Tagged Questions

A sequence of morphisms where the image of one is the kernel of the next. It is useful in abstract algebra (ring/module/group theory) and homology theory in particular. Do not use this tag for sequences of numbers; use (sequences-and-series) instead.

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If $i$ is an inclusion why is the induced $i_*$ an epimorphism

Given the following exact homology sequence of a pair. This is in Example 2 (page 134) from Munkres. This is where I am always stuck computing homology using exact sequence. I cannot grab the last ...
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Prove $\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$

We took this idea from Simon Plouffe see here $$\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$$ Can anyone prove this identiy? We found this ...
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Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
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Module of finite length $\implies$ finite direct sum of indecomposable modules

I'm trying to solve the following question. Let $M$ be an $R$-module of finite length (i.e, both Artinian and Noetherian). Prove that it is isomorphic to a finite direct sum of indecomposable ...
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Why functors that preserve cokernels are right exact?

"Let $F$ be a functor from $A-$modules to $A-$modules that preserves cokernels. Then $F$ take exact short sequences into right exact short sequences." I found a mistake in my proof. I also can't find ...
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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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If $G$ is cyclic of order $2$ show $|H^1(G,\mathbb{Z})|=2$.

Let $G$ be the cyclic group of order $2$ acting by inversion on $\mathbb{Z}$. Show $|H^1(G,\mathbb{Z})|=2$. A hint is provided: if $E=\mathbb{Z} \rtimes G$ then every element in $E - \mathbb{Z}$ has ...
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How to determine probabilities according to a position on a list

I'm working on a SQL system in which i have to retrieve a "winning" user from a users list. The winner is determined randomly, but the users should have more chances if their coeficient number is ...
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Why isn't $1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$ a split extension? [duplicate]

Let $\mathbb{Z}_n$ be the cyclic group of $n$ elements and $1$ the trivial group. I'm looking at the following extension: $$1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$$ Because every ...