Questions tagged [modules]
For questions on modules over rings.
4
votes
1answer
77 views
Tensor product of modules over a monoid in a monoidal category
Assume $\mathcal{C}$ is a monoidal category, with unit $I$. Given a monoid object $M$, I'd like to talk about modules over $M$, but couldn't find any reference. This might seem quite a stretch, but it ...
6
votes
1answer
174 views
Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?
Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no ...
0
votes
0answers
39 views
Restriction of fractional ideal sheaf to irreducible component is torsion-free
I translate the question into commutative algebra:
Let $R$ be a one-dimensional, reduced ring (which is also finite free over some PID since the considered curve corresponding lies finitely over the ...
3
votes
1answer
49 views
example of a non-finitely generated co-Hopfian module over a commutative QF ring
Can anyone provide an example of such a module? or show that no such module exists? For semisimple rings, we have co-Hopfian if and only if finitely generated. Perhaps the fact that QF rings (...
0
votes
0answers
61 views
Localization of Hom module in an advanced setting (0-dualizing module)
Let $A,\,B$ be noetherian rings such that and let $M$ be an $A$-module. Let $g:A \to B$ be a ring homomorphism which makes $B$ into a finite free $A$-algebra. Now we can regard the $A$-module $\...
0
votes
0answers
94 views
is the homology of a free complex again free?
Let $k$ be a field and $R$ be a finitely generated graded $k$-algebra (e.g. a polynomial ring in some variables).
Let $M$ be a finitely generated graded vector space so the graded module $\widetilde{M}...
12
votes
1answer
220 views
Tilting Objects in BGG Categories $\mathcal{O}$
Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
1
vote
1answer
78 views
Existence of symplectic basis
Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic ...
1
vote
0answers
72 views
Generating the annihilator ideal up to finite index
Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...
1
vote
1answer
95 views
modules whose every submodule is a homomorphic image
Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$.
Can we characterize all ...
2
votes
0answers
30 views
Homotopy monoidal structure from Hirsch Algebra
In the paper available: https://escholarship.org/uc/item/77r5k6cb
The differential is defined on page 13. Is this the wrong differential? Where are the $E_{p,q}$ maps coming from the twisted tensor ...
9
votes
2answers
268 views
Divided power algebra is artinian as a module over the polynomial ring
I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow.
In the paper Homological algebra on a complete intersection, with ...
8
votes
1answer
91 views
Injective indecomposable modules over Laurent polynomial rings
What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...
1
vote
0answers
68 views
The multi-set monad and modules
I am trying to analyze the category of algebras for the finite free commutative monoid monad, aka the finite multiset monad. This monad is frequently described as having a multiplication that takes a ...
4
votes
0answers
61 views
On ideals in Noetherian rings, isomorphic to the trace of some finitely generated module
Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$.
Let $I$ be a ...
1
vote
0answers
24 views
$U$-reflexive modules in Morita duality
Let $_RU_S$ define a Morita duality between $R$ and $S.$ Exercise 25 in Lam,"Lectures on Modules and Rings," asks one to show that if $R$ is left-Artinian, then the $U$-reflexive $R$-modules are ...
0
votes
0answers
115 views
Is $R$ finitely generated?
Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n(n\in\Bbb N)$ and suppose $S$ is finitely generated, if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, ...
5
votes
1answer
99 views
Equivalence of idempotents and projective modules over nonunital rings
For a nonunital ring $R$ (or "rng") one has to be a little bit careful when considering the category of (left or right) $R$-module and, furthermore, the standard equivalent definitions of projective ...
3
votes
1answer
77 views
Dimension of a module over a left-Ore domain
If $R$ is a domain, and $M$ a (left) $R$-module, what are the different notions of dimension of $M$ and their respective assets, what do they measure?
I found out that if $\dim_RM$ is the cardinal of ...
0
votes
2answers
146 views
Noetherian module, over Noetherian ring, which is isomorphic to its double dual [duplicate]
Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$.
Then is the natural map $j:M \to M^{**}$ defined as $j(m)(...
2
votes
0answers
31 views
Numerator-cancellable Modules
I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following :
Let $R$ be a ...
0
votes
0answers
39 views
Cassels Frohlich, Module index
In the book “Algebraic Number Theory” written by Cassels and Frohlich, module index is defined.
Let R be Dedekind domain, K be its quotient field, U be a n-dimensional vector space over K, and L,M be ...
1
vote
0answers
53 views
n-Gorenstein algebras and tilting modules
Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$...
11
votes
1answer
139 views
Group rings such that every (countably generated) module has a maximal submodule
Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma.
I am interested in the following question, with two variants.
...
3
votes
1answer
162 views
Higher Extension Group Question
Suppose we have an associative unital ring $R$, and we have an $R$-module $M$ with a length 3 socle filtration, i.e. write
$$soc(M) \text{ for the socle of } M,$$
$$soc^2(M) \text{ for the preimage ...
5
votes
3answers
370 views
Irreducible representations and invariant subspaces
Consider two invertible n-by-n matrices, $n>2$, $X$ and $Y$ over a finite field $k$ (say for simplicity $k=\mathbb Z/ \mathbb Z_2$). Is there any reasonable way to check that there is no proper ...
