- Any nonempty collection of closed sets has a minimal element
Discuss relationship between algebraic set and its coordinate ring in terms of Noetherian property
Use correspondence ideals - closed sets
Define: Irreducible space
A set/topological space is called irreducible if it is nonempty and not a union of two proper nonempty closed subsets
Prove: Any Noetherian space is a finite union of irreducible subspaces
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
Define: Irreducible space, affine algebraic variety, quasi-affine variety
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.
Discuss the maps I and Z and 5 properties
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?
- If T1 < T2, then Z(T1) > Z(T2)
- If Y1 < Y2, then I(Y1) > I(Y2)
- I(Y1 U Y2) = I(Y1) intersect I(Y2)
- I(Z(a)) = rad a
- 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]
Discuss and prove Weak Nullstellensatz
Max ideals <=> points
if working over algebraically closed field B30
Discuss and prove Strong Nullstellensatz
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
Discuss dictionary between algebraic sets and coordinate rings
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
Examples of algebraic sets. Irreducible or not?
- A^1 finite complement (irreducible since only proper closed subsets are finite, yet it is infinite)
- A^2 complements of points and curves
- A^n irreducible (corresponds to 0 ideal - a prime ideal)
- Determinantal Varieties
- Nonempty open subset of irreducible space is irreducible and dense
- If Y is irreducible subset of X, then Y closure in X is also irreducible
B19-20
Define: affine curve, surface, hypersurface
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).
Define: affine coordinate ring
Discuss correspondence between coordinate rings and finitely generated k-algebras which are domains.
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.
Define: dimension of topological space, height of prime ideal, Krull dimension
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.
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)
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
Discuss how to use dimension theory of Noetherian rings to get results about dimension of affine sets
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
Define: primary ideal, coprimary module
examples?
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.
What is Lasker-Noether Thm? Proof
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.
Discuss quotients of varieties by groups. Examples?
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
- 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
- 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.
- Binary quantics
pg 45-50
Discuss when the ring of invariants is finitely generated. Proofs?
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.
Discuss cyclic quotient singularity
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
Discuss Parameter spaces and Moduli Spaces
Examples?
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].
Survey different topological definitions of dimension
- Number of parameters needed to define a point: fails space filling curves
- 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
- Transfinite dimensions
Survey different algebraic definitions of dimension
Idea: A set of high dimension has MANY functions on it
- 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 - 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
- Gelfand-Kirillov Dimension: nope
- 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
- Min # of elements of system of parameters: nope
- Homological dimension: define by asking when do various homology groups vanish
Exercise 1.2
Exercise 1.3
Exercise 1.5
Exercise 1.6
Exercise 1.7
Exercise 1.1
Exercise 1.8
Exercise 1.9
Exercise 1.10
Exercise 1.11
Exercise 1.12
