Could someone help me to find:
$R_0$ a local noetherian ring, $R$ a graded algebra over $R_0$ generated by $x_1,...,x_n$ of degree 1, and
$M$ graded $R$-module, finitely generated, such that $M$ is not flat over $R_0$ but it is $R_0$-flat for every localisation $M_f$ with $f\in R_1$?
The problem comes from the fact that if you look at the groups $M_d$ of the graded module you have two situations:
- If $M$ is $R_0$-flat then $M_d$ is free over $R_0$ for every $d\in\mathbb{Z}$.
- If $M_f$ is $R_0$-flat for every $f\in R_1$ then $M_d$ is free over $R_0$ for $d\gg0$.
I've tried $R_0:=K[[t]]$, $R:=R_0[y]$ and $M:=\frac{R}{ty^2}\oplus R$. I can't see clearly the property in 2).
Another thing, is there any geometric intuition behind this? I mean the fact that if you take $\tilde{M}$ for a graded module as the sheaf of $Proj(R)$ module this is not a cathegory equivalence respect to the cathegory of graded $R$ module, becouse the sheaf $\tilde{M}$ depends on $M_d$ for $d>>0$.
Also any reference for the second observation would be great.
Thanks