2
votes
0answers
35 views

Free objects in the category of dg modules

Suppose that $A$ is a dg algebra, does the category of dg modules over $A$ where morphisms are degree zero maps that commute with differential have a free object ( in general)? I have been reading a ...
3
votes
0answers
36 views

A variant of projective objects?

Let $\mathcal{C}$ be an additive category. Is there a common name for objects $P \in \mathcal{C}$ with the property that $\hom(P,-) : \mathcal{C} \to \mathsf{Ab}$ is right exact, i.e. preserves all ...
7
votes
2answers
109 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 ...
1
vote
2answers
69 views

Extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ (Hilton & Stammbach III.1.2)

Question is to compute $E(\mathbb{Z}_p,\mathbb{Z})$ i.e., equivalence classes of extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ By an extension of $A$ by $B$ i mean an $R$ module $E$ such that ...
2
votes
1answer
60 views

Commuting with kernels implies left exactness in Abelian category

I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as ...
0
votes
2answers
50 views

Non-bijective isomorphism in a category of of sets.

I have been commanded on homework to find a non-bijective isomorphism in a category whose objects are sets, whose morphisms are set maps, and composition is the usual function composition. So our ...
2
votes
1answer
52 views

Do any 2 morphisms from objects $X$ to $Y$ define a chain homotopy equivalence?

I was curious about one thing: Let $A$ and $B$ be abelian categories with enough projectives, let $X$, $Y$ be objects in $A$ and let $P_{\bullet} \rightarrow X$, $P'_{\bullet} \rightarrow Y$ be ...
2
votes
0answers
41 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
0
votes
0answers
38 views

In an abelian category,every morphism can be written as composition of epi and mono.

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism. After many ...
1
vote
1answer
41 views

Additive, covariant functor commutes direct limits, then it commutes with direct sums?

Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ ...
2
votes
1answer
38 views

Derived Functors and nice Resolutions

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following: "Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor ...
4
votes
1answer
87 views

Composition of bicartesian squares

A commutative square is called bicartesian when it is both pull-back and push-out. In an abelian category, consider two pull-back squares $(X)$ and $(Y)$: $$ \begin{array}{ccccc} A & ...
3
votes
2answers
77 views

Definition of Hochschild homology in terms of Tor functor (bar resolutions)

I had 2 kind of dumb questions about the definition of Hochschild homology in terms of the Tor functor: 1 - Let $R$ be a $k$-algebra and $M$ an $R$-bimodule, let $H_*(R,M)$ be the Hochschild homology ...
1
vote
1answer
47 views

When does a functor commute with colimits?

Is it true that an additive functor between abelian categories commutes with colimits if it's right-exact and commutes with (arbitrary) direct sums? If yes, does someone know a good source of a ...
1
vote
2answers
98 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
4
votes
1answer
94 views

Two definitions of homology

Let $f,g$ be arrows in an abelian category such that the composite $gf$ is defined and is given by the zero arrow. I shall try to find a definition for the quotient $\ker g /\operatorname{im} f$, ...
2
votes
1answer
51 views

Action of the functor Ext$_1(-,-)$ on extensions

Suppose we have an exact sequence of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & L & \overset{f}{\longrightarrow} & M & \overset{g}{\longrightarrow} & E & ...
2
votes
1answer
108 views

showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
1
vote
1answer
75 views

Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra

How to prove that a chain complex is a projective object in $ {Ch} $ (chain complexes of $R$-modules) iff it is a split exact complex of projectives? A chain complex of projectives means a chain ...
4
votes
0answers
49 views

Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L $ and $M$ in ...
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 ...
7
votes
0answers
79 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
2
votes
1answer
84 views

Additive functor is exact $\iff$ quasi-ismorphisms preserved?

While reading Weibel's Homological Algebra, on pg. 391 he considers an additive functor $F:\mathcal{A}\to\mathcal{B}$ between abelian categories, and writes "If $F$ is not exact, then the induced ...
2
votes
1answer
78 views

Conditions to ensure the chain homotopy category $K(\mathcal{A})$ is abelian?

