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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. 

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