Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.
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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 $R\text{-Mod}$. ...
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1answer
60 views
Finite generation of Hom between cyclic and artinian module
Let $R$ be a Noetherian ring with unit, and $I$ be a nonzero ideal of $R$. Let $M$ be an artinian $R$ module. Is $Hom(R/I, M)$ finitely generated?
This is tagged as duplicate, but it is not. Here $M$ ...
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2answers
81 views
Property of modules via exact sequences
Suppose $A\neq 0$ is a commutative ring with $1$. Let $L, M, N$ be $A$-modules such that the sequence
$$0\longrightarrow L\overset{\alpha}{\longrightarrow} M\overset{\beta}{\longrightarrow} ...
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0answers
39 views
How to define the natural map on the second page of a spectral sequence?
I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
3
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1answer
47 views
trefoil knot and meridian/longitudinal cycles
I hope this is a simple question...
For the trefoil knot 3_1, whose knot group is given by a presentation of the fundamental group, $\pi_1(M) = \langle a,b: aba = bab \rangle$, where the meridian and ...
3
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2answers
39 views
On chain complex morphisms
The following seems quite obvious to me. Nevertheless I would like to have another opinion.
Suppose $(A_\bullet,d_A)$ and $(B_\bullet,d_B)$ are chain cmplexes, such that
$d_A$ is the trivial ...
2
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3answers
41 views
Is there a terminological difference between “sequence” and “complex” in homology theory
Suppose you are given something like this:
$\dots \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \dots$
People tend to talk about "chain complexes" but about "short exact sequences". Is ...
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8answers
624 views
Intuitive meaning of Exact Sequence
I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all.
Can anyone explain them for me? ...
2
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1answer
34 views
Embedding into a morphism of distinguished triangles
Everything in this question happens in a triangulated category $\mathbf{D}$. I am trying to prove that in a diagram like this
$$
...
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0answers
28 views
Motivating the definition of right derived functors in the context of derived categories.
Let $A$ and $B$ be abelian categories and let $F : A \to B$ be an additive functor. Let $K^+(F) : K^+(A) \to K^+(B)$ be the induced functor on the corresponding homotopy categories of left bounded ...
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1answer
102 views
How *should* we have known to invent homological algebra?
Previously I asked How did we know to invent homological algebra?, because I was under the misapprehension that concrete examples of long exact sequences had been a major motivation for developing ...
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375 views
How did we know to invent homological algebra?
Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought.
(Original question ...
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1answer
50 views
Characterization of injective objects in abelian categories
In this link it is proved that in an abelian category $\mathcal C$ we have that $f:A\rightarrow B$ is mono iff the sequence $0\rightarrow A\rightarrow B$ is exact, where the arrow from $A$ to $B$ is ...
5
votes
1answer
100 views
Explanation of example 3F.7 in Hatcher
The section I am refering to is the following example on page 314 of Hatcher's Algebraic Topology:
I'm a bit confused by his statement about relations and can't quite see what he is trying to say. ...
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2answers
66 views
Question on $\mbox{Ext}^1$
I have 2 questions, one of them concerning the isomorphicity of quotient groups (rings), and the other is on $\mbox{Ext}^1$. It's pretty long, but somehow related to each other. So I just kinda put ...
8
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1answer
85 views
Can it happen that the image of a functor is not a category?
On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a ...
5
votes
1answer
77 views
What is the integral homology of $\mathrm{GL}_2(\mathbb{Z}[i])$?
I am currently trying to compute homology groups of general linear groups over the ring of integers of an imaginary quadratic number field. As I would like to check my results I would like to know if ...
5
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1answer
78 views
Mistake in Popescu's book “Abelian Categories with Applications to Rings and Modules”
Corollary 5.5 a) in chapter 1 on page 13 in Popescu's book "Abelian Categories with Applications to Rings and Modules" says:
Let $F\colon C\rightarrow C^\prime$ be a functor and $G$ be a full and ...
5
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1answer
49 views
If $M \simeq N$ in ${\tt stmod}(G)$ will $M \oplus \text{(proj)} \simeq N \oplus \text{(proj)}$ in ${\tt mod}(G)$?
Let $G$ be a finite group and ${\tt stmod}(G)$ the stable module category for $G$, i.e., the category whose objects are $G$-modules and whose morphisms are $G$-module homomorphisms modulo those that ...
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0answers
33 views
Left-invertible $R$-module homomorphisms.
I am trying to understand the proof of the following statement
Let $\varphi: M\to N$ be an $R$-module homomorphism. Then it has a left-inverse if and only if the sequence
$$
0\rightarrow ...
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5answers
63 views
Split-Lemma for chain complexes
Suppose $k$ is a field and $A$, $B$ and $C$ are chain complexes of $k$-vector spaces, i.e., objects in $\mathbf{Ch}(k\text{-}\mathbf{Vect})$. Is there
are chain complex version of the split lemma, ...
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0answers
52 views
What are V(f) and D(f) in real practice of EGA
https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!355
would like to do and understand what is V(f) and D(f)
where D(f) = SpecA - V(f)
in the following diagram, it said V(f) is subset of p ...
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1answer
38 views
Does the analog of homological algebra studying maps where, say, $d \circ d \circ d = 0$ have a name?
I don't have an application in mind or anything; I'm just curious.
We can think about homological algebra as the study of endomorphisms $d$ such that $d \circ d = 0$. Most of homological algebra ...
3
votes
3answers
110 views
Proving that free modules are flat (without appealing projective modules)
Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first ...
