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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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What does $C_{-1}(X)$ mean?

I recently came across the formula that the Euler characteristic is equal to $$\sum\limits_{i=-N}^N (-1)^i\dim C_i(X)$$ For this to make sense, $C_{-1}(X)$ would have to exist. What would that be? What is a $-1$-simplex? A $1$-simplex with an odd…
user67803
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Why is $\operatorname{Tor}(\mathbb{Z}_m,\mathbb{Z}_n) \cong \mathbb{Z}_{\operatorname{gcd}(m,n)}$?

The definition of $\operatorname{Tor}$ I am using is: Let $K \to F\to A\to 0$ be a free resolution of $A$ and $B$ an abelian group, then $\operatorname{Tor}(A,B) := \ker (f \otimes 1_B)$ if $f$ is the map $f\colon F \to A$. What is the free…
fish_monster
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Ext functor and Projective Module

Let $R$ be a ring and $P$ is an $R$-module. The statement are equivalent: 1-$P$ is projective. 2-For every $R$-module $N$ and for $i\geq 1$ , $Ext^i_R(P,N)=0$ 3-For every $R$-module $N$ , $Ext^1_R(P,N)=0$ 4-For every finitely presented $R$-module…
pink floyd
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show that functor $L_0T$ is right exact

Show that $L_0T$ is right exact.Here T is an additive functor from category of $\Lambda$-modules to abelian group. I try to prove it,but I think the condition is too little.Any hints?
Daniel Xu
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Verifying that $T_0 = F, T_n = 0$ is a universal $\delta$-functor

I am trying to prove the following: An Introduction to Homological Algebra - C. A. Weibel (1994) Exercise 2.1.2) If $F~:A\to B$ is an exact functor, show that ${{T}_{0}}=F$ and ${{T}_{n}}=0$ for $n\ne 0$ defines a universal $\delta $-functor (of…
math-ha01
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Merge operation in homological algebra?

I provide you with a definition for the Merge operation in one standard textbook on the minimalist program in linguistics: Merge: "basic structure-building mechanism. Merge takes two elements A and B and forms a two-membered set labeled C. C can…
Javier Arias
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