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Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

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Why is $ H_{3}(\mathfrak{sl}_{2}) = k $

This is my first post here so I apologise in advance if I've done anything against the rules. My question is with regards to an answer given here Exercise 7.7.3 in Weibel (computation of $H^3(\...
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Show that $M$ admits a projective cover

I was asked to show. Let $R$ an Artinian ring and $M$ a left $R$-module finitely generated. Show that $M$ admits a projective cover. My Attempt: I take any $L$ $R$-module projective finitely ...
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spectral sequence and a comparison theorem

I am wondering how the comparison theorem can be useful for resolving extension problem. Here is a quote from the book An Introduction to Homological Algebra by Weibel. (Weibel) Comparison Theorem ...
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New symbol for an exact sequence

A sequence $A\to B \to C$ is exact if $\operatorname{im} f = \ker g$, where $ f:A\to B$ and $g:B\to C$. Why is there not a symbol to denote such a sequence? Which one would you suggest?
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Serre spectral sequence of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2=\mathbb{R}P^\infty$

As the title shows, we have a fibration of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2\sim\mathbb{R}P^\infty$. I am trying to check my understanding of Serre spectral sequence with ...
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Computing Ext for a complex of modules, help with a proof in Stacks Project

I am stuck on a step in the proof of Lemma 15.66.2 here. Let $R$ be a commutative ring with identity and let $K^{\bullet}$ be a complex of $R$-modules. I am stuck on the following sentence: "Choose a ...
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Could we define injective modules or projective modules for topological modules?

Topological Modules are defined here Wikipedia. My question is can we define notions like injective modules or projective modules? Can we define $Tor$ and $Ext$ functors? I have tried hard to find ...
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Hom(G,-) functor is exact?

Given an exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ we know that the induced sequence $0 \rightarrow Hom(G,A)\rightarrow Hom(G,B)\rightarrow Hom(G,C)$ is exact. But ...
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Long exact sequence of cohomology group “without” Snake lemma

Let a short exact sequence $$ 0 \to L \to M \to N \to 0 $$ is a short exact sequence of $G$-modules, then a long exact sequence is induced: $$ 0\longrightarrow L^G \longrightarrow M^G \...
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Functoriality of Lyndon-Hochschild-Serre spectral sequences in coefficients.

It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long ...
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1answer
26 views

The direct sum of any family of objects

Suppose in an Abelian category $\mathscr C$, the direct sum of any family of objects exists, then is $\bigoplus_{i\in\varnothing}A_i$ equal to 0 or meaningless?
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functoriality of Grothendieck spectral sequence

I am looking for a reference which treats the functoriality of the Grothendieck spectral sequence for elements of the derived category of an abelian category.
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How does one tensore two exact sequences

This is related to Ravi Vakil's notes on Foundations of Algebraic Geometry. In 15.3.B, we are asked to check that if quasicoherent sheaves $F$ and $G$ are globally generated at a point $p$, then so is ...
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1answer
38 views

What's the meaning of “decreasing filtration”?

Page 152-153, "Algebraic Geometry" by Lei Fu. The condition (e) of the definition of spectral sequence is listed as follows: (e) A family of objects $H^n(n\in \Bbb Z)$ in $\mathcal C$ and each $H^...
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Group actions on spectral sequences of group cohomology

Suppose I have a group extension $1 \rightarrow N \rightarrow H\rightarrow K\rightarrow 1$, and we have a group $G$ which acts on $H$, and $K$ by automorphisms and it does not have action on $N$. ...
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How is the differential induced by $d_C$ on $\Omega C$ defined for $(C,d_C)$ is a dga coalgebra?

Again I am confused about something regarding the cobarconstruction of a dga coalgebra $(C,d_C)$. The cobar construction of $C$ is the dga algebra $(T(s^{-1}\bar{C}),d_1+d_2)$ where $d_2$ is induced ...
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Topological $K$ functor : reduced splitting

In pg40, Hatcher's K theory he states, The restriction of vector bundles to a basepoint $x_0 \in X$ defines a homomoprhism $K(X) \rightarrow K(x_0) \cong \mathbb Z$ which restricts to an ...
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Cap product Hochschild (co)homology

Let $A$ be an associative algebra and $M$ be an $A$-bimodule. Then we can form the Hochschild cochains $C^\bullet(A,A)$ and chains $C_\bullet(A, M)$ and define a pairing (cap product) $$C^\bullet(A,A)\...
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Singular chain complex as a graded algebra

Let $X$ be a topological space and denote $S_*(X)$ the singular chain complex of $X$. There is a chain map (The Eilenberg-Zilber map) $$E: S_*(X)\otimes S_*(X) \rightarrow S_*(X \times X)$$ which ...
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How to prove the sufficient and necessary conditions of pullback square?

