0
votes
1answer
28 views

Show that Q, as a Z module, is a direct summand in a direct product of copies of Q/Z.

Prove:Q, as a Z module, is a direct summand in a direct product of copies of Q/Z. This is a problem from P.J.Hilton&Stammbach's Homological Algebra. If this is true, then there exists a ...
3
votes
1answer
55 views

Why we can consider both modules as modules over $R_{(p)}$? (Bruns and Herzog, Theorem 1.5.9)

I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over $R_{(p)}$ in this theorem? ... i know that ...
2
votes
1answer
37 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 ...
2
votes
1answer
62 views

Is any right exact sequence of modules induced by free modules?

Let $R$ be a ring and let $M \to N \to K \to 0$ be an exact sequence of $R$-modules. Is there an exact sequence of free modules $A \to B \to C \to 0$ and a commutative diagram $$\begin{array}{c} M ...
0
votes
1answer
35 views

Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
1
vote
1answer
29 views

Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
1
vote
2answers
52 views

Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
1
vote
1answer
23 views

Hom($P$, $R$) $\neq 0 $ if $P$ is a nonzero projective left $R$-module (Rotman)

I've found this exercise, number $3.11$ from Introduction to homological algebra. Prove that $\operatorname{Hom}(P, R) \neq 0 $ if $P$ is a nonzero projective left $R$-module. Any hint?
1
vote
0answers
17 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
2
votes
0answers
39 views

Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
2
votes
0answers
102 views

Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
0
votes
1answer
52 views

Doubt about the tensor product

Suppose $M$ is an abelian group and $F(X)$ is the free abelian group over $X$. Is it true that any element of $M\otimes F(X)$ can be written as a finite sum $$m_{1}\otimes x_{1}+ \cdots+m_{n}\otimes ...
0
votes
0answers
44 views

Problem about tensor products and modules over a group algebra

Can anyone help me with hints for these problems? I would appreciate it a lot. 1) Supose $G$ is a group and $k$ is a field. If $U,V$ and $W$ are $kG$-modules then $\operatorname{Hom}_{kG}(U\otimes ...
4
votes
1answer
47 views

Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion.

Given an integral domain $R$, and a left torsion $R$-module $A$ (i.e. $\forall{a}\in A,\exists{r}\in R$ such that $ra=0$) how would you show that $\mathrm{Tor}_n^R(A,B)$ is also a torsion $R$-module?
5
votes
1answer
114 views

How to calculate $Ext(M,N)$?

I am confused about the calculation of $\text{Ext}(M,N)$. If $N$ is a fixed module and if we consider the projective resolution $$\cdots \to C_1 \to C_0 \to M \to 0,$$ then $\text{Ext}_n(M,N)$ is the ...
3
votes
1answer
73 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 ...
2
votes
1answer
29 views

When are maps between Hom sets induced?

I'm trying to better understand $R$-module homomorphisms, and I know that say, an $\, f:M\to N$ induces $\, f_*:Hom_R(V,M)\to Hom_R(V,N)$ or $\, f^*:Hom_R(N,V)\to Hom_R(M,V)$. What I'm wondering is, ...
4
votes
0answers
67 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
0
votes
0answers
47 views

Global dimension of endomorphism rings

Does anybody have an idea on how the global dimension of the endomorphism ring of a module over a (nice enough) ring is related to the global dimension of the endomorphism ring of its projective ...
3
votes
1answer
52 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 ...
2
votes
1answer
26 views

Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
1
vote
2answers
34 views

Showing that an epimorphism to a free module of finite rank splits

Let $M$ be an $R$-module and let $F$ be a free $R$-module of finite rank. Let $\phi : M \to F$ be an epimorphism. Then show that $M$ has a submodule $F' \cong F $ such that $M=F' \oplus \ker\phi$. ...
3
votes
0answers
45 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
2
votes
1answer
57 views

Homology of Chain Complexes from Free Resolution

Suppose I have an $R$-module $M$ and a free resolution $$ \ldots \to F_2 \to F_1 \to M \to 0. $$ I apply an additive functor $f$ in $R$-$\mathbf{Mod}$ to the free resolution to get $$ \ldots \to ...
8
votes
1answer
73 views

An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are ...
6
votes
1answer
91 views

Is it true that Tensor product of injective modules is injective?

Is it true that if $M$, $N$ are injective modules over a commutative ring $R$ (with identity) then $M\otimes_R N$ is also injective ?
0
votes
1answer
60 views

Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
0
votes
1answer
61 views

When do we have $m\otimes n = 0$ [duplicate]

Let $M$ and $N$ be $R$-modules ($R$ a commutative ring with identity). Let $m \in M$ and $n \in N$. Is there any necessary and sufficient condition to have $m\otimes n = 0$ (as an equation in ...
6
votes
1answer
119 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
3
votes
1answer
181 views

A problem about an $R$-module that is both injective and projective.

Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$. This is exercise 7.52 of Rotman's Advanced ...
4
votes
1answer
98 views

On a commutative diagram

Let a commutative diagram: \begin{array}{ccccccccc} 0 & \longrightarrow & A & \overset{f}{\longrightarrow} & B & \overset{g}{\longrightarrow} & C & \longrightarrow & ...
1
vote
1answer
72 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
47 views

If $A \cong A^*$, is every projective module also injective?

Suppose $A$ is a finite-dimensional algebra over $k$. Assume further that $A \cong A^* = \text{Hom}(A,k)$ as $A$-modules. My question is: is every finite dimensional projective module over $A$ ...
0
votes
1answer
104 views

decomposition of cokernel

Lets work over $\mathbb Z$ and represent maps $f:\mathbb Z^l \longrightarrow \mathbb Z^n$ as matrices. For the following matrices write an equivalent diagonal matrix. I want to write a decomposition ...
1
vote
2answers
56 views

Injective modules under change of rings

Let $R$ be a ring with identity, $I$ an ideal and $M$ a left injective module with $IM= 0$. How can I show that $M$ is an injective $\frac RI$ module?
8
votes
1answer
132 views

A direct product of projective modules which is not projective

I am looking for an elementary example of a family $\{M_\alpha\}_\alpha$ of projective $R$-modules whose direct product is not projective. The simplest example that I know is the $\Bbb{Z}$-modules, ...
5
votes
2answers
174 views

characterization of projective/injective/flat modules via $\operatorname{Hom}$ and $\otimes$

Let $R$ be a commutative unital ring and $M$ an $R$-module. Then $M$ is projective iff $\operatorname{Hom}(M,-)$ is exact, injective iff $\operatorname{Hom}(-,M)$ is exact, and flat iff $M\otimes-$ is ...
2
votes
1answer
61 views

Existence of injective hull

Is that true that every module over a ring has an injective hull? The term "has an injective hull" appears in several different contexts, some of them say that modules over a ring has this property ...
4
votes
1answer
56 views

Weak flat condition?

Let $R$ be a unit ring (not necessarily commutative). Then it is clear that for a right $R$-module $M$ we have: $M$ is flat $R$-module $\Rightarrow$ for any left $R$-module $E$ with $E\otimes_{R}M=0$ ...
2
votes
2answers
160 views

finitely generated & finitely related = finitely presented module?

Let $R$ be a ring $M$ an $R$-module. How can I prove that if $M\cong R^n/N$ for some $n\!\in\!\mathbb{N}$ and some submodule $N\leq R^n$ and if $M\cong R^{(I)}/\langle u_1,\ldots,u_m\rangle$ ...
1
vote
0answers
47 views

Rejects and injectives

Let $A$ be any ring and consider modules on the left. It is well known that the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then: $Tr(P,A)P=P$, for $P$ projective; ...
5
votes
2answers
159 views

$p$ prime, $P = \left\{ \frac{m}{p^e} \middle| m, e\in \mathbb{Z} \right\}$. Prove that $\mbox{Ext}(P; \mathbb{Z}) \cong \mathbb{Z}^{(p)}/\mathbb{Z}$

I don't know why the book Homology by Saunders Mac Lane is wwaaayyy tttoooo hard to digest. :((( This is like the third time I read this book, but still not clear is everything, and to tell the ...
5
votes
1answer
121 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 ...
2
votes
1answer
119 views

Submodules of $\operatorname{Hom}_R(M,N)$ with $R$ a commutative ring.

Is there a way to characterize the submodules of the $\operatorname{Hom}_R(M,N)$? $M,N$ are arbitrary $R$-modules and $R$ a commutative ring, to assure that $\operatorname{Hom}$ will be an ...
2
votes
1answer
82 views

Finite generation of Hom between cyclic and artinian module

Let $R$ be a Noetherian ring with unit, and $I$ be a nonzero ideal of $R$. Let $M$ be an artinian $R$ module. Is $\operatorname{Hom}(R/I, M)$ finitely generated? Thanks.
6
votes
2answers
156 views

Property of modules via exact sequences

Suppose $A\neq 0$ is a commutative ring with $1$. Let $L, M, N$ be $A$-modules such that the sequence $$0\longrightarrow L\overset{\alpha}{\longrightarrow} M\overset{\beta}{\longrightarrow} ...
5
votes
2answers
95 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 ...
5
votes
1answer
80 views

If $M \simeq N$ in ${\tt stmod}(G)$ will $M \oplus \text{(proj)} \simeq N \oplus \text{(proj)}$ in ${\tt mod}(G)$?

Let $G$ be a finite group and ${\tt stmod}(G)$ the stable module category for $G$, i.e., the category whose objects are $G$-modules and whose morphisms are $G$-module homomorphisms modulo those that ...
3
votes
3answers
446 views

Proving that free modules are flat (without appealing projective modules)

Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first ...
7
votes
1answer
135 views

Question on Projective Dimensions

$\require{AMScd}$I have a question regarding a claim in A first course of homological algebra by Northcott. I think it's very easy, since the author didn't provide a proof, and just kind of claimed ...