Section 110.35 (05FZ): Zero dimensional local ring with nonzero flat ideal

110.35 Zero dimensional local ring with nonzero flat ideal

In [Lazard] and [Autour] there is an example of a zero dimensional local ring with a nonzero flat ideal. Here is the construction. Let

$k$ be a field. Let

$X_ i, Y_ i$ ,

$i \geq 1$ be variables. Take

$R = k[X_ i, Y_ i]/(X_ i - Y_ i X_{i + 1}, Y_ i^2)$ . Denote

$x_ i$ , resp.

$y_ i$ the image of

$X_ i$ , resp.

$Y_ i$ in this ring. Note that

$x_ i = y_ i x_{i + 1} = y_ i y_{i +1} x_{i + 2} = y_ i y_{i + 1} y_{i + 2} x_{i + 3} = \ldots$

in this ring. The ring

$R$ has only one prime ideal, namely

$\mathfrak m = (x_ i, y_ i)$ . We claim that the ideal

$I = (x_ i)$ is flat as an

$R$ -module.

Note that the annihilator of

$x_ i$ in

$R$ is the ideal

$(x_1, x_2, x_3, \ldots , y_ i, y_{i + 1}, y_{i + 2}, \ldots )$ . Consider the

$R$ -module

$M$ generated by elements

$e_ i$ ,

$i \geq 1$ and relations

$e_ i = y_ i e_{i + 1}$ . Then

$M$ is flat as it is the colimit

$\mathop{\mathrm{colim}}\nolimits _ i R$ of copies of

$R$ with transition maps

$R \xrightarrow {y_1} R \xrightarrow {y_2} R \xrightarrow {y_3} \ldots$

Note that the annihilator of

$e_ i$ in

$M$ is the ideal

$(x_1, x_2, x_3, \ldots , y_ i, y_{i + 1}, y_{i + 2}, \ldots )$ . Since every element of

$M$ , resp.

$I$ can be written as

$f e_ i$ , resp.

$h x_ i$ for some

$f, h \in R$ we see that the map

$M \to I$ ,

$e_ i \to x_ i$ is an isomorphism and

$I$ is flat.

Lemma 110.35.1.slogan There exists a local ring $R$ with a unique prime ideal and a nonzero ideal $I \subset R$ which is a flat $R$ -module

Proof. See discussion above. $\square$

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