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How to compute Tor$_1$ ($\mathbb{Q} /\mathbb{Z}$, $\mathbb{Q} /\mathbb{Z}$).

I am having trouble finding a projective resolution of $\mathbb{Q} /\mathbb{Z}$.

BetaY
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1 Answers1

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I'm not sure what tools you have at your disposal, but you could try this: The short exact sequence $0 \to \mathbb{Z}\to \mathbb{Q}\to \mathbb{Q}/\mathbb{Z} \to 0$ induces a long exact sequence of Tor-modules. In this long exact sequence is a piece which looks like $$\mathrm{Tor}_1^\mathbb{Z}(\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) \to \mathrm{Tor}_1^\mathbb{Z}(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z})\to \mathbb{Z}\otimes\mathbb{Q}/\mathbb{Z}\to \mathbb{Q}\otimes \mathbb{Q}/\mathbb{Z}$$ The first and last terms are both $0$ so $$\mathrm{Tor}_1^\mathbb{Z}(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z})\cong \mathbb{Z}\otimes \mathbb{Q}/\mathbb{Z}\cong \mathbb{Q}/\mathbb{Z}$$

froggie
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  • We haven't learned anything about the long exact sequence of Tor-module. I wonder if there is other ways? I will have a look at the long exact sequence though. – BetaY Nov 09 '15 at 13:47