Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 1
1. (a) Suppose that f is ane. For x X , we may determine f 1 (x) by pulling back
along the canonical map Spec (x) X . We may thus assume that X = Spec(k )
for k a eld. In this c
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 3 (due Wednesday, May 1)
Solve the following problems, and turn in the solutions to four of them. Note: no classes
April 2226 because Ill be out of town.
1. Let R be a ring.
(a) Let M be a nite
Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 3
1. (a) Let K be the kernel of f . By hypothesis, there is another surjection f : F M
with F a nite free R-module such that ker(f ) is nitely generated. Put F =
F F and f = f f :
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 2 (due Wednesday, April 17)
Solve the following problems, and turn in the solutions to four of them.
1. (a) It was shown in class that if R S is a formally smooth morphism of rings, then
S/R is
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 1 (due Wednesday, April 10)
Solve the following problems, and turn in the solutions to four of them. As usual, please
document any collaboration and cite all external references. These now incl
Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 2
1. (a) We rst check that S/R is free on the generator dx. On one hand, it is clear that
dx generates S/R since S is a quotient of S = Fp [x]. On the other hand, we have a
1
p2
w
Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 4
1. Suppose that R is perfect and noetherian. For I an ideal of R, put I 1/p = cfw_x1/p : x I ;
this is again an ideal of R. We must then have I = I 1/p , as otherwise the sequen
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 4 (due Wednesday, May 8)
Solve the following problems, and turn in the solutions to four of them. (Note that there
is a second page!)
1. In case you need any convincing that nonnoetherian rings
Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 7
1. The map GL(m) Spec Z ZJ VJ is the multiplication map on functors of points. The
map VJ GL(m) Spec Z ZJ is dened on functors of points to send a matrix A = (xij )
to the pair
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 7 (due Wednesday, June 5)
Solve the following problems, and turn in the solutions to four of them, including at most
two of 13 and at most one of 78.
Notation for problems 13: x positive intege
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 6 (due Wednesday, May 29)
Solve the following problems, and turn in the solutions to four of them. No homework
due Wednesday, May 22 due to qualifying exams.
1. Let f : Y X be a nite surjective
Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 6
1. We may assume that U = Spec(A) is ane. Since f is ane, U = f 1 (U ) is also
ane; write it as Spec(B ). Choose elements s1 , . . . , sm of B which form a basis of
Frac(B ) ove
Math 203C (Algebraic Geometry), UCSD, spring 2013
Solutions for problem set 5
1. (a) We have X/k = P3 /k O(4) = O because the canonical sheaf on Pn is O(n 1).
k
k
To compute H 1 (X, OX ), we may write it as H 1 (P3 , f OX ) for f : X P3 the
k
k
closed imm
Math 203C (Algebraic Geometry), UCSD, spring 2013
Problem Set 5 (due Friday, May 17)
Solve the following problems, and turn in the solutions to four of them. Throughout this
problem set, let k be an algebraically closed eld (of arbitrary characteristic un