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Let $R$ be a unital commutative noetherian ring.
Let $x_1,x_2$ be two non zero divisors of $R$.
Is it real that $\frac{R_{x_1x_2}}{R_{x_1}+R_{x_2}}\cong_{_{R-mod}}\frac{R_{x_1}}{R}\otimes_R\frac{R_{x_2}}{R}$?
The problem is that i have some difficult to imagine why is it so, and i don't know how to do it.
I don't know if in some cases there is something like $\frac{M}{M_1}\otimes_R\frac{N}{N_1}=\frac{M\otimes_R N}{M_1\otimes_R N_1}$ for $M,N,M_1,N_1$ $R-module$.
But i am not convinced that even if i have something like that i can conclude.
I am convinced it should works becouse for example if one takes $K[|x,y|]$ (ring of formal power series over $K$) and localise to $x$ and $y$ in both cases obtains the sub vector space of $k[x^{-1},y^{-1}]$ formed by strictly negative powers.

Any suggestion or hint would be welcome, thanks.

yo yo
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  • First, $R$ is not necessarily an $R$-submodule of $R_x$. Think of $R=k[x]/(x^2)$. here $x$ is nilpotent in $R$, so $R_x=0$. So it doesn't make much sense to think of $R_x/R$. – Evans Gambit Aug 03 '21 at 05:34
  • Yes ok thanks... Take $x_1,x_2$ non zero divisors. – yo yo Aug 03 '21 at 09:43

1 Answers1

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Question: "Any suggestion or hint would be welcome, thanks."

Answer: If $S:=Spec(A), f,g\in A$ and $D(f), D(g) \subseteq S$ it follows

$$D(f)\times_S D(g) \cong D(f) \cap D(g) \cong D(fg)$$

hence

$$A_f\otimes_A A_g \cong A_{fg}.$$

There are inclusions of open subschemes $D(f), D(g) \subseteq S$, and "taking intersections" correspond to "forming the fiber product over $S$":

Form of basic open set of affine scheme: The intersection of two basic open sets.

hm2020
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