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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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. ...
2
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1answer
42 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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23 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 ...
6
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1answer
128 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
5
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1answer
116 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 ...
2
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0answers
58 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 ...
3
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29 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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52 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.$(A)\geq 2$ ...
2
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1answer
27 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 ...
1
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1answer
35 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 ...
0
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1answer
50 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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1answer
60 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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0answers
24 views

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 ...
2
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0answers
62 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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0answers
43 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
44 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. ...
5
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1answer
51 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 ...
0
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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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1answer
57 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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2answers
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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33 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 ...
2
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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 ...
3
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2answers
65 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
53 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 ...
1
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1answer
44 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
35 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
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1answer
78 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
55 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
37 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
32 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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58 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 ...
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1answer
20 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
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0answers
55 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 ...
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31 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 ...
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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 $n\geq 0$, ...
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69 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
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1answer
65 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 ...
3
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0answers
38 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 ...
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2answers
72 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 ...
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1answer
37 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
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1answer
53 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
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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 ...
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1answer
31 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
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1answer
47 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 $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
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46 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 ...
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1answer
26 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
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1answer
81 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 ...