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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Equivalences between categories $\mathcal{K}^b(\text{Injectives})$ and $\mathcal{D}^b(\mathcal A)$ if $\mathcal{A}$ has enough injectives

I have the following question: Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough ...
3
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1answer
35 views

Homotopy category of chain complexes as a localization

For an abelian category $\mathcal{A}$, define the homotopy category of chain complexes $\mathcal{K}(\mathcal{A})=\mathcal{C}(\mathcal{A})/\mathcal{I},$ where $\mathcal{C}(\mathcal{A})$ denotes the ...
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40 views

Chains (Syntax, Morphology, Algebra) [on hold]

I am providing you with a brief paper which discusses the definition and role of chains in the syntax and morphology of natural language(s): http://www.aclweb.org/anthology/Y10-1018 Section 3 is ...
3
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1answer
57 views

On finite generation of certain $\operatorname{Ext}$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a ...
2
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32 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
3
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34 views

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
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18 views

Functoriality of group homology

I understand that group homology $H_*(-)\colon \mathsf{Grps} \to \mathsf{Ab} $ is a functorial. In Weibel's homological algebra, there is an argument in 6.7 show this by using that $H_*(G;-)\colon ...
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1answer
61 views

Are there any theorems about functors that reflect exactness?

Suppose $F:\mathbf{A}\to \mathbf{B}$ is an additive functor between two abelian categories, we say $F$ is exact iff it preserves short exact sequences. Is there a name for a functor $F$ that ...
4
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1answer
60 views

A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor ...
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1answer
48 views

Extensions of short exact sequences and second cohomology group

Let $G=\mathbb{I}_{p}=<g>$ be the cyclic group of order $p$, where $p$ is a prime and $A=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ a $G-$ module with the action $g^{n}(x,y)=(x+ny,y)$. I want to show ...
3
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33 views

Non-split chain complex which is chain-homotopy equivalent to its homology sequence

This is exercise 1.4.4 from Weibel. Consider the homology $H_*(C)$ of chain complex $C$ as a chain complex with zero differentials. It is easy to show that if C is split, then there is a chain ...
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1answer
39 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
2
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1answer
46 views

Group homology with coefficients vanish

Say $G$ is a group and $M$ is a $\mathbb ZG$-module with the property that $H_i(G;M)=0$ for all $i\ge 0$. Does this happen besides $M=0$?
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0answers
39 views

Is Hom(F, I) an FP-injective module? [closed]

This question can be in the advanced abstract algebra, I think this is right. In fact, I have deleted the tage-abstract algebra. I am confused for the "off-topic", I hope it is right now. Let f be a ...
4
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2answers
30 views

Existence of non-split sequence

Let $G$ be an abelian group such that $G$ contains non-zero elements of finite order. Why there exists some short exact non-split sequence: $0 \rightarrow \mathbb{Z} \rightarrow H \rightarrow G ...
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0answers
18 views

Syzygies in geometry or topology?

I am interested in knowing about the application of Hilbert's Syzygy Theorem (or, for that matter, of the concept of syzygy itself) in geometry or topology, that is, in the fields that have to do with ...
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3answers
65 views

Merge operation in homological algebra?

I provide you with a definition for the Merge operation in one standard textbook on the minimalist program in linguistics: Merge: "basic structure-building mechanism. Merge takes two elements A and B ...
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23 views

Hochschild-Serre spectral sequence for not normal subalgebra

I am trying to understand lemma 2.26 from http://www.math.ru.nl/~solleveld/scrip.pdf I am coserned about calculation of $E^{p, q}_1$. If $\mathfrak{h}$ is Lie ideal than everything is fine. But here ...
2
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0answers
57 views

Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram ...
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1answer
37 views

A necessary and sufficient condition for contravariant auto-equivalence on module categories

I have a problem about the condition of contravariant auto-equivalence on module categories. Let $R$ be a algebra over a field. Let $\mathcal{C}$ be a abelian subcategory of $R$-modules, and assume ...
3
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34 views

The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in ...
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1answer
33 views

Non-integral blocks of category $\mathcal{O}$ over $\mathfrak{sl}_2$ are semisimple.

Hi: I have a problem as follows. Consider the category $\mathcal{O}$ of $\mathfrak{g}: = \mathfrak{sl}_2(\mathbb{C})$. Let $r\in \mathbb{C}$ but $r\notin\mathbb{Z}$. Let $s_\alpha$ be the simple ...
3
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1answer
58 views

Ext groups due to Yoneda: why is this class zero

Consider category of $\mathbb{K}[x]$ modules. Let $\mathbb{K}$ be trivial $\mathbb{K}[x]$ module i.e. $x$ acts by zero. Easy to see that $Ext^2 (\mathbb{K}, \mathbb{K}) = 0$. But there is exact ...
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14 views

When are all Gorenstein projective also pure-injective?

