Hower, Martinez , Moorhead
Exercise 5.6
Show that every extension of an abelian group B of bounded exponent by a torsion-free abelian group F splits.
Consider the exact sequence 0 Z Q. All torsion free abelian groups are at as Z-modules, so
tensoring with
math 8714: Topics in Algebra
April 8, 2010
Assignment V
Matt Jones, John Tuley
5 Give an example to show that the Universal Coefficient Theorem for Homology may fail for complexes that
are not flat.
Solution. Consider the chain:
0
0
Q~Z
with Q~Z in the n
Lizzi, Wakeeld
Homological Algebra
1
Homework 5 Problem 4
Problem 1.
Show that if mA = cfw_0, then mExt1 (A, B) = cfw_0 and that if nB = cfw_0, then nExt1 (A, B) =
Z
Z
cfw_0.
Show that if A is uniquely m-divisible (i.e., the map a ma is an isomorphism of
Problem 3. (Gern, Selker). Let D be an abelian group. Then D is divisible i Ext1 (Q/Z, D) = 0
Z
Proof. (). Suppose that D is divisible. Then, by Baers criterion, D is injective so Ext1 (Q/Z, D) =
0.
(). Suppose that Ext1 (Q/Z, D) = 0. Thus also ext1 (Q/Z,
HOMOLOGICAL ALGEBRA HOMEWORK V
KELLER, PRATARELLI
Problem 2 Let A and B be R-modules and let A be a submodule of A. For a homomorphism
: A B, dene the obstruction of to be () where : Hom(A , B) Ext1 (A/A , B) is the
connecting homomorphism. Show that : A
HOMOLOGICAL ALGEBRA: HOMEWORK 5
MATTHEW MOORE AND BRYCE CHRIESTENSON
1) Atiyah-MacDonald 7.16: Let A be a Noetherian ring and M a nitely generated
A-module. Then the following are equivalent:
(1) M is a at A-module;
(2) Mp is a free Ap -module for all pri
HOMOLOGICAL ALGEBRA HOMEWORK IV
GERN, KELLER
Problem We saw in the last section that if R = Z (or more generally, if R is a principal ideal
domain), a module B is at i B is torsionfree. Here is an example of a torsionfree ideal I that is
not a at R-module
TorR (R/I, R/J)
=
1
J R
IJ
IJ
I
0 I R R/I 0
R/J
TorR (R/I, R/J) I R R/J R R R/J R/I R R/J 0
1
R R R/J R/J
: I/IJ I R R/J :
=
I R R/J I/IJ (i + IJ) = 1 (i + J) (j (r + J) = jr + IJ i, j I
r R
R IJ J
R
Lizzi, Moorhead
Exercise 4.5
A short exact sequence 0 A B C 0 is pure exact if it remains exact under tensoring with an
arbitrary module. Show that C is at i any short exact sequence ending in C is pure exact.
Suppose that C is at, let 0 A B C 0 be a SES
math 8714: Topics in Algebra
March 17, 2010
Assignment IV
4 Let 0
John Tuley, Nathan Wakefield
A B C 0 be an exact sequence of R-modules.
(a) Show that if C is flat, then A is flat iff B is.
Solution. Suppose that C and A are both flat. We follow the pro
Topics In Algebra, Homework 5
Yingwei Li, Charlie Scherer
April 7, 2010
Problem (5). Let p0 , p1 , p2 , . be the sequence of prime numbers.
a) t
nN
Zpn =
nN
Zpn .
b) If D is dened by the short exact sequence
0
Zpn
nN
then there is some extension of D by
math 8714: Topics in Algebra
May 3, 2010
Assignment VI
Yingwei Li, John Tuley
1 Calculate H 2 Z4 , Z2 .
Solution. We apply Theorem 6.2.2, which tells us that
H 2 Z4 , Z2 Z2 Z4 ~NZ2 ,
where Z2 Z4 is the subgroup of Z2 fixed under the action of Z4 (in other
HOMOLOGICAL ALGEBRA
HOMEWORK ASSIGNMENT V
Read pages 73-90.
PROBLEMS
1. (Chriestenson, Moore) Exercise 7.16 of Atiyah-Macdonald.
2. (Keller, Pratarelli) Let A and B be R-modules and let A be a submodule of A. For a homomorphism : A B dene the obstruction
HOMOLOGICAL ALGEBRA
HOMEWORK ASSIGNMENT IV
Read pages 30-73.
PROBLEMS
1. (Martinez, Moore) Suppose T is a right exact functor and some
derived functor LTn , n > 0, is also right exact. Show that LTm is the zero
functor for all m n.
