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May 3rd, 2018, 12:32 AM   #1
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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 $a \otimes( \frac{1}{p^k}+\mathbb Z)$
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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