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
and linear maps between base extensions. Then we will
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 induction that for x > 0 and k 0, the k th derivative f (
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. Consider the set S = (Rcfw_0)cfw_A, B.
For any two positive re
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 1 (B) = B f (X).
Therefore, if f is surjective, then 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 let 1 , . . . , n be its dual basis in V .
Suppose [gij
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 which the level set f 1 (c) is a regular submanifold of R2
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 y/x + x/y on R3 . Compute
(X) and d.
Solution.
By denit
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 Theorem, a powerful result useful for calculating the
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 fundamental
group construction
1 : cfw_spaces cfw_groups ; X 1 (X
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 , . . . , vn ]
is identied with [v0 , . . . , vk , . . . , vn ] by
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
the manner in which covering spaces can be used prove so
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-connected covering space between X and X/G is isomorphi
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 with
great success towards homology computations on si
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 introduced later, we want functors from the category of to
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 one 0-cell, one 2-cell and one 4-cell. Hence
the cellula
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) = Z, H1 (R) =
g Z, Hn (R) = 0 for n > 1. Also we know H0
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 examples of manifolds. Rn is certainly an n-manifold.
Also
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.
multiplicative notation
G, H, K, N
gh
e=1
g 1
gn = 1
f (gh
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 product cell structure and cellular homology.
There are cell
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 Higher Direct Images of Sheaves
39
9 Flat Morphisms
42
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 2, this gives y = 0 and 2 x3 - x = 0. Since Hx, yL lie
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 @x0 , , xn D.
Then for any f e IHY L, f is homogeneous
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 get the same ideal. So we may reduce to the case in wh
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 containing Y ; since X is closed X = ZHAL for some ideal A.
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 that f (P ) = 0
for all P Z(I) in Pn , then f q I for so
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, P is a prime ideal and P = I(Z(f ).
Now k[x, y]/P k[x].
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 open subset of A1 .
6.1b We can assume Y = A1 \cfw_a1 , .
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. Futhermore, if l(D) = 2, dim|D| = 1, thus |D| denes a
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 proper subset of A1 has invertible elements not in k and o i