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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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$X $ is projective in $ \mathcal{A} $ for any projective $(X, t) $ in $ \mathcal{A}[T]$

I'm reading Gelfand's Methods of homological and here's a problem: Moreover, as the functor $X \mapsto(X, 0) $ represents $ \mathcal{A} $ as a full subcategory of $ \mathcal{A}[T]$, $X $ is projective in $ \mathcal{A} $ for any projective $(X, t) $…
Peanica
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Exact sequence extracted from projective resolution

Let $\textbf P$ be projective resolution of a module. Why is the sequence $P_{n+1}\xrightarrow{\partial_{n+1}}P_n\xrightarrow{\overline{\partial}}B_{n-1}\rightarrow 0$ exact, where $B_{n-1}=B_{n-1}(\textbf P)$ and $\partial_n$ factors as…
Madara Uchiha
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Definitions of split injection and split surjection

Let $f: C_\bullet \to D_\bullet$ be chain map between chain complexes. What does it mean for $f$ to be a split injection? And a split surjection? I am aware of the definitions of split monomorphism and split epimorphism in an arbitrary category.…
fcm
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making sense of hom-set between chain complex and abelian group

Let $\textbf C$ be a chain complex of right $R$-modules, let $A$ be a left $R$-module and let $B$ be an abelian group. How to make sense of the $\mathrm{Hom}(\textbf {C}\otimes_R A,B)$ and $\mathrm{Hom}_R(\textbf {C},\mathrm{Hom}(A,B))$?
Madara Uchiha
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Homology of total complex of a double complex

The following question arose while studying chapter 12.5 of the book Categories and Sheaves by Kashiwara and Schapira, abbreviated in the following by [KS]. Let $X$ be a double complex (with cohomological convention, i.e. with differentials of…
asdq
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Where to find worked out examples on Exact Sequences in Homological Algebra?

I am having trouble finding worked out computational examples using exact sequences. I have searched the net some amounts looking for them - as well as looking at books such as Hatchers Algebraic Topology as well as Rotman's Intro. to Homological…
Relative0
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Homomorphism of chain complexes induces homotopy/isomorphism of chain complexes

I was given the following exercise, but the person who gave me the exercise wasn't sure about some of the details (such as signs). Let $(C, \partial)$ be a chain complex, and $h: (C, \partial) \rightarrow (C, \partial)$ satisfy $h^2 = 0$. Then $(C,…
Gurkeerat Chhina
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Quasi-isomorphism to free chain complex

Let $C$ be a chain complex over a principal ideal domain $R$. How can I construct a chain complex $F$ of free $R$-modules which is quasi-isomorphic to $C$? Edit: It is well known how to do this with a chain complex concentrated in degree $0$, and we…
co-co-nuts
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Associated graded object

I have a question, how do I prove the following proposition? A and B are filtered chain complex , $f: A \rightarrow B $ is a filtered chain map if indicated mapping on the associated graded object of $f: A \rightarrow B $ be quasi-isomorphism…
fateme
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Is every homology map induced by a chain map

If I have a chain map $f:C\to D$, I know that there is an induced map $f_\ast: H(C)\to H(D)$ on homology. Is the other way around also true: If I start with a map $g:H(C)\to H(D)$ with $C$ and $D$ chain complexes, can I always find a chain map $C\to…
mike
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homotopy equivalence induces homotopy equivalence of Hom complexes?

Let $f\colon A\to B$ be a homotopy equivalence of chain complexes (with $g$ the weak inverse and chain homotopies $\phi\colon {\rm id}_D\to f\circ g$ and $\psi\colon g\circ f\to{\rm id}_C$). Then the associated chain map between…
S.Gau at Math
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Given a short exact sequence show that F is surjective if and only h is injective.

If $ A \stackrel{f} {\longrightarrow} B \stackrel{g} {\longrightarrow} C \stackrel{h} {\longrightarrow} D$ is an exact sequence, prove that $f$ is surjective if and only if $h$ is injective. Proof ($\longrightarrow$): Suppose $f$ is surjective.…
Alexander King
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Proving that there is an exact sequence

This is what I got so far before getting stuck Proof: Show that $im \iota \subseteq ker f$. Let $g \in im \ \iota$. Then there is some $n \in ker\ f$ such that $g = \iota(n)$. If $f: M \rightarrow N$ is surjective, then $im \ \iota = M$. Since $g…
Alexander King
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Derived functor of derivation?

Let $R$ be a commutative ring with 1 and $A, B$ be $R$-algebras. If $N$ is a $B$-module and $\phi:A\to B$ is an $R$-algebra homomorphism, then $N$ admits as $A$-module structure via $\phi$. Now we can easily check that there exists an exact…
Seewoo Lee
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Truncation of an injective resolution

This piece comes from the proof of corollary 10.5.11 in Weibel's book on homological algebra. There we start with a cochain complex $X^{\bullet}$ and we are trying to construct a quasi-isomorphism $X^{\bullet} \to X'^{\bullet}$ with $X'^{\bullet}$…
Bananeen
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