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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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Definition of mapping cone in Weibel's book

Let $f: B_{\bullet} \rightarrow C_{\bullet}$ be a map between two chain complexes. In Weibel's H-book, he defined the mapping cone $\text{cone}(f)$ of $f$ to be the chain complex with $B_{n-1}\oplus C_{n}$ in degree $n$ and differential given…
Yining Zhang
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About an example of split chain complex of vector spaces

I can't understand an example Weibel gives in his Introduction to Homological Algebra, page 15. So let $C.$ be any chain complex of vector spaces over a field. We use the following decomposition: $$C_n=Z_n\oplus B_n'$$ where $B_n'\cong…
bateman
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Derived tensor product

Let $A$ be a commutative ring, $M$ and $N$ $A$-modules. Is the derived tensor product $M[0]\otimes^L N[0]$ isomorphic to $M\otimes_A N$? I know that the derived tensor product is supposed to be a "redefinition" of the usual tensor product.
Richard Pink
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Quasi-isomorphism of Complexes

Let's $(K^{\bullet}, d^{\bullet})$ is the complex over field $A$ (i.e. all $K^{i}$ are vector spaces over this field) and $(L^{\bullet}, {\delta}^{\bullet})$ such that $$L^{i}=H^{i}(K)~\text{and all}~{\delta}^{i}=0.$$ Why this two complexes $K$ and…
Aspirin
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The Ext-functor and independence of resolution

Recall that $\text{Ext}_A^1(M, N)$ is in one-to-one correspondence with equivalence classes of extensions $$0 \to N \to - \to M \to 0.$$ (Ignore Baer sums for now, use them in your answer if strictly necessary.) Moreover one can show that…
Andrew Thompson
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Finding missing entries to commutative $3 \times 3$ diagram with exact rows and columns

Assume that the following diagram of abelian groups has exact rows and columns. Can you determine the missing entries and maps? Give short reasoning. $$ \require{AMScd} \begin{CD} {} @. 0 @. 0 @. 0 {} {} …
Eric
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Constructing bijection between $\mathrm{Ext}_{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z},A)$ and $A/mA$

I'm reading Mac Lane's Homology and get stuck at the proof of proposition $1.1$ chapter $3$. This proposition states that there exist bijection $$ \eta:\mathrm{Ext}_\mathbb{Z}(\mathbb{Z}/m\mathbb{Z},A)\to A/mA:[E]\mapsto a+mA $$ where $a$ is…
Norbert
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Split exact sequences

Let $0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$ be a split-exact short exact sequence of $R$-modules, where $R$ is any ring. Let $T$ be an additive functor from $R$-modules to abelian groups. Then is it true that we still get a…
Alan Lee
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exercise 6.15 from joseph rotman's introduction to homological algebra .

(i) If $f:A \rightarrow A^{'}$ is a chain map, there is an exact sequence $$0 \rightarrow A^{'} \overset{i}{\rightarrow} M(F) \overset{p}{\rightarrow} A^{+} \rightarrow 0$$ where $i_{n}:A_{n}^{'} \rightarrow A_{n-1}\bigoplus A_{n}^{'}$ is given by…
kpax
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Using Tor to find the Torsion submodule

Say $R$ is an integral domain with field of fractions $F$. I need to show that, for any $R$-module $B$, $Tor_1^R(F/R, B)\cong t(B)$, where $t(B)$ is the torsion submodule of $B$. So say $$\cdots\to P_2\to P_1\to P_0\to B$$ is a projective…
Nishant
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If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also

EDIT: The original title was: If a morphism of diagrams of complexes is composed of quasi-isomorphisms, is the induced arrow a quasi-isomorphism? Let $J$ be a small category and $C$ be the category of complexes over an abelian category. If $F,G:J\to…
Bruno Stonek
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In the existence of a short exact sequence, the projective dimension of $B$ is less than the larger of projective dimensions of $A$ and $C$

If there is an exact sequence of $R$-modules $0 \rightarrow A \stackrel{\alpha}{\longrightarrow} B \stackrel{\beta}{\longrightarrow} C \rightarrow 0$, then $\mathrm{pd}(B) \leq \mathrm{max}\{ \mathrm{pd}(A), \mathrm{pd}(C) \}$. Here,…
ShinyaSakai
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Question on differential modules

Let $A,B$ be differential modules with differentiation homomorphism $d$ (such that $d^2=0$). Then let say that $g$ is an epimorphism from $A$ into $B$. Then is it possible for an induced homomorphism $H(g):H(A)\to H(B)$ to not be an epimorphism?…
generic properties
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Can we define the homology of the homology chain complex

Let $(C,\partial)$ be a chain complex where each $C_i$ is an $R$-module (R being a given ring). We know that the quotients $H_i(C,\partial)=\ker(\partial_i)/Im(\partial_{i+1}$ are also $R$-modules. I wonder if the family $\{H_i\}_{i\geq 0}$ forms a…
palio
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Quotient of a chain complex by two quasi-isomorphic chain subcomplexes

If you take the two quotients of a chain complex by two quasi-isomorphic chain subcomplexes, are the results quasi-isomorphic as well? I think it can be proved by making use of long exact sequences and the five-lemma, but I cannot find a proper…
WJL
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