Some questions:
1) This is proposition $3.5$ , page $39$ of Atiyah's and Macdonald's book.
Let $M$ be an $A$-module. Then $S^{-1}A \otimes_{A} M \cong S^{-1}M$ as $S^{-1}A$-modules.
So the idea is to use the universal property of the tensor product. The mapping $S^{-1}A \times M \rightarrow S^{-1}M$ given by $(a/s,m) \mapsto am/s$ is $A$-bilinear so we get an $A$-linear map $f: S^{-1}A \otimes_{A} M \rightarrow S^{-1}M$ given by $f((a/s) \times m) = am/s$.
Then the rest of the proof shows the map is injective and surjective. My question is, can't we simply give the inverse? let $g: S^{-1}M \rightarrow S^{-1}A \otimes_{A} M$ given by $g(m/s) = 1/s \otimes m$.
Then we have:
$(g \circ f)((a/s) \otimes m)=g(f(a/s))=g(am/s)=1/s \otimes am = a/s \otimes m$.
Similarly for $f \circ g$.
2) This is exercise $4$ (same book) page $44$: let $f: A \rightarrow B$ be a ring homomorphism and let $S$ be a multiplicatively closed subset of $A$. Let $T=f(S)$. Show that $S^{-1}B \cong T^{-1}B$ as $S^{-1}A$-modules.
My question here: why $S^{-1}B$ makes sense? I thought that we always needed that the multiplicatively closed set is a subset of the ring $B$, here $S \subseteq A$, why we can take the localization then?
3) Let $M$ be an $A$-module let $\textrm{Supp}(M)=\{P \in Spec(A) : M_{P} \neq 0\}$, the support of $M$.
I want to compute $\textrm{Supp}(\mathbb{Z} \otimes \mathbb{Z}/2\mathbb{Z})$ (viewed as a $\mathbb{Z}$-module).
I know that in general $\textrm{Supp}(M_{1} \oplus M_{2}) = \textrm{Supp}(M_{1}) \cup \textrm{Supp}(M_{2})$.
So two doubts here:
$\textrm{Supp}(\mathbb{Z}) = \{0\} \cup \{(p) : \textrm{p is prime}\}$ right? because if we localize at such prime ideals we don't get the trivial module.
On the other hand I think $\textrm{Spec}(\mathbb{Z}/2\mathbb{Z}) = \{(0)\}$ and if we localize $\mathbb{Z}/2\mathbb{Z}$ at $(0)$ we get again $\mathbb{Z}/2\mathbb{Z}$ because $\mathbb{Z}/2\mathbb{Z}$ is a field right?
So in conclusion $\textrm{Supp}(\mathbb{Z} \otimes \mathbb{Z}/2\mathbb{Z}) = \{(0)\} \cup \{(p): \textrm{ p is prime}\}=\textrm{Spec}(\mathbb{Z})$.
Is this OK?
Thanks in advance
\textbf{EDIT}: Sorry I meant $\textrm{Supp} (\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z})$.