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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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Counterexample to, if $f$ is acyclic then $kerf$ and $cokerf$ acyclic.

This is the second half of exercise 1.3.5 in An Introduction to Homological Algebra by Weibel, it simply asks if this statement is true of false and I believe it is false but cannot construct a counterexample. Ler $f:A \rightarrow B$ be a morphism…
Michael N
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The definition of syzygies - free or projective?

For a module $M$ there is always a surjection $F \to M$ with $F$ free. As free modules are projective, there is always a surjection $P \to M$ with $P$ projective, and one may form the first syzygy $\Omega^1(M) := \text{ker}(P \to M)$. Iterating this…
Andrew Thompson
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Definition of bar resolution

I am reading Weibel's book about homological algebra. In section 6.5, the bar resolution, he uses some notation I really do not understand. So given G, a group, what is $[g_1\otimes...\otimes g_n]$? And what is $[g_1|...|g_n]$?
user198206
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Two-sided bar construction for algebras: $B_*(M,R^e,R)$ is quasi-isomorphic to $M\otimes_{R^e}B(R,R,R)$

Let $k$ be a commutative ring. Let $R$ be a $k$-algebra. Weibel defines a notion of "relative" Tor in his intro to homological algebra book. For a right $R$-module $M$ and a left $R$-module $N$, he defines $\mathrm{Tor}_*^{R/k}(M,N)$. If you look at…
Bruno Stonek
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How to compute Tor$_1$ ($\mathbb{Q} /\mathbb{Z}$, $\mathbb{Q} /\mathbb{Z}$).

How to compute Tor$_1$ ($\mathbb{Q} /\mathbb{Z}$, $\mathbb{Q} /\mathbb{Z}$). I am having trouble finding a projective resolution of $\mathbb{Q} /\mathbb{Z}$.
BetaY
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Homological Algebra Problem

suppose $A^{'}$ is sub module of $A$ and $$0\rightarrow A^{'}\overset{(d^{-1})^{'}}{\rightarrow}(I^{0})^{'} \overset{(d^{0})^{'}}{\rightarrow} (I^{1})^{'}\rightarrow \ldots $$ is injective resolution for $A^{'}$ . 1) show that we can make injective…
slopephoupha
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Definition of chain homotopic.

Let $C_{\bullet}, D_{\bullet}$ be two nonnegatively graded chain complexs of $R$-modules with maps $d^C,d^D$ respectively($d^C_n: C_{n+1} \to C_n$), and let $f,g: C_{\bullet} \to D_{\bullet}$ be two chain maps. Then, a chain homotopy from $f$ to $g$…
user117449
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Ext$_R^n(Q,A)=0=$Tor$_n^R(Q,A)$ where $Q$ is the field of fractions of a domain $R$

I am currently working through a problem in Rotman: Let $R$ be a domain and let $Q=$Frac$(R)$. If $r\in R$ is nonzero and $A$ is an $R$-module for which $rA=0$, prove that for all $n\geq 0$, $\mathrm{Ext}_R^n(Q,A)=0=\mathrm{Tor}_n^R(Q,A)$. I…
1234
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Cohomology of a split cochain complex

I have $A$ and $B$ two graded vector spaces, and $D: A \oplus B \to A \oplus B$ with $D(a + b) = d_0(a) + d_1(a) + d_0(b)$ for $a \in A$ and $b \in B$, where $d_0 : A \to A$, $d_0 : B \to B$ and $d_1 : A \to B$ all of degree +1 (cohomological…
jeanmfischer
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example of hom of direct sum

I have a question . Can anyone give me examples for $Hom(B,\oplus A_j)$ not isomorphic to $\oplus Hom(B,A_j)$ or $\prod Hom(B,A_j)$ as abelian groups? Here $A_j$ and B are both modules. I have read $Hom(B,\prod A_j)$ is isomorphic to $\prod…
user146507
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Does $Ext^n(A,C)=0$ imply $Ext^{n+1}(A,C)=0$

I'm studying a bit of homological algebra and I'm now studying about the projective dimension of an $R$-module $M$. This is how it is defined: Since the category $R-\operatorname{Mod}$ has enough projectives, for any $R$-module $M$ we can write a…
math.n00b
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When does homology commutes with arbitrary direct sums

Is it necessary to have the criteria that the direct sum of a collection of monics is a monic, to show that homology commutes with arbitrary direct sums? Because when I tried to prove the result, I really didn't use this criteria and I don't see how…
user170873
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a group homology computation

I was assigned to compute the group homology of $\mathbb{Z}^k$ with $\mathbb Z$ as coefficient ring(with the trivial action): $H_*(\mathbb{Z}^k, \mathbb{Z})$. I know that $H_*(\mathbb{Z}^k, \mathbb{Z}) = {\rm Tor} _*…
S. Ha
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RHom and Koszul complexes.

Let $A$ be a Koszul algebra over a field $k$ that is both left and right finite. One can consider the its Koszul complex $X$ as a dg $ A^!$-$A$ bimodule. I want to show: For all $ i \in \mathbb{Z}$, $ RHom_A(X, X \langle -i \rangle [i]) $ has its…
Anette
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Ext, Extensions and homomorphisms between them

Say that we have $R$-modules (let's assume over a commutative ring $R$). Consider extensions of a module $A$ by a module $C$, so that we have short exact sequences: $0 \rightarrow B \rightarrow^i E \rightarrow C \rightarrow 0.$ We say that two…
Shaf_math
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