Lemma 109.37.1. There exist examples of

1. a flat finite type ring map with geometrically irreducible complete intersection fibre rings which is not of finite presentation,

2. a flat finite type ring map with geometrically connected, geometrically reduced, dimension 1, complete intersection fibre rings which is not of finite presentation,

3. a proper flat morphism of schemes $X \to S$ each of whose fibres is isomorphic to either $\mathbf{P}^1_ s$ or to the vanishing locus of $X_1X_2$ in $\mathbf{P}^2_ s$ which is not of finite presentation, and

4. a proper flat morphism of schemes $X \to S$ each of whose fibres is isomorphic to either $\mathbf{P}^1_ s$ or $\mathbf{P}^2_ s$ which is not of finite presentation.

Proof. See discussion above. $\square$

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