Questions tagged [modules]

For questions on modules over rings.

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Functors between module categories that comes from restriction

Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod $ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space. ...
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144 views

Flatness over a local noetherian ring

Let $(R,\mathfrak m)$ be a local noetherian ring, and $M$ an arbitrary $R$-module. Suppose that $\mathrm{Tor}_1(M,R/\mathfrak m)=0$. Does it follow that $M$ is flat? The answer is positive when $M$ ...
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90 views

How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
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0answers
50 views

Faithfull flatness of a module containing the ring as a direct summand

Let $R$ be a not necessarily commutative ring, and let $M$ be a projective left $R$-module. Question. If $R$ is a direct summand of $M$ as a left $R$-module, then is it true that $M$ is faithfully ...
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1answer
136 views

Is the rank of free module spectra unique?

Given a commutative ring, the rank of a free module is unique. This is the well known statement that commutative rings have invariant basis numbers. Does an analogue of this property hold for free ...
4
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1answer
343 views

A similar construction to Ext, can we describe it better and does it have any use?

Let $R$ be a ring and $\text{Mod}\,R$ the category of $R$ modules. For two $R$-modules $X,Y$ one can define $\text{Ext}_R^n(X,Y)$ as follows. We take an injective resolution $0\rightarrow Y\rightarrow ...
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3answers
968 views

Is every additive, left exact functor isomorphic to a hom functor?

Let $A$ be an Artin algebra, $\text{mod}\,A$ the category of finitely generated $A$-modules and $\text{Ab}$ the category of abelian groups. Is every additive, covariant, left-exact functor $F:\text{...
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42 views

submodule between free modules with rank $d$ on a valuation ring

Let $A$ be a valuation ring (not necessary DVR) and $M_{1}, M_{2}$ two free modules with rank $d$. Assume that $M_{1}\subset M_{2}$ and $M$ is a submodule $M_{1}\subset M\subset M_{2}$. Then Is $M$ ...
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0answers
47 views

When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?

$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...
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28 views

Making a generating set of a section of a graded polynomial $R$-module coming from a quotient into a basis of a quotient by higher degree polynomial

Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $...
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1answer
194 views

Is the class of isomorphism types of a module category always a set?

Let $A$ be a ring and $\text{mod} A$ the category of finitely generated (right) modules over $A$. Is the class of isomorphism types of $\text{mod} A$ always a set? In particular, is it the case if $A$ ...
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85 views

Surjection from finite rank free $R$-module to finitely generated $R$-module and basis associated to generator set

Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ ...
5
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1answer
224 views

Must the inclusion of an indecomposable module in the direct sum of two copies always split?

We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module and let $f:X \longrightarrow X \oplus X$ a monomorphism. Must $f$ always be a split monomorphism?
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1answer
112 views

Indecomposable modules such that the radical is a submodule of the socle

We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module such that the radical $\text{rad} \,X$ is a submodule of the socle $\text{soc}\,X$. What can we say ...
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1answer
237 views

On the annihilator of a module

Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$? Remark. The annihilator of a ...
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68 views

Filtration over tensor product

Let $$ M \supset M_1 \supset \ldots \supset M_n \supset \ldots \text{ and } N \supset N_1 \supset \ldots \supset N_n \supset \ldots$$ be exhaustive decreasing filtrations of modules over a commutative ...
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81 views

Extending finitely many primitive elements of $\mathbb{Z}^{n+1}$ to bases with non-trivial intersection

Given a finite number of primitive elements $v_1,\dots,v_k\in\mathbb{Z}^{n+1}$ (i.e. the gcd of the entries of each $v_i$ is $\pm1$), is it always possible to find an element $v\in\mathbb{Z}^{n+1}$ ...
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92 views

Isomorphism of modules over an algebra in a monoidal category $\mathcal{C}$

I'll re-ask this because the other question was very poorly formulated by me. If $\mathcal{C}$ is a monoidal category, then the definition of a left module category over $\mathcal{C}$ is the same as a ...
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45 views

Existence of non-zero pseudo-null submodules

Let $p$ be a rational prime, and let $\Lambda_d$ be the Iwasawa algebra in $d$ variables, i.e. $\Lambda_d = \mathbb{Z}_p[[T_1, \ldots, T_d]]$. Let $A$ be a finitely generated and torsion $\Lambda_d$-...
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135 views

Associated point of coherent sheaf

$\DeclareMathOperator\Ann{Ann}\DeclareMathOperator\Ass{Ass}$That's a question about a proof I found in E. Sernesi's Deformations of algebraic schemes on page 188: The sheaf $F$ is assumes to be ...
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1answer
395 views

When is the category of finitely presented modules abelian?

Let $R$ be an associative ring with identity and $\mathrm{mod}R$ be the category of finitely presented $R$-modules. I would like to know when the category $\mathrm{mod}R$ is abelian. I know that if $R$...
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1answer
221 views

Making use of extra symmetries; more examples?

TL; DR. In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I ...
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54 views

Decomposition an $A$-module to irreducible ones

Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra. Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
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101 views

Loewy structure of $S_4$

How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
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1answer
267 views

For what modules is the endomorphism ring a division ring?

