Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 1
1. Identify R4 with the space of quaternions cfw_a + bi + cj + dk : a, b, c, d R. Then the
function (a + bi + cj + dk ) i(a + bi + cj + dk ) = b + ai d + cj denes an everywhere
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 4
1. (a) To get a map f : V Pd1 , it must be the case that for each closed point P of
k
X , the elements of V do not all have positive order of vanishing at P (so that we
get a va
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 4 (due Wednesday, February 6)
Solve the following problems, and turn in the solutions to four of them. You may (and
should) use the Riemann-Roch theorem! Throughout this problem set, let k be a
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 3 (due Wednesday, January 30)
Solve the following problems, and turn in the solutions to four of them.
1. Let X be a scheme. Let F be a quasicoherent sheaf on X which is locally nitely
generate
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 3
1. The property of upper semicontinuity may be checked locally on X , so we may assume
at once that X = Spec(R) is ane, so that F M for M = F (X ). The upper
=
semicontinuity pr
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 2
1. (a) We may view Xi Xj as the open subscheme of Xi consisting of those points x
for which fj does not belong to the maximal ideal of the local ring OX,x = OXi ,x .
But if we i
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 2 (due Friday, January 25)
This problem set is due later than usual because there will be no lectures on Friday,
January 18 or Monday, January 21. Solve the following problems, and turn in the
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 5
1. By hypothesis, there exist f1 , . . . , fn R generating the unit ideal such that M is
nitely generated over D(fi ); that is, the module Mfi over Rfi is nitely generated.
Sinc
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 5 (due Wednesday, February 13)
Solve the following problems, and turn in the solutions to four of them.
1. Let R be a commutative ring. Let M be an R-module such that the quasicoherent sheaf
M
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 8 (due Wednesday, March 6)
Solve the following problems, and turn in the solutions to four of them.
1. (a) Let X be an arbitrary locally ringed space (not necessarily a scheme) and let
R be a r
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 9
1. (a) Let k be a eld and let U be the complement of the origin in A2 = Spec k [x, y ].
k
Then
O(U ) = k [x, y, y 1 ] k [x, x1 , y ] = k [x, y ]
but the induced map U Spec k [x,
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 9 (due Friday, March 15)
Solve the following problems, and turn in the solutions to four of them.
1. (a) Give an example of a scheme in which the intersection of two open anes fails to
be ane.
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 8
1. (a) Given a morphism X Spec(R) of locally ringed spaces, pullback of global
sections denes a map R O(X ). In the other direction, given a homomorphism
R O(X ), we dene a map
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 7 (due Wednesday, February 27)
Solve the following problems, and turn in the solutions to four of them, including no
more than two of problems 57. (Surgeon generals warning: too much abstractio
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 6
1. There are enough proofs of the snake lemma posted online that it hardly seems necessary to include one here. See for example http:/planetmath.org/?op=getobj&id=
5578&from=obj
Math 203B (Algebraic Geometry), UCSD, winter 2013
Problem Set 6 (due Wednesday, February 20)
Solve the following problems, and turn in the solutions to four of them. If and only if
youve never seen the snake lemma before, please submit problem 1 as one of
Math 203B (Algebraic Geometry), UCSD, winter 2013
Solutions for problem set 7
1. If f : X Y is nite and Y = Spec(A) is ane, then f OX is a OY -algebra whose
underlying OY -module is quasicoherent and nitely generated. By a previous home
work problem, it h