All Questions
Tagged with abelian-categories homology-cohomology
46 questions
2
votes
0
answers
47
views
does an exact sequence of filtered complexes induce an exact sequence of spectral sequences
Let $(A_i, d_i, F_i)$ be a filtered differential complex (with bounded filtration) for $i=1,2,3$. Assume we have filtered chain maps $i:A_1\to A_2$ and $j:A_2\to A_3$ such that
$$0\to A_1 \overset{i}{\...
- 3,332
0
votes
1
answer
50
views
Meaning of "quasi-isomorphism" of diagram indexed by the negative integers
I've been given this exercise:
Fix a commutative unital ring $R$ and a diagram $$A^\bullet = \cdots \to A_{-2} \to A_{-1}\to A_{0}$$ of
objects and maps in $\mathrm{Ch}(R \mathrm{-Mod})$ indexed by ...
- 35
0
votes
1
answer
71
views
Reference for Hurewicz's theorem in algebraic topology
Let me begin by stating that under no circumstances am I an expert in algebraic topology. That being said, some years ago I came across Hurewicz's theorem relating the (co)homology groups to the ...
- 240
0
votes
1
answer
57
views
Definition of homology group as quotient in chain complex
I am working through some theory about abelian categories and complexes from "An Introduction to Homological Algebra" by Rotman. I don't understand one of the sections which I will explain ...
- 833
0
votes
1
answer
66
views
Snake Lemma Weibel 1.3.2
I'm reading An introduction to homological algebra by Charles A. Weibel and I'm trying to prove the snake lemma 1.3.2.
Using this diagram (don't have enough points to embbed it in my question) it says ...
- 130
1
vote
1
answer
95
views
definition of chain homotopy
We can define the notion chain homotopy in the category of chain of $R-$modules. But do we have any definition of chain homotopy in any abelian category?
Analogously I can define the notion of chain ...
- 391
0
votes
1
answer
124
views
Homology functor preserves coproduct
(I am aware of the existence of this question).
Hello, there's a step of a proof that $H_n$ preserves coproducts in an abelian category that I do not understand, despite it looking fairly simple...
...
- 355
0
votes
0
answers
86
views
Special short exact sequences of chain complexes
For simplicity let's say I am working in the category of chain complexes of $R$ modules. I have a chain map $i:C\rightarrow D$, which I know to be a weak equivalence (i.e. $i_*$ is an isomorphism of ...
- 4,163
0
votes
0
answers
111
views
Well-definedness of Cohomology
In an abelian category, cohomology can be defined as kernel of a cokernel and cokernel of a kernel, as follows,
My problem is why are they the same? I expect this will be some image=coimage type ...
- 325
1
vote
0
answers
56
views
Isomorphism between the hom complexes $\mathrm{Hom}(A,B[n])$ and $\mathrm{Hom}(A,B)[n]$
Let $A$ and $B$ be chain complexes of $k$-modules. Is there an isomorphism or a homotopy equivalence between the hom complexes $\mathrm{Hom}(A,B[n])$ and $\mathrm{Hom}(A,B)[n]$?
I tried with the map $...
- 79
1
vote
2
answers
145
views
Quasi-isomorphisms are stable under homotopy base change in $\operatorname{Ch}(\mathcal{A})$
Edit: People are commenting that what I'm trying to do follows from "general theory of model categories" or "general theory of categories of fibrant objects". I have no idea at all ...
2
votes
1
answer
51
views
Factorization of quasi-isomorphism is also a quasi-isomorphism
Let $\mathcal{A}$ be an abelian category, $ X_\bullet
\overset{f}{\hookrightarrow} Y_\bullet \overset{g}{\hookrightarrow} Z_\bullet$ be in $Ch(\mathcal{A})$ such that $gf:X_\bullet \hookrightarrow Z_\...
- 195
6
votes
2
answers
2k
views
Short Exact Sequence of Complexes Induces Long Exact Sequence of Homology Groups
I am following Lang's Algebra on General Homology Theory and wanted to try proving the short exact sequence of complexes
$$\require{AMScd}
\begin{CD}
0 @>{}>> A @>{f}>> B @>{g}>...
- 1,498
0
votes
1
answer
154
views
Morphism from image to kernel for a complex of objects in an abelian category.
I'm back again with another basic category theory question. I promise I will get my head around this - one day.
Consider a complex (so $f_{n}\circ f_{n-1}=0$), of objects in an abelian category:
$$
\...
- 1,647
1
vote
1
answer
159
views
Weibel Lemma 1.6.2.
The following is (a part of) Lemma 1.6.2. from Weibel's Homological Algebra.$\newcommand{\C}[1]{\mathcal{#1}}\DeclareMathOperator{\coker}{coker}\newcommand{\md}[1]{{\left\lvert #1 \right\lvert}}\...
