Tagged Questions

Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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if $F_{\bullet}$ is a complex and $r$ an integer, what is $F_{r-\bullet}$?

While reading the paper Some results and questions on the Castelnuovo-Mumford regularity, by Marc Chardin, I encountered in the proof of Theorem 5.1 the notation $F^N_{r-\bullet}$. To provide some ...
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Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ ...
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Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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Definition of $Hom(A,B)$

I have lots of confusion about definition of $Hom(A,B)$. I would like to ask several questions with my thoughts. Hopefully I could solve my problem. -Firstly, my book write that if $A$ and $B$ is ...
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Direct limit of a direct system looking like a cochain complex of objects.

I would like to ask you about a special kind of direct systems $(A_i, f_{i}^{j} )_{ i,j \in ( I , \leq ) }$ looking like a cochaîn complex $(A_i , f_{i}^{j} )_{ i,j \in ( \mathbb{N}^* , \leq ) }$ ...
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Kähler differential over a field

I have been working with Kähler differentials, and I have $\Omega^1_{B/k}$, where B is a commutative $k$-algebra, and $k$ is a field. I was wondering that for $d(b)=0$, does this imply that $b\in k$? ...
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Variation on localization of Tor

It is known that Let $R$ be a commutative ring with unit and $S \subset R$ a multiplicative sistem. If $M$ and $N$ are $R$-modules there is a isomorphism of $S^{-1}R$-modules: ...
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Long exact sequence to short exact sequence [duplicate]

$$\dots\xrightarrow{p_*}\pi_{n+1}(B)\xrightarrow{\partial}\pi_n(F)\xrightarrow{\text{inc}_*}\pi_n(E)\xrightarrow{p_*}\pi_n(B)\xrightarrow{\partial}\pi_{n-1}(F)\xrightarrow{\text{inc}_*}\dots$$ From ...
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$\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
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I'm searching for $X$ and $V$ such that the evaluation map $f:H^n(X,A;V)\to \hom_\mathbb{Z}(H_n(X,A;\mathbb{Z}),V)$ is not injective

Let $V$ be a $\mathbb{Z}$-module $(X,A)$ a pair of topological spaces. We define the pairing-map in singular co- homology$$H^n(X,A;V)\times H_n(X,A;\mathbb{Z})\to V$$ ([\xi ],[\alpha])\mapsto \xi ...
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Contractibility of an exact chain complex

How can one prove that an exact (acyclic) chain complex of projective modules that is trivial in negative degrees is contractible? I would appreciate some nudges in the right direction more than ...
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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 ...