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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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
58 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 ...
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43 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
37 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 ...
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1answer
83 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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22 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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69 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
42 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 ...
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102 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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52 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 ...
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1answer
47 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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41 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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42 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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35 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 ...
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2answers
42 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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1answer
60 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
52 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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49 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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33 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 ...
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1answer
73 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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19 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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21 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.
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102 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 ...
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1answer
73 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
62 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.
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1answer
74 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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85 views

counterexample to “symmetric” nine lemma

Consider a commutative diagram of $R$-modules ...
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1answer
81 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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101 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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1answer
30 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 ...
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1answer
87 views

$R$ noetherian, $I$ injective $R$-module $\Rightarrow$ $S^{-1}I$ is injective over $S^{-1}R$

I am trying to prove that if $R$ is a noetherian ring, $S$ a multiplicative part and $I$ an injective $R$-module, then $S^{-1}I$ is an injective $S^{-1}R$-module. So far I thought: I reduce to check ...
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Over an Artin algebra, does Gorenstein projective and Gorenstein flat modules coincides?

In general, Gorenstein projective and Gorenstein flat modules have no obvious relations. But I find that nobody discusses Gorenstein flat modules when the ring is an Artin algebra. So I believe that ...
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37 views

Constant projective dimension of $R/I^i$ for all $i$.

Let $R$ be a local Noetherian ring and $I$ an $R$-ideal. What can we say about the ideal $I$ if the projective dimension of $R/I^i$ for $i \ge 1$ is a finite number which is independent of $i$, i.e., ...
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1answer
96 views

Zero dimensional Gorenstein ring

Let $(R,\mathfrak m)$ be a zero dimensional Gorenstein ring and $\mathfrak q$ be an $\mathfrak m$-primary ideal of $R$. Then TFAE: 1) $\mathfrak q$ is irreducible, 2) $(0:\mathfrak q)$ is principal, ...
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54 views

Yoneda Embedding into Left Exact Functors

I think I am very confused about something. I've been reading a bit about the Mitchell embedding theorem, and I read that the proof first embeds a given small abelian category $\mathscr{A}$ into the ...
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27 views

Homologie of Lie algebra with coefficients in tensor product of modules

I'd like to prove that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(M,N).$ My idea was that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(k,M\otimes N)$. ...
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1answer
89 views

A non flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0$ for all $n\ge 1$

I want to find a non-flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0 \,\, \forall n\ge 1$, where $R=k[x,y]/(xy)$ and $k$ is field.
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1answer
82 views

Projective resolution of k over R=k[x,y]/(xy)

I want to prove that $\operatorname{Tor}_{n}^{R}(k,k)=k\oplus k,\,\,\forall n\ge 1$. I found the projective resolution $$ R^4\longrightarrow R^3\longrightarrow R^2\longrightarrow R ...
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0answers
55 views

Homological algebra and Grothendieck Topologies

I have recently became familiar with the theory of Grothendieck Topologies and Cech cohomology for sheaves over a site. It seems that many of homological concepts in algebra, can be formulated in ...
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Question on adjoint functors

Can someone provide me an enlightenment on the following three statements? (I stumbled on them at the part dealing injective modules in a text of homological algebra.) 1) Let $F \dashv G \colon ...
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Understanding the Definition of the Tensor Product of Chain Complexes

The tensor product of chain complexes (of $R$ modules) $C_\bullet ,D_\bullet$ is defined as $$(C_\bullet \otimes D_\bullet )_n = \bigoplus_{i+j=n} C_i \otimes_R D_{j}$$ I understand this definition ...
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Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long ...
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1answer
102 views

Motivation for Definition of Derived Category

On the $n$Lab entry about derived categories, I read the derived category of an abelian category $\mathsf A$ is the localization of $\mathsf{Ch}_\bullet (\mathsf A)$ at the quasi-isomorphisms. My ...
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1answer
35 views

Extension of nonisomorphic simple objects

Let $X$ and $Y$ be two nonisomorphic simple objects in an abelian category. Are all extensions of $X$ by $Y$ trivial? ( $\mathrm{Ext}^1(X,Y)=0$ ?)
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Homology of the complement of a real hypersurface

Consider a real algebraic set $Z(f) = \{x \in \Bbb{R}^n\,|\, f(x) = 0\} \subset \Bbb{R}^n$ (not necessarily irreducible). I'm thinking about wether the (Euclidean) closure of a connected component of ...
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34 views

Graph Theory with Homology Diagram

In homological algebra we end up playing around with diagrams a lot. For example, when we prove that a whole diagram commutes (like a chain map between two chain complexes) we simply prove that each ...
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1answer
36 views

Chain complexes as a model category?

I'm reading a paper called Model Categories and Simplicial Methods by Paul Goerss and Kristen Schemmerhorn, and they show that cofibrations have lifting property with respect to acyclic fibrations for ...
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58 views

If $f= \mathrm{ker}\,g$, then $g = \mathrm{coker}\,f$?

I didn't understand a step in the proof of Proposition 5.92 from Rotman's Introduction to Homological Algebra (2nd Ed.) where he says: "there is a morphism $g: B\to C$ [in a given abelian category ...
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1answer
53 views

Is there a non-projective submodule of a free module?

Is there an example of a commutative ring $R$ with identity such that there exists a free $R$ module $M$ that has a non-projective submodule? I tried experimenting with modules over $\mathbb Z$ but ...