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.

3,132 questions
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Two questions about the Hom functors

Given a locally small category $\mathcal{C}$, Wikipedia defines the Hom functors as At my lectures (for the more specific case of $\mathcal{C}=\operatorname{R-Mod}$ -- the category of R-modules for ...
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How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?

Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
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Torsion-less module over commutative ring whose injective hull is Hopfian

Let $M$ be a module over a commutative ring (with unity) $R$. Let $E_R(M)$ denote the injective Hull of $M$ . If $M$ is torsion-less (i.e. $\cap_{f\in M^*=Hom_R(M,R)} \ker f=(0)$ ) and $E_R(M)$ is ...
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Tor commutes with direct limits

Weibel has shown that the filtered colimit functor $$\varinjlim :(RMod)^I \rightarrow (RMod)$$ where $I$ is a filtered category, is exact. In corollary 2.6.16, pg 58 he claims Corollarry 2.6.16. ...
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Homology of solvable Lie algebras

Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the ...
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An infinite product of fields is not a semisimple ring

I want to show that an infinite product of fields is not a semisimple ring. I know Artin-Wedderburn Theorem, but I wonder can I explain it without using this theorem? Any help will be appreciated. ...
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Left ideal $I$ is a direct summand of $R$ iff $I = Rr$, $r^2 = r$

I am stuck on proving a left ideal $I$ of a ring $R$ is a direct summand of $R$ if and only if $I = Rr$ with $r^2 = r$. Could you help me with that? Any help will be very appreciated. Thanks!
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Snake lemma for derived functors

Assume we are in some abelian category with enough injectives, and we are given a short exact sequence: $$0\longrightarrow A'\longrightarrow A\longrightarrow A''\longrightarrow 0$$ And a left exact ...
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Module of arbitrary projective dimension

Given any natural number $n$, does there a ring $A$ and an $A$-module $M$ such that projective dimension of $M$ is $n$? I am think this statement should be true but I don’t know how to find such $A$ ...
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Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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Projective dimension over restriction of scalars

Let $A$ and $B$ be two rings with a ring homomorphism $f: A\to B$ such that $B$ has finite projective dimension over $A$. Is it true that any module which has finite projective dimension as $B$-module ...
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Does equivalence of derived categories preserve boundedness?

Let $\mathcal{C}$ be an abelian category and consider the derived category $D(C)$. Suppose that $F: D(\mathcal{C}) \to D(\mathcal{C})$ is an auto-equivalence. My question: must $F$ preserve the ...
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Any $f': A \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ an $R$ module homo can be lifted to $F': B \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ [duplicate]

I have posted a question here: Any $f': A \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ can be lifted to $F': B \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ but I didn't have many views and after 36 hours I got ...
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The projective dimension of modules in a short exact sequence

Let $0\to K\to P\to A\to 0$ be a short exact sequence of right modules with $P$ projective and $A$ not projective. Suppose $\text{pd}A<\infty$ and $\text{pd}K<\infty$, where $\text{pd}$ is the ...
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Unique $\mathbb{R}$-linear map between tensor products

I am reviewing materials in tensor products and I got stuck on this one, and I am never comfortable with "showing there exists a unique linear map" type of question. Let $V$ be a real vector space ...
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Any $f': A \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ can be lifted to $F': B \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$

let $\psi: A \to B$ be an injective $R$ module homomorphism, and it is given that any $f: A \to M$ $\mathbb{Z}$ module homomorphism can be lifted to a $\mathbb{Z}$ module homomorphism $F: B \to M$ s.t ...
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Characterizing a module of Kahler differentials

Consider the $\mathbb C$-algebra $R=\mathbb C[x,y,z]/(z(y^2-x^3)-1)$. How to prove that the module of Kahler differentials $\Omega_{R/\mathbb C}$ of $R$ over $\mathbb C$ is a free $R$-module of rank 2?...
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Fiber integration is independent of the operations involved.

In Definition 2 of Fiber Integration nlab post, the author claimed that the operation is indpendent of the choices involved. How is this so? The post itself is quite long. I think it is easier ...
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$A\otimes \mathbf{Q}\cong B\otimes \mathbf{Q}$ implies $A$ and $B$ are $\mathcal{C}$-isomorphic

I am trying to solve exercise 211 on Davis-Kirk: Let $\mathcal{C}$ be the class of torsion abelian groups. Show that for any abelian groups $A,~B$, $A\otimes \mathbf{Q}\cong B\otimes \mathbf{Q}$ ...
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Mapping spaces for chain complexes

For any model category $M$, there is a mapping space $Map(X,Y)$ for two objects $X,Y$ of $M$ such that $\pi_0(Map(X,Y)) = Hom(X,Y)$ in $Ho(M)$. Chain complexes over a ring $R$ have a model structure (...
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$H^2(G;\mathbb Z) \cong H^1(G;\mathbb {C}^*)$.

$H^2(G;\mathbb Z) \cong H^1(G;\mathbb {C}^*)$. Where $G$ is a finite group and $G$ acts trivially on $\mathbb Z, \mathbb C^*$. I have really tried hard but still I couldn't solve it , any help will ...
Help understanding a theorem about group homology $H_0$
I'm self-studying homological algebra. I have problems on understanding a theorem about $H_0$. First, I don't know where the bottom row of the commutative diagram comes from, see the red line in ...
Let $C$ be a chain complex over a principal ideal domain $R$. How can I construct a chain complex $F$ of free $R$-modules which is quasi-isomorphic to $C$? Edit: It is well known how to do this with ...