1
vote
0answers
73 views
Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
6
votes
1answer
163 views
intersection of free/affine submodules, comparison with vector spaces
If $W_1,W_2 \subset V$ are finite-dimensional $k$-vector spaces of dimensions $d_1, d_2 \leq d$, respectively, then $d_1 + d_2 > d$ suffices to guarantee $W_1 \cap W_2 \neq \{0\}$. There are ...
-2
votes
2answers
338 views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
9
votes
1answer
317 views
Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
1
vote
0answers
107 views
A question concerning a short exact sequence with an action
Let $A$ and $D$ be two non-trivial abelian groups and $B,C$ be two non-abelian groups. Also, let $C$ is a free group and acts on $A,B,D$.
Let $0\to A \xrightarrow{f}B\xrightarrow{g}C\to 0$ be a ...
2
votes
1answer
193 views
A-infinity modules
Using: https://arxiv.org/pdf/math/9910179.pdf as a reference...
My question involves spelling out explicitly the comment in 4.2 -
"Equivalently, the datum of an $A_\infty$-structure on a graded ...
4
votes
1answer
67 views
Given a representation-infinite algebra, when is every AR component infinite?
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
3
votes
1answer
84 views
Extensions of modules of type $FP_n$
Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely ...
4
votes
1answer
619 views
local ring all whose non-maximal ideals are finitely generated
Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ?
It is easy to see ...
7
votes
1answer
247 views
Is there a converse to the Brauer–Nesbitt theorem?
$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in ...
8
votes
3answers
624 views
$R$ a DVR with fraction field $K.$ What are the $R$-submodules of $K^n?$
It is known that if $R$ is a DVR with fraction field $K,$ then the $R$-submodules of $K$ are $0,K,x^nR,$ with $n$ any integer and $x$ a generator of the maximal ideal of $R.$ I was wondering if there ...
0
votes
0answers
63 views
Isomorphisms between different completions of a module
Given a $R$-module $A$, suppose that $A$ has two different unbounded filtrations
$\cdots \subseteq M_{i+1} \subseteq M_i\subseteq M_{i-1}\subseteq \cdots $
$\cdots \subseteq N_{i+1} \subseteq N_i\...
5
votes
1answer
167 views
Surjectivity of a map on inverse limits
(The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
0
votes
0answers
40 views
Solving linear systems for integer values in MAGMA
Say we are given a quaternion algebra D over a number field F as well as a maximal $\mathcal{O}_F$-order $\Delta$ $\subseteq$ D and say we have a $\mathbb{Z}$-basis $\omega_1, . . . , \omega_n$ for $\...
2
votes
1answer
124 views
Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ?
Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or ...
2
votes
1answer
81 views
When can every countably generated submodule of a a non-countably generated projective module be contained in a countably generated direct summand ?
One answer to this Lemma on infinitely generated projective modules shows that every finitely generated module of a non-countably generated projective module is contained in a countably generated ...
7
votes
1answer
173 views
Adjoints of scalar extension and scalar coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts):
$h^*$: Scalar extension by ...
2
votes
1answer
129 views
On minimal generating sets of certain submodules
All our rings are commutative with unity.
For an $R$-module $M$ and a submodule $N$ of $M$ and ideal $I$ of $R$, let $(N:I):=\{m\in M : Im \subseteq N\}$. Let $\mu (M)$ denote the least cardinality ...
4
votes
1answer
99 views
Some examples of vertex algebra modules
Recently I'm learning the vertex modules. In the paper, there are a lot of abstract theory about the module theory,for instance the $C_{2}-$cofinite conditions and associated variety. I hope to find ...
0
votes
0answers
57 views
On the set of radical ideals which is the radical of an ideal generated by atleast a fixed cardinal no. many generators
Let $R$ be a commutative ring with unity and $\alpha$ be an infinite cardinal. For any ideal $J$ of $R$, let $\mu(J)$ denote the minimal cardinality among generating sets of $J$. For $I$ an ideal of $...
0
votes
0answers
32 views
Flag variety and Schubert cells for free modules
Let $R$ a principal domain, and $M$ a free module of rank $d$ on $R$, with
a fixed basis $(m_1, \ldots, m_d)$.
The definition of flags translates directly to this setting:
a flag of $M$ would simply ...
6
votes
3answers
159 views
Is the category of symmetric bimodules over a commutative ring closed under extensions?
Let $A$ be a commutative ring, and consider the category of bimodules over $A$.
An $A$-bimodule $M$ is called symmetric if $a\cdot m = m \cdot a$ for all $a \in A$, $m \in M$.
Is the category of ...
1
vote
0answers
30 views
Hilbert functions of graded modules generated by mapped generators
I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...
3
votes
0answers
135 views
Integral domain $R$ with fraction field $K$ such that for every $u \in K$, the subring $R[u]$ of $K$ is flat $R$-module
Let $R$ be an integral domain with fraction field $K$. If for every $u \in K$, the subring $R[u]$ of $K$ is a flat $R$-module, then is it true that $R$ is a Prufer domain ?
If $R$ were moreover a ...
