TENSOR PRODUCTS II
KEITH CONRAD
1. Introduction
Continuing our study of tensor products, we will see how to combine two linear maps
M M and N N into a linear map M R N M R N . This leads to at modules
Problem Set #1
Math 453 Dierentiable Manifolds
Assignment: Chapter 1 #2, 5, 6, 7, 8
Chapter 2 #1, 3, 5
Clayton J. Lungstrum
January 22, 2013
Exercise 1.2
Let
f (x) =
e1/x
0
: x>0
.
: x0
(a) Show by in
Problem Set #4
Math 453 Dierentiable Manifolds
Assignment: Chapter 5 #1
Chapter 6 #1, 2, 4
Clayton J. Lungstrum
February 13, 2013
Exercise 5.1
Let A and B be two points not on the real line R. Conside
Problem Set #5
Math 453 Dierentiable Manifolds
Assignment: Chapter 7 #1, 5, 7, 8, 9, 11
Clayton J. Lungstrum
February 20, 2013
Exercise 7.1
Let f : X Y be a map of sets, and let B Y . Prove that f (f
Problem Set #2
Math 453 Dierentiable Manifolds
Assignment: Chapter 3 #1, 2, 4, 5, 7, 10, 11
Clayton J. Lungstrum
January 30, 2013
Exercise 3.1
Let e1 , . . . , en be a basis for a vector space V and l
Problem Set #7
Math 453 Dierentiable Manifolds
Assignment: Chapter 9 #1, 2, 3, 5
Clayton J. Lungstrum
March 6, 2013
Exercise 9.1
Dene f : R2 R by
f (x, y) = x3 6xy + y 2 .
Find all values c R for whic
Problem Set #3
Math 453 Dierentiable Manifolds
Assignment: Chapter 4 #1, 2, 3, 4, 5, 6
Chapter 5 #3
Clayton J. Lungstrum
February 1, 2013
Exercise 4.1
Let be the 1-form zdx dz and let X be the vector
MATH 607
Solutions to
Homework Problems
Homework # 1: Hints
Bredon, Sec. 1.1, Problem 2. Observe that
f (x) = dist(x, x0 ) dist(x, y) + dist(y, x0 ) = dist(x, y) + f (y) = f (x) f (y) dist(x, y)
and s
FUNDAMENTAL GROUPS AND THE VAN KAMPENS
THEOREM
SAMUEL BLOOM
Abstract. In this paper, we dene the fundamental group of a topological
space and explore its structure, and we proceed to prove Van-Kampens
1
SMSTC Geometry and Topology 20112012
Lecture 5
The Seifert van Kampen Theorem
Andrew Ranicki (Edinburgh)
10th November, 2011
2
Introduction
Topology and groups are closely related via the fundamenta
MATH 215B HOMEWORK 4 SOLUTIONS
1. (8 marks) Compute the homology groups of the space X obtained from n by
identifying all faces of the same dimension in the following way: [v0 , . . . , vj , . . . , v
FUNDAMENTAL GROUPS AND COVERING SPACES
ETHAN JERZAK
Abstract. In this paper, I will briey develop the theory of fundamental
groups and covering spaces of topological spaces. Then I will point toward
t
Algebraic Topology, Fall 2013
Selected Solutions for HW 6.
Hatcher 1.3.24.
Suppose we have a covering space action of a group G on a path-connected, locally pathconnected space X. Show:
a) Every path-
CUP PRODUCTS IN COMPUTATIONAL TOPOLOGY
JONATHAN HUANG
Abstract. Topological persistence methods provide a robust framework for
analyzing large point cloud datasets topologically, and have been applied
CHAPTER 6
Singular Homology
1. Homology, Introduction
In the beginning, we suggested the idea of attaching algebraic objects to topological spaces in order to discern their properties. In language int
Cup Products
March 7, 2012
Example
Consider the spaces X = CP 2 and Y = S 2 S 4 . The natural question
whether X and Y are homotopy equivalent arises. We know that as CW
Complexes, both X and Y have o
HW 10 SOLUTIONS, MA525
Hatcher 2.2 Problem 29
The Mayer-Vietoris sequence gives
Hn (Mg ) Hn (R) Hn (R) Hn (X) Hn1 (Mg )
The compact space R deformation retracts on a wedge of g circles. So H0 (R) =
72
CHAPTER 5
Manifolds and Surfaces
1. Manifolds and Surfaces
Recall that an n-manifold is a Hausdor space in which every point
has a neighborhood homeomorphic to a open ball in Rn .
Here are some exa
14
Finite abelian groups
When dealing exclusively with abelian groups it is customary to use additive
notation. An abelian group whose operation is written as addition is called
an additive group.
mul
Final Exam
Algebraic Topology
December 13, 2006
You may apply theorems from the course, but please give the name or statement of the theorem.
1. Compute the homology groups of S 3 S 5 by using a produ
Contents
2 Cohomology of Sheaves.
1
3 Cohomology of a Noetherian Ane Scheme
6
4 Cech Cohomology
12
5 The Cohomology of Projective Space
21
6 Ext Groups and Sheaves
27
7 The Serre Duality Theorem
38
8
Math 130a - Problem Set 4
Evan Dummit
5.1a. Singular points occur when both partial derivatives are equal to zero. Differentiating gives 4 x3 - 2 x = 0 and 4 y3 = 0, so
since the characteristic is not
Math 130a - Problem Set 2
Evan Dummit
2.10a. Clearly, q-1 [email protected] , , xn DL = 8Ht x0 , , t xn <, t e K <, since the points in the latter set are precisely those which are mapped
onto the projective point
Math 130a - Problem Set 3
Evan Dummit
2.17a. Consider the coordinate ring AHY L. By hypothesis this ideal is prime, so we may dehomogenize with respect to any of the
variables and then rehomogenize to
Math 130a - Problem Set 1
Evan Dummit
1a. Let y e Y ; then y annihilates IHY L so y e ZHIHY LL. Also, ZHIHY LL is closed by the definition of Z , so ZHIHY LL Y . Conversely,
let X be a closed set cont
Hartshorne, Exercises I.2: Solutions
Various Authors
September 25, 2006
1. The homogeneous Nullstellensatz: If I S is a homogeneous ideal, and
if f S is a homogeneous polynomial with deg f > 0, such t
Hartshorne, Exercises I.1: Solutions
Peter LeFanu Lumsdaine and Claire Tomesch
September 7, 2006
1. (a) Let f = y x2 k[x, y], and consider P (f ). Since k[x, y] is a UFD
and f is irreducible therein,
Hartshorne, Chapter 1.6 Answers to exercises.
REB 1994
6.1a By 6.7, Y is isomorphic to an open subset of some projective space, and therefore to a proper open
subset of P 1 , and therefore to some ope
Solution to Math256a section IV.3 (H.Zhu, 1994)
3.1) One direction follows easily from 3.3.4. We show that D is not very
ample when deg(D) < 5. Now suppose D is very ample, then l(D) = l(D
P Q) + 2 2
Hartshorne, Chapter 1.3 Answers to exercises.
REB 1994
3.1a Follows from exercise 1.1 as 2 ane varieties are isomorphic if and only if their coordinate rings are.
3.1b The coordinate ring of any prope