Tagged Questions

8
votes
2answers
125 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 ...
10
votes
2answers
92 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
50 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 ...
7
votes
1answer
77 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
4
votes
3answers
123 views

Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter 0, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint operator, then $\mathcal{F}$ is ...
3
votes
2answers
89 views

Coproducts and products of modules

I've just looked at the book "A Course in Homological Algebra" ( by Hilton and Stammbach) . They show the universal property of the direct sum (coproducts) using injections and the universal property ...
2
votes
1answer
100 views

The relation between retraction and coproduct in R-Mod

Let $X$,$Y$,$Z$ be objects of R-Mod. ($\oplus$ denotes the direct sum or coproduct of modules). The question is, are the following statements equivalent? $X$ is a retraction of Y. ...
0
votes
1answer
46 views

Images of simple modules under exact endofunctors

I'll be quite fuzzy with the prerequisites, as I don't know myself in what generality the statement I want to understand holds. Let $\mathcal{C}$ be a category of modules over a non-commutative ring ...
2
votes
1answer
62 views

Dual notion of free module

I was just introduced to the notion of projective module (through Matsumura, Commutative ring theory), and there it is said that the reason that there is no easy description for injective modules ...
4
votes
1answer
70 views

Defining the image of a map

Let $\mathcal{A}$ be an abelian category. For a morphism $f:A\rightarrow B$, its image is usually defined in the following way: a map $i:I\rightarrow B$ is called an image of $f$ if it is a kernel of ...
2
votes
1answer
116 views

Additive functor over a short split exact sequence.

$0\to A'\stackrel{f}{\longrightarrow} A \stackrel{g}{\longrightarrow} A''\to 0$ is a short split exact sequence, where $A'$, $A$, $A''$ are $R$-modules, and $T$ is an additive functor from ...
0
votes
1answer
27 views

$F(M)$ as a $\text{Hom}_{R_1}(M,M)$-module for some functor $F:R_1-\text{mod}\rightarrow R_2-\text{mod}$

I'm trying to generalize one aspect of the accepted answer in: $\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$ Is the following statement correct? If: $R_1$ and $R_2$ are rings; ...
2
votes
0answers
78 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
2
votes
1answer
67 views

Injective Cogenerators in the Category of Modules over a Noetherian Ring

Let $R$ be a Noetherian ring and let $\mathcal{A}$ be an injective $R$-module. The injectivity of $\mathcal{A}$ is equivalent to the exactness of the functor $Hom_R(-,\mathcal{A})$, i.e. whenever we ...
4
votes
1answer
127 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
5
votes
1answer
104 views

If $I$ is injective and $P$ is projective, is $\operatorname{Hom}(P,I)$ injective?

In Bourbaki's Algebra, there is the following exercise ($A$ is an arbitrary ring with unity): Suppose that $I$ is an $(A,A)$-bimodule which is injective as a left $A$-module and a right $A$-module ...
0
votes
0answers
47 views

What is the cohomological dimension of a functor?

Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a functor between abelian categories. Could anyone explain what the cohomological dimension the functor $F$ is? We may need some additional condition to ...
5
votes
1answer
116 views

When are $R$-mod and $R$-mod-$R$ equivalent?

Let $R$ be a commutative ring with 1. Under what conditions are $R$-mod (the category of $R$-modules) and $R$-mod-$R$ (the category of $R$-$R$ bimodules) equivalent as categories?
2
votes
0answers
97 views

A natural homomorphism of dual modules which is not a monomorphism?

I've been mixing up my reading with a little category theory. When thinking about dual modules, this idea popped up. Suppose $M$ and $N$ are modules over some commutative ring $A$. Suppose $M^\vee$ ...
8
votes
4answers
402 views

Reference request: compact objects in R-Mod are precisely the finitely-presented modules?

Let $R$ be a ring. According to this MO question, the modules $M \in R\text{-Mod}$ such that $\text{Hom}(M, -)$ preserves all filtered colimits (the compact objects) are precisely the ...
4
votes
2answers
169 views

Can we compare limits over different diagrams?

Fix an index (small) category $I$. Let's say our category $\mathcal{C}$ has limits of type $I$. In this case, $\varprojlim_I:\mathcal{C}^I \to \mathcal{C}$ is a functor. Let's say our category ...
1
vote
1answer
77 views

Equivalent module categories

Let $A$ and $B$ be rings and let $A\text{-mod}$ and $B\text{-mod}$ be their abelian module categories. Let $F:A\text{-mod}\to M\text{-mod}$ and $F':B\text{-mod}\to A\text{-mod}$ be functors which ...
0
votes
1answer
109 views

Right exactness of the Hom functor in the first argument for free modules

Let $0 \rightarrow E_1 \stackrel{\sigma}{\rightarrow} E_2$ be an exact sequence of free $A$-modules, $A$ being a commutative ring. Let $F$ be a free $A$ module. Then is it true that the sequence ...
7
votes
1answer
197 views

Question about the Dual Statement for Injective Modules

It's well-known that there are several equivalent definitions of a projective $A$-module $P$: Given an exact sequence $M \xrightarrow{f} M' \to 0$ and a morphism $P \xrightarrow{g} M'$, ...
3
votes
1answer
250 views

Calculating Hom(A,B)

I have been studying modules and homological algebra as of late but somehow I have missed how to calculate Hom(A,B) for abelian groups, modules and Hom(A,_)/Hom(_,B) for exact sequences. I have no ...
3
votes
1answer
201 views

Is restriction of scalars a pullback?

I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback". I don't understand why this might be the case. In particular, consider a ...
6
votes
2answers
551 views

Learning to think categorically (localization of rings and modules)

I've been reading Ravi Vakil's notes on algebraic geometry notes and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal ...