Fix a prime $p$. Let $\mathbb Z_p$ be ring of $p$-adic integers.
What are the examples of finitely generated and flat $\mathbb Z_p$-algebra of characterisc $p$?
What are the examples of Noetherian and flat $\mathbb Z_p$-algebra of characteristic $p$ ?
there is a canonical homomorphism $\mathbb Z_p \to \mathbb Z_p/p \mathbb Z_p:=\mathbb F_p$. So $\mathbb F_p$ is $\mathbb Z_p$-algebra. It is finitely generated. But $\mathbb F_p$ does not seem flat $\mathbb Z_p$-algebra. Similarly $\mathbb F_p[x_1, \cdots, x_n],~\mathbb F_p[[x_1, \cdots, x_n]]$ are also finitely generated $\mathbb Z_p$-algebra. But none of them seems flat as $\mathbb Z_p$-algebra. The same is true for the 2nd question as well, i.e., $\mathbb F_p, ~\mathbb F_p[x_1, \cdots, x_n],~\mathbb F_p[[x_1,\cdots,x_n]]$ are all Noetherian $\mathbb Z_p$-algebra but doesn't seem flat.
Are there examples of the above two questions ?
Thanks