Give, for every prime number p ≥ 3 {\displaystyle {}p\geq 3} , an example of an F p [ t ] {\displaystyle {}{\mathbb {F} }_{p}[t]} -algebra S {\displaystyle {}S} , an ideal I ⊆ S {\displaystyle {}I\subseteq S} and an element f ∈ S {\displaystyle {}f\in S} such that f ∉ I ∗ {\displaystyle {}f\not \in I^{*}} in S F p ( t ) {\displaystyle {}S_{{\mathbb {F} }_{p}(t)}} , but f ∈ I ∗ {\displaystyle {}f\in I^{*}} in S ⊗ F p [ t ] F p [ t ] / m {\displaystyle {}S\otimes _{{\mathbb {F} }_{p}[t]}{\mathbb {F} }_{p}[t]/{\mathfrak {m}}} for every maximal ideal m {\displaystyle {}{\mathfrak {m}}} .
Is this possible for the ring S = F p [ t ] [ x , y , z ] / ( z 4 − x y ( x + y ) ( x + t y ) ) {\displaystyle {}S={\mathbb {F} }_{p}[t][x,y,z]/(z^{4}-xy(x+y)(x+ty))} ?
![{\displaystyle {}{\mathbb {F} }_{p}[t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e938046e47cbe5ec7261053992e56a0e6233b939)
![{\displaystyle {}S\otimes _{{\mathbb {F} }_{p}[t]}{\mathbb {F} }_{p}[t]/{\mathfrak {m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9bdab2a9630ecd1c4edbe31b745c14fd37c57d)
![{\displaystyle {}S={\mathbb {F} }_{p}[t][x,y,z]/(z^{4}-xy(x+y)(x+ty))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b391f7b9bdbe9ba3db80f35aa2e10f2bd437d07e)