Let $\mathscr C$ be an Abelian category and $W,X,Y,Z$ objects in $\mathscr C$. How to prove the following lemma? Lemma: The square $$ \require{AMScd}\begin{CD} W @>u >> X \\ @VVV @VVV \\...
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reverse of a sequence in homological algebra

Say I have a sequence $C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow... \rightarrow C_n$ In math terminology, what is the "reverse" of this sequence? Is it $C_n \leftarrow C_{n-1} \leftarrow ... \...
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How to show $f^{-1}(D)/f^{-1}(C)\simeq D/C$ by the language of Abelian category?

Let $\mathscr C$ be an Abelian category and $f:A\to B$ an epimorphism in $\mathscr C$. Let $g:C\to D$ and $h:D\to B$ be monomorphisms in $\mathscr C$, thus $C,D$ are subobjects of $B$. Let $f^{-1}(...
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References about universal extension

Let $\Lambda$ be a finite dimensional $k$-algebra over an algebraically closed field $k$. $T$ is a tilting module, then for an $\Lambda$-module $X$, I have seen in a material that "consider the ...
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Two questions about the functor $DHom_{\Gamma}(-,I)$

Let $\Gamma$ be an Artin algebra with standard duality $D$. $I$ is the injective envelop of $\Gamma$ as $\Gamma$-module. Let $S$ be a set consisting of $\Gamma$-modules such that $X \in S$ iff there ...
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Proof in Lee's Introduction to Topological manifolds

In his book, Lee has what appears to be a very simple proof of the fact that if $f:I\to X$ is a path in $X$, then $f^{-1}\sim -f$ where $f^{-1}(t)=f(1-t).$ He takes $\Delta_2$ to be the simplex with ...
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1answer
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Does the differential of an augmented dga algebra fix the augmentation ideal?

I am reading about the bar/cobar construction in the book Algebraic Operads. The differential on the bar construction of a augmented dga algebra $A$ is a sum of two differentials $d_1+d_2$ where $d_1$ ...
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1answer
36 views

Derived functor of derivation?

Let $R$ be a commutative ring with 1 and $A, B$ be $R$-algebras. If $N$ is a $B$-module and $\phi:A\to B$ is an $R$-algebra homomorphism, then $N$ admits as $A$-module structure via $\phi$. Now we can ...
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1answer
39 views

How do continuous maps between spaces induce homomorphisms between Cech cohomologies?

Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ a continuous map. For a given open cover $\mathcal{V}$ of $Y$ and an abelian group $G$ I know how $f$ induces a group homomorphism from $H^{...
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Is there an established definition of the simple complex associated to a triple (or higher) complex?

The simple complex associated with a commutative double complex has a well-established definition. The $i$th term is the sum of terms in the couple complex with indices summing to $i$, and the maps ...
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P is a projective module. Prove that there exists $F$ free such that $P \oplus F \cong F$

P is a projective module. Prove that there exists $F$ free such that $P \oplus F \cong F$ I am trying the following idea: Since $P$ is projective it is the direct summand of a free module $F_0 = P\...
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Group/Hochschild homology is an invariant

I am wondering about how to prove two statements I found in my lecture notes on homological algebra, which are stated without proof: 1) Let G be a group, M a G module, and let $H_n(G,M)$ denote the n-...
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1answer
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Understanding $\operatorname{Ext}_R^1(M,A)$ as obstructions to lifting homomorphisms

Let $R$ be a ring. Let's work in the category of left $R$-modules. There is a way of looking at $\operatorname{Ext}_R^1(M,A)$ as classifying extensions of $M$ by $A$ up to equivalence. But, if we ...
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Prove that $\mathbb{Z}_2 \otimes_\mathbb{Z}$ is not exact.

Prove that $\mathbb{Z}_2 \otimes_\mathbb{Z}$ is not exact. I am trying to find an exact sequence such that if we apply the functor above we don't get an exact sequence. It is known that the functor ...
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1answer
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Proving $Ind^{G}_{H}M:=M\bigotimes_{RH}RG$, where $M$ is projective, is projective.

