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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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
377 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
152 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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182 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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1answer
224 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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1answer
73 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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251 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
148 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 ...
4
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1answer
617 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
151 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
273 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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494 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
155 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 $N$: $$\ldots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow N \rightarrow 0.$$ ...
5
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1answer
215 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 ...
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1answer
130 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
163 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
284 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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489 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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2answers
200 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
290 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
462 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
68 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
129 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.
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1answer
391 views

Help with short exact sequence

I have the following short exact sequence: $$\ker(f)\rightarrow G\rightarrow \mathbb{Z}$$ Where the first map is inclusion and the second map is $f$. And $G$ is an abelian group. I'd like to known if ...
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1answer
185 views

On the injective dimension of a module

Let $R$ be a ring and $M$ an $R$-module then inj dim $M\leq i\in\mathbb{N}$ if and only if $\mathrm{Ext}^{i+1}(N,M)=0$ for every cyclic module $N$. The implication from left to right is obvious, I'm ...
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286 views

On the bounded derived category of a finite dimensional algebra with finite global dimension

Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) ...
4
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1answer
165 views

Why are projective modules contained in this class of modules?

Suppose $A$ noetherian and define $G(A):=\{M: M$ is an $A$-module reflexive and Ext$^i_A(M,A)=$Ext$^i_A(M^*,A)=0$ for $i\geq1\}$ Why are projective modules contained in this class? Of course if ...
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2answers
251 views

Finite length module and graded local duality

In the proof of Theorem 20.18 in Eisenbud Commutative Algebra, the following fact is stated: If $S=k[X_1,\ldots,X_r]$ and $N$ is a finite length graded $S$-module, then ...
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1answer
228 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
12
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1answer
522 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
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2answers
188 views

What does $Tor^{R}_n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
5
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1answer
179 views

Mackey functor question regarding Greenlees and May

Let $G$ be a finite group. In their paper Some Remarks On the Structure of Mackey functors , Greenlees and May define a functor: $R: GMod \rightarrow M[G]$ where $GMod$ is the category of finite ...
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0answers
163 views

Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions

I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$ $0 ...
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3answers
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Cokernels - how to explain or get a good intuition of what they are or might be

When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over ...
9
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1answer
522 views

The construction of the localization of a category

I was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand. Let me remind you the definition of the localization of a category: ...
20
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1answer
1k views

When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then ...
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0answers
191 views

Homology and Homotopy group

$E, F, B$ are topological space, $B$ path connected. If we have given a long exact sequence.. $$\cdots\to \pi_1(F)\to \pi_1(E)\to \pi_1(B)\to\cdots$$ what will the relationship of $H_1(F,\mathbb R)$, ...
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1answer
179 views

Is there a quasi-isomorphism between a complex of sheaves and its Godement resolutions?

I have a doubt, I read somewhere that the Godement resolution of a sheaf $\mathcal{F}$ is a quasi-isomorphism $\mathcal{F} \rightarrow C^\bullet(\mathcal{F})$. Just right off the bat when I read ...
5
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2answers
186 views

Localization and Extension of modules

Let $R$ be a commutative ring and $S$ be an $R$-algebra. Assume that $S$ is finitely generated as an $R$-module. Let $M$ and $N$ be finitely generated $S$-modules and $\mathfrak{m}$ a maximal ideal ...
3
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1answer
401 views

Computing the homology of a torus by relative homology of a cylinder

I was trying to compute the homology of a torus with the long exact sequence for relative homology formed quotient, inclusion, and boundary $ \dots\to\tilde{H}_n(A)\overset{i_*}{\to} ...
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1answer
170 views

Relation between group zero-cohomology and the dual of group zero-homology

Let $\Gamma$ be a group and $A$ be an abelian group and let's take group zero-homology and zero-cohomology, $H_0(\Gamma,A)$, $H^0(\Gamma,A)$. Is there any relation between $H^0(\Gamma,A)$ and ...
3
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0answers
191 views

Adapted classes of objects and left (right) exact functors

I had a question about adapted classes of objects, I was confused by the definition and how it relates to left exact functors. Let $\mathcal{A}$ be an abelian category with enough injectives, let $F: ...
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0answers
41 views

Inequality of numerical invariants of complex algebraic surfaces?

Let $S, T$ be algebraic surfaces over $k=\mathbb{C}$, and $\phi: S \longrightarrow T$ a surjective morphism. Furthermore we have the numerical invariants: \begin{align*} q(S) &:= \dim H^1(S, ...
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1answer
86 views

Exact sequence of four sheaves in Beauville: associated l.e.s.?

This question is about an exact sequence of four sheaves on a smooth projective surface $S$ over $k=\mathbb{C}$, to be found in Beauville: complex algebraic surfaces, theorem I.4, page 3 (second ...
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0answers
276 views

Dual sequence and its exactness

I am reading Lang's Algebra and trying to fill in the gaps in my mathematical background while I train for the quals. So I came across the following exercise (chapter 20, ex. 26 in the third edition). ...
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3answers
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Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
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Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
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1answer
84 views

Is it true, that $H^1(X,\mathcal{K}_{x_1,x_2})=0$? - The cohomology of the complex curve with a coefficient of the shaeaf of meromorphic functions…

Let X be complex curve (complex manifold and $\dim X=1$). For $x_1,x_2\in X$ we define the sheaf $\mathcal{K}_{x_1,x_2}$(in complex topology) of meromorphic functions vanish at the points $x_1$ and ...