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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How to prove $E$ is an injective module?

I was stuck with the seemingly simple homework problem: A $R$-module is injective if and only if every exact sequence $$0\rightarrow E\rightarrow B\rightarrow R/I\rightarrow 0$$ splits.Here $I$ is an ...
3
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1answer
90 views

Calculation of dimension of Socle

Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $$\dim_k \operatorname{Soc}(S/t^3S)=2?$$.($\dim_k$ means dimension as $k$-vector space.) background: ...
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1answer
75 views

Injective modules with trivial group action

If I have a divisible abelian group (i.e $\mathbb{Z}$-injective) $D$ and I take an arbitrary group $G$, and then I give $D$ the trivial $G$ action, then will $D$ be an injective ...
6
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629 views

Dimensions of vector spaces in an exact sequence

I've read the following formula in wikipedia: Given finite dimensional vector spaces $V_i$ and an exact sequence $\cdots\rightarrow V_i\rightarrow V_{i+1}\rightarrow\cdots$, we have $$ \sum_{n\in ...
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1answer
118 views

What is the reason direct product of injective modules are injective, while direct sum not necessarily?

In reviewing my algebra class material, I "discovered" a strange phenomenon, that the direct product of injective modules is injective, while if $R$ is not noetherian then the direct sum is not ...
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79 views

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of ...
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226 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
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96 views

Inductive vs projective limit of sequence of split surjections

Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of abelian groups, the connecting homomorphisms of which are ...
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144 views

Morphisms in the Category of Epimorphisms

Consider the category of epimorphisms $\mathcal E$ in a given abelian category, where epimorphisms are objects, and morphisms of this category are pairs of arrows which make its objects commute. That ...
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692 views

Proving that Hom$(N,-)$ is left exact.

Say we have a ring $R$ and let $A,B,C$ be $R$-mod. Then prove $\hom_R(N,-)$ is left exact (where $N$ is some fixed $R$-mod). Basically we want to show that given that we know that $0\rightarrow ...
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75 views

Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
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248 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} ...
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65 views

Resolutions over finite dimensional algebra

Let $A$ be a finite dimensional algebra over a field $k$ and global dimension of $A$ is finite. I want to study $A$ as a bimodule i.e. as $A^e=A \otimes A^{op}$-module. It is easy to see that ...
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3answers
140 views

$\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ $\Rightarrow$ $R$ is a PID

Is the following true: If $R$ is a commutative unital ring with $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$, then $R$ is a PID. If yes, how can one prove it? Since $0$ is a prime ...
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113 views

For a finitely generated abelian group are the homomorphisms into the integers finitely generated

If $L$ is a finitely generated abelian group, then is $Hom(L,\mathbb{Z})$ a finitely generated abelian group? Thank you
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118 views

Exact sequences and Hom

I have a exact sequence $$1 \longrightarrow A \overset{\phi}\longrightarrow B \overset{\psi}\longrightarrow C \longrightarrow 1$$ of commutative rings $A,B,C$ and the the exact sequence is such that ...
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1answer
71 views

Stabilizer of $\operatorname{Aut}(B)$ acting on $0\to A\rightarrow B\rightarrow C\rightarrow 0$

I am reading an article on elementary homological algebra and have a trouble understanding one statement. Let $R$ be a ring and $A,B,C$ modules over $R$. Let $S$ be a set of exact sequences of the ...
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98 views

On complexes of projective modules

How can I prove the following statement? Let $\beta: B\rightarrow C$ be a quasi-isomorphism of complexes of $R$-modules. If $P$ is a complex of projective $R$-modules which is bounded below, then ...
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253 views

Derived Functors

Why do we say "This functor is left exact, but not right exact" instead of "This functor preserves limits, but not colimits". It seems more natural to base the theory of derived functors on the ...
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1answer
170 views

On injective resolutions

Let $h:M \rightarrow N$ be a module homomorphism and $((F,e);\alpha)$ be a right resolution of $M$ and $((I,d);\beta)$ be an injective resolution of $N$. If $f,g:(F,e) \rightarrow (I,d)$ are morphisms ...
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2answers
203 views

Computing $\operatorname{Ext}^1_R(R/x,M)$

How to compute $\operatorname{Ext}^1_R(R/(x),M)$ where $R$ is a commutative ring with unit, $x$ is a nonzerodivisor and $M$ an $R$-module? Thanks.
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Continuous homomorphisms

I was reading JS Milne's book on Arithmetic duality theorems and he states on page 105 that for a finitely generated torsion-free G-module (G is actually a galois group) M we have ...
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0answers
65 views

Exactness Properties of Schur Functors

The title says it all: What are the exactness properties of Schur Functors? Thanks!
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264 views

Exactness of short exact sequences

Let $A$ be a ring and let $P$ be a projective $A$-module. Then, the exactness of the sequence: $$0\longrightarrow M_1 \overset{f}{\longrightarrow}M_2\overset{g}{\longrightarrow}M_3\longrightarrow 0 ...
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176 views

Reflection of Exact Sequences

Consider the category of $R$-modules. I am trying to see how i can express a short exact sequence in terms of kernels and cockerels, and how this description can be used to prove that a conservative ...
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168 views

Kernel of Zero Morphism in Abelian Categories

Consider an abelian category $C$. Let $f:M \rightarrow N$ be a zero morphism, i.e. the zero element of the abelian group $Mor_C(M,N)$. What is the kernel of $f$? Applying the definition, i get that it ...
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1answer
347 views

Injective Morphisms, Monomorphisms and Left Invertible Morphisms in Abelian Categories

