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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What is the trivial module functor?

In Weibel's book on homological algebra, he mentions the trivial G-Module on page 160. By this, does he mean the the functor $\mathcal{F}: \text{G-Mod} \to \text{G-Mod}$ by making $G$ act trivially on ...
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Pullback and Kernel

We consider everything in the category of groups. It is known that monomorphisms are stable under pullback; that is, if $$\begin{array} AA_1 & \stackrel{f_1}{\longrightarrow} & A_2 \\ ...
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Total complex homology exact sequence

I'm been trying to do this problem (Problem 5.1.1) from Weibel's Introduction to Homological Algebra but I can't really see how to finish it. The statement of the problem is summarized as follows: ...
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14 views

Quasi-isomorphisms are localizing in the homotopy category of cochain complexes

I'm having trouble grokking the proof of the above fact in Gelfand-Manin, Theorem 4 of III.4, page 161-162. I don't think it makes sense to copy out everything here, I'll just assume you have a copy. ...
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62 views

Analogue in algebra for characteristic classes?

By Swan's Theorem, we know that projective modules over a ring are an algebraic analogue of vector bundles over a base space. Is there some sort of cohomology theory of rings (or modules? or schemes, ...
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37 views

Can a chain subcomplex be a direct summand but it's 'origin' is not? (group homology)

Let $H\!\leq\!G$ be finite groups and $C_\ast(H)\!\leq\!C_\ast(G)$ their bar complexes (each $C_k(G)$ is a free $R$-module with basis $(G\!\setminus\!\{1\})^k$). Is it possible that $H$ is not a ...
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Prove that $M$ is a complex.

Let $f:(A,d) \rightarrow (A^{'},d^{'})$ be a chain map. For each $n$ define $$M_{n}=A_{n-1} \oplus A^{'}_n$$ and $\Delta_{n} :M_{n} \rightarrow M_{n-1}$ by $$\Delta_{n}:(a_{n-1},a_{n}^{'}) ...
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duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...
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26 views

a question about covariant and countravariant exact functor .

Let $T:‎_{R}‎\mathfrak{M}‎‎ \rightarrow ‎_{R}\mathfrak{M}‎$ be an exact (covariant) functor. For each $n \in \mathbb{Z}$ and every complex $A$ of R-modules, prove that $H_{n}(TA) \cong TH_{n}(A) $. ...
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21 views

Koszul complex and locally free resolution

Let $V$ be an $n$-dimenational vector space. We consider the tautological sequence on the Grassmannain $Gr_{k}(V)$ $$ 0 \to \Gamma \to V \times Gr_k(V) \to Q \to 0,$$ and the projection $p:V \times ...
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166 views

Forgetful functor from R-modules to abelian groups?

I am trying to see, if the forgetful functor from $\mathbb{Z}[X]$-modules to abelian groups is exact and in case it is not exact, is it left or right exact. In general, i understand the definition of ...
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exercise 6.15 from joseph rotman's introduction to homological algebra .

(i) If $f:A \rightarrow A^{'}$ is a chain map, there is an exact sequence $$0 \rightarrow A^{'} \overset{i}{\rightarrow} M(F) \overset{p}{\rightarrow} A^{+} \rightarrow 0$$ where $i_{n}:A_{n}^{'} ...
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1answer
40 views

The Verdier Quotient

In A.Neeman's book and D.Murfet's notes I have been reading about the construction of the Verdier quotient of a triangulated category, $\mathscr{T}$, by some triangulated subcategory $\mathscr{C}$. In ...
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22 views

Show that C is a split exact chain complex if and only if the identity map on C is null homotopic.

Show that C is a split exact chain complex if and only if the identity map on C is null homotopic. any hint or reference or idea will be great,thank you very much.
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44 views

Let $f$ be a morphism of chain complexes. Show that if $ker(f)$ and $coker(f)$ are acyclic, then $f$ is a quasi-isomorphism.

