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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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Cokernels in Co-Cartesian Diagram are isomorphic.

We assume the category is abelian. Let the below be a co-cartesian diagram. A co-cartesian diagram is just the dual of a cartesian diagram. \begin{array}{ccc} A& \xrightarrow{f} & B \\[3pt] \downarrow {a} & & \downarrow{b} \\ C&…
random123
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Chain Complex operation

In my homework, it asks me to show that the operation $-\otimes_{k} V$ sends exact sequence to exact sequence. What does the operation mean in terms of the map? For example if the map from $C_{n}$ to $C_{n+1}$ was $c \mapsto c'$, does the new map…
BetaY
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Functor between chain complexes $\overset{?}{\implies}$ preserves zero-objects?

Let $\mathsf A,\mathsf B$ be two abelian categories. Any functor $F:A\longrightarrow B$ which preserves zero objects lifts to a functor $\mathsf{Ch}(A)\longrightarrow \mathsf{Ch}(B)$. Furthermore, if it is additive, it lifts to a 2-functor since it…
user153312
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Quotient of tensor product

I am doing some work on $Tor$, and maybe Odin too (joke!). Been calculating it for some and it's interesting to say the least. Well I have reached a step I feel intuitively is the case but I am not certain so I bring it up here to see if my hunch is…
Zelos Malum
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$2\mathbb{Z}_8\otimes \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4\otimes \mathbb{Z}_4$

I am working on a problem and after much work I have gotten out an answer $2\mathbb{Z}_8\otimes \mathbb{Z}_4$ which I decided to take on a detour and work with in my mind, mostly to see if it is isomorphic to $\mathbb{Z}_4\otimes \mathbb{Z}_4$ which…
Zelos Malum
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$M\oplus N$ is free, is $M\oplus 0_N$ free

Just a quick question, as the title says let $M,N$ be $R$-modules, if $M\oplus N$ is free, is then $M\oplus 0$ also free? My initial feeling is "no" because it would be isomorphic to $M$ then which need not be free.
Zelos Malum
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Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$?

Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$ and $H_n(A_\bullet/2A_\bullet)=\mathbb{Z}/2\mathbb{Z}$?
user270468
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Figure out Middle step of an exact sequence

I am doing some homework and it involves filling out a commutative diagram, I just want to discuss some of my thoughts and see what's wrong and most importantly, why. One of the sequences in the diagram, we're given that they are all exact in rows…
Zelos Malum
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Split-injective iff split-surjective

I was reading Keith Conrad's notes here and was wondering if there is any way to only prove (1) $\iff$ (2) which comes out as Let $0 → N → M→ P → 0$ be a short exact sequence of R-modules. The following are equivalent: (1) There is an $R$-linear map…
Maurice
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pd$(\sum\limits_{\alpha\in A}M_\alpha)=\sup\limits_{\alpha\in A}\{$pd$(M_\alpha)\}$

I came across the following problem in Rotman's Advanced Modern Algebra: 11.69. If $\{M_\alpha\}_{\alpha\in A}$ is a family of left R-modules, prove that pd$(\sum\limits_{\alpha\in A}M_\alpha)=\sup\limits_{\alpha\in A}\{$pd$(M_\alpha)\}$, where pd…
1234
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Split exact sequences of vector spaces

The question is on page 2, exercise 1.1.3. For the proof that $\{ C_n \}$ is a chain complex I only need to show that $(i\circ p)\circ (i\circ p) = 0$ where $i$ is the inclusion map, and $p$ is the projection, if I am not mistaken this follows from:…
MathematicalPhysicist
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Chain morphism into a subcomplex homotopic to identity

Let us assume we have a chain complex $(X_\bullet,\partial_\bullet)$ of vector spaces and a subcomplex $(Y_\bullet,\partial_\bullet)$. Let us furthermore assume that there exists a morphism $f_\bullet : (X_\bullet,\partial_\bullet) \rightarrow…
shuhalo
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Fundamental lemma of homological algebra via acylic models?

The fundamental lemma of homological algebra discusses the extension of arrows to chain maps from a projective to an arbitrary resolution, and the uniqueness-up-to-homotopy of such an extension. Indeed, if $\mathsf{A}$ is an abelian category and…
Arrow
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Direct limit and constant are adjoint functors

I have a question. Why $(\varinjlim, | |)$ is an adjoint pair of functors? Here the definition of constant direct system || is: For any I, fix a module A and set $A_i=A$, all $i\in I$, and $\phi_j^i=1_A$ for all $i\leq j$. The adjoint pair of…
cali
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Two split exact sequence implies another one is split?

$$\begin{array}{ccccccccc} &&0&&0&&0\\ &&\downarrow &&\downarrow && \downarrow\\ 0 & \to & \mathbb{Z}_2\{a\} & \to & \mathbb{Z}_2\{a\} & \to & 0 & \to & 0\\ & &\downarrow & & \downarrow …
user26170
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