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
0answers
17 views

Isomophisms of modules

I'm reading a book about homological algebra. There is one exercise with whom I have a problem. Show that for any $\mathbb{Z}$-module $M$ and any $q\in \mathbb{Z}$ we have I) $ \ \ ...
2
votes
0answers
25 views

“Associative” law for $Hom$ useful in computing $Ext$.

Setting: let $R$ be a ring, $f: R \to S$ a ring homomorphism, $A$ a $R$-module and $B$ a $S$-module. Sometimes, when I compute by hand some $Tor$ groups, I use the property of tensor product: $ A ...
0
votes
0answers
20 views

Counting chain maps

Let $\mathbb{K}$ be a field and let $C_{\cdot}$ and $K_{\cdot}$ be bounded chain complexes with coefficients in $\mathbb{K}$. Then the set of chain maps $f_{\cdot}:C_{\cdot}\to K_{\cdot}$ is a ...
3
votes
2answers
50 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
0
votes
1answer
39 views

How many projectives and injectives exist in a path algebra?

I do not know an efficient way to determine whether a quiver representation is projective or injective. The definitions and properties such as "Projectives are summands of free modules", etc do not ...
1
vote
1answer
32 views

Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
1
vote
1answer
33 views

Why is the $\text{End}_A(M)$-module $\text{Hom}_A(N,M)$ finitely generated?

Let $A$ be an Artin algebra and let $M,N$ be some finitely generated modules in mod(A). Why is then the $\text{End}_A(M)$-module $\text{Hom}_A(N,M)$ finitely generated? Thanks for the help.
2
votes
1answer
30 views

Relating different Ext groups

If $G$ is a group, $H$ is a normal subgroup, and $A$ and $B$ are $G$-modules, are there any general theorems that relate Ext$_G(A,B)$ to Ext$_H(A,B)$?
4
votes
1answer
70 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
2
votes
1answer
46 views

Homology of a graph.

Let $\Gamma$ be a graph with $V$ vertices and $E$ edges. If we orient the edges, we can form the incidence matrix of the graph. This is a $V\times E$ matrix whose $(i j)$ entry is $+1$ if the edge ...
1
vote
1answer
33 views

Weibel definition 1.4.1. understanding the indexes on splitting maps

The book says: Definition 1.4.1. A complex $C$ is a called split if there are maps $s_n : C_{n+1} \to C_{n+1}$ such that $d = dsd$. The maps $s_n$ are called splitting maps. If in addition $C$ ...
1
vote
1answer
27 views

Do we have a “short five lemma” for any two of the isomorphisms?

$\require{AMScd}$ The "short" Five Lemma concerns the famous form of exact commutative diagram: $$\begin{CD}0@>>>A@>>>B@>>>C@>>>0\\&@VV\simeq ...
0
votes
0answers
34 views

Split exact sequences of vector spaces

The question is on page 2, exercise 1.1.3. For the proof that $\{ C_n \}$ is a chain complex I only need to show that $(i\circ p)\circ (i\circ p) = 0$ where $i$ is the inclusion map, and $p$ is the ...
0
votes
1answer
19 views

A homological algebra question.(Chain map).

In Robert Ash's notes a chain map is defined by the next relation: $f_{n-1}\circ d_n = d_n\circ f_n $; while in Charles Weibel's book on page 2, it's defined as follows: $u_{n-1}\circ d_n = d_{n-1} ...
2
votes
0answers
51 views

when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
0
votes
0answers
26 views

Question about cohomology of free abelian group

Let $A$ and $B$ be finite abelian groups and suppose that $B$ acts on $A$. Now, suppose we have two surjective homomorphisms $f,g:\mathbb{Z}^n\twoheadrightarrow B$ for some $n\in \mathbb{N}$. This ...
1
vote
1answer
27 views

Ext$_R^n(Q,A)=0=$Tor$_n^R(Q,A)$ where $Q$ is the field of fractions of a domain $R$

I am currently working through a problem in Rotman: Let $R$ be a domain and let $Q=$Frac$(R)$. If $r\in R$ is nonzero and $A$ is an $R$-module for which $rA=0$, prove that for all $n\geq 0$, ...
2
votes
0answers
61 views

Singular homology: Change of coefficients

Let $f: X \to Y$ be a map of topological spaces which induces isomorphisms $H_*(f;\mathbb{Z})$ on singular homology with $\mathbb{Z}$-coefficients. Show that $f$ induces isomorphisms ...
4
votes
1answer
58 views

Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
2
votes
0answers
33 views

Extend a map to a 1-cocycle

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where ...
3
votes
2answers
65 views

Complete and unabridged proof of the theorem of acyclic models

Can someone indicate me where I can find a complete and unabridged proof of the said theorem? By "complete and unabridged" I mean not writing something like "details are left to the reader as an ...
0
votes
1answer
33 views

Every projective $R$-module $P$ is free

I have come across a theorem which states that if the underlying ring $R$ is a principal ideal domain then every $R$-module $P$ which is projective is free also. But the problem is I have encountered ...
5
votes
1answer
51 views

Natural Transformation: Direct Products

I have result that tells me $$\displaystyle \varphi : \text{Hom}_R \bigg(A, \prod_{i \in I} B_i \bigg) \to \prod_{i \in I} \text{Hom}_R(A, B_i)$$ is a $Z(R)$-isomorphism. The next result tells me that ...
2
votes
0answers
10 views

Pontryagin duality for torsion abelian groups.

I am trying to prove Pontryagin duality for torsion abelian groups. It might appear that this question is a duplicate of this, but I assure you its not. Rather if the linked question had all the ...
-1
votes
1answer
68 views

Example of flat module but not torsion free [closed]

I want an example of flat module but not torsion free. Does it exist? Please hint me. Thanks. Torsion submodule: if $R$ is a domain and $M$ is an $R$-module, then its torsion submodule is ...
0
votes
1answer
28 views

Hom Functor Preserves Direct Products

I'm looking at a statement in Rotman's 'Introduction to Homological Algebra' which I'm having a problem with: Theorem 2.30.i: There is a $Z(R)$-isomorphism $$\varphi : \text{Hom}_R \bigg( A, ...
2
votes
1answer
39 views

Can we see directly from the cocycle condition that 2-cocycles are symmetric?

Let $A$ be an abelian group and let $C$ be a cyclic group. All central extensions of $C$ by $A$ are abelian because in any such extension $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
-1
votes
0answers
55 views

how to show naturally isomorphic

I have a homological exam on Saturday , and I have some problem to understand of naturally isomorphic.my problem . the end of this theorem must proof naturally isomorphic $T_n $and ...
0
votes
0answers
34 views

short exact sequences of complexes and triangles in the homotopy category

Suppose I start with an abelian category $\mathcal{A}$, form its category of complexes $C(\mathcal{A})$ and consider a short exact sequence in this category: $$0 \to A^{\bullet} \to B^{\bullet} \to ...
0
votes
1answer
25 views

In an SES of chain complexes in an abelian category two of complexes exact implies the third is exact.

Consider a short exact sequence of chain complexes: $$0_{\cdot} \rightarrow A_{\cdot} \xrightarrow{f} B_{\cdot} \xrightarrow{g} C_{\cdot} \rightarrow 0_{\cdot}$$ If any two of ...
6
votes
1answer
73 views

Applications of diagram lemmas

I'm currently reading Theo Bühler's survey on exact categories about which he says This article is written for the reader who wants to learn about exact categories and knows why. Very few ...
1
vote
1answer
42 views

Weibel's book, Page 8. $\text{Tot}(C)$. What is the sum of the horizontal and vertical differentials in a bicomplex?

... define the total complexes $\text{Tot}(C) = \text{Tot}^{\Pi}(C)$ and $\text{Tot}^{\oplus}(C)$ by $\prod_{p+q = n} C_{p,q}$, and $\bigoplus_{p + q = n}C_{p,q}$. The formula $d = d^h + d^v$ ...
4
votes
0answers
47 views

Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
1
vote
1answer
30 views

What does “universal w.r.t. this property” mean? (kernel of a morphism in an additive category)

In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property. This is ...
2
votes
0answers
22 views

How do you form differential maps in a quotient complex? (Weibel pg. 5)

They say "...In this case we can assemble the quotient modules $C_n / B_n$ into a chain complex $$ \cdots \xrightarrow{d} C_{n+1}/B_{n+1} \xrightarrow{d} C_{n}/B_{n} \xrightarrow{d} \cdots $$ But ...
2
votes
0answers
30 views

homology commutes with direct product of chain complexes. Direct proof

This is an attempt to prove that direct product of chain complexes commutes with homology (exercise in Weibel's book). I've had some success since I've proved that $Z_n(\prod_{\alpha \in A} C_{\alpha ...
0
votes
1answer
20 views