Let $k$ be a field, $A$ a $k$-algebra of finite length and $M$ an $A$-module of finite length. When does it happen, that $\text{End}(M)$ is a division ring? Notice if $M$ is simple, then it happens ...
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29 views

Rearrangements of Jordan–Holder quotients

Suppose we have a composition series $$0=M_n\subset M_{n-1}\subset\cdots\subset M_1\subset M_0$$ of $R$-modules, or one of groups: $$1=G_n\lhd G_{n-1}\lhd\cdots\lhd G_1\lhd G_0$$ (not necessarily ...
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1answer
107 views

Finitely presented modules admitting projective covers

A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ...
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0answers
74 views

Thin representations for quiver algebras

A representation $M$ of a quiver is called thin when $M$ has a dimension vector consisting only of 0 or 1 entries. When $A=kQ$ is a path algebra for a tree $Q$, then there is the nice result that ...
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1answer
86 views

Integral monoid rings and Ore conditions

Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$. I have two, ...
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1answer
154 views

Top and bottom composition factors of $M$ are isomorphic

Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...
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350 views

A question about pullbacks of $C^\infty_M$-modules

First, let me state a definition: Let $M$ be a smooth manifold and suppose $\mathcal{E}$ is a sheaf of $C^\infty_M$-modules. Given a point $x \in M$ let $I_x$ denote the vanishing ideal at $x$. We ...
5
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1answer
161 views

Simple quotients of a triple tensor product

Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V_1\otimes V_2\otimes V_3$. Is it ...
3
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1answer
122 views

MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra

Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$. Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
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1answer
148 views

Naive question on tensor product

Let $A, B$ be $\mathbb{C}$-algebras, which are also integral domains. Suppose there is an injective ring homomorphism $f:A \to B$. Assume further than $f$ is a finite morphism in the sense that $f$ ...
2
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1answer
118 views

If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?

EDIT: The question was originally about general Noetherian rings instead of PID's. Thanks to YCor for pointing out how wrong this was in the comments below (1 2 3). Question 1: Let $R$ be a PID. ...
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38 views

Definition of modular Howe correspondence

Let $(G,G')$ be a pair of mutually centralized subgroups of a symplectic group $Sp_n(\mathbb{F}_q)$ (called a dual pair), and let $\omega_{G,G'}$ be the restriction of the Weil representation (with ...
2
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1answer
180 views

Endomorphism rings of infinitely generated free modules generated by idempotents?

Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
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0answers
49 views

Invariants of a $kG$-module via its comoposition series, when does $M^P \supsetneq N^P$ hold for a $p$-group for $N\subseteq M$ maximal?

Let $G$ be a finite group, $k$ a field, $M$ a $kG$-module, $M^G$ the invariants of $M$ under $G$, $P$ a Sylow $p$-subgroup of $G$ where $p = \text{char}(k)$, $N$ a maximal submodule of $M$ and $S$ the ...
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0answers
73 views

Cone of a morphism of complexes that are concentrated in degree $0$ and $1$

Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...
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0answers
18 views

Weakening s-unitality

Recall that a (non-unital, non-commutative) ring $R$ is left s-unital if for every $r\in R$, we have $r\in Rr$. Consider the following conditions: There is a nonzero integer $m$ such that for all $r\...
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1answer
213 views

When does $\operatorname{Ext}_C^1(M,N_i)=0$ imply $\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0$?

Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\...
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0answers
56 views

On interpolation of endomorphisms

Let $R$ be a commutative ring and let $M$ be a $R$-module. Fix a non zero $m\in M$. We are given a family of endomorphisms of $M$ $(f_i)_{i \in I}$ and two functions $\alpha : I\times I \to I$, and $\...
4
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0answers
63 views

Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

Let $R$ be any ring and let $A\to B\to C\to [1]$ and $A'\to B'\to C'\to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:A\to A'$, $g:B\to B'$ and $h:C\to C'$ be morphism of ...
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2answers
742 views

Classification of subgroups of finitely generated abelian groups

A finitely generated abelian group $A$ is isomorphic to a direct sum of cyclic groups. I am interested in an extension of this result on couples of abelian groups $(A,B),$ where $B$ is a subgroup of $...
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0answers
139 views

Is it possible to compute the basis of this module?

Let $A$ be a polynomial algebra in $n$ variables over field $\mathbb{F}$ of characteristic zero which is algebraically closed. Assume that $a_1,\ldots, a_n, b_1,\ldots, b_n\in A$ are such that $a_1b_1+...
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0answers
52 views

Why does this cluster tilting object form a local slice?

I am reading the paper "Cluster automrphisms", here is the link: http://prospero.dmat.usherbrooke.ca/ibrahim/publications/Cluster_Automorphisms.pdf In the proof of lemma 3.1 I am stuck: For ...
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0answers
222 views

How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
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1answer
83 views

References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series. Let $R$ be an associative unital ring and $...
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0answers
41 views

What is the maximal weight submodule of $\text{Hom}_{\mathfrak{g}}(M,N)$?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. For ...
3
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1answer
99 views

Mackey theory for semidirect products: equivalence between constructions for modules

I am learning about the so-called "Mackey Machine" for unitary irreps of semidirect products of locally compact groups. Let $G = N \rtimes H$ where $N$ is a closed normal abelian subgroup and $H$ is a ...

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