2. (Hower, Jones, Selke
HOMOLOGICAL ALGEBRA
HOMEWORK ASSIGNMENT III
Read pages 5-29.
Terminology: exact functor = functor preserving exact sequences. Additive functor
= a functor F : C D between additive categories such that the induced function
F : HomC (A, B) HomC (F (A), F (B
HOMOLOGICAL ALGEBRA
HOMEWORK ASSIGNMENT II
Read pages 424-431. Terminology: Ab-category = preadditive category.
PROBLEMS
1. (Selker, Wakeeld)
(a) Show that Z2 is a cogroup in the category of abelian groups.
(b) Show that Z2 is not a cogroup in the categor
HOMOLOGICAL ALGEBRA
HOMEWORK ASSIGNMENT I
Read pages 1-5, 417-424.
PROBLEMS
1. (Chriestenson, Gern, Harper) Exercise 1.1.6 of Weibel.
2. (Hower, Jones) Exercise A.1.3 of Weibel. (Change the word isomorphic to equivalent.)
3. (Keller, Li) First part of exe
Problem 6. (Martinez, Selker). Given a homomorphism : G G one can convert any
G-module into a G module by restriction of scalars. Show that such a function induces an additive
homomorphism on n-cochains:
: C n (G, A) C n (G , A) : f f n .
Show that maps
Tyson Gern, Adam Lizzi
Homework 6, Problem 5
May 3, 2010
Problem 5. Describe all central extensions of Z2 Z2 by Z2 up to equivalence. Explain why
some inequivalent extentions have isomorphic middle factors.
Solution. Before we start on the problem at hand
Topics In Algebra, Homework 6
John Hower, Charlie Scherer, Nathan Wakeeld
April 28, 2010
Problem (4). Let E and F be elds such that E = F ( ) for some F and suppose that G := Gal(E/F )
is non-trivial (this happens if and only if is a non-square in F and E
HOMOLOGICAL ALGEBRA HOMEWORK VI
KELLER, MOORHEAD
Problem 3
Suppose that G = cfw_1, g has two elements, and that A is a G-module.
(a) Show that for n > 0
H n (G, A) =
ker(g + 1)/im(g 1) n odd;
ker(g 1)/im(g + 1) n even.
(b) Let A = C and let G = cfw_1, g w
ASSIGNMENT 6 - QUESTION 2
CHRIESTENSON, PRATARELLI
Exercise 2. Calculate H 1 (Aut(A), A) when A is the Klein group.
Proof. Note that A = Z2 Z2 and Aut(A) = S3 . This is because A has three nonidentity elements all of order two, so they can be permuted in
Topics In Algebra, Homework 4
Nick Pratarelli, Charlie Scherer
March 18, 2010
Problem (3). Suppose a, b, m are positive integers with a, b|m so that Za and
Zb are Zm -modules.
1. TorZm (Za , Zb ) Zc with c = gcd(a, b).
=
0
2. For n > 0, TorZm (Za , Zb ) Z
Problem 2. (Hower, Jones, Selker).
(a) Show that any Z4 -module is isomorphic to one of the form
and .
Z2
Z4 for some
Z
(b) Explain how to determine the isomorphism type of T orn 4 (A, B) for any Z4 -modules A and
B.
Solution:
(a) Let X M be maximally i
Homological Algebra
Homework 2 - Question 3
Chriestenson, Hower
Problem: Imagine the integer polynomial ring in proper class of non-commuting variables:
Z cfw_x | an ordinal
Let C be the category R-modules, by which we mean the category whose objects are
Topics In Algebra, Homework 2
Andrew Moorehead, Charlie Scherer
February 11, 2010
Problem (2). In the following every category is taken to be a full subcategory
of, Ab, the category of abelian groups.
a) The category of torsion abelian groups is an abelia
Problem 1 (Selker, Wakeeld). (a) Z2 is a cogroup in the category of abelian groups
(b) Z2 is not a cogroup in the category of all groups.
(c) Z2 is not a cogroup in the category of groups of exponent 4.
Proof. (a) Because coproducts exist in the category
Nathan Wakeeld, Josh Wiscons
Homework 1, Problem 8
Let C := O, M ; , id, dom, cod be a small category and S the category of sets. As C is
small, M is a set, so for each A O, HomsTo(A) := cfw_f M : cod(f ) = A is also a set.
Hence, we may dene a map F : C
Topics In Algebra, Homework 1
Charlie Scherer, Kevin Selker, John Tuley
January 27, 2010
Problem (7). Let O be the ordered pair functor from Set to itself (O(A) = A A, O(f ) = f f ). Let U
be the unordered pair from Set to itself, (U (A) = cfw_x, y : x, y