Chapter 1.3 Flashcards
For quasi-affine variety Y in A^n, define: f: Y –> k regular at a point, f regular
Quasi-projective variety?
A function f:Y –> k is REGULAR AT A POINT p in Y if there is an open neighborhood U with p in U < Y, and polynomials g,h in A = k[x1,…,xn], s.t. h is nowhere zero on U and f = g/h on U. Also say LOCALLY REGULAR AT P. We say f is REGULAR on Y if it is regular at every point of Y
Now if Y is a quasi-projective variety, f: Y –> k is REGULAR AT A POINT p in Y if there is an open neighborhood U with p in U < Y and homogeneous polynomials g,h in S = k[x0, … , xn] of the same degree s.t. h is nowhere zero on U and f = g/h on U. (Note that quotient is a well-defined function since same degree). Again, f is REGULAR on Y if it is regular at every point of Y
Prove: A regular function f: Y –> k is continuous when k is identified with A^1 in the Zariski topology
What can be said about two regular functions f, g on a variety Y which agree on some nonempty open subset U of Y?
Pf. Show f^-1 of a closed set is closed. A closed set of A^1 is a finite set of points, so it is sufficient to show that f^-1(a) is closed for any a in k. Check locally…pg 15
If f = g on U, then f = g on Y – the set of points where f - g = 0 is closed and dense, hence equal to Y.
Define the category of varieties. Show category
Let k be a fixed algebraically closed field. A VARIETY OVER k is any affine, quasi-affine, projective, or quasi-projective variety. If X, Y are two varieties, a MORPHISM phi: X –> Y is a continuous map s.t. for every open set V < Y and for every regular function f: V –> k, the function f o phi : phi^-1(V) –> k is regular.
pg 16
Define: O(Y), local ring of P on Y O_p, function field, rational functions on Y
Show local ring is local, function field is field.
How are these 3 rings related?
Let Y be a variety. We denote by O(Y) the ring of all regular functions on Y where U is an open subset of Y containing P, and f is a regular function on U. We identify pairs <u> = if f = g on U int V.</u>
O_p is local - max ideal is the set of germs which vanish at p. If f(p) != 0, 1/f regular in some neighborhood of P.
If Y is a variety, we define the FUNCTION FIELD K(Y) of Y as follows: an element of K(Y) is an equivalence class of pairs <u> where U is a nonempty open subset of Y, f is a regular function on U, and where we identify two pairs <u> and if f = g on U int V. The elements of K(Y) are called RATIONAL FUNCTIONS ON Y.</u></u>
K(Y) is a field: Any two nonempty open sets intersect… pg 16
O(Y) –> O_p –> K(Y) inclusions</u></u></u>
Discuss (with proof) how the O(Y), O_p and K(Y) relate to the affine coordinate ring A(Y) of an affine variety Y
We have:
- O(Y) = A(Y)
- For each p in Y, let mp < A(Y) be the ideal of functions vanishing at p. Then p –> mp gives a 1-to-1 correspondence between the points of Y and the maximal ideals of A(Y)
- For each p, O_p = A(Y)_mp and dim O_p = dim Y
- K(Y) is isomorphic to the quotient field of A(Y), and hence K(Y) is a finitely generated extension field k of transcendence degree = dim Y.
pg 17, B117
Let U_i < P^n be the open set defined by x_i != 0. Prove the mapping phi_i:U_i –> A^n is an isomorphism of varieties.
We showed previously that this is a homeomorphism 2.2. (Show again). Just check regular functions are the same on any open set… pg 18
Discuss (with proof) how the O(Y), O_p and K(Y) relate to the projective coordinate ring S(Y) of a projective variety Y.
First, we define S_(p) to be the subring of elements of degree 0 in the localization of S.
We have:
- O(Y) = k
- For any p in Y, let mp < S(Y) be the ideal generated by the set of homogeneous f in S(Y) s.t. f(p) = 0. Then O_p = S(Y)_mp
- K(Y) = S(Y)_((0))
pls 18-19
Discuss ringed spaces
These are topological spaces with sheaf of rings
Topological Manifolds > C^1 > C^2 > … > C^inf (smooth) > C^omega (analytic) > smooth algebraic varieties over R or C
Floppy to Rigid. Major change between analytic and smooth. B114-115
Show the twisted cubic is isomorphic to P1
Key Idea: Cover cubic by open affine sets Ui. Choose functions on each Ui, check that they are the same on intersections
Cor. Exhibit isomorphic projective varieties (twisted cubic and P^1) with different graded coordinate algebras.
B 121 -123
Show A^2 - {0} not affine
Compute ring of regular functions … B123
Discuss the group of automorphisms of an algebraic set
In general these are almost always trivial. Arbitrary algebraic sets are very ugly, complicated, not symmetric.
Affine and Projective spaces have lots of symmetry
Affine: Euclidean group + lots of others. Headache for n>1. Jacobian conjecture
Projective: PGL(2,k)
GAGA
What is Ax Grothendick theorem? Method of proof?
Theorem. If you have a regular map f: V –> V over an algebraically closed k, then f INJECTIVE => f SURJECTIVE.
Pf Idea. Very easy for finite fields - everything finite number of points where injective always implies surjective.
True over any algebraic extension of finite field <– polynomials finite all coef and solutions lie in some finite field.
So it is true for algebraic closures of finite fields. From model theory we know any 1st order statement true for alg closures of finite fields is true for algebra closed field in char 0.
Discuss with proof equivalence of categories affine varieties and integral domains
Pg 19
Exercise 3.1
Exercise 3.2
Exercise 3.4
Exercise 3.5
Exercise 3.6
Exercise 3.8
Exercise 3.11
Exercise 3.13
Exercise 3.15
Exercise 3.16
Exercise 3.17
Exercise 3.19
Exercise 3.21
