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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.

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Weibel 4.1.2. Projective , injective and flat dimension of $A$

If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an exact sequence, show that $p d(B) \leq \max \{p d(A), p d(C)\}$ with equality except when $p d(C)= p d(A)+1$. $id (B) \leq \max \{i d(A), i d(C)\}$ with equality except when $i…
Ryze
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$M$ is projective iff $\operatorname{Ext}_R ^1 (M,P)= 0 $ for $P$ projective?

We know that an $R$-module $M$ is projective iff $\operatorname{Ext}_R ^1 (M,N)= 0 $ for every $R$- module $N$. Is it true that: $M$ is projective iff $\operatorname{Ext}_R ^1 (M,P)= 0 $ for every projective $R$- module $P$ ?
Junior Katidis
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Homology module of projective resolution

I do not understand our first example for a homology module. Let $A, B$ be $R$-modules ($A$ a left one, $B$ a right one), $0 \to P_1 \to P_0 \to A \to 0$ a projective resolution (exact). Then consider $P_1 \to P_0 \to A \to 0$ exact. Since ${}…
user7427029
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Example of non-hereditary cotorsion pair?

$\DeclareMathOperator{\Ext}{Ext}$ Let $\mathcal{A}$ be an abelian category and $(\mathcal{D},\mathcal{E})$ be a cotorsion pair, i.e. classes of objects of $\mathcal{A}$, such that $D\in \mathcal{D}$ if and only if $\Ext^1(D,E)=0$ for all $E\in…
Rene Recktenwald
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Why is $Tor_0^R(M,N)=0$?

Suppose we have the following free resolution of a module $N$ $$0\dots\to F_2\to F_1\to F_0\to N\to 0 $$ By definition, this free resolution is exact. Now we tensor this by a module $M$. We get $$\dots\to M\otimes F_2\to M\otimes F_1\to M\otimes…
Anju George
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differential on total chain complex

There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(\partial b)_{p,q}=\partial'b_{p+1,q}+\partial''b_{p,q+1}$,but I don't have any clues what $b_{p+1,q},b_{p,q+1}$ are.
Daniel Xu
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On a question in page 42 of Cartan and Eilenberg's Book "Homological algebra"

In page 42 of their book, there is a setence "in this case, it can be easily seen that the homomorphism "$\overline{u}$:Ker(T(M)$\rightarrow$ T(P))$\rightarrow$ T(M)" is defined by inclusion". I can not check it, can someone who read that book…
nick
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How does the splitting in the Künneth theorem work?

In the algebraic Künneth theorem, see https://ncatlab.org/nlab/show/Künneth+theorem theorem 2.2 We assume we have two complexes, one of which is free over some PID. Now this gives some short exact sequence whose left map is $\oplus{H_k(C) \otimes…
davik
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Let $R=\mathbb{Z}[x]$. Calculate $\operatorname{Hom}_R(R/(2x),R/(4))$

Let $R=\mathbb{Z}[x]$. Calculate $\operatorname{Hom}_R(R/(2x),R/(4))$ as an $R$-module. My guess is that it's somehow isomorphic to $R/(2)$, as $2$ seems to be the greatest common divisor of $2x$ and $4$, however this is just a guess. In…
njlieta
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How does one define projective dimension in terms of the Ext functor?

Would it be $pd_{R}M:=sup\lbrace i\vert Ext^{i}_{R}(M,N)\neq 0:N$ is an $R$-module$\rbrace$ or $pd_{R}M:=sup\lbrace i\vert Ext^{i}_{R}(M,N):N\neq 0$ is a free $R$-module$\rbrace$ or something else entirely?
Rhoswyn
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Lim^1 of a tower

Let $B$ be a (torsion free) $\mathbb{Z}$-module. Consider the following tower: $$T=\dots\to B\to B\to B$$ where each map is the multiplication by $p\in\mathbb{Z}$. What is $\underset{\mathrm{\leftarrow}}{lim}^1 T$? Using $p$-adic expansion, I can…
Gregg
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Homological algebra - exact sequences, injectivity and surjectivity

Prove that $\displaystyle A \overset f \longrightarrow B \overset h \longrightarrow C \overset g \longrightarrow D$ is exact iff $f$ is surjective and $g$ injective، where h= 0 I know that it is exact if $\operatorname{im}f=\ker h$ and…
Aaaaaa
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reverse of a sequence in homological algebra

Say I have a sequence $C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow... \rightarrow C_n$ In math terminology, what is the "reverse" of this sequence? Is it $C_n \leftarrow C_{n-1} \leftarrow ... \leftarrow C_1$?
user352102
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Proving $Ind^{G}_{H}M:=M\bigotimes_{RH}RG$, where $M$ is projective, is projective.

Let $H\leq G$ be a subgroup, $R$ a ring and $M$ be a projective $H$-module. Prove that the induced module $Ind^{G}_{H}M:=M\bigotimes_{RH}RG $ is projective.
Rhoswyn
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Short Exact Sequence proof confusion

I'm reading through Rotman's text "An introduction to Homological Algebra" and i'm really lost on one of his examples. In part's (i) he uses $i_*(f)=0$ and then $if(x)=0 $ for all $x \in X$. What's confusing me is how he uses $if(x)=0$ for all $x…
Alexander King
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