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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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A question of a notation in homological algebra

I heard this definition: Given a map f : M → N of modules, and projective resolutions P• → M and Q• → N, a map of projective resolutions covering f is a map g: P• → Q• of chain complexes such that $H_0(g)$ ≈ f. But I don't understand what $H_0(g)$…
6666
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a question about derived functor tor

let $R$ be a $k$-algebra,$k$ is a commutative ring, ($R$ is flat $k$ module),B is $R$ module,any C is $k$ module then prove: $R\bigotimes_k {\mathrm {Tor}}_n^k（B,C）\cong {\mathrm {Tor}}_n^R(B,R\bigotimes_kC)$ What I don't understand is when $n$…
user363992
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Chain complex of free modules

Let $X$ be a chain complex of free R modules. Let $Z_n=\ker(d_n)$ and $B_n=\operatorname{Im}(d_{n+1})$ where $d_n :\to X_n \to X_{n-1}$. Then how can I conclude that the short exact sequence $0 \to Z_n \to X_n \to B_{n-1} \to 0$ is split? How can we…
budi
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Generalization of Universal Coefficient Theorem

Suppose we are in an abelian category $\mathscr{A}$. Given a fixed monomorphism $A \overset{i}{\hookrightarrow} B$, and an object $C$, I would like to express concisely the notion of the group of maps $A \to C$ modulo those which extend to $B$. I…
Eric Auld
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Right exact functor applied to epimorphism of cohomology is still epimorphism?

Let $\mathcal A,\mathcal B$ be abelian categories and $F$ an additive, right exact functor $\mathcal A\rightarrow\mathcal B$. Suppose I have a morphism of chain complexes (in positive degrees) $C^{\,\boldsymbol{\cdot}}\rightarrow…
Tomo
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If $\text{Ext}_R^1(A,I) = 0$ for all $A\in ob(_R\text{Mod})$, then $I\in ob(_R\text{Mod})$ is injective.

Let $\text{Ext}^1(A,I)=0$ for all $A\in ob(_R\text{Mod})$, then $I\in ob(_R\text{Mod})$ is injective. I got stuck by this problem. Any ideas?
Thyrð Åkyrïkr
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Are the derived functors of an additive functor additive?

Are the derived functors of an additive functor additive? Does this follows formally from the definition?
sumit
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Why is $\mathbb{Z}/p^2\mathbb{Z}$ indecomposable in the homotopy category of chain complexes

I want to understand the accepted answer to this question. The answer is supposed to work for the homotopy category of chain complexes of abelian groups too. (i.e. it shows that that category is not exact). As mentioned in the comments the argument…
Ward Beullens
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Exact sequences of pairs

Take a triple of topological spaces $(X, A, B)$ consists of a topological space $X$ and two subspaces $A,B$ with $B \subseteq A \subseteq X$. Why is the following sequence of pairs exact? $$ 0 \rightarrow (A,B) \xrightarrow{i} (X, B)…
vanderlylic
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Several problems in homological algebra

I met so many problems when I study homological algebra by myself. Thus, I really would like to see the answers. Hopefully, everyone can help me (my big thanks). 1) When we create torsion functor from a resolution of M illustrated…
Thế Long Lê
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How does the universal coefficient theorem give a map $H_k(M;\mathbb{Q})\to H_k(M;\mathbb{C})$?

On the wikipedia page for Hodge cycles, it is stated that the universal coefficient theorem gives us a map $$H_k(M;\mathbb{Q})\to H_k(M;\mathbb{C})$$ But I don't see how. From what I know we would only obtain short exact sequences $$0\to…
user2520938
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Request for a $\mathrm{Hom}$ functor example

Let $$ P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{d_0} M \to 0$$ be an exact sequence of $R$-modules. Consider $$ (*) \hspace{1 cm} P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \to 0$$ that is, the sequence with $M$ removed.…
Rudy the Reindeer
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The Ext-functor and inverible modules

I need some help regarding an argument from the proof of Proposition 4.2.1 in John Rognes article Galois Extensions of Structured Ring Spectra. We are supposed to prove that $\text{Ext}_{R[G]}^s(R,T)=0$ for $s \neq 0$ if $R \rightarrow T$ is a…
Nelly L
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Homological Algebra - Tor

I am trying to prove the following: If A and B are abelian groups with mA = 0 = nB, where (m, n) = 1 , Then $Tor_{1}^{\mathbb{Z}}\left( A,B \right)=0$. Conclude that, in this case, exactness of $0\to D\to C\to B\to 0$ implies exactness of $0\to…
photaetraicr
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How does Tor behave with respect to restriction of scalars?

Let $A$ be an augmented commutative $k$-algebra, where $k$ is a commutative ring. Let $\varepsilon:A\to k$ be the augmentation and $\eta:k\to A$ be the unit. Let $M$ be a right $A$-module. Is it true that…
user46225
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