1
vote
1answer
40 views

Example of a commutative square without a map between antidiagonal objects?

In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> N @>>> M\\ @. @VVV @VVV \\ 0 @>>> X ...
7
votes
2answers
110 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
0
votes
0answers
53 views

Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
6
votes
0answers
62 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 ...
5
votes
1answer
40 views

If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive

Let $$ F : \text{A-Mod} \to \text{A-mod} $$ be an additive functor. Then if $0$ is the zero-object $F(0) $ is the zero object. Why this is true ? The definition of additive functor that I know is ...
0
votes
1answer
36 views

In a semi-simple module, any submodule is a direct factor?

I need help understanding the following : In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple) (i) How is this equivalent to the ...
5
votes
1answer
71 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
3
votes
0answers
45 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 ...
1
vote
1answer
64 views

Pullback in morphism of exact sequences

Suppose we have the following morphism of short exact sequences in $R$-Mod: $$\begin{matrix}0\to&L&\stackrel{f'} \to& M'&\stackrel{g'}\to &N' & \to 0\\ ...
1
vote
1answer
57 views

Pushout of an injective map is injective

This is an exercise from Rotman , Introduction to homological algebra. Given a pushout diagram in $R$-Mod $$\begin{array} AA & \stackrel{g}{\longrightarrow} & C \\ \downarrow{f} & & ...
0
votes
1answer
38 views

Applying an additive covariant functor to a sequence

Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
2
votes
1answer
76 views

Newbie into categorical proofs

Let $$ F:Mod_A \to Mod_B $$ an aditive , exact and covariant functor and $$ M ∈ Mod_A $$ and $$M_1 , M_2 $$ submodules of M . Show that $$ F( M_1\cap M_2)=F(M_1)\cap F(M_2 ) $$ $$ F( M_1+ ...
2
votes
1answer
55 views

Exercise from Rotman, direct limit of quotient modules

Suppose $$0 \to U \to V \to V/U \to 0$$ is an exact sequence of left $R$-modules. Let $\lbrace U_i, \alpha^{i}_j\rbrace$ be a direct sequence of submodules of $U$, where $$\alpha^{i}_j : U_i \to U_j ...
2
votes
0answers
40 views

Should a left module be an enriched functor or enriched presheaf?

In this page http://ncatlab.org/nlab/show/module#InEnrichedCategory, They defined the left module over a monoid object A as an enriched presheaf. However, consider the modules over a ring, the left ...
2
votes
1answer
48 views

Do covariant functors preserve direct sums?

Suppose $T$ is a covariant functor from the category $R$-mod to Ab (the category of abelian groups) Is is necessary that $T(B \oplus C) \cong T(B) \oplus T(C) $ ? Does the answer change if we ...
0
votes
1answer
28 views

Extending morphism between derived functors

Let N,M be objects in a left exact functor $F:A\rightarrow B$ between abelian categoire's source, most importantly say there is an isomorpihsm $\psi: F(M)\rightarrow R^dF(N)$ is it possible to extend ...
2
votes
1answer
56 views

equivalence in category

First gives some definitions, and then the property that I am confused. $A$, $B$ are both $R$-module, and $C$, $D$ an (additive) abelian group, consider the category $M(A,B)$ whose objects are all ...
2
votes
1answer
39 views

Origin of the term “module” for profunctors

Why do they call profunctors "modules"? How do they exactly relate to modules over rings?
3
votes
1answer
39 views

Existence of Boundary Homomorphisms for Cohomology

I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of $P_0 = ...
0
votes
0answers
33 views

Comodules as a functor category

Let C be a comonoid in some preadditive monoidal category $\mathfrak{C}$, then how can we express the category of C-comodules, in terms of some sort of functor category? I mean is there a similar ...
5
votes
2answers
147 views

Relations between monoids and modules?

What is the relation between monoids and modules? Are they completely different algebraic structures, or is there a kind of inclusion relation like "elements of a module are also elements of a ...
2
votes
1answer
49 views

Action of the center of an enriched category on a module as an enriched functor

If $M$ is a left-module over a ring $R$, then $M$ carries an action of $Z(R)$, the center of $R$, since it is a subring of $R$, by restriction of scalars. Since $Z(R)$ is commutative we may view this ...
1
vote
0answers
96 views

Almost continuous and almost cocontinuous functors

I would like to consider Hom-like functors $H:\mathcal{A}\rightarrow\mathcal{B}$ defined between abelian categories or, more specifically, between module categories, in the following sense: $H$ is ...
2
votes
1answer
77 views

Coproduct diagram for tensor product

Is it true that in the category $R$-Mod of $R$-modules (and $R$-module homomorphisms), the diagram $$ M \longrightarrow M \otimes_{R} N \longleftarrow N,$$ where the arrows are the maps $m \to m ...
2
votes
1answer
23 views

Transferring Exactness

If $$\begin{matrix}0&\rightarrow&A&\rightarrow&B&\rightarrow&C&\rightarrow&0\\ &&\downarrow&&\downarrow&&\downarrow\\ ...
4
votes
1answer
64 views

Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
3
votes
1answer
66 views

Direct product commutes with direct sum?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
0
votes
1answer
48 views

Coproducts and direct products

Is the arbitrary direct sum of modules a submodule of their coproduct? Ie is $\underset{i \in I}{\coprod} M_i \cong \underset{i \in I}{\bigoplus} M_i$... if not then if each $M_i$ where to be ...
1
vote
1answer
29 views

Cotensor and counit?

