Questions tagged [modules]

For questions on modules over rings.

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What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?

Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...
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3 votes
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56 views

Modules with special properties

Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $End_A(M)$ annihilates the socle $soc_A(M)$ of $M$. Note that $M$ is ...
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Semi simplicity over commutative algebras over non-algebraically closed fields

I have already posted this on stackexchange I have a question: If k is an arbitrary field then is it true that if $M$ a finite dimensional $k[x, y]$ is semisimple as a $k[x]$ module and also as a $...
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3 votes
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97 views

Are there (co-)homological obstructions to the extendability of a homomorphism?

Let $k$ be a commutative ring and $A \subset B$ an extension of $k$-algebras. Can we associate to this extension a (co-)chain complex so that its (co-)homology $H$ allows for statement such as "...
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2 votes
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367 views

Minimal number of generators of kernel of a matrix of polynomials over $\mathbb Q$ and $\mathbb Z$

$\DeclareMathOperator\im{im}$I have asked some related questions before on math.SE here and on MathOverflow here (answered here). This post is self-contained. Let $R' = \mathbb Q[x_1,\dotsc,x_n]$, and ...
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0 votes
0 answers
51 views

Freeness of injective hull of finite module over Gorenstein Artinian ring

This is Lemma 4.1 in Brochard, Khare, Iyengar: Wiles defect and criteria for freeness. I have managed to understand the proof, except for the following part: Why is $F$, the injective hull of a finite ...
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1 answer
233 views

Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module

I have asked a related question on math.SE here, but the notation is a bit different. As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-...
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1 vote
1 answer
48 views

Hilbert - Samuel multiplicity of $B$ when there is a surjection $A \rightarrow B$

The setting is: Let $A, B$ be commutative, Noetherian, local rings, $\phi:A \rightarrow B$ a surjective homomorphism. Both rings also come with surjections $\lambda_A, \lambda_B$ to a DVR $\mathcal{O}$...
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1 vote
1 answer
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Pontrjagin dual of modules [closed]

I am not sure whether this question is appropriate to appear here. If not, I apologize for that. Given an $R$-module $M$, we define its Pontrjagin dual as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. ...
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$\operatorname{Ext}^1$ isomorphic to quotient ring

I'm reading this paper: Brochard, Iyengar and Khare: Wiles defect for modules and criteria for freeness. In lemma 4.5, there is an isomorphism $\operatorname{Ext}_A^1(k,A) \cong \frac{I_A}{\varpi I_A} ...
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  • 141
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1 answer
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Nondegenerate pairings versus perfect pairings for finitely generated projective modules

Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing $$ \langle -,-\rangle:M \otimes_R N \to R $$ is non-degenerate if, for all $n \...
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0 answers
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$\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones

I am trying to refine my understanding of derived categories. Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ ...
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A bi-variate polynomial interpolation question

Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity ...
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1 vote
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Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring

Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
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1 answer
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Absolutely irreducible representation and splitting field

Let $A$ be a finite-dimensional algebra over a field $F$. A representation $M$ of $A$ is called absolutely irreducible if $M\otimes_FE$ is irreducible as a representation of $A\otimes_FE$ for all ...
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7 votes
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232 views

Split epimorphism of modules - does the finite case imply the infinite case?

Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
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1 vote
1 answer
64 views

Epimorphism going out of an inverse limit into a finite dimensional module

Let $k$ be a field and $A$ a finite dimensional $k$-algebra. Given a sequence of inclusions $M_1 \subseteq M_2 \subseteq \dots$ of $A$-modules consider the direct limit $M:= \bigcup_{i=1}^\infty M_i$. ...
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1 vote
0 answers
57 views

Is it possible to describe a $k$-basis for $M\otimes_{kH}N$ when $M$ is a $k[G\times H]$-module and $N$ is a $k[H\times K]$-module?

