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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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Each $(R,S)$-bimodule is a left $R \otimes_k S^{op}$-module

I am trying to understand and fill the gaps of Rotman's proof (in his homological algebra text). This approach is different from this post ("Post 1") and is more complete than this one ("Post 2"). I understand everthing about the proof except…
IsaacR24
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$Hom_\mathbb Z$ and direct products

This is problem 2.25(ii) from Rotman's Homological Algebra Text. 1. Problem Let $p$ be a prime, $B_n$ be a cycle group of order $p^n$, and $A = \bigoplus_{n=1}^\infty B_n$. Show that $$ \mathcal S := Hom_\mathbb Z(A, A) \ncong \bigoplus Hom_\mathbb…
IsaacR24
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Basic confusion about $Tor_n$ and projective resolutions

Osborne's text has the below basic example. 1. Example 9 $R = \mathbb Z_4$, $A = B = \mathbb Z_2$ A projective resolution of $\mathbb Z_2$ is $$ \cdots \rightarrow \mathbb Z_4 \xrightarrow{\times 2} \mathbb Z_4 \xrightarrow{\times 2} \mathbb Z_4…
IsaacR24
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Short exact sequence with four terms

We want to build an exact sequence $$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow D \longrightarrow 0 $$ such that $A \oplus C \cong B \oplus D$ and other in which that property does not hold. Is it possible to obtain…
Paolo Jove
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Homotopies of short exact sequences of chain complexes

Suppose we have a morphism between short exact sequences of chain complexes valued in an abelian category $\mathcal{D}$, where the left and middle vertical arrows are chain homotopy equivalences $\require{AMScd}$ \begin{CD} 0 @>>> A @>>> B @>>> C…
abhi01nat
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Short exact sequence of chain complex with non-trivial connecting homomorphism

Title is pretty straightforward, it is almost a duplicate of Example of Short exact Sequence of chain complexes but I was unable to understand the answers. I have tried many complexes, and unable to find one where the connecting homomorphism. If…
Shean
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Definition of an induced homotopy in Hilton-Stammbach's book A Course in homological algebra

Studying Hilton-Stammbach's (2nd Ed), on page 170, it starts from a pair of chain maps, $\varphi, \varphi^{'}: C \rightarrow C' $, for which there is a homotopy, $\Gamma$. Next, a map $\Gamma_{\sharp}: C \otimes_ {\lambda} D$ ( with $\Lambda $ a…
user754647
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What chain complex is denoted by a single $R$?

I was reading about Koszul Complex and I am fairly new to the topic. So lemme start with a ring $R$, not necesarilly commutative and let $x\in R$ be central. Denote $K(x)$ to be the chain complex $0\to R\to R \to 0$ and the map from $R$ to $R$ is…
UnsinkableSam
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A little confusion about extensions $E(-,-)$ and $\mathrm{Ext}(-,-)$

If we want to calculate $E(\mathbb{Z}/p\mathbb{Z},\mathbb{Z})$, i.e. equivalence classes of short exact sequences $\mathbb{Z}\rightarrow E\rightarrow\mathbb{Z}/p\mathbb{Z}$, we have…
Anonymous Coward
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Does the functor $Hom_R^\bullet(I^\bullet, -)$ preserve quasi isomorphisms?

This question has raised from my current research; the terminology and notation comes from either of C. Weibel's "introduction to homological algebra" or "Methods of Homological algebra" by S. Gelfand and Y. Manin. Let $I^\bullet$ be an exact…
H. Ali
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Composition of quasi-isomorphisms

If $f \circ g$ and $f$ are quasi-isomorphisms, is $g$ a quasi-isomorphism ?
M. Di
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Condition for the tensor product functor $(-)\otimes B$ of chain complexes to be exact

I am interested in the unbounded case and looking for sufficient conditions on the modules of the complex or the ring for this to hold. Or even restricting the functor to some unbounded subcategory. We know that since $(-) \otimes B$ is a left…
Bjorn
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Projective presentation of an $R$-module

Let $R$ be a commutative ring with unit, and let $A, B$ be $R$-modules. Into the book "A Course in Homological Algebra" of Peter J. Hilton and U. Stammbach, the authors present the functor $Ext_R(A, B)$, by starting from a projective presentation of…
Rick88
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Knowing the middle of module of a short exact sequence

This question actually comes up when I am working in algebraic topology. If I am given a short exact sequence (we may consider this as a short exact sequence of $\mathbb{Z}$-modules) $$0\to \mathbb{Z}^2\to X\to\mathbb{Z}^7\to 0$$ for instance, can I…
Ivan So
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What does $d=d^h+d^v$ mean?

When one defines $\text{Tot}(C) = \text{Tot}^{П}(C)$ a total complex one usually writes "the formula $d=d^h+d^v$ defines map $d$: $\text{Tot}^{П}(C)_n \rightarrow \text{Tot}^{П}(C)_{n-1}$". How one gets this map $d$? (What does $d=d^h+d^v$ mean…
kelly
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