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26 2. Exact Sequences

This document lists 10 exercises related to exact sequences of modules. The exercises ask the reader to prove properties of exact sequences, construct exact sequences from commutative diagrams, and give examples of situations where exactness does not imply surjectivity of maps between homomorphism groups.

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Benito Camelas
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261 views2 pages

26 2. Exact Sequences

This document lists 10 exercises related to exact sequences of modules. The exercises ask the reader to prove properties of exact sequences, construct exact sequences from commutative diagrams, and give examples of situations where exactness does not imply surjectivity of maps between homomorphism groups.

Uploaded by

Benito Camelas
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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26 2.

EXACT SEQUENCES

Exercises
Exercise 2.1. Show that
0 !M !0
is exact if and only if M is the zero module.
Exercise 2.2. Let f : M ! N be an R-module homomorphism.
Show that there is an exact sequence
f
0 ! ker f ! M ! N ! coker f ! 0
of R-modules.
Exercise 2.3. Complete the proof of the Five Lemma (Theorem
2.1): show that if f2 and f4 are surjective, and if f5 is injective, then
f3 is surjective.
Exercise 2.4. Let
M1 M2 M3 0
f1 f2

N1 N2 N3 0
be a commutative diagram of R-modules with exact rows. Show that
there exists a unique R-linear map f3 : M3 ! N3 making the resulting
diagram commute.
Exercise 2.5. Consider a commutative diagram of R-modules
↵
0 M E N 0
idM f idN

0 M E0 N 0
in which both rows are short exact sequences. Deduce from the snake
or five lemma that f must be an isomorphism. Show that f is an iso-
morphism without using the snake or five lemma. (Hint for surjectivity:
given y 2 E 0 choose an x 2 E with same image as y in N . Show that
there is an z 2 M with f (↵(z) + x) = y.)
Exercise 2.6. Give an example of a diagram as in Theorem 2.2,
for which the ‘snake map’ d : ker f3 ! coker f1 is non-zero.
EXERCISES 27

Exercise 2.7. Let R be a ring and let


M1 M2
↵1 ↵2

N1 N2
be a commutative diagram of R-modules, in which the two horizontal
maps are injective. Show that there exists an R-module E and an exact
sequence
0 ! ker ↵1 ! ker ↵2 ! E ! coker ↵1 ! coker ↵2
of R-modules.
Exercise 2.8. Let R be a ring, let
(2) 0 ! M1 ! M2 ! M3
be an exact sequence of R-modules, and let N be an R-module. Show
that there is an exact sequence of abelian groups
0 ! HomR (N, M1 ) ! HomR (N, M2 ) ! HomR (N, M3 ).
Give an example to show that the exactness of 0 ! M1 ! M2 !
M3 ! 0 need not imply that the map HomR (N, M2 ) ! HomR (N, M3 )
is surjective.
Exercise 2.9. Let R be a ring, let
M1 ! M2 ! M3 ! 0
be an exact sequence of R-modules, and let N be an R-module. Show
that there is an exact sequence of abelian groups
0 ! HomR (M3 , N ) ! HomR (M2 , N ) ! HomR (M1 , N ).
Give an example to show that the exactness of 0 ! M1 ! M2 !
M3 ! 0 need not imply that the map HomR (M2 , N ) ! HomR (M1 , N )
is surjective.
Exercise 2.10. Let I and J be left ideals in a ring R. Show that
there are exact sequences
0!I \J !I J !I +J !0
and
0 ! R/(I \ J) ! R/I R/J ! R/(I + J) ! 0
of R-modules.

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