## 109.37 Finite type, flat and not of finite presentation

In this section we give some examples of ring maps and morphisms which are of finite type and flat but not of finite presentation.

Let $R$ be a ring which has an ideal $I$ such that $R/I$ is a finite flat module but not projective, see Section 109.32 for an explicit example. Note that this means that $I$ is not finitely generated, see Algebra, Lemma 10.108.5. Note that $I = I^2$, see Algebra, Lemma 10.108.2. The base ring in our examples will be $R$ and correspondingly the base scheme $S = \mathop{\mathrm{Spec}}(R)$.

Consider the ring map $R \to R \oplus R/I\epsilon$ where $\epsilon ^2 = 0$ by convention. This is a finite, flat ring map which is not of finite presentation. All the fibre rings are complete intersections and geometrically irreducible.

Let $A = R[x, y]/(xy, ay; a \in I)$. Note that as an $R$-module we have $A = \bigoplus _{i \geq 0} Ry^ i \oplus \bigoplus _{j > 0} R/Ix^ j$. Hence $R \to A$ is a flat finite type ring map which is not of finite presentation. Each fibre ring is isomorphic to either $\kappa (\mathfrak p)[x, y]/(xy)$ or $\kappa (\mathfrak p)[x]$.

We can turn the previous example into a projective morphism by taking $B = R[X_0, X_1, X_2]/(X_1X_2, aX_2; a \in I)$. In this case $X = \text{Proj}(B) \to S$ is a proper flat morphism which is not of finite presentation such that for each $s \in S$ the fibre $X_ s$ is isomorphic either to $\mathbf{P}^1_ s$ or to the closed subscheme of $\mathbf{P}^2_ s$ defined by the vanishing of $X_1X_2$ (this is a projective nodal curve of arithmetic genus $0$).

Let $M = R \oplus R \oplus R/I$. Set $B = \text{Sym}_ R(M)$ the symmetric algebra on $M$. Set $X = \text{Proj}(B)$. Then $X \to S$ is a proper flat morphism, not of finite presentation such that for $s \in S$ the geometric fibre is isomorphic to either $\mathbf{P}^1_ s$ or $\mathbf{P}^2_ s$. In particular these fibres are smooth and geometrically irreducible.

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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