Weibel exercise 1.1.2. the $n$th homology module is a functor from category Ch-Mod$(R)$ to Mod-$R$

Ch-Mod$(R)$ is the category of $R$-module chain complexes. How do you turn a homology module into a functor? Thanks for teaching.
0
votes
0answers
16 views

Weibel's book exercise 1.1.2. Cycles get sent to cycles by chain complex homs $u : C_{\cdot} \to D_{\cdot}$

A morphism of chain complexes is a family of homs $u_n : C_n \to D_n $ such that $u_{n-1} d_n^{(C)} = d_n^{(D)} u_n$. Weibel's book says that cycles "get sent to cycles". To me that means that ...
1
vote
1answer
32 views

Is this a typo in Weibel, page 1?

It says a morphism $u : C_{\cdot } \to D_{\cdot}$ of chain complexes is a family of homomorphisms $u_n : C_n \to D_n$ such that $u_{n-1} d_n = d_{n-1} u_{n}$, but shouldn't it just be that $u_{n-1} ...
0
votes
0answers
25 views

How do I write a correct answer to Weibel exercise 1.1.1.?

Exercise 1.1.1. Set $C_n = \Bbb{Z}/8$ for $n \geq 0$ and $C_n = 0$ for $n \lt 0$. Let $d_n : x \pmod{8} \to 4x \pmod{8}$ Compute the homology modules of the chain complex $C_{\cdot}$. I got that ...
0
votes
1answer
12 views

Does $C < G$ imply $H_n(C,A) < H_n(G,A)$?

Suppose to have two groups $C$ and $G$ (not necessarily abelian) such that $C < G$ (subgroup, not necessarily proper). Let's fix an abelian group $A$ such that it is a trivial $G$-module (and ...
1
vote
0answers
21 views

$S$ subring of $R$. Is a projective objects in $R$-$\bmod$ still projective in $S$-$\bmod$?

Let $R$ be a ring (not necessarily commutative and not necessarily with unit). Recall the definition of $R$-$\bmod$ as an abelian group $A$ on which $R$ acts on the left respecting the following ...
-1
votes
0answers
39 views

question about $\textrm{Tor}$ functor

I read this fact but I don't know why is true when $R$ not necessarily commutative hold? anybody can hint me?thanks a lot;
1
vote
1answer
22 views

Is zeroth homology right adjoint to taking homotopy type of projective resolution?

Let $\mathsf A$ be an abelian category and $\mathsf{K(A)}$ be the homotopy category of chain complexes over $\mathsf A$. Let $P_\bullet,Q_\bullet$ be projective resolutions of $A,B\in \mathsf A$ ...
5
votes
1answer
57 views

Nonspliting short exact sequence

The short exact sequence $0\rightarrow \mathbb Z \stackrel{\alpha}{\longrightarrow} \mathbb Z \oplus \mathbb Q \stackrel{\beta} {\longrightarrow} \mathbb Q \rightarrow 0$ is splits because we have ...
0
votes
0answers
15 views

Chain morphism into a subcomplex homotopic to identity

Let us assume we have a chain complex $(X_\bullet,\partial_\bullet)$ of vector spaces and a subcomplex $(Y_\bullet,\partial_\bullet)$. Let us furthermore assume that there exists a morphism ...
2
votes
0answers
39 views

Fundamental lemma of homological algebra via acylic models?

The fundamental lemma of homological algebra discusses the extension of arrows to chain maps from a projective to an arbitrary resolution, and the uniqueness-up-to-homotopy of such an extension. ...
2
votes
0answers
29 views

Relationship between acyclic models and universal $\delta$-functors

(An elementary version of) The acyclic models theorem more-or-less says that natural transformations between the zeroth homology of a free functor taking values in $\mathsf{Ch}^+_\bullet(\mathsf A)$ ...
4
votes
0answers
74 views

$\operatorname{Ext}^n$: computation verification

I would like someone to verify my computation of $\operatorname{Ext}^n$. Problem: Let $p$ be a prime, $k$ a field of characteristic $p$, $G = \langle x \mid x^p = 1 \rangle$, $B = kG$, $S = k(1 ...
0
votes
1answer
16 views

Centralizer acting on the homology of a subgroup

Let $H\subset G$ be a subgroup. Let $E_*G$ be a free (right) $\mathbb ZG$-resolution of the trivial representation $\mathbb Z$. Because $E_*G$ is then also a free $\mathbb ZH$-resolution of the ...