For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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14
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114 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
10
votes
0answers
229 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
9
votes
0answers
67 views

Why are noetherian and artinian modules important?

As a TA I was recently asked to give the students an introduction to two (quite related) concepts that are new to me, noetherian and artinian modules. I intend to prove the characterisation theorem ...
8
votes
0answers
253 views

An elegant description for graded-module morphisms with non-zero zero component

In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the ...
7
votes
0answers
66 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
7
votes
0answers
341 views

Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
6
votes
0answers
225 views

Elementary divisors theorem for Dedekind domains (Exercise in Lang's Algebra)

Exercise 13 (b) of Chapter III in Lang's Algebra is as follows. Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and ...
6
votes
0answers
80 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
6
votes
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275 views

Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem

For any matrix A with entries in a PID, there exist invertible matrices P and Q such that B = PAQ, where B is in Smith normal form. This theorem is usually proved by using elementary row/column ...
5
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42 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
5
votes
0answers
43 views

Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
5
votes
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61 views

On semisimple rings

Let $R$ denote a ring with unity. I know that, if $R$ is semisimple, then every $R$-module is semisimple. In particular the class of indecomposable $R$-modules coincides with the class of simple ...
5
votes
0answers
169 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...
5
votes
0answers
103 views

Category of Chain Complexes of $R$-modules

So I have a couple of questions: 1- Formally speaking, what is a "quotient of a chain complex" of $R$-modules? 2- I want to show that any chain complexes of $R$-modules $C_\bullet $ is the ...
5
votes
0answers
98 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
5
votes
0answers
183 views

Dual modules and first cohomology

Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module. What hypotheses are needed on $G$, $M$ to ...
4
votes
0answers
28 views

Integer such that there is a $k$-algebra isomorphism for any two algebras.

Is there an integer $\ell = \ell(m, n) \ge 1$ such that for any $k$-algebras $A$ and $B$ there is a $k$-algebra isomorphism $\text{M}_m(A) \otimes_k \text{M}_n(B) \cong \text{M}_\ell(A \otimes_k B)$?
4
votes
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101 views

Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & ...
4
votes
0answers
38 views

Too Many Members in a Finitely Generated Module are Linearly Dependent

I am new to module theory and as of now am not very comfortable with the subject. So can somebody please check whether my claim and its proof is okay? Consider the following statement: Let $M$ be ...
4
votes
0answers
54 views

Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
4
votes
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53 views

Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
4
votes
0answers
69 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
4
votes
0answers
56 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
4
votes
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79 views

Property of free submodules for a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
4
votes
0answers
159 views

The dimension of vector space $F^{X}$.

Here's the problem: Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and ...
4
votes
0answers
49 views

Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
4
votes
0answers
70 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
4
votes
0answers
71 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
4
votes
0answers
139 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 ...
4
votes
0answers
70 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) ...
4
votes
0answers
115 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is ...
4
votes
0answers
70 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
4
votes
0answers
121 views

Finitely presented modules and torsion

Suppose $R$ is a Dedekind domain and $A$ a $(m \times n)$-matrix of rank $r$ over $R$. $A$ induces an $R$-module homomorphism $\varphi:R^m \to R^n$ via $x \mapsto xA$ giving rise to an exact sequence ...
4
votes
0answers
187 views

When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules: ...
4
votes
0answers
69 views

Decomposing the group algebra $\mathbb{R}[Q]$ as a product of matrix rings

I am trying to decompose $\mathbb{R}[Q]$ as a product of matrix rings ($Q$ is the group of quaternions) By Maschke's theorem, $\mathbb{R}[Q]$ is semisimple. I will begin by decomposing it as a left ...
4
votes
0answers
161 views

Comparison of positive elements and Hilbert C*-modules

I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras. Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
4
votes
0answers
29 views

An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
4
votes
0answers
85 views

Separability of finitely generated projectives over commutative ring

A class $\mathcal{C}$ of $R$-modules is called -separative if $A \oplus A \simeq A \oplus B \simeq B \oplus B$ implies $A \simeq B$ for each $A,B \in \mathcal{C}$ -cancelative if $A \oplus C \simeq ...
4
votes
0answers
42 views

Normal Form problem (in a module over a PID)

Let $A,B$ be $n\times n$ matrices with entires in a PID $D$ and $\det AB\neq 0$. Suppose diag$\{a_i\}$, diag$\{b_i\}$, and diag$\{c_i\}$ are normal froms for $A$, $B$, and $AB$. In particular, ...
4
votes
0answers
379 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
4
votes
0answers
122 views

Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
4
votes
0answers
259 views

Zero divisors in modules?

Let $R$ be a ring. I find myself considering $M=R^n$, as an $R$-module. If $a$ is not a zero divisor in $R$, it holds that $\forall x \in M: ax=0 \Rightarrow a=0 \vee x=0$ . For what kinds of ...
4
votes
0answers
87 views

Decompose a module $M$ of the form $N \times N$, where $N$ is simple

Let $M$ be a $\mathbb{C}[G]$-module of the form $M=N\times N$, where $N$ is simple. How to conclude that $M$ has infinitly many direct sum decompositions into two copies of $N$ ? This is what I have ...
4
votes
0answers
129 views

Why has the category of all discrete $G$-modules not enough projectives when $G$ is profinite?

I found this article about Galois Cohomology. In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is the category of all discrete $G$-modules, $\mathbf{C}_G$ doesn't have enough ...
3
votes
0answers
22 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
3
votes
0answers
31 views

Does $M \otimes_R N = 0$ for a non-unital ring $R$ if there are ideals $I,J \lhd R$ such that $MI+JN = 0$ and $I+J = R$?

Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$. This is very easy to prove if the ring is unital as you may write ...
3
votes
0answers
56 views

Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
3
votes
0answers
49 views

How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here ...
3
votes
0answers
68 views

Every module over a division ring is free?

I am currently trying to answer the following true/false question: True or False: Every module over a division ring $R$ is free. I know the result would be true if $R$ is a field (ie a ...
3
votes
0answers
61 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...