Let $H\leq G$ be a subgroup, $R$ a ring and $M$ be a projective $H$-module. Prove that the induced module $Ind^{G}_{H}M:=M\bigotimes_{RH}RG $ is projective.
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Construction of “etale associated-sheaf” in Rotman's

On pg 282 Prop 5.68 Rotman makes the following construction given a presheaf $P$ of abelian groups over a space $X$. The construction is as follows: For $P^{et}:= (E^{et}, p^{et},X)$. I am ...
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1answer
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Showing that effaceable delta-functors are universal

I am attempting to prove Theorem 1.3A in III.1 of Hartshorne's algebraic geometry, which says that $\delta$-functors $T=(T^i)_{i\geq0}$ with each $T^i$ effaceable/erasable for $i\geq 1$, are universal....
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Finding a chain map from an $F$-acyclic resolution to an injective resolution which is a monomorphism in each degree

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F:\mathcal{A}\to\mathcal{B}$ be a left-exact additive functor to $\mathcal{B}$ another abelian category If $M$ has an $F$-...
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1answer
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How does coassociativity of a coalgebra $C$ imply that the derivation on $\Omega C$ is a differential?

I am trying to show that $d²=0$ where $d$ is the derivation on $T(s^{-1}\bar{C})$ induced by the map $s^{-1}\bar{C}\to T(s^{-1}\bar{C})$ defined by $$s^{-1}x\mapsto -\sum (-1)^{|x_{(1)}|}s^{-1}x_{(1)}\...
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1answer
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Additive functor on a long exact sequence

If $$0\to X_0\to X_1\to X_2\to X_3\to \dotsb$$ is exact. Why does an additive, left-exact covariant functor $G$ gives us an exact sequence: $$0\to G(X_0)\to G(X_1)\to G(X_2)$$ I know that by left-...
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108 views

Why is the etale map an open map?

In Proposition 5.59 (page 276) of his book An Introduction to Homological Algebra, Rotman states that an etale map is always an open map on sheaf space. (5.59iii) Proposition 5.59 Let $\mathcal{S}...
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free chain complex is acyclic iff contracting homotopy explanation

I am aware of this post. The following is a slight generalization, A free chain complex $(A_*, \partial)$ is acyclic iff it has a contracting homotopy. In that we dont' require $A_n=0$ for $n<...
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Functor preserving long exact sequences

Let $\mathcal{C}$ and $\mathcal{D}$ be abelian categories. An exact functor $F:\mathcal{C}\to\mathcal{D}$ preserves exactness of short exact sequences: $$0\to A\to B\to C\to 0$$ goes to $$0\to F(A)\...
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Applying a functor to a homotopy of chain maps yields an isomorphism of homology groups?

I'm reading through the section on derived functors in Lang's Algebra and I came across this explanation of the natural isomorphism $H^i(F(I_M)) = H^i(F(J_M))$ for two different injective resolutions $...
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1answer
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Any references for Wall's Lemma of exactness?

At title goes, I read the following lemma in Bredon's book (GTM139 Topology and Geometry Page 189). But I find that I have never seen it in any other texts on algebraic topology or homological algebra....
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1answer
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Relation between the $2^n$-Bockstein homomorphisms

The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\mathbb{Z}/2)$$ is associated to the short exact sequence $$0\to\mathbb{Z}/2\to\mathbb{Z}/{2^{n+1}}\to\mathbb{Z}/{2^...
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Enough injectives in the category of chain complexes

So I am a little confused about a question I thought would be obvious. Let $\mathcal{A}$ be an abelian category and let $\text{Ch}(\mathcal{A})$ be the category of chain complexes. Is it true that $\...
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A question related to group cohomology and spectral sequences

It is actually a follow-up question of a mathoverflow question. I don't quite understand the answer there. I tried to compute the group cohomology of $H^n(\mathbb{Z}_4,\mathbb{Z})$ via the Lyndon-...
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2answers
47 views

Prove that in the diagram $A_1 \rightleftarrows_{i_{1}}^{\pi_1} A \leftrightarrows_{i_{2}}^{\pi_2}$ we have $A \cong A_1 \oplus A_2$

Let $A_i , A_2, A$ be left $\mathrm{R} -$ modules. If in the diagram below $$A_1 \rightleftarrows_{\pi_{1}}^{i_1} A \leftrightarrows_{\pi_{2}}^{i_2} A_2$$ we have that $\pi_1 i_1 = 1_{A_1}$ and $\...
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1answer
67 views

Why is homology functor additive?

Let $\mathcal{A}$ be an abelian category and $\text{Ch}_*(\mathcal{A})$ the category of chain complexes $A_\bullet$ of objects in $\mathcal{A}$. We let $$H_i(A_\bullet):=\text{Coker}(\text{im}_{d_{i+1}...
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1answer
48 views

Is Whitehead lemma true for super Lie algebras?

Classical Whitehead lemma states that if $\mathfrak g$ is a finite-dimensional complex Lie algebra and $M$ is a finite-dimensional $\mathfrak g$-module, then first cohomology group $H^1(\mathfrak g, M)...