Let $f$ be a morphism of chain complexes. Show that if $ker(f)$ and $coker(f)$ are acyclic, then $f$ is a quasi-isomorphism. Is the converse true? I am self reader of homology algebra and I stuck in ...
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3answers
106 views

Undergrad level presentation on homological algebra and some related topics

I'm a TA of an introductory course about modules, category theory and homological algebra and the students have to do a 2 hour long presentation as a final exam. There's one student who really likes ...
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48 views

Showing regularity by the Auslander-Buchsbaum formula

Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ with $\operatorname{gl.dim}(R) < \infty$. According to this Wikipedia article it follows from the ...
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62 views

Koszul Homology vs Koszul Cohomology

Let $R$ be a ring and $x \in R$. The Koszul complex $K_\bullet(x)$ is then $0 \rightarrow R \stackrel{x}{\rightarrow} R \rightarrow 0$. Given $x_1,\dots,x_n \in R$ the Koszul complex ...
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12 views

Reference request for Homology Gysin sequence.

I am trying to study the Homology Gysin sequence (not cohomology). I am interested in finding references that either use, or explain the Homology Gysin sequence, especially if it gives descriptions ...
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17 views

which condition is for testing whether is invariant functor

Assume F is functor Tor(Hom(X,Y), Hom(F(X), F(Y))) = 0 is F injective (faithful) or surjective (full) or bijective (fully faithful) ? if question 1 is not bijective, which condition such as function ...
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Injective resolution of complexes equivalent to regular definition

Let A be an abelian category and let $A \in $ A. Denote by InjA the category of injective objects of A. We denote by $A\langle0\rangle$ the complex concentrated in degree zero. I define an injective ...
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53 views

Tensoring an exact sequence of $R$-modules with $R/x$

Let $R$ be a commutative ring with an $R$-module $M$, and let $x \in R$ be an $M$-regular element. Then tensoring any short exact sequence $0 \to B \to A \to M \to 0$ with $R/x$ yields a short exact ...
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29 views

Misunderstanding in Cartan-Eilenberg?

In Cartan Eilenberg's Homological algebra, page 13 it says: If $\Gamma$ is a principal ideal ring, then each ideal $I$ of $\Gamma$ is isomorphic with $\Gamma$, thus $I$ is free and $\Gamma$ is ...
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33 views

$\operatorname{left.fin.dim}(A)=0$ if and only if $\operatorname{soc}(A_A)$ contains an isomorphic copy of every simple right $A$-module

I've been trying to find an (easy) example to show that there exists an Artin algebra $A$ such that $\operatorname{right.fin.dim}(A)\neq\operatorname{left.fin.dim}(A)$, where ...
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1answer
95 views

Elementary motivations for free resolutions

Let $M$ be a finitely generated module over a Noetherian ring $R$ which admits a finite free resolution $0 \to F_n \to \dots \to F_0 \to M \to 0$. There is no doubt that knowing such a resolution is ...
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1answer
41 views

vanishing of an Ext-Functor for a finite graded module of positive grade over a polynomial ring

Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $M$ be a finitely-generated, graded $S$-module, such that there exists a homogeneous $M$-regular element $\xi ...
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39 views

If $F$ is a left exact functor is $A$ acyclic iff $F$ preserves exactness of every SES $0\to A\to B\to C\to 0$?

If $F:\mathscr{A}\to\mathscr{B}$ is a left exact functor between abelian categories where $\mathscr{A}$ has enough injectives, is it true that $A$ is an acyclic object iff $F$ preserves exactness of ...
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Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
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38 views

examples of additive categories which have morphism that has no kernel and morphism has no cokernels.

can you tell me examples of additive categories which have morphism that has no kernel and morphism has no cokernels. if you tell me reference which provide this kind of examples it will be ...
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1answer
21 views

the category of divisible abelian groups, $\pi:\mathbb{Q} \rightarrow \frac{\mathbb{Q}}{\mathbb{Z}}$ is monic but not one to one.

show that in the category of divisible abelian groups, natural mapping $\pi:\mathbb{Q} \rightarrow \frac{\mathbb{Q}}{\mathbb{Z}}$ is monic but not one to one. if you give me hint,Idea or reference ...
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37 views

To check d^2= 0 in the standard complex of Lie superalgebras.

For a Lie superalgebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $V$ we can define the cohomology $H^i(\mathfrak{g}, V)$ with coeffiecient in $V$ to be the cohomology space of the following complex: ...
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28 views

In general Morita context, does P and Q being progenerators imply that it is an isomorphism?

