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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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$T$ and $S$ are universal homological $\delta$-functors and $T_0\simeq S_0$. Is $T\simeq S$?

A homological $\delta$-functor is a collection of additive funcors $T_n:\mathcal A\to \mathcal B$ ($n\geq 0$) with $\delta_{n,A,B,C}:T_n(C)\to T_{n-1}(A)$ defined for each short exact sequence $A\to B\to C$ such that $...\to T_{n+1}(C)\to T_n(A)\to…
Yuz
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A family of chain maps and chain homotopy

Assume everything is over a field of characteristic zero and one has a family of chain maps $f_t\,:\,V_\bullet \rightarrow W_\bullet$ where $t\in [0,\,1]$. I was wondering if there is a way to construct a chain homotopy between $f_0$ and $f_1$? One…
Yining Zhang
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Weibel Exercise 2.1.2, checking as morphism of $\delta$-functor

Suppose $T$ is a $\delta$-functor and $T_0=F$ for a exact functor $F$ and $T_i=0$ for $i>0$. The aim of the question is to show that $T$ is universal, meaning if there is another $\delta$-functor $S$ and we have a natural transformation $f_0:S_0\to…
Ivan So
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Relation between two monoidal products on chain complexes

In this question I work in the category of chain complexes of Abelian groups concentrated in degree $\geq -1$ (Most of the complexes I am interested in are augmented) There is a standard notion of tensor product of chain complexes. I have been…
Patrick Nicodemus
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Hom$(A,C)$ is not exact?

Let $A$ be arbitrary and consider a complex $C$ of the form $$0 \to C_0 \underset{f_0}{\to} C_1 \underset{f_1}{\to} C_2 \to \cdots$$ Prove that if $C$ is exact and all the $C_i$ are free, then $\text{Hom} (A,C)$ is not exact. I'm having trouble…
Dominated Convergence Theorem
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Why is $\mathrm{Ext}^n_R(Q,A)=0$?

Let $R$ be a domain and $Q$ its field of fractions. Let $A$ be an $R$-module such that $rA=0$ for some $r \in R \backslash\{0\}$. Why $\mathrm{Ext}^n_R(Q,A)=0$ for all $n \geq 0$? For $n=0$, $Ext^0_R(Q,A) \cong Hom_R(Q,A)$, if $f : Q \rightarrow…
Rick88
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If the homotopic category is abelian, then it is semi-simple.

Let $\sf{A}$ be an abelian category and consider its homotopic category $\sf{K}(\sf{A})$. I am writing some notes about homological algebra and, since the reader usually wonders why this isn't generally an abelian category, I want to prove right…
Gabriel
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Finite projective dimension and vanishing of ext on f.g modules

Let $A$ be a commutative noetherian ring. Suppose $M$ is a finitely generated $A$-module. Let $n>0$ be an integer. It is well known that if $Ext^n(M,N) = 0$ for all $A$-modules $N$, then $M$ has a finite projective dimension. What happens if we…
the L
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Hochschild cohomology and extension (counterexample?)

Let $k$ be a field, $R$ a unital, associative $k$-algebra and $M$ an $R$-$R$-bimodule; it is a classical fact that $\mathrm{HH}^2(R, M)$ classifies Hochschild ($k$-algebra) extensions of $R$ by $M$. Anyway it seems strange to me not to require some…
Lorenzo Cecchi
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Ext of the rationals Q is a vector space over the rationals

How to see that Ext$_Z^1$(Q, A) is a vector space over Q (where Q is the rationals) for any abelian group A? Any help would be appreciated!
scsnm
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Is There a List of Errors in Cartan and Eilenberg's "Homological Algebra"?

Is there a list of errors in Cartan and Eilenberg's Homological Algebra posted somewhere accessible? The article by Flanders, "Satellites of half exact functors, a correction," Proc. Amer. Math. Soc. 15 (1964), 834–837, points out an error in one…
Tri
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Showing a collection of maps is a projector

Suppose I have a collection of maps defined as follows: for $d_{n}:C_{n} \rightarrow C_{n-1}$ and $s_{n}: C_{n} \rightarrow C_{n+1}$ I have : $t_{n}=1-f'_{n} -f_{n}$ , where $f_{n}=s_{n-1}d_{n}$ and $f'_{n}=d_{n+1}s_{n}$. Furthermore I am given…
Morettin
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Confusion about the differential of the mapping cone

Let $f:B\to C$ be a map of chain complexes and let $C(f)$ denote the mapping cone of $f.$ In Weibel's book, he defines the differential of $C(f)$ by the formula \begin{align*} d_{C(f)}:C(f)_n&=B_{n-1}\oplus C_n\to…
D. Brogan
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Category of cochain complexes is abelian

It is well known that if $\mathbf{A}$ abelian, then so is $\mathcal{Cochain}(\mathbf{A})$, the category of cochain complexes in $\mathbf{A}$. Clearly $\mathcal{Cochain}(\mathbf{A})$ has all finite (co)products and (co)kernels, but it is unclear how…
Sid Caroline
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Chain map factoring through 0 map on Homology

Suppose you have chain complexes $A,B,C,D$. And say you have maps $f:A \to B, g:B \to C, h:C \to D$, but only $g$ is a chain map. However, $h\circ g\circ f$ is a chain map even though $f$ and $h$ weren't chain maps. Now suppose that $g$ induces the…
Onkar Singh Gujral
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