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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.

A chain complex is a sequence of abelian groups, vector spaces, or modules, with linear maps connecting them which compose to zero.

Homological algebra is the study of chain complexes and their homology groups.

5114 questions
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Left exactness of Hom

For the exact complex $A \to B \to C$, does the complex $$\text{Hom}(M,A) \to \text{Hom}(M,B) \to \text{Hom}(M,C)$$ fail to be exact? If not, why do we need the exactness of $0 \to A \to B$ to get the exactness at $\text{Hom}(M,B)$? Please teach…
Pierre MATSUMI
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Why do some modules have finite projective dimension and some don't?

Why do some modules have finite projective dimension and some don't (if we consider modules over the same ring)? What factors does it depend on?
Kelly
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Coeffaceable - universal $\delta$ functor

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories. A functor $F:\mathcal{A} \to \mathcal{B}$ is coeffaceable if for every $A$ there is a surjection $u:P \to A$ such that $F(u) = 0$. If $F_*$ is a homological $\delta$-functor such that each…
Riya Sharma
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Two morphisms between complexes which are not homotopic but induce the same morphisms of cohomology

Let $\mathcal{A}$ be an abelian category and $f, g \colon X^\bullet \to Y^\bullet$ be two morphisms of complexes in $\mathcal{A}$. Suppose that the morphisms between cohomology induced from $f$ and $g$ are identical in all degrees. Then, my question…
kaede
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Confusion on proof of Five Lemma

I have a question about the proof of the Five Lemma. For the sake of simplicity, I will talk about the proof provided on Wikipedia. In the proof, there are two things I am confused about: this line "Then $t(n(c)) = p(h(c)) = t(c′)$. Since t is a…
junglekarmapizza
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Is a double complex commutative or anti-commutative?

The definition of a double complex on Weibel's an Introduction to Homological Algebra is ...... the maps $d^h$ go horizontall, the maps $d^v$ go vertically, and eah square anticommutes. While that on nLab is ......each row and each column is a…
Yuz
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Question about map induced by Ext

I didn't quite understand some definitions on some homology topics. Let $R$ be a commutative ring, $A, A', C$ $R$-modules, consider the map $f^*: Ext^1_R(C,A) \rightarrow Ext^1_R(C,A') $ induced by $f : A \rightarrow A'$ how is the map defined? In…
Rick88
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Motivation for anticommutativity, double complex (Weibel's An Introduction to Homological Algebra)

I'm new to homological algebra, and I've come across this: A double complex (or a bicomplex) in $\mathcal A$ is a family $\{C_{p,q}\}$ of objects of $\mathcal A$, together with maps $$d^h: C_{p,q}\rightarrow…
cansomeonehelpmeout
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Standard free resolution of a Hopf algebra : ?$\exists$ an explicit chain homotopy?

Consider a Hopf algebra $H$ over $\mathbb C$ with $\varepsilon \colon H\to \mathbb C$ being its counit map. I am trying to figure out why the standard resolution of $\mathbb C$, seen as a right $H$-module via the counit map, is indeed a resolution.…
Ivan Bartulovic
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Sufficent condition for a Quasi-isomorphism to be a chain equivalence?

Let $f:C\rightarrow D$ be a chain map between chain complexes over a PID $R$. It's given that the chain map $H_q(f)$ is an isomorphism for every $q$ and that for every $q$, $C_q,D_q$ are free $R$-modules. Must $f$ be a chain equivalence ? If I…
Amr
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Projective resolution of Laurent polynomials

Let $A=\mathbb C[t_1,t_1^{-1},\ldots,t_n,t_n^{-1}]$ be the ring of Laurent polynomials in $n$ variables. I am looking for a projective resolution of $A$ as an $A$-bimodule that be as explicit as possible. Thanks in advance.
Francisco
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Does a surjective chain map induce a surjective chain map in homology? (AGAIN)

I have the same question posted here (namely, if $f:\mathcal{C}\to \mathcal{D}$ is a surjective chain map between chain complexes then can I be sure that the induced in homology $f_\ast:H(\mathcal{C})\to H(\mathcal{D})$ is surjective too?) There,…
Derso
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$E \otimes_k A $ is central simple $E$-algebra

Theorem $9.120$ from Rotman's homological algebra states that if $A, B$ are central simple $k$-algebra, then so is $A \otimes _k B$. In the proof of $9.133$ it is stated that if $A$ is an central simple $k$-algebra and $E/k$ is a field extension…
scsnm
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Are homology groups of a chain complex isomorphic to that of free chain complex?

Given a chain complex $A_\bullet\in\mathrm{Ch(\mathbf{Ab})}$, are there exist some chain complex $A'_\bullet\in\mathrm{Ch(\mathbf{Ab})}$ whose homology groups are all isomorphic to that of $A_\bullet$ such that $A'_p$ are all free abelian groups? I…
Yuta
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Are chain complexes chain equivalent to free ones?

Given a chain complex $A_\bullet\in\mathrm{Ch(\mathbf{Ab})}$, are there exist some chain complex $A'_\bullet\in\mathrm{Ch(\mathbf{Ab})}$ which is chain equivalent to $A_\bullet$ such that $A'_p$ are all free abelian groups?
Yuta
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