2
votes
1answer
72 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+ ...
0
votes
1answer
27 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
21 views

Transferring Exactness

If $$\begin{matrix}0&\rightarrow&A&\rightarrow&B&\rightarrow&C&\rightarrow&0\\ &&\downarrow&&\downarrow&&\downarrow\\ ...
8
votes
1answer
80 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 ...
3
votes
0answers
54 views

Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
1
vote
0answers
21 views

About a (so very) strong version of flatness

Let $A, B$ rings. WHat are $(A, B)$-bimodule $M$ such that $(-)\otimes_A M: Mod_A \to Mod_B$ preserve all (small) limits? A second question is: the characterization of bimodule ${}_AM_B$ such ...
4
votes
1answer
133 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 ...
1
vote
1answer
134 views

What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
0
votes
0answers
70 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 ...
1
vote
1answer
87 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 ...