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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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Does every l.e.s. "in homology" come from a s.e.s. of complexes?

Given a long exact sequence of the form $$ \dots\to A'_n \to B'_n \to C'_n \,\xrightarrow{\omega_n}\, A'_{n-1} \to B'_{n-1} \to C'_{n-1}\to \dots\qquad (*) $$ is there a way to recover a short exact sequence of complexes $\mathcal…
fosco
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15
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An example of computing Ext

I've been looking for less trivial examples of computing Ext than finitely generated abelian groups, which tends to be the standard example (and often the only example). Here's an interesting exercise I found in some notes: Let $M = \mathbb{C}[x,y]…
Zach Conn
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Additive functors preserve split exact sequences

How can I prove that additive functors preserve split exact sequences?
Berry
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$\mathbb{Z}/2\mathbb{Z}$ coefficients in homology

I don't see the point in using homology and cohomology with coefficients in the field $\mathbb{Z}/2\mathbb{Z}$. Can you provide some examples for why this is useful?
fosco
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How *should* we have known to invent homological algebra?

Previously I asked How did we know to invent homological algebra?, because I was under the misapprehension that concrete examples of long exact sequences had been a major motivation for developing homological algebra. I've now learned to my…
Daniel McLaury
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12
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Is this "snake lemma" true in derived category?

Suppose there is a diagram of cochain complexes $$ \begin{array}{c} 0 & \to & X_1\phantom\alpha & \to & Y_1\phantom\beta & \to & Z_1\phantom\gamma & \to & 0 \\ & & \downarrow\alpha & & \downarrow\beta & & \downarrow\gamma \\ 0 & \to &…
Jz Pan
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Weibel 1.4.1: Show that acyclic bounded below chain complexes of free R-modules are always split exact.

I can't quite figure out this problem Weibel 1.4.1: Show that acyclic bounded below chain complexes of free R-modules are always split exact. I can see that at the end of the chain, we have $$\cdots \to C_1 \xrightarrow{d_1} C_0 \to 0$$ with $C_0$…
SorcererofDM
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Weibel IHA Exercise 1.2.5

I have started to work through 'An introduction to homological algebra' by Weibel and spend more time than I want going in circles on exercise 1.2.5. The exercise states the following: Proof in an elementary way (not with spectral sequences or the…
bbnkttp
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Does finite projective resolution imply finite free resolution?

Suppose that $R$ is a ring (commutative, if it simplifies things), and that $M$ is a (left) $R$-module. Then $M$ has a projective resolution of length $n$ if and only if $\operatorname{Ext}_R^m(M,-)$ vanishes for all $m>n$. Additionally, if $M$…
Aaron
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Exercise 2.4.2 In Weibel An Introduction to Homological Algebra

I'm having an issue with exercise 2.4.2 in Weibel's homological algebra book. Namely, it asks you to prove that if $U:\cal{B}\rightarrow\cal{C}$ is an exact functor, then $$U(L_iF)\simeq L_i(UF)$$ Where $L_iF$ is the left derived functor of…
ottoak
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Pontrjagin duality for profinite and torsion abelian groups

I'm having trouble proving exercise 6.11.3 of "Introduction to homological algebra" by Weibel. I need to show that the category of torsion abelian groups is dual to the category of profinite abelian groups. It also gives a hint to show that $A$ is a…
KevinDL
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Intuition behind tor and ext functors

I am self-learning basic homological algebra. I am in the part where they construct derived functors to compute cohomology. I have come across these two sequence of functors "tor" and "ext", which are derived functors of "tensor" and "hom"…
Grobber
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Does the adjoint relationship between Tensor and Hom functors give an adjoint relationship between Ext and Tor?

The question is as in the title; I don't really have a more specific question, I am more interested in a "yes, here is an example of how it is useful," a "yes, but it really isn't useful," or a "no, here is why." The context is we are covering…
TomGrubb
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9
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Why a free resolution rather than merely a projective one?

Suppose we have a module over a ring. We choose a projective resolution which allows us to define derived functors. In particular, sometimes we can choose our resolution to be free. I always hear the term "free resolution" used, and that indicates…
Eric Auld
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Proving Two Complexes are Not Quasi-Isomorphic

In Richard Thomas' paper "Derived Categories for the Working Mathematician" he mentions (page 6) that the two complexes $$ \begin{align*} C^\bullet&= \mathbb{C}[x,y]^{\oplus 2}\xrightarrow{(x,y)}\mathbb{C}[x,y]\\ D^\bullet&=…
Brian Fitzpatrick
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