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.

learn more… | top users | synonyms

0
votes
1answer
19 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 $R\text{-Mod}$. ...
2
votes
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 $M$ ...
5
votes
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} ...
2
votes
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
votes
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
votes
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
votes
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 ...
23
votes
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
votes
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 $$ ...
1
vote
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 ...
8
votes
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 ...
23
votes
3answers
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 ...
0
votes
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. ...
5
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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, ...
0
votes
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 ...
3
votes
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 ...
7
votes
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 ...
2
votes
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?
1
vote
0answers
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 ...
1
vote
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 ...
3
votes
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 ...
3
votes
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
votes
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 ...
2
votes
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 ...
2
votes
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, ...
1
vote
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 ...
1
vote
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. ...
0
votes
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 ...
1
vote
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 ...
1
vote
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? ...

1 2 3 4 5 10