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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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Computing $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$

I'm trying to find an abelian group $B$ such that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},B)$ is non-zero. My first guess was just to choose $B=\mathbb{Z}$. Using the following argument, I deduced that…
Paul Gilmartin
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Functor preserves kernels iff it's left exact

I'm trying to understand the proof to a statement in Rotman's 'Introduction to Homological Algebra': Proposition 5.25, p. 240: Let $F :_R\text{Mod} \to \text{Ab}$ be a covariant functor. Then $F$ preserves kernels iff $F$ is left exact. It's my…
Robert Cardona
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Coeffaceable implies universal $\delta$-functor

My question is essentially about Grothendieck's Tohoku paper Proposition 2.2.1 but in the context of coeffaceable instead of effaceable. Grothendieck's paper does not give much suggestions to my questions other than the words standard technique. Let…
Xiao
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is this exact sequence a special case of the Snake lemma?

I encountered the following exact sequence a while ago, and wondered if it was a special case of the Snake lemma. It looks like it would be, but I don't quite see how... The context is that $A,B$ are operators on a Hilbert space $H$ (but it holds in…
Tony
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Flatness and tensor product of rings

Let $R_1$ and $R_2$ be two subrings of a ring $R$ (not necessarily commutative) which commute in $R$ so that we have a ring homomorphism $R_1\otimes_\mathbb{Z} R_2\rightarrow R$ and $R$ is a module over $R_1\otimes_\mathbb{Z}R_2$. Assume also that…
Riccardo de Rossa
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Calculating $ \operatorname{Ext}(\mathbb{Z}/2, \mathbb{Z}/2)$ over $\mathbb{Z}$

Calculating $ \operatorname{Ext}(\mathbb{Z}/2, \mathbb{Z}/2)$ over $\mathbb{Z}$: I just need someone to confirm that I have calculated $ \operatorname{Ext}(\mathbb{Z}/2, \mathbb{Z}/2)$ correctly. I know this is an easy, common calculation, however I…
Daven
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How do I understand the module structure on Yoneda Ext?

Suppose $R$ is a (commutative) ring, and $M$ and $N$ are (finitely generated) $R$-modules. Then I know each $\mathrm{Ext}_R^i(M, N)$ has the structure of an $R$-module. On the other hand, via Yoneda's description of Ext, each $\varepsilon \in…
Eric Canton
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Bicartesian squares of abelian groups

A commuting square is called bicartesian if it is both a pullback and a pushout. I would like to show that given any diagram of abelian groups $A \stackrel{f}{\twoheadrightarrow} B\stackrel{\beta}{\hookrightarrow} C$, I can always embed this as…
jmracek
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Small Question on the Tor functor

Suppose that I have an $A$ - module $N$ with $A$ commutative and I take a projective resolution of $N$: $$\ldots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow N \rightarrow 0.$$ Suppose $M$ is some other $A$ - module. Now why is it the…
user38268
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Definition of contractible chain complex

A relatively simple question. A book I'm reading states "a complex homotopic to the zero complex is called contractible"... but I don't understand the statement. I know what it means for chain maps to be homotopic, but not chain complexes…
Cookie Monster
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Infinite Sum Axioms in Tohoku

In his Tohoku paper, section 1.5, Grothendieck states the following axioms that an abelian category might satisfy: AB4)Infinite sums exist, and the direct sum of monomorphisms is a monomorphism. AB5)Infinite sums exist, and the and if $A_i$ (indices…
tceps
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A chain complex is split if and only if it splits as a direct sum.

This is the first part of Exercise 1.4.2 in An Introduction to Homological Algebra by Weibel. The first part is showing that a chain complex, $C$, with boundaries $B_n$ and cycles $Z_n$ in $C_n$ is split if and only if there are $R$-module…
Michael N
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Is homology of a chain complex a universal delta-functor?

Let $\mathcal{A}$ be an abelian category and let $Ch(\mathcal{A})$ be the category of homologicaly, non-negatively graded chain complexes in $\mathcal{A}$. The sequence of homology functors $H_n:Ch(\mathcal{A})\to \mathcal{A}$ is a (in fact, the…
KotelKanim
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Direct sum of complexes

How can I figure out the classical construction (direct sum, product, pullbacks, and in general direct and inverse limits) in the category made by chain complexes and chain maps (of abelian groups or any abelian stuff)? Because of this category is…
fosco
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Definition of exact split complex

I am reading "an introduction to homological algebra by Charles A.Weibel" and the author deifnes split exact complex to be exact complex $\{C_n,d_n:C_n\to C_{n-1}\}$ such that there exists a sequence of maps $s_n:C_n\to C_{n+1}$ such that…
omar
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