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4
votes
1answer
105 views

Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor: $$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$ left-adjoint to the (forgetful) embedding: $$...
1
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1answer
160 views

Indecomposable weirdos (cnt.)

This post is a continuation of Weirdos but algebraic. Logically, the quoted post could follow the present one rather than precede it. Question Does there exist an indecomposable weirdo which is ...
1
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0answers
54 views

On submodules of vector fields

I don't know much about modules aside from their basic definition and that they are more complicated than vector spaces. I am asking this question because I wish to have a more "algebraic" ...
0
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0answers
69 views

Certain morphism between graded modules

Say we have a morphism $f=(f_{i,j}) : \oplus_{i}M_{i} \rightarrow \oplus_{j}N_{j}$ (The direct sum of graded modules is finite) and $f_{i,j}$ are morphisms of graded modules but not necessary with the ...
1
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0answers
44 views

Existence of nontrivial transfinite divisibility in $R$-modules

Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...
0
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1answer
51 views

Can we extract an injective envelope from a monomorphism?

Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we ...
3
votes
1answer
67 views

Effect of extending scalars on maps of modules

Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \...
1
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1answer
93 views

Geometric meaning of colocalization of modules?

Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
4
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1answer
103 views

Global splitting field for algebras

Let $A$ be a finite dimensional algebra. A field $K$ is a splitting field for an indecomposable $A$-module $M$ in case the local algebra $End_A(M)/(rad(End_A(M))$ is 1-dimensional. $K$ is called a ...
4
votes
1answer
124 views

Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
2
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0answers
209 views

Module structure for $\mathbb{Z}$

I am interested to know which module structures we can define in the additive group of integers $\mathbb{Z}$. It is easy to prove that $\mathbb{Z}$ does not admit a vector space structure. (For more ...
4
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1answer
179 views

Linear algebra over non-commutative semirings

I'm reading up on linear algebra over semirings, and I'm wondering why people seem to stop short of showing an equivalence between linear transformations between free modules and matrices. It seems ...
5
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1answer
124 views

Projective module which splits off sequence of submodules, but not the sum

Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that: $X$ is projective, $X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...
3
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0answers
123 views

The category $\text{add}M$ for a module $M$

Let $A$ be a $K$-algebra and consider the category $\text{mod}A$ of finitely generated left $A$-modules. If $M$ is in $\text{mod}A$, then $\text{add}M$ denotes the full subcategory of $\text{mod}A$ ...
2
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1answer
165 views

Left module which cannot be made into a bimodule?

Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...
2
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0answers
44 views

Non-singular rings which are Rickart

A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$. It turns out that a ring $R$ is right Rickart iff every ...
0
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1answer
279 views

A lemma on a sequence of three morphisms

Few months ago, I proved that when there is three morphisms of modules over a commutative ring with zero composition, i.e., a sequence $$A \xrightarrow{\alpha} B \xrightarrow{\beta} C \xrightarrow{\...
1
vote
1answer
128 views

Injective hull of quotient of modules

Let $R$ be a Noetherian commutative ring. If $N\leq M$ are $R$-modules, then is $E(M)$ isomorphic to a submodule of $E(N)\oplus E(M/N)$? Here $E(M)$ denotes the injective hull of $M$.
4
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2answers
263 views

Adjoints for radical and socle functors

Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
0
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1answer
130 views

Is a coslice (slice under) category a full subcategory of it ambient category?

Let $\operatorname{C}$ be a category and $c\in \operatorname{C}$ an object. Consider the coslice (sometimes called slice under) category ${\operatorname c}/{\operatorname C}$. My question is whether ${...
9
votes
1answer
188 views

Is an overring of an order reflexive as a module over the order?

Let $R$ be a one-dimensional Noetherian domain with fraction field $K$, let $\tilde{R}$ be the integral closure of $R$ in $K$, and assume that $\tilde{R}$ is finitely generated as an $R$-module. (In ...
3
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0answers
135 views

Is there a constructive proof of Baer's Criterion?