For an artin algebra of finite global dimension, each Gorenstein projective module is projective then is pure-injective. Are there any other examples having this property? That is, all Gorenstein ...
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3answers
63 views

Exact sequence with flat module tensored by module stays exact

The following theorem is given in Liu proposition 1.2.6: Let $A$ be a ring. Let $0\to M^\prime\to M\to M^{\prime\prime}\to 0$ be an exact sequence of $A$-modules. Let us suppose that ...
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1answer
34 views

Kernel of a ring homomorphism involving group rings over the integers

Consider the group ring $\mathbb{Z}[\mathbb{Z}]$; it consists of Laurent polynomials with integer coefficients. Let $n>1$ be a positive integer. I want to find kernel of the ring homomorphism ...
3
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2answers
53 views

Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
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0answers
41 views

Projective resolution

I just started to learn homological algebra and I find it quite hard, so I am sorry if the question is unclear and confused. In fact I am confused. Let $K^-(A)$, where $A$ is an abelian category with ...
1
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1answer
37 views

Degree of an antipodal map

Let $f:S^n\to S^n$ a continuous map, $n>0$; we consider the induced homomorphism $f_* : H_n(S^n)\to H_n(S^n)$, and, recalling $H_n(S^n)\simeq\mathbb Z$, define $deg(f)\doteq f_*(1)$. I'm asked to ...
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1answer
37 views

Exact short sequence vs exact long sequence?

could anyone explain me what exactly the difference between an exact long sequence and an exact short sequence is? I think it pertains to homology theory, right?
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26 views

cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get ...
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32 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
2
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2answers
40 views

Why is dh$(k(x))=$dim $X$?

Let $X$ be an integral Noetherian scheme. Let $x\in X$ be a regular closed point of $X$. Then Huybrechts and Lehn in his book, says that dh$(k(x))=$dim $X$. Here dh$(k(x))$ refers to the ...
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45 views

Quillen groupoid of a groupoid.

For any category $\mathcal{C}$ we can define its Quillen's groupoid, denoted $\mathcal{Q}(\mathcal{C})$, as the category which have the same objects than $\mathcal{C}$ and the arrows between two ...
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1answer
35 views

topological graph theory and the first Betti number

I am confused by a statement: in Wikipedia, In topological graph theory the first Betti number of a graph G with n vertices, m edges and k connected components equals $$m - n + k.$$ I am ...
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1answer
43 views

Locally finite type space relate homology and cohomology

This is certainly an easy question... Why does a map of spaces $f:X\rightarrow Y$ which induces an isomorphism in cohomology $f^*:H^*(Y)\rightarrow H^*(X)$ induces an isomorphism in homology ...
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0answers
24 views

Does the tensor product of a finitely presented module and a flat module always finitely presented?

If M is an R-module which admits a degreewise finite projective resolution (i.e., a projective resolution P of M such that each Pi in P is finitely generated projective) and N a flat R-module, does ...
4
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1answer
49 views

Depth of a module over local ring and vanishing of Ext functor

I'm studying depth of $A$-modules, where $A$ is a noetherian ring, in Matsumura's Commutative Algebra text and I'm experiencing some trouble understanding the proof of a basic result. I think all of ...
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0answers
17 views

let $G$ be a group with $cd(G)=m$ ,let $U$ be a subgroup of $G$ of finite index in $G$ ,show that $cd(U)=m$ .

let $G$ be a group with $cd(G)=m$ and $U$ be a subgroup of $G$ of finite index in $G$. Show that $cd(U)=m$ . $cd(G)$:a group $G$ has cohomological dimension$\leq n $ ,denoted by $cd(G)\leq n $ if ...
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20 views

if $H^{n+1}(G,A)=0$ for all $G$-module $A$ ,then $H^{k}(G,A)=0$ for all $k>n$ and for all $G$-modules $A$.

if $H^{n+1}(G,A)=0$ for all $G$-module $A$ ,then $H^{k}(G,A)=0$ for all $k>n$ and for all $G$-modules $A$. any hint or idea or references to study will be great,thanks.
3
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1answer
85 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
2
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1answer
65 views

Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me ...
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1answer
37 views

Quotient objects as constructions from subobjects?

A quotient object of an object $A$ is usually denoted $A/B$ (we're talking about equivalence classes of epis). It seems that in categories like $\mathsf {Grp}$ and $\mathsf {Ab}$ one can associate ...
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28 views

if $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$.

If $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$. It will be great if you help me with this. Any hint or guidance will be great. Thanks.
4
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1answer
54 views

Calculating the second cohomology group for trivial group action

Let $G$ be a finite group acting trivially on $\mathbb{R}^*$. How can I compute $H^2(G,\mathbb{R}^*)$? It seems that direct calculations are somewhat hopeless, but the answer should be simple anyway.
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1answer
65 views

counterexample to “symmetric” nine lemma

Consider a commutative diagram of $R$-modules ...
3
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1answer
49 views

Commutative diagram with exact sequences as columns and rows

Suppose that we have the following commutative diagram of groups and homomorphisms $$\newcommand\twoheaduparrow{\mathrel{\rotatebox{90}{$\twoheadrightarrow$}}} \begin{array} A & A_3 & ...
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2answers
66 views

Retract and homology

I have this problem in Hatcher's book : Show that if $A$ is a retract of $X$ then the map $H_{n}(A)\rightarrow H_{n}(X)$ induced by the inclusion $A\subset X$ is injective. I think I have ...
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40 views

Surjectivity in universal coefficient theorem, please delete

I am stuck proving the morphism $h$ in the universal coefficient theorem for cohomology $$0 \rightarrow \mathrm{Ext}(H_{n-1}(X),G) \rightarrow H^n(X,G) \stackrel{h}{\rightarrow} \mathrm{Hom}(H_n(X),G) ...
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1answer
27 views

Existence of certain homomorphism on cochaincomplexes

I found the following problem online. I'm not sure if this is easy or not as I'm not sure how one defines the class of an element in $H^p$. Let $C=\bigoplus_{p\in\mathbb Z}C^p$, $C^\prime$ ja ...