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 May 3rd, 2018, 12:32 AM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 104 Thanks: 1 Element in a tensor product Let $p$ be a prime number and $M_p$ be the $Z$-module defined by $M_p=A_p/\mathbb Z$ where $A_p=\left\{a\in \mathbb Q\;|\;p^na\in \mathbb Z\;\text{for some}\; n\ge 0\right\}$. Let $A$ be a $\mathbb Z$-module. Every element of $A\otimes_{\mathbb Z}M_p$ may be expressed in the form $a \otimes( \frac{1}{p^k}+\mathbb Z)$ for some $a \in A$ and some $k\ge 0$. We denote $Tp(C)$ is the $p$ component of the torsion group of $C$ and $M_p=T_p(\mathbb Q/\mathbb Z)$. Please help me to show the following equivalence: >Show that $a \otimes( \frac{1}{p^k}+\mathbb Z)=0$ in $A\otimes_{\mathbb Z}M_p$ if and only if $a \in T_p(A)+p^k A$. Hint: Recall that we found an explicit exact sequence $\left\{0\right\}\to K \overset{\iota}\to F \overset{\pi} \to M_p \to \left\{0\right\}$ where $F$ and $K$ are free abelian groups. It will be helpful to tensor this sequence with $A$.) Last edited by mona123; May 3rd, 2018 at 12:35 AM. Tags element, product, tensor Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post honzik Abstract Algebra 1 November 23rd, 2015 03:37 PM guynamedluis Real Analysis 1 March 13th, 2012 11:10 AM pascal4542 Abstract Algebra 0 February 12th, 2010 10:30 AM otaniyul Linear Algebra 0 October 30th, 2009 06:40 PM riemannsph12 Abstract Algebra 0 May 13th, 2009 05:50 PM

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