It is known that the chain homotopy category $K(\mathcal{A})$ for an abelian category $\mathcal{A}$ need not be abelian. For example, $K(\mathrm{Ab})$ is not even abelian. Are there any known ...
2
votes
1answer
88 views

Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
1
vote
1answer
40 views

Acyclic resolution but not projective

Suppose $\mathfrak{C}$ is an abelian category which does not have enough projectives and we're interested in computing the right derived functors of some covariant functor $F$. If however, every ...
1
vote
1answer
44 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
2
votes
2answers
31 views

Extending monics in a commutative diagram

Given a commutative diagram in a Grothendieck category $\mathscr{A}$ \begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & ...
3
votes
2answers
66 views

Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and ...
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 ...
7
votes
1answer
91 views

Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
5
votes
0answers
62 views

Best approximation to an adjoint functor

I have the following question. Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best ...
2
votes
0answers
102 views

Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
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 ...
3
votes
1answer
86 views

Projective object in the category of chain complexes

I have the following sequence of projective $\mathbb{Z}$-modules: $\cdots \rightarrow 0 \rightarrow \mathbb{Z} \overset{\times 2}\rightarrow \mathbb{Z} \rightarrow 0 \rightarrow \cdots $ This is ...
1
vote
2answers
75 views

kernel of a monic morphism

Problem Suppose $\mathscr{C}$ is an arbitrary category with zero object. $A$ and $B$ are two objects of $\mathscr{C}$. Let $f\in Mor_\mathscr{C}(A,B)$. It's given that $f$ is monic. I need to show ...
5
votes
1answer
171 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 ...
15
votes
1answer
436 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
6
votes
0answers
145 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
0
votes
0answers
42 views

Is the category of chain complexes complete and cocomplete in small?

Does the category of chain complexes (let's say of modules over some ring) have all small limits and colimits? What I understand is that the category of chain complexes is certainly finitely ...
1
vote
1answer
132 views

Proof of the five lemma

How to do this using the snake lemma? this is an exercise in Lang's Algebra book. It should somehow be obvious, but I don't see it
1
vote
1answer
89 views

Direct limit and products

Any of your comments (or if you know a resource which could be handy) regarding this problem would be appreciated: Show that finite products commute with filtered direct limits. Got no idea how to ...
6
votes
1answer
152 views

The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
3
votes
1answer
65 views

On the definition of an exact sequence in an abelian category

I am slightly confused about the notion of exactness in a general abelian category (I want to stay clear of anything related to the Mitchell embedding theorem). Here are two definitions that I have ...
-1
votes
1answer
126 views

Monic and epic implies isomorphism in an abelian category? [duplicate]

Is it true that monic and epic implies isomorphism in an abelian category?
2
votes
1answer
71 views

How to prove exactness implies complex?

In an abelian category, there are notions of exact sequence and complex. Since the objects there may not be abelian groups, the definition of exact sequence and complex are all complicated. And the ...
0
votes
0answers
34 views

Finding coproduct of category(specified in the question!) [duplicate]

I asked a question few minutes ago, and when I saw the answer to my question, I found that I had explained my question wrongly (so the answer was not what I wanted to know). So I decided to write new ...
1
vote
2answers
107 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
2
votes
1answer
138 views

Understanding Equivalence of Categories

An equivalence of two categories $\mathcal{C},\mathcal{D}$ consists of a pair of functors $F:\mathcal{C} \rightarrow \mathcal{D}$, $G:\mathcal{D} \rightarrow \mathcal{C}$ and natural isomorphisms $FG ...
2
votes
1answer
73 views

Right exactness on a dense subcategory

Let $F : C \to D$ be a $k$-linear functor between cocomplete $k$-linear categories, which preserves directed colimits (in particular arbitrary direct sums). Let $C' \subseteq C$ be a dense full ...