Mathematics Stack Exchange
Mathematics Stack Exchange

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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Constructing the cone of a morphism

When constructing the cone of a morphism $f:M\rightarrow N$ (also called the mapping cone), a new complex $M[1]$ is constucted from $M$ via $$M[1]^i=M^{i+1}$$ with differentials $$d_{M[1]}^i=-d_M^{i+1}.$$ My question is why is there a change of…
user568543
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A question about projective modules

These is an equivalent relation about projective modules. P is projective , (1)P is a direct summand of free module (2)If P is a quotient of the R-module M, then P is isomorphic to direct summand of M. I am confused here, what does it mean that P…
user53800
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Question on finding free resolutions.

Let $A=\mathbb{Q}[x,y,z]$ and let $M=A/(x,y)$,$N=A/(x,y)^3$ be two $A$-modules. I am supposed to compute a free resolution for the two modules respectively and then compute $\text{Tor}^{A}_i(M,N)$ using the two different resolutions. I am struggling…
user117449
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Weibel Exercise $1.1.4$ About $\hom(\mathbb{Z}/n, C)$

I am reading Weibel's Introduction to Homological Algebra. I have come across this exercise: $$ \text{If $C$ is a chain complex and $H_n(\hom(\mathbb{Z}/n, C)) = 0$, then $H_n(C) = 0$.} $$ It seems that this is false as stated. I can take the…
J126
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Can we obtain the following two isomorphisms?

Let $R$ be a ring. $M,N$ be two submodules of $W$. Let $$0 \longrightarrow M\longrightarrow M+N\longrightarrow H\longrightarrow 0$$ and $$0 \longrightarrow N\longrightarrow M+N\longrightarrow L\longrightarrow 0$$ be two split short exact…
Junling Zheng
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Finding free resolution of module.

Let $A= \mathbb{Q}[x]/(x^2)$ and define the module $M$ by $M=A/(x)$. I am supposed to compute $\text{Ext}_{A}^n(M,A)$. First we start by finding a free resolution for $M$ by $$\mathbb{Q}[x]/(x^2) \to A/(x) \to 0$$ where the first arrow is just the…
user117449
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$\mathbb{Z}_n$ is a $\mathbb{Z}_{n^2}$ projective module?

$\mathbb{Z}_n$ is a $\mathbb{Z}_{n^2}$ projective module? I try to apply the definition...
Problemsolving
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Can two non-zero extension class in $\mathrm{Ext}^1$ give zero extension class in $\mathrm{Ext}^2$?

Given two short exact sequences of modules $$0\to N\to P\to R\to 0$$ $$0\to R\to Q\to M\to 0$$ Denote their extension classes by $e_1\in \mathrm{Ext}^1(R,N)$, $e_2\in\mathrm{Ext}^1(M,R)$. Suppose $e_1\neq 0,e_2\neq 0$, is it possible that $e_2\cup…
user93417
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Evaluation map of internal hom of chain complexes

In the category of chain complexes over some abelian category, besides the ordinary external $\hom(C^*, D^*)$ which contains chain maps, there is the internal hom which is the chain complex $\hom^i(C^*, D^*) = \prod_j \hom(C^j, D^{i+j})$ with the…
Bubaya
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Resolution of an algebra

I try to understand the resolution of an algebra as described in https://projecteuclid.org/download/pdf_1/euclid.ijm/1255378502. I don't understand how the multiplication is defined after adjoining a variable. Given, for instance, the exterior…
ilil
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References needed for Theorem

I am studying homological algebra by my self from Basic Homological algebra by M.Scott Osborne , I am having trouble to understand Theorem 3.9 page 56. My question is where can I find this Theorem in another reference like Rotman or others. Any…
Team
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One question about homological algebra

Let $A$ be an Artin algebra, and $0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$ a short exact sequence with $pd N<\infty$ in $\text{mod}-A$, where $pd N$ is projective dimension of $N$. Then why $\Omega^{pd N}(L) \cong \Omega^{pd N}(M)$?…
jin zhang
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understanding how a cohomlogy functor can be a right derived functor of a left exact functor

My supervisor has been telling me about how you can understand a cohomology functor as a right derived functor of a left exact functor. We were talking about two specific examples, the cohomology of a topological space with respect to a sheaf, and…
Rupert
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Need help understanding why $0 \to A_3 \to S_3 \to \{1,-1\}\to 0$ is not split exact.

I can see that $S_3$ is not the direct sum of $A_3$ and $\{1,-1\}$, so that the sequence doesn't split. However, we were told that there still exist a map $\beta:\{1,-1\} \to S_3$ such that $g \beta=I_{\{1,-1\}}$, where $g:S_3 \to \{1,-1\}$ is the…
Mark
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Schanuel's Lemma in Rotman book

The statement is: Given exact sequences $0\rightarrow K\overset{i}{\rightarrow} P\overset{\pi}{\rightarrow} M\rightarrow 0 $ and $0\rightarrow K'\overset{i'}{\rightarrow} P'\overset{\pi'}{\rightarrow} M\rightarrow 0$ where $P$ and $P'$ are…
Soulostar
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