Let $(A,B,P,Q,f,g)$ be a general Morita context (that is $A$,$B$ rings, ${}_AP_B$ and ${}_BQ_A$ and $f:Q\otimes_AP\rightarrow B$, $g:P\otimes_BQ \rightarrow A$ bimodule morphisms that satisfy the ...
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1answer
33 views

Is there any way to show that an equation of this form splits?

If I have this exact sequence $\mathbb{Z}^2 \rightarrow B \rightarrow \mathbb{Z}^n\rightarrow \mathbb{Z}$, does it split? If so how do I know.
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1answer
26 views

Proving that some property on a chain complex of groups implies isomorphism between direct sums of these groups.

Let $C_*$ be a chain complex such that every $C_i$ is a torsion-free finitely generated abelian group, with $C_i=0$ for every $i<0$ and every $i>N$ for some sufficiently large integer $N$. If ...
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2answers
108 views

Elementary way to show the exact sequence $0 \to M \to \mathbf Z^2 \to \mathbf Z \to 0$ implies $M = \mathbf Z$

I am computing the singular homology of spheres by induction. In the process, I have come across the following short exact sequence $$0 \to H_1(S^1) \to \mathbf Z^2 \to \mathbf Z \to 0.$$ I wonder ...
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1answer
32 views

Quasi-isomorphism from “almost acyclic” complex to its homology

The following is an exercise in the book Representation Theory of Finite Reductive Groups by Cabanes and Enguehard. Let $\mathcal{A}$ be an abelian category. Let $X$ be a complex of objects of ...
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1answer
43 views

Ext groups for fraction field and a module annihilated by an element

Suppose $Q$ is the field of fractions for a domain $R$ and $A$ is an $R$-module such that $rA = 0$ for some $0 \ne r \in R$. Why is it the case that $\text{Ext}_R^n(Q,A) = 0$ for all $n \ge 0$? I ...
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1answer
19 views

On the relations between rank and torsion of homology and cohomology of a CW pair.

I am reading Massey's book on algebraic topology and on the chapter of universal coefficient theorem of cohomology, there is this exercise 4.1 that I don't know how to solve. Let (X,A)be a pair such ...
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2answers
66 views

Free objects in the category of dg modules

Suppose that $A$ is a dg algebra, does the category of dg modules over $A$ where morphisms are degree zero maps that commute with differential have a free object ( in general)? I have been reading a ...
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A variant of projective objects?

Let $\mathcal{C}$ be an additive category. Is there a common name for objects $P \in \mathcal{C}$ with the property that $\hom(P,-) : \mathcal{C} \to \mathsf{Ab}$ is right exact, i.e. preserves all ...
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Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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1answer
25 views

Basic idea for finding critical point via Morse theory

Please what is the basic idea for finding critical point via Morse theory and critical groups? Thank you
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1answer
56 views

Exercise 1.1.3 in Charles Weibel’s book “An Introduction to Homological Algebra”

I am trying to teach myself some homological algebra and I got stuck right at the start with Exercise 1.1.3 from the book “An Introduction to Homological Algebra” by Charles Weibel. Exercise 1.1.3 ...
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119 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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45 views

Difference between two concepts of homotopy for simplicial maps?

I learn from Gelfand and Manin's Methods of Homological Algebra, Exercise 2 for I.4 that two maps $f,g\colon X\to Y$ between simplicial sets $X,Y$ are simply homotopic (maybe usually called simplicial ...
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69 views

About the definition of homology

can someone explaine me this definition of Homology: "The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact." I know that homology measure the number ...
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Extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ (Hilton & Stammbach III.1.2)

Question is to compute $E(\mathbb{Z}_p,\mathbb{Z})$ i.e., equivalence classes of extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ By an extension of $A$ by $B$ i mean an $R$ module $E$ such that ...
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185 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in ...
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20 views

Is there an explicite description for injective sheaves?

I want to find a criterion for sheaves of modules to be injective. It would be great if one can such a criterion for sheaves of modules over a ringed space. But an answer for sheaves of abelian groups ...
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1answer
74 views

Commuting with kernels implies left exactness in Abelian category

I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as ...