Tagged Questions
1
vote
0answers
48 views
Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.
I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following.
We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
2
votes
2answers
47 views
Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$
Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $.
I ...
0
votes
1answer
55 views
exact sequence problem
Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} ...
1
vote
1answer
43 views
hom and exact sequence
Let $$ 0 \longrightarrow \operatorname{Hom}(M,Β_1) \stackrel{f^*}\longrightarrow \operatorname{Hom}(M,Β) \stackrel{g^*}\longrightarrow \operatorname{Hom}(M,Β_2) $$ be an exact sequence for any ...
0
votes
1answer
34 views
Applying an additive covariant functor to a sequence
Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
1
vote
2answers
75 views
Help proving a short exact sequence
Show the following sequence is an exact sequence of $\mathbb Z$-modules when $n$ is a positive integer such that $n=rs$:
$$ 0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0. $$
...
0
votes
0answers
39 views
Cokernels of Modules and Exact Sequences
Suppose we have n-dimensional free modules over a dvr $T_1$, $T_2$, $T_3$ (i.e. $T_i \cong R^n$) such that
$$0 \to T_1 \to T_2 \to T_3 \to 0$$
is exact. Suppose further that $E_1,E_2,E_3$ are ...
0
votes
0answers
30 views
A proposition on exact sequence of inverse limit (Lang, Algebra, p. 165)
I am trying to understand this proof.
My only question is that what are the vertical maps here?
0
votes
1answer
25 views
Equivalence between exact sequence of module and its induced one.
Let $X,X',X''$ be $A$-modules and denote by $\mbox{Hom}_A(X',X)$ the set of $A$-homomorphisms of $X'$ into $X$.
Proposition 2.1 in the Lang's Algebra text states the following:
A sequence
$$
X' ...
0
votes
1answer
55 views
Short Exact Sequences
Let $M \ge N \ge P$ be R-modules. Prove that there exist natural (not depending on choices) R-homomorphisms $N/P \to M/P$ and $M/P \to M/N$ for which the sequence $0 \to N/P \to M/P \to M/N \to 0$ is ...
3
votes
0answers
69 views
exact sequence and modules proposition.
I have problems to prove the following proposition:
Let's consider $$0 \rightarrow L \stackrel{\alpha}{\rightarrow} M
\stackrel{\beta}{\rightarrow} N \rightarrow 0$$ an exact sequence of modules
and ...
2
votes
1answer
52 views
A Sort of Exact Sequence
I have not given a lot of thought to this question: It may be very easy or very hard or somewhere in between.
Suppose we have a sequence of modules and morphisms which looks like
$ \ldots \to A_1 ...
1
vote
1answer
45 views
Exact Sequences of R-Modules
In "A Course in Ring Theory by Passman" it is mentioned, "But the kernel of the combined epimorphism $P\rightarrow B\rightarrow C$ is clearly equal to $E$". I don't understand this part. How can the ...
0
votes
1answer
33 views
Why these two propositions have different requirements
Proposition 2.18 is similar to 2.19.
Why we need $N$ flat in 2.19?
What's the difference between 2.18 and 2.19?
3
votes
1answer
90 views
Showing $M\cong M'\oplus M''$ given an exact sequence
I am struggling with the following question:
$R$ is a ring. $$M'\overset{f}{\longrightarrow} M\overset{g}{\longrightarrow} M''$$ are homomorphisms of $R$-modules such that for any $R$-module $N$, the ...
1
vote
1answer
41 views
If an $A$-module $M$ is locally finitely presented (resp. related) then $M$ is finitely presented (resp. related)
In this question I want to ask for a better proof than the one I am about to give for the statement with finitely presented, and inquiry if the statement is also true for the notion of finitely ...
3
votes
1answer
74 views
Determinant of long exact sequence
Let the following be a long exact sequence of free $A$-modules of finite rank:
$$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$
I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where ...
1
vote
1answer
44 views
Constructing an exact sequence?
Let $R=\mathbb{Z}[X,Y]$. I'm trying to construct an exact sequence of $R$-modules
$$
0\to R\stackrel{f}{\to} R\oplus R\stackrel{g}{\to} R\stackrel{h}{\to}\mathbb{Z}\to 0
$$
where $h(p(X,Y))=p(0,0)$.
...
5
votes
2answers
117 views
How to prove surjectivity part of Short Five Lemma for short exact sequences.
Suppose we have a homomorphism $\alpha, \beta, \gamma$ of short exact sequences:
$$
\begin{matrix}
0 & \to & A & \xrightarrow{\psi} & B & \xrightarrow{\phi} & C & \to & ...
1
vote
1answer
82 views
Exact sequence of $R$-modules
Let $0\longrightarrow N\overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}L\longrightarrow0$
be a short exact sequence of $R$-modules. Prove that this chain splits iff
$f(N)$ is direct ...
0
votes
1answer
70 views
How do you prove a piece of the Short Five Lemma?
Let $\alpha, \beta, \gamma$ be a homomorphism of short exact sequences, in that order. Then if $\alpha, \gamma$ are injective, then so is $\beta$.
Let the sequeces be:
$$
\begin{matrix}
0 & \to ...
2
votes
4answers
153 views
Short exact sequences of modules.
My understood definition: $0 \to A\stackrel{\psi} \to B\stackrel{\phi} \to C \to 0$ is a short exact sequence of modules iff $\psi$ is injective, $\phi$ is surjective, and $\ker \phi = \psi(A)$, which ...
2
votes
1answer
79 views
If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is exact and $B\simeq A\oplus C$ as a $R$-module, does this sequence split?
Suppose $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence that $B\simeq A\oplus C$ as a $R$-module. Does this short exact sequence split?
I think the answer is no, ...
1
vote
2answers
102 views
Exact sequence of modules
Let $R$ be a ring. Let's say we have an exact sequence of $R$-modules
$$0\rightarrow P\rightarrow R^2 \overset{f}\rightarrow R\rightarrow 0,$$
where $P\cong\ker(f)$.
Because of $R$ beeing a free ...
5
votes
1answer
129 views
Equivalence between Ext and Hom
This is a question from Homology by Saunders Mac Lane. This is problem 5 page 76. I've been struggling to solve this problem for like more than a day, but still nothing valuable comes across my mind ...
5
votes
2answers
99 views
Question on $\mbox{Ext}^1$
I have 2 questions, one of them concerning the isomorphicity of quotient groups (rings), and the other is on $\mbox{Ext}^1$. It's pretty long, but somehow related to each other. So I just kinda put ...
3
votes
2answers
231 views
Characterization of short exact sequences
The following is the first part of Proposition 2.9 in "Introduction to Commutative Algebra" by Atiyah & Macdonald.
Let $A$ be a commutative ring with $1$. Let $$M'
...
1
vote
1answer
71 views
Which Short Exact Sequences Can I Extract From A Doubly Infinite Exact Sequence?
I know how if we have a short exact sequence of $R$ modules,
$0 \rightarrow A_1 \rightarrow A_2 \rightarrow A_3 \rightarrow 0$ , we can deduce properties about the known modules from the unknown ...
3
votes
1answer
353 views
Exact sequences and functor Hom
For an abelian group $G$ we denote by $G^*$ the $\mathbb{Z}$-module $\text{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ -- group of all $\mathbb{Z}$-module homomorphisms from $G$ to ...
16
votes
4answers
388 views
Non-isomorphic exact sequences with isomorphic terms
I love it when an undergraduate catches me out. I'm lecturing a first course in (not necessarily commutative) rings (with 1) and I've spent the last few weeks doing basic module theory. I defined a ...