Let $\mathcal{C}$ be an abelian category. A morphism $f:X \rightarrow Y$ is called injective if its kernel is zero. $f$ is called monomorphism if whenever $f \circ g=0$, for $g:Z \rightarrow X$, then ...
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1answer
146 views

Factoring a morphism via its Co-image and Image in Abelian Categories

This question refers to the Lemma 3.10 of the chapter of Homological Algebra of the Stacks Project. In particular, the lemma states that any morphism $f:x \rightarrow y$ can be factored uniquely as ...
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2answers
112 views

cohomology of an exact sequence

$$0\to M\to Q_1\to Q_2\to\dots\to Q_i \to N\to 0$$ exact sequence, then $$H^n(N)\cong H^{n+i}(M)$$
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1answer
180 views

$\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$

I'm trying to read (part of) "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)). Here are some basic assertions I need ...
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220 views

Bicartesian squares of abelian groups

A commuting square is called bicartesian if it is both a pullback and a pushout. I would like to show that given any diagram of abelian groups $A \stackrel{f}{\twoheadrightarrow} ...
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72 views

Trace map $Ext^i(E,E)\rightarrow H^i(X,O_X)$

Let $X$ be a scheme (or complex manifold if you like) and $E$ be a sheaf on $X$. I would like to know the definition of so-called trace map $$Ext^i(E,E)\rightarrow H^i(X,O_X)$$ for $1\le i\le \dim X$. ...
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1answer
247 views

Why is zero this map between Exts?

Let $(R,\mathfrak m,k)$ be a local ring of depth $d$ and $u:F_1\rightarrow F_0$ a homomorphism of finite free modules such that $\operatorname{Im}u\subset \mathfrak mF_0$. Then this map induces the ...
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1answer
145 views

The Gorenstein dimension of a ring

I'm studying on these notes. I have a question about page 64, the remark. A local ring is Gorenstein if and only if the Gorenstein dimension of the residue field is finite. Of course if the ...
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1answer
549 views

Is the tensor product of two torsion-free modules always non-zero?

Let $R$ be a commutative domain and let $M$ and $N$ be torsion-free $R$ modules. I would like to know whether or not $M\otimes_{R}{N}$ is always non-zero. Now, I know this is true in the finitely ...
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1answer
140 views

Non-abelian simplicial cohomology

Is there a theory of simplicial cohomology with coefficients in a non-abelian group ? I've found next to nothing on Google so far... I'm interested in particular in the cohomology of graphs with ...
3
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1answer
233 views

Homology of pair (A,A)

Why is the homology of the pair (A,A) zero? $$H_n(A,A)=0, n\geq0$$ To me it looks like the homology of a point so at least for $n=0$ it should not be zero. How do we see this?
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482 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
5
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2answers
150 views

Small Question on the Tor functor

Suppose that I have an $A$ - module $N$ with $A$ commutative and I take a projective resolution of $N$: $$\ldots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow N \rightarrow 0.$$ ...
4
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1answer
195 views

Colimits in that category of short exact sequences of abelian groups

I'm wondering whether the category whose objects are short exact sequences of abelian groups, and whose morphisms are commutative diagrams of such short exact sequences, is cocomplete. Working ...
5
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1answer
123 views

Global dimension of quasi Frobenius ring

Let $R$ be a quasi-Frobenius ring (so $R$ is self-injective and left and right noetherian). I want to prove that $lD(R)=0$ or $\infty$, where $lD(R)$ denotes the left global dimension. I'm ...
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1answer
150 views

Question on minimal free resolution

Let $M$ be a finitely generated module over a polynomial ring $R$ over a field $k$. Let $F_{\bullet}$ be a minimal free resolution of $M$ : $$0\longrightarrow F_p \longrightarrow ....\longrightarrow ...
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1answer
267 views

Four Lemma(proof)

I am proving four lemma: I want to show that if the rows are in the commutative diagram are exact and m and p are surjective, and q is injective, then n is surjective. See the following link. ...
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413 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
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196 views

Let $G$ be any abelian group and $a\in{G}$. Show there exists a homomorphism $f:G\rightarrow{\mathbb{Q}/\mathbb{Z}}$ such that $f(a)\neq{0}$.

Let $G$ be any abelian group and $a\in{G}$. Show there exists a homomorphism $f:G\rightarrow{\mathbb{Q}/\mathbb{Z}}$ such that $f(a)\neq{0}$. I can prove this question (I think) if I use the fact ...
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1answer
272 views

Computing $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$

I'm trying to find an abelian group $B$ such that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},B)$ is non-zero. My first guess was just to choose $B=\mathbb{Z}$. Using the following argument, I ...
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3answers
405 views

Modules with projective dimension $n$ have not vanishing $\mathrm{Ext}^n$

Let $R$ be a noetherian ring and $M$ a finitely generated $R$-module with projective dimension $n$. Then for every finitely generated $R$-module $N$ we have $\mathrm{Ext}^n(M,N)\neq 0$. Why? By ...
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1answer
64 views

Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$

Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that means that all proj. indecomposable $A$-modules and all inj. indecomposable $A$-modules ...
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1answer
69 views

Equivalent properties on the vanishing of Bass numbers

I'm studying on this notes. I'm finding some difficulties on proposition 12 on page 15. Let me recall what we are trying to prove: At first we are trying to prove that if inj ...
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2answers
121 views

Extention of vector bundles on projective line: $Ext^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))=$??

I want to know the value $Ext^1({\mathcal O_{\mathbb{P}^1}}(n),{\mathcal O_{\mathbb{P}^1}}(m))$ for integer m, n.