2
votes
0answers
40 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
44 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
93 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
89 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
88 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
71 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
47 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
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 ...
6
votes
0answers
70 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
83 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
72 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
85 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
38 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
43 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
30 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
65 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
86 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 ...
0
votes
0answers
26 views

Exactness of derived functor [duplicate]

Is the right derived functor of a left exact functor left exact also? If now, can anything be said about its exactness in general?
5
votes
0answers
61 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
91 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
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
1answer
81 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
69 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
157 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
415 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
138 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
128 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
130 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
51 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
109 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
70 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
33 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
105 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
131 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
47 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 ...
3
votes
2answers
91 views

Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
1
vote
0answers
36 views

Equivalence and complexes homotopically-minimal

Let $A$ and $B$ be two finite-dimensional algebras over a field $k$ and $G\colon \mathcal{K}^{-}(\mathcal{P}_A) \to \mathcal{K}^{-}(\mathcal{P}_B)$ be an (triangulated) equivalence. By [Krause-05], a ...
4
votes
1answer
50 views

Complete abelian categories with projectieve generators are fully abelian.

This is my first time on stackexchange so if you need more detail from me , please ask. I was reading the book "Abelian Categories : An Introduction to the Theory of Functors" by Peter Freyd , and I ...
0
votes
2answers
54 views

Do $\operatorname{Hom}( - , R)$ and $ - \otimes_R B$ commute when applied to $A\cong R^d?$

Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that $$ \operatorname{Hom} (A \otimes_R B , R) \cong ...
3
votes
1answer
56 views

Proof that derived functors don't depend on choice of resolution.

Can somebody help me out with this? Let $X$ be an object in an abelian category $A$ with enough injectives, let $0 \rightarrow X \rightarrow M^{\bullet}$ be an injective resolution , let $0 ...
2
votes
2answers
127 views

Is taking cokernels coproduct-preserving?

Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms. Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the ...
3
votes
0answers
78 views

Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
2
votes
1answer
393 views

k-linear category

Let $C$ be a additive category and $k$ is a commutative ring. $C$ is called $k$-linear if the morphism sets $C(x,y)$ have the $k$-module structures for all $x,y\in Obj(C)$ and the compositions of ...
8
votes
2answers
129 views

Etymology of Tor and Ext

The names of the important functors Tor and Ext seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from.
5
votes
2answers
76 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
1
vote
1answer
62 views

Derived functors - how is natural transformation between $L_0T$ and $T$ constructed?

For simplicity's sake, consider the categories $R\text{-Mod}, S\text{-Mod}$ of left $R$-modules and left $S$-modules, respectively, and let $\mathcal{F}$ be some precovering class in $R\text{-Mod}$. ...