This is a question about a discrepancy I have between two notions of flatness.
- A ring map $\mathbb{Z}\longrightarrow \mathbb{Z}[i]$ is flat if $\mathbb{Z}[i]$ is a flat $\mathbb{Z}$-module. Since it is a free $\mathbb{Z}$-module, it is flat.
- As maps of schemes, $\operatorname{Spec}\mathbb{Z}[i]\longrightarrow \operatorname{Spec}\mathbb{Z}$ is flat only if its fibres have equal cardinality (in the finite fibred maps case). According to this "intuitive" constraint for flatness, we see that primes which are $1\bmod 4$ have fibres of size 2, while primes that are $3\bmod 4$ have fibres of size 1. Shouldn't this mean that the map of schemes is not flat?