Suppose $k$ is a field for Let $M$ be a finitely-generated a $k[G\times H]$-module and let $N$ be a finitely-generated $k[H\times K]$-module. Then in particular, $M$ and $N$ are finite-dimensional $k$-...
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1 vote
0 answers
54 views

When some idempotent ideals belong to the Gabriel filter of ideals for a hereditary torsion theory

Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under ...
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modules over principal ideal rings

Let $R$ be a commutative principal ideal ring (not necessarily Artinian) and let $M$ be a finitely generated $R$-module. Is $M$ a direct sum of cyclic $R$-modules? (i.e. a generalization of the theory ...
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2 votes
1 answer
101 views

Structure theorem for finitely generated $\Lambda$-modules - uniqueness part

In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}_p[[T]]$. If $M$ is a finitely generated torsion ...
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3 votes
1 answer
181 views

Split monomorphisms of modules - does the finite case imply the infinite case?

Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...
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1 vote
0 answers
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Exterior algebra of free modules over Hopf algebras

Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action $$h \...
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2 votes
1 answer
145 views

On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
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8 votes
1 answer
399 views

For every ring R, is there a block-diagonal canonical form for a square matrix over R?

This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring $R$. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily ...
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7 votes
2 answers
107 views

Rings of finite uniserial type

If $R$ is a ring and $M$ an $R$-module, $M$ is uniserial if its lattice of submodules is a chain. Over an Artinian $R$, the chain will be finite. From what I understand, deciding when two uniserial ...
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5 votes
0 answers
78 views

Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules

Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
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5 votes
0 answers
204 views

Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$

Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional. There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
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8 votes
0 answers
245 views

Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question: ...
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2 votes
0 answers
93 views

The "matrix direct sum" monoid modulo unitary equivalence

Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
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2 votes
0 answers
100 views

Is this concept of a left-abelian category studied?

A category is abelian if it is preadditive and it has a zero object, it has all binary biproducts, it has all kernels and cokernels, and all monomorphisms and epimorphisms are normal. Now we ...
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1 vote
0 answers
66 views

When is a bounded complex of $RG$-modules contractible?

If we have a $p$-modular system $(K, \mathcal{O},k)$, let $R = k$ or $\mathcal{O}$ and $G$ a finite group. When is a bounded complex of $RG-RG$-bimodules $\Gamma$ contractible? I've seen this response ...
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2 votes
1 answer
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When the annihilator of each nonzero submodule is prime

Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...
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1 vote
0 answers
37 views

A "spectral theorem" to SVD reduction for every commutative *-ring

Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is ...
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5 votes
1 answer
208 views

Concept of an exact ideal of a module category

Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
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21 votes
1 answer
2k views

Reference request: a tale of two mathematicians

I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so: When P. Gabriel presented the theorem in a conference [sometime around ...
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3 votes
1 answer
220 views

RIng that is flat over a subring as a right module but not as a left module

What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module. The following question is my motivation: Faithful flatness for ...
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4 votes
0 answers
287 views

A projective module over a domain that is not faithfully flat?

Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
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2 votes
2 answers
153 views

Module complements to rings embedded in a projective module

Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module. A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-...
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1 vote
0 answers
77 views

Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
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1 vote
0 answers
92 views

Finding an injective envelope containing another injective envelope

Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
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4 votes
1 answer
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$\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$

Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that $$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
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3 votes
1 answer
176 views

Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
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3 votes
1 answer
101 views

On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...
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  • 81
4 votes
1 answer
187 views

Infinite linearly independent set in finitely generated module

Let $R$ be a (commutative, otherwise the answer is easy, see the comment below) ring and let $M$ be a finitely generated $R$-module. Is it possible that $M$ admits an infinite linearly independent set?...
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  • 3,556
0 votes
0 answers
49 views

Tensor products and intersections of modules

Is it true that $A\otimes_{\Bbbk} B \cap B \otimes_{\Bbbk} A = B\otimes_{\Bbbk} B$ if $B \subset A$ are $\Bbbk$-modules over a ring $\Bbbk$?. It works for $\Bbbk$ a field. Does it work in any other ...
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2 votes
0 answers
91 views

Is there a category of "chains of modules" that behaves well with taking direct limits?

I came up with the following definition of a category of certain "chains of modules" and want to know if this concept is already known and studied. Let $R$ be ring. An object in our category ...
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2 votes
1 answer
83 views

Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable

Crossposted on StackExchange on July 28 (no answer so far). Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
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4 votes
1 answer
160 views

Is there essentially unique notion of module over monoidal stable $\infty$-categories?

There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...
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4 votes
0 answers
122 views

Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras

Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...
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