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1answer
90 views
Question on Projective Dimensions
$\require{AMScd}$I have a question regarding a claim in A first course of homological algebra by Northcott. I think it's very easy, since the author didn't provide a proof, and just kind of claimed ...
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1answer
32 views
Is every additive monofunctor between abelian categories left exact?
Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?
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38 views
Split extensions and Ext functor
We consider the following exact sequences, first is a proyective resolution of $C$ and second is an extension $\xi$ of $A$ by $C$:
$P_2\xrightarrow {d_2}{P_1}\xrightarrow{d_1}P_0\rightarrow ...
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1answer
39 views
$f$ can be extended iff $\partial f = 0$
If
$0\rightarrow{A'}\rightarrow{A}\rightarrow{A''}\rightarrow{0}$
is an exact sequence of modules, then there exists an exact secuence
...
4
votes
1answer
53 views
Is there a nice list of spectral sequences that don't come from particular constructions?
When you first learn about rings, it's important to have examples of, say, a PID which is not a Euclidean domain, a UFD which is not a PID, and so forth, to help build intuition and provide test ...
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0answers
48 views
What are some important examples of differential objects that aren't naturally graded?
[By a "differential object" I mean an object $A$ in some abelian category $\mathcal{A}$ together with a morphism $d : A \to A$ such that $d \circ d = 0$. By a "differential module" I mean a ...
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0answers
79 views
Properties of quotient categories.
Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
4
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1answer
83 views
Relating the Künneth Formula to the Leray-Hirsch Theorem
I am reading through Bott & Tu's Differential Forms in Algebraic Topology, which very early on discusses the Künneth formula and the Leray-Hirsch theorem for smooth principal bundles. The proof of ...
3
votes
1answer
101 views
Does taking the direct limit of chain complexes commute with taking homology?
Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree 0) $$C_i=0\rightarrow C^{0}_{(i)}\rightarrow ...
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1answer
39 views
Projective resolution of tensor product
Let $M,N$ are $R$ modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain projective resolution $M\otimes N$ using $P^\bullet, Q^\bullet$. If i understand correctly homology ...
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0answers
48 views
Finite projective dimension and vanishing of ext on f.g modules
Let $A$ be a commutative noetherian ring.
Suppose $M$ is a finitely generated $A$-module.
Let $n>0$ be an integer. It is well known that if $Ext^n(M,N) = 0$ for all $A$-modules $N$, then $M$ has ...
3
votes
2answers
75 views
Show that $[l_1 \cdot l_2 \cdot l_3 ] = [l_1 + l_2 + l_3] \in H_1(X)$ The first Homology group of X
Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) =
l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$).
Show that ...
2
votes
1answer
67 views
Another basic short exact sequence problem
In the following commutative diagram of R-modules, all of the rows and columns are exact. Prove that $K$ is isomorphic to $L$.
\begin{array}{ccccccccccc} &&&&&&&&0 ...
2
votes
0answers
28 views
Support of a direct sum of local cohomology modules
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following:
$$\operatorname{Supp}(\bigoplus_{j\ge ...
2
votes
2answers
33 views
Set of Homomorphisms as an $R-$ module
$\require{AMScd}$
I'm reading A first course of homogocial algebra by D.G. Northcott, and I don't quite get the Example 1 on page 25. Here's what it says:
Example 1
Let the module $A$ belongs to ...
3
votes
1answer
46 views
All local cohomology modules being zero
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$?
The converse of ...
8
votes
2answers
219 views
Vanishing of a certain Tor
I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes
and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
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1answer
21 views
Submodules of homology modules
I have been dealing with certain subgroups of group cohomology, and the following general question comes to my mind. Suppose $C$ is a chain complex of $R$-modules and $H_n(C)$ its $n$-th homology ...
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1answer
35 views
Group extension reference request
I'm looking for a reference for the following "well known" result
Let $C$ be an abelian group and $G$ a finite group, and let $$0 \rightarrow C \rightarrow W \rightarrow G \rightarrow 0$$ be a ...
3
votes
2answers
41 views
Covariant functor, and left exact
I'm reading A first course of Homological Algebra by Northcott, and there is something that the author said it was straightforward. But for some reason, I just don't see the straightforwardness of it.
...
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0answers
35 views
How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?
I am trying to prove that the global dimension of the polynomial ring $F[x_1,...,x_n]$, where $F$ is a field , is exactly $n$. And by Koszul Complex, I know its global dimension is greater than or ...
3
votes
1answer
45 views
Finite Projective Dimension implies non vanishing Ext
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$?
Can't we write the free module as a direct ...
3
votes
1answer
65 views
$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true:
$$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$
The ...
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1answer
21 views
What is the relation between graded modules and finitely generated modules
The reason I ask this question is I found two different statements about Hilbert's syzygy theorem from Jacobson's Basic Algebras 2nd and Wikipedia. Please have a look at the following pictures. The ...
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2answers
57 views
Questions about projective modules.
Let $P$ be a projective module and $M$ a submodule of $P$. We know that $M$ is also a projective module. Can we conclude that $P=M\oplus N$ for some module $N$? Thank you very much.
3
votes
1answer
65 views
Property of Hom-functor
How to prove $$\operatorname{Hom}_{R}(A,\operatorname{Hom}_{\mathbb{Z}}(R,B))\cong \operatorname{Hom}_{\mathbb{Z}}(A,B)$$ where $R$ is a commutative ring, $A$ an $R$-module and $B$ an abelian group?
...