- 396
2
votes
1
answer
726
views
Fundamental Lemma of homological Algebra, what is $H_0f$ supposed to mean?
I am currently attending a lecture in homological algebra, where we discussed the fundamental lemma of homological algebra (i will just cite the relevant part of the theorem):
Let $\mathcal{A}$ be an ...
- 2,486
2
votes
0
answers
88
views
Derived functor of $H^0$
Let $R$ be a commutative unitary ring and $\mathrm{Mod}_R$ be the category of $R$-modules and $C := \mathrm{Ch}_{\geq 0}(\mathrm{Mod}_R)$ be the category of chains of $R$-modules which are zero in ...
- 437
5
votes
0
answers
652
views
Naturality of connecting homomorphisms
I'm reading Weibel's "An Introduction to Homological Algebra" page by page. I'm confused about theorem 1.3.1:
Theorem 1.3.1 Let $0\to A.\xrightarrow{f} B.\xrightarrow{g}C.\to 0$ be a short ...
- 514
1
vote
1
answer
68
views
Condition for an element in Ext be zero
Consider an abelian category $\mathcal{A}$ and two objects $A,B$ of $\mathcal{A}$. It is straightforward that an element $\eta \in \text{Ext}^{1}(A,B)$ of the form $0 \to B \xrightarrow{f} X \...
- 33
2
votes
0
answers
131
views
Can equivalent abelian categories have non-equivalent derived categories?
This is a point of stupid confusion for me.
Let $s:\mathcal{A}\to\mathcal{B}$ be an equivalence of abelian categories. Does this functor induce a triangulated equivalence $\overline{s}: \mathbb{D}^b(\...
- 335
2
votes
0
answers
59
views
On the categorical definition of Chain Homology
I am having trouble with the general definition of Chain homology. Given an Abelian category $\textbf{A}$, I have defined the category of Chain Complexes $\text{Ch}_\bullet(\textbf{A})$ as usual. ...
- 1,359
5
votes
1
answer
265
views
Grothendieck category with a generator has injective hulls (envelopes): a subtlety in Freyd's book on abelian category
First, Freyd proves that an object in a Grothendieck category is injective if and only if it has no proper essential extensions. For each object $A$ of the category, he chooses a monomorphism $e_A\...
- 4,648
1
vote
1
answer
121
views
Chain Complex morphism: Arbitrary maps instead of Homomorphism?
Given two chain complexes $\{ G_i \}$ and $\{ H_i \}$ we usually define a morphism of chain complex as a family of homormophisms $\{ f_i \}$ such that the diagram commutes:
$$
\begin{matrix}
\dots &...
- 8,772
6
votes
1
answer
233
views
Weibel: spectral sequence of a filtration
Let $$\dots \subseteq F_{p - 1}(C) \subseteq F_p(C) \subseteq F_{p + 1}(C) \subseteq \dots$$ be a filtration of a chain complex in an abelian category. In his book Introduction to Homological Algebra, ...
- 4,648
1
vote
1
answer
380
views
Properties of a middle resolution of a Horseshoe Lemma
I understand the proof of Horseshoe lemma as it is presented in, e.g. Weibel's book. However both Weibel and these notes note an additional property which is at the bottom of my screenshots here:
...
- 4,648
2
votes
1
answer
78
views
Spectral sequence of a filtration: a possible mistake
$\require{AMScd}$The following is taken from these notes by Daniel Murfet.
Let $ \cdots \subseteq F^{p + 1}(C) \subseteq F^p(C) \subseteq F^{p - 1}(C) \subseteq \cdots$ be a filtration of a complex $...
- 4,648
2
votes
1
answer
178
views
Exactness in category theory
In MacLane's 'Category Theory for the working mathematician' there is a definition of exactness (page 200):
'A composable pair of arrows $f: a\rightarrow b$ and $g: b\rightarrow c$ is exact at b if ...
- 1,190
1
vote
1
answer
32
views
A complex with prescribed cohomology
Let $\mathcal A$ be an Abelian category (I am happy to assume that $\mathcal A\cong \mathrm{Mod}(R)$ for some ring $R$ if that helps). Given the following long exact sequence in $\mathcal A$
$$
0\to ...
- 2,024
3
votes
2
answers
41
views
Are the two morphisms $T^i(C)\to T^{i+1}(A)$ the same for each $i$?
For short exact sequence $0\to A\to B\to C\to 0$,
we have $T^i(C)\to T^{i+1}(A)$ such that the long sequence $$0\to T^0(A)\to T^0(B)\to T^0(C)\to T^1(A)\to \cdots$$ is exact.