Baer's Criterion states than one can check injectivity of an $R$-module on inclusions of ideals. The proof, however, strikes me as very nonconstructive: it employs both Zorn's Lemma and LEM. Does ...
0
votes
1answer
120 views

Notion of module over commutative post lie algebra

Let $ (S, \{. \}, [.]) $ be an algebra over a vector space endowed with two bilinear maps $ \{. \}, [.] : S \times S \rightarrow S$ and satisfying some compatibility conditions. In example, if S is ...
-4
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1answer
97 views

Finitely generated module [closed]

If a finitely generated module $M$ injected in a free module, then would the image of $M$ be a free finitely generated module?
7
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2answers
559 views

Beauville-Laszlo for schemes

For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...
2
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0answers
65 views

Lattices with trivial coinvariants for finite groups

Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank. Question: Is there a finite group $G$ and a $\mathbb{Z}...
6
votes
3answers
516 views

Dual of a bimodule

For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module. Note: ...
4
votes
1answer
461 views

Parabolic Kazhdan-Lusztig Conjecture

In this post, Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^*$, which is under the name: Kashiwara and Tanisaki - Characters of ...
9
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0answers
173 views

What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...
7
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1answer
246 views

Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...
4
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0answers
139 views

Row rank and column rank of matrix with entries in a commutative ring

Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed in "Ranks of Modules" one can say that the row rank of $A$ is ...
0
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0answers
175 views

Partial monoid in the category of categories of modules: The spotty nature of monad composition

It seems that I am working on a conjecture in category theory. In particular, I am curious about the spotty nature of the composition of monads on Set. I am guessing that there is a category, $\...
2
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0answers
80 views

Isoartinian and isosimple modules

I'm reading this article by A. Facchini and Z. Nazemian, in wich they discuss modules with chain conditions up to isomorphism. A couple of the main concepts are the following: Definition We say that ...
1
vote
1answer
102 views

Hom functor and arbitrary coproducts

Let $M$ be a finitely generated $R$-module. It's easy to check that, in this case, $\mathbf{Hom}(M,-)$ preserves infinite sums. Now suppose that $M$ is projective. Is the reciprocal true? That is, if ...
2
votes
0answers
90 views

Fock space for distinguishable particles

I am proposing that a category, $\mathcal{C_{\mathcal{Fd}}}$, of Fock Spaces for distinguishable particles is, in fact, a suitably defined tensor product in the category of categories of Modules. In ...
-2
votes
1answer
138 views

Module such that every finitely generated submodule is semisimple [closed]

Is there an example of a module $M$ (over a commutative ring) that is not free, and such that each of its finitely generated submodule is semisimple (i.e. such that any submodule of any finitely ...
0
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1answer
35 views

Equivalent condition for distributive property of $\mathcal{S}(M)$ [closed]

$\textbf{Definitions:}$ Let $M$ be an $A$-module. The collection $\mathcal{S}(M)$ of all submodules of $M$ is partially ordered by the inclusion relation. The collection $\mathcal{S}(M)$ is said to ...
1
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0answers
92 views

Composition of monads induces tensor product in the category of modules

I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the ...
5
votes
2answers
466 views

Zero tensor product over a complex algebra?

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space. Q1: Can this ...
3
votes
1answer
297 views

When does the forgetful functor from modules to vector spaces have a right adjoint?

Given any algebra $R,$ when does the forgetful functor $R\text{-}Mod \rightarrow Vec$ have a right adjoint? Does this imply any finiteness conditions on R? Is there a book/paper discussing this? I'...
0
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0answers
80 views

Structures like vector spaces but closed under heterogeneous products

The category of (pseudo-)Euclidean vector spaces (vector spaces with a nondegenerate but not necessarily positive-definite quadratic form) is not closed under products because $R^n$ over $R$ and $Z_2^...
2
votes
0answers
89 views

What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...
2
votes
1answer
228 views

Flatness of submodules of free modules

Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group. If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...
7
votes
2answers
443 views

Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$. Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...
7
votes
1answer
322 views

Global to local principle for f.g. $\mathbb{Z}[x]$ modules

In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
4
votes
0answers
125 views

When does the canonical $t$-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...
2
votes
1answer
143 views

Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it. Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
2
votes
1answer
110 views

A weak Schur's lemma for non-semisimple finite dimensional algebras

Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
5
votes
0answers
105 views

Any f.p. faithful simple module over a primitive group ring?

Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$. There ...
3
votes
0answers
116 views

What is the $Ass(Ext^p_R(M,R))$?

Let $R$ be a Noetherian commutative local ring, $M$ a finitely generated $R$-module with $p=pd M<\infty$ (projective dimension of $M$). What is the relation between $Ass(Ext^p_R(M,R))$ and $Ass(M)$?...

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