If M is a C-bicomodule, then considering C as a $C$-bicomodule also, is $M \square_C C \cong C$, where $\square_C$ is the cotensor product in $^C\mathscr{M}^C$.
2
votes
1answer
29 views

Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
3
votes
1answer
25 views

cofree comodules and embedding

For an $R$-coalgebra C, is it possible for every C-comodule M to be embeded into a C-comodule of the form $\underset{i \in I}{\bigoplus} C$?
2
votes
1answer
47 views

Cofree objects and subobjects

In a category with cofree objects, are all objects necessarily subobjects of a cofree object? Particularly, if the category of interest is a comodule category would this be true?
2
votes
1answer
43 views

Are cofree comodules Quasi-finite?

Are cofree comodule quasi-finite, where by quasi-finite I mean there is a left adjoint to the cotensor functor?
2
votes
1answer
55 views

Duals of a finite dimensional eveloping coalgebra

Let $C^e$ be the enveloping $k$-coalgebra of a $k$-coalgebra $C$ and denote by ${C^e}^{\star}:=\mathrm{Hom}\,_{k}(C^e,k)$. Then is ${C^e}^{\star} \cong {C^{\star}}^e$?
1
vote
0answers
49 views

Where is the mistake? (derived functors )

Assume $pd(M) =n \leq \infty$ for a left $R$-module. I then have to show there exists a free module $F$ such that $Ext_{R}^{n}(M,F) \neq 0 $. I have tried these steps and obtained a contradiction: ...
8
votes
1answer
81 views

An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are ...
5
votes
0answers
74 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
1answer
172 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
1
vote
2answers
47 views

Why does the center of $R$-$\mathbf{mod}$ consist of multiplication maps induced from central elements of $R$?

The center of a category $C$ is the class of natural transformations from $1_C$ to $1_C$. In $R-\textbf{mod}$, I have been able to show that the morphisms $\eta_M(c):x\mapsto cx$ for $x\in M$ and $M$ ...
2
votes
3answers
98 views

$0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
4
votes
0answers
49 views

$M\times N$ Doesn’t Have a Module Structure

In Keith Conrad's notes (page 4) is written: For two $R-$modules $M$ and $N$ , $M\oplus N$ and $M\times N$ are the same sets, but $M\oplus N$ is an $R-$module and $M\times N$ doesn’t have a ...
3
votes
0answers
60 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 ...
0
votes
2answers
107 views

Show that $\operatorname{Hom}_R(M, -)$ is a functor from the category of $R$-modules to the category of abelian groups.

Show that $\operatorname{Hom}_R(M, -)$ is a functor from the category of $R$-modules to the category of abelian groups. Let $F$ be the functor. Every $R$-module is also an additive abelian ...
0
votes
1answer
104 views

Localization of modules as adjunction

Usually, the localization of a $R$-module $M$ by a multiplicative subset $S \subseteq R$ with $1 \in S$ is categorically defined as the initial object of the full subcategory $\mathbf C$ of $M \, ...
2
votes
3answers
176 views

Intuitive Explanation of Hom Functor Property

I am new to category theory and keep running across it while studying algebra. Whenever I see the $\mathrm{Hom}$ functor mentioned (in the context of modules), two of its basic properties are listed: ...
4
votes
1answer
137 views

Does “maximal submodule <=> simple quotient module” generalize to abelian categories?

Does the statement "If $A$, $B$ are modules over a commutative ring $R$, then $B$ is a maximal submodule of $A$ if and only if $A/B$ is a simple module" generalize to the setting of abelian ...
2
votes
1answer
174 views

A example of a monoidal non symmetric category of $R$-bimodules

It is well know that if $R$ is a commutative ring, then the category of modules $_R\operatorname{Mod}$ is monoidal symmetric. (Edit. I wrote the category of $R$-bimodules, as F. Muro explained this ...
8
votes
2answers
158 views

What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
13
votes
2answers
160 views

Questions about Rickards proof that $D^b_\mathtt{sg}(A) \equiv \mathtt{stmod}(A)$

Setup: Let $A$ be a self-injective algebra (so projective = injective for modules) and let $D^b(A)$ and $K^b(A)$ be the bounded derived category and the full subcategory consisting of the perfect ...
0
votes
1answer
65 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...