# Tagged Questions

**2**

votes

**0**answers

193 views

### Tensor product of pullbacks of abelian categories

Does the tensor product of abelian categories commute with pullbacks? In more detail:
Let $k$ be a field. We consider $k$-linear small abelian categories ...

**3**

votes

**1**answer

240 views

### Projectives and Injectives in Functor Categories

Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose ...

**2**

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**0**answers

56 views

### Quotient categories and essential extension

Let $A$ be a right Noetherian positively graded ring. Let $Gr(A)$ be the category of right graded $A$-modules, and $Tors(A)$ be the full subcategory of $Gr(A)$ of torsion modules. Let $QGr(A)$ be the ...

**5**

votes

**0**answers

212 views

### Extension to a right exact functor

Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...

**3**

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**0**answers

175 views

### Cisinski-Deglise model structure on chain complexes

Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the ...

**7**

votes

**2**answers

474 views

### Properties of quotient categories.

I asked this on math.stackexchange.com, but didn't get any answer.
Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre/thick/dense ...

**1**

vote

**0**answers

110 views

### Why are the left exact functors from an abelian category to abelian groups cocomplete and have a injective generator?

Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{C}, \mathcal{B})$ the category of left exact functors between abelian categories.
What is the ...

**3**

votes

**2**answers

295 views

### Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?

What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch?
There is ...

**9**

votes

**1**answer

409 views

### Examples of applications of the Freyd-Mitchell embedding theorem.

The Freyd-Mitchell embedding theorem states the following:
Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor ...

**7**

votes

**2**answers

352 views

### Recovering an abelian category out of its derived category

I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.
In Wikipedia it has been stated that since ...

**1**

vote

**0**answers

295 views

### Additive functors preserving quasi-isomorphism

Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories (with enough injectives and projectives) and $K^\cdot, L^\cdot$ objects of $\textrm{Ch}(\mathcal{A})$. ...

**11**

votes

**5**answers

1k views

### Cov. right-exact additive functors that don't commute with direct sums?

Background
Recently, I have been writing up some notes on derived functors and I came across the Eilenberg-Watts theorems [1], which essentially explain why it is hard to find derived functors ...

**10**

votes

**1**answer

568 views

### Higher “Cartan-Eilenberg” Resolutions

I am a number theory graduate student learning a bit of homological algebra, and I am curious about higher complexes in abelian categories. I apologize if my post is slightly vague as I am not an ...

**7**

votes

**2**answers

739 views

### The composition of derived functors - commutation fails hazardly?

Hello,
When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...

**12**

votes

**3**answers

1k views

### Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that ...

**1**

vote

**2**answers

397 views

### Tot and colimits

This must be a well-known exercise with spectral sequences, but I don't know a reference for it. I'm trying to figure out when does $Tot$ commute with colimits.
More precisely, let $X$ be a double ...

**9**

votes

**2**answers

399 views

### Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories?

Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can fail to be abelian categories.
Poking around on the internet has ...

**5**

votes

**1**answer

905 views

### Is the chain homotopy category, K(Ab), an Abelian category? By Ab, I mean the category of Abelian groups.

Let A be an Abelian category.
From this category, we can form the chain complex category Ch(A). The objects of Ch(A) are chain complexes of objects of A. The morphisms of Ch(A) are chain maps. ...

**4**

votes

**3**answers

928 views

### Existence of projective resolutions in abelian categories

It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." ...

**19**

votes

**2**answers

762 views

### Is there an infinity × infinity lemma for abelian categories?

Many people know that there is a (3×3) nine lemma in category theory. There is also apparently a sixteen lemma, as used in a paper on the arXiv (see page 24). There might be a twenty-five lemma, as ...