For short exact sequence $...
- 2,716
3
votes
0
answers
108
views
Question on why a particular quasi-isomorphism between complexes doens't have an inverse
My question is on the example below, taken from page 4 of http://www.math.wisc.edu/~andreic/publications/lnPoland.pdf.
I'm not familiar enough with this stuff yet to understand why the quasi-...
user525033
3
votes
1
answer
76
views
Is a "subfunctor generated by x" really a subfunctor?
I am reading Freyd's Abelian Categories, and Essential Lemma 7.12 says:
Let $\mathcal{A}$ be an abelian category, and $Ab$ be the category of abelian groups. Let $M \rightarrow E$ be an essential ...
- 905
15
votes
4
answers
1k
views
Homological algebra using nonabelian groups
Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice ...
- 3,216
1
vote
1
answer
89
views
Computing Ext for a complex of modules, help with a proof in Stacks Project
I am stuck on a step in the proof of Lemma 15.66.2 here. Let $R$ be a commutative ring with identity and let $K^{\bullet}$ be a complex of $R$-modules. I am stuck on the following sentence:
"Choose a ...
- 3,693
0
votes
1
answer
228
views
How to show a morphism $f^•: A^•\to B^•$ of complexes induce the morphism of cohomology objects $H^i(A^•)\to H^i(B^•)$?
Let $\mathscr C$ be a Abelian category, how to show a morphism $\varphi^•: A^•\to B^•$ of complexes induce the morphism of cohomology objects $H^i(A^•)\to H^i(B^•)$?
$A^•:\qquad \cdots\to A^{i-1}\...
- 2,716
2
votes
1
answer
50
views
Injective objects in a category
Let $C$ be a category and $x\in$ Ob$(C)$. Assume that $I\in$ Ob$(C)$ is injective in $C$, and there are morphisms $f:x\rightarrow I,~ g:I\rightarrow x$ such that $gf =1_x$. Is it true that $x$ is ...
- 63
1
vote
1
answer
971
views
Exact additive functor preserves homology
Let $R$ and $A$ be rings, $T:{}_{R}\text{Mod}\rightarrow {}_A\text{Mod}$ be an exact additive functor and $(C_\bullet,d_\bullet)$ a chain complex in ${}_{R}\text{Mod}$. Prove that
$$H_n(TC_\bullet,...
- 653
4
votes
1
answer
203
views
Non-trivial extensions with trivial connecting homomorphism in long exact sequence?
Let $X$ be a smooth projective curve over an alg. closed field $k$. Consider an exact sequence of locally free sheaves:
$$0 \to \mathcal{E}_1 \to \mathcal{E}_2 \to \mathcal{E}_3 \to 0$$
This ...
- 4,899
1
vote
1
answer
76
views
If an object of an abelian category is split in the derived category, then must it be split?
Let $\mathcal{A}$ be an abelian category, and $A\in\mathcal{A}$ an object. Let $D(\mathcal{A})$ be its derived category, and let $A'$ denote the image of $A$ in $D(\mathcal{A})$. If $A' = B'\oplus C'$ ...
user355183
1
vote
1
answer
120
views
A particular exact sequence in an abelian category
Let $\mathcal{A}$ be an abelian category and let $(X^\bullet, d^\bullet)$ be a chain complex in $\mathcal A$. I want to show that there exists an exact sequence $$0\to \text{im}(d^{i-1})\to \ker(d^i)\...
- 4,212
2
votes
1
answer
176
views
Homotopy of chain complexes (category theoretic proof)
I know the usual proof of the fact that if a morphism between chain complexes $f$ is homotopic to zero then it induces the $0$ map on cohomology.
I was wondering if there is an easy proof of this ...
- 3,795
3
votes
0
answers
49
views
An alternative of projective dimension in triangulated categories
Does anybody know anything about an alternative of the notion of projective dimension (defined in abelian categories with enough projective objects) in triangulated categories?
- 31
1
vote
0
answers
134
views
Easy characterization of Cohomology in an Abelian Category
It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still...
Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
- 4,506
0
votes
1
answer
150
views
Are the hom sets in the category of varieties abelian groups?
This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
- 6,828
42
votes
3
answers
2k
views
Why do universal $\delta$-functors annihilate injectives?
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\...
- 93.3k
29
votes
2
answers
3k
views
Meaning of "efface" in "effaceable functor" and "injective effacement"
I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
- 132k
42
votes
3
answers
5k
views
How to define Homology Functor in an arbitrary Abelian Category?
In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient
Ker d / Im d
where d as usual denotes the differentials, indexes skipped for simplicity.
How ...
Waldi