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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A problem about isomorphism in module theory

For a sequence of $R$-modules like this:$A \overset{f}{\rightarrow} B \overset{g}{\rightarrow}C$ such that $\mathrm{Im} f \subset\ker g$ and $B/\mathrm{Im}f \cong B/\ker g$. Then $\mathrm{Im}f=\ker ...
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30 views

Properties of isomorphism in module theory [on hold]

I have two exercises, but I can't solve them: a. If $X, A, B $ are $R$-modules with $A \subset B \subset X $, prove that if $X/A \cong X/B$ then $A=B$ b. If $X/A \cong X$ then $A=0$
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100 views

Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
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1answer
64 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
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the functor Ext repairs the inexactitude of Hom on the right

For a short exact sequence of R -modules: $ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$. Prove that: the following sequence is exact $0 \rightarrow \mathrm{Hom} ...
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1answer
95 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
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2answers
31 views

Double complex with exact rows

Let $(K^{p,q},\delta,d)$ be a double complex of modules. We assume that $\delta$ of degree $(1,0)$, $d$ has degree $(0,1)$ and $d$ and $\delta$ commute. Since $d$ and $\delta$ commute, then $\delta$ ...
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1answer
66 views

Colimits and tensor product ground rings

Is it true that $\varinjlim (M \otimes_{A_i} N) = M \otimes_A N$ where $A = \varinjlim A_i$ and $M$ and $N$ are $A$-modules? Take maps $f : A_j \rightarrow A_k$ and $m : M \otimes_{A_j} N \rightarrow ...
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49 views

When is a homomorphic image of a polynomial ring self injective [closed]

Is the ring $\dfrac{\mathbb{Z}_3[x, y]}{(x^2y)}$ self injective?
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33 views

Projective and injective resolutions of cyclic modules

Let $R$ be a ring and $M$ a cyclic $R$-module. It is well-known that always exist projective and injective resolutions of $M$. Is it any method to construct explicitly these resolutions which have the ...
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37 views

Koszul complex and isomorphism of graded algebras

I'm reading an article about noncommutative geometry and I'm trying to prove the following theorem Let $R$ be a commutative ring and assume that $I$ is ideal generated by regular sequence ...
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1answer
27 views

Something wrong with proof: left adjoint functor preserves projectives

First a remark, I skipped the hypothesis "left adjoint to an exact functor" on purpose because the sketch of argument I wrote down I didn't use this, at least according to me. I know that there ...
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1answer
52 views

Is each $F$-acyclic resolution homotopic to a projective resolution?

Here is an excerpt from some notes I stumbled upon online: From what I understand, the "elementary proof" is just the fundamental lemma of homological algebra which says the homotopy type of chain ...
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89 views

Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
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1answer
38 views

Long exact sequence in homology: naturality=functoriality?

In every book I've looked, the "naturality" of the long exact sequence in homology simply says that every arrow between short exact sequences translates into an arrow between the long exact sequences ...
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2answers
65 views

Hatcher Exercise 2.2.38

I'm struggling to show exactness at $C_n\oplus D_n$. Let's take $(x,y)\in C_n\oplus D_n$ in the kernel of $C_n\oplus D_n\to E_n$, i.e. the pushforwards $x', y'$ into $E_n$ resp. satisfy $x' + y' = ...
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1answer
47 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
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1answer
36 views

Equivalence between derived categories preserve distinguished triangles

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!
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19 views

Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct ...
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121 views

Extension of group with Ext$^{1} (A, B) = 0.$

Are there any infinite torsion free abelian groups $A$ and $B,$ with $A$ is not projective and $B$ is not divisible but $$\text{Ext} ^{1}(A, B) = 0.$$ Thanks
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105 views

In what kinds of categories is a monic epi an isomorphism?

Is there a general description of categories $\mathscr{C}$ in which all monic epis are actually isomorphisms? In general, monic epis need not be isomorphisms. For example, in the ...
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53 views

Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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27 views

Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in ...
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45 views

Dominant dimension $\geq 2$ implies a certain double centralizer property

let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: $\bullet$ $Ae$-dom.dim.$(A)\geq 2$ ...
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25 views

Question concerning a faithful module over an Artinian ring

Let $A$ be an Artinian ring and $M$ in $\operatorname{\mathfrak{mod}} A$. Is it true that $M$ is faithful if and only if there is an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for ...
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1answer
34 views

Short exact sequences and different extension

Let $A = \mathbb Z$ and $B = \mathbb Q.$ Then Ext$(A, B)$ gives the set of all equivalent extensions of $A$ by $B.$ I have few questions. Is this sequence $0\rightarrow \mathbb Z ...
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1answer
49 views

Any epi with codomain $P$ is split implies $P$ is projective [duplicate]

I'm struggling to prove that if any epi with codomain $P$ splits, then $P$ is a projective object. The converse direction I proved by factoring the identity to give a right inverse of the pi. How can ...
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58 views

short exact sequence of algebras over a field

Let $A,B,C$ be algebras over a field $F$ ($F=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime). The height of $A$ is defined to be $$ \mathrm{height}(A)=\sup_{a\in A}\inf\{n(a)\in \mathbb{N}\mid a^{n(a)+1}=0 ...
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41 views

Short exact sequence and extension

Let $$0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 ~~~~~(1)$$ be a short exact sequence of abelian groups. Suppose $$0\rightarrow X^{'} \rightarrow Y^{'} \rightarrow Z^{'} \rightarrow 0 ...
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pd$(\sum\limits_{\alpha\in A}M_\alpha)=\sup\limits_{\alpha\in A}\{$pd$(M_\alpha)\}$

I came across the following problem in Rotman's Advanced Modern Algebra: 11.69. If $\{M_\alpha\}_{\alpha\in A}$ is a family of left R-modules, prove that pd$(\sum\limits_{\alpha\in ...
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60 views

Interaction of functors and homology in abelian categories

I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense ...
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40 views

Specific case of tensor-hom adjunction

I'm currently working on a project, for which I need various bits of category theory which I've not seen much of before and do not know in detail, so I would like some confirmation (and probable ...
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1answer
42 views

Ext groups and Isomorphism

Let $A$ and $B$ be abelian groups. Let $\mathrm{Ext}(A, B) = 0$, and let $C$ and $D$ be group with $C\cong A$ and $D\cong B$. Does this imply $\mathrm{Ext}(C, D) = 0$? Thanks in advance.
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22 views

Two short exact sequnce are isomorphic extension but not split [duplicate]

Is there some example of two short exact sequences which are isomorphic but not equivalent? Specially I am looking for a short exact sequence having extension unique up to isomorphism but not split. ...
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1answer
47 views

Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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1answer
29 views

Module of homomorphisms as injective module

We know that for any $R$-module exist injective $R$-module $\overline{M}$ such that there is inclusion $i:M\rightarrow \overline{M}$, where we treat $M,\overline{M}$ as $\mathbb{Z}$-modules. Show ...
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56 views

Tensor product of modules and being torsion free

$(i\otimes_{\mathbb{Z}} 1_N):M'\otimes_{\mathbb{Z}} N \rightarrow M\otimes_{\mathbb{Z}} N$ is a monomorphism for every monomorphisms $i:M'\rightarrow M$ iff $\mathbb{Z}$-module $N$ is ...
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69 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) $ \ \ ...
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32 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 ...
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0answers
50 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 ...
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2answers
60 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 ...
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1answer
48 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 ...
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1answer
33 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 ...
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1answer
39 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.
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1answer
34 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$_H(A,B)$?
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76 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
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1answer
50 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 ...
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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 $C$ ...
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1answer
30 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 ...
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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 ...