# separability is required for integral closures to be finitely generated

The parent theorem assumed that $L$ was a separable extension^{} of $K$. Here is an example that shows that separability is in fact a necessary condition for the result to hold.

Let $k=(\mathbb{Z}/p\mathbb{Z})({b}_{1},{b}_{2},\mathrm{\dots})$ where the ${b}_{i}$ are indeterminates^{}.

Let $B$ be the subring of $k[[t]]$ (power series in $t$ over $k$) consisting of

$$\left\{\sum _{i=0}^{\mathrm{\infty}}{c}_{i}{t}^{i}\right\}$$ |

such that $\{{c}_{0},{c}_{1},{c}_{2},\mathrm{\dots}\}$ generates a finite extension^{} of $(\mathbb{Z}/p\mathbb{Z})({b}_{1}^{p},{b}_{2}^{p},\mathrm{\dots})$. Then every element of $B$ is a power of $t$ times a unit (first factor out the largest power of $t$. What’s left is $a(1+{c}_{1}t+{c}_{2}{t}^{2}+\mathrm{\cdots})$; its inverse^{} is ${a}^{-1}(1-{c}_{1}t-\mathrm{\cdots})$). Hence the ideals of $B$ are powers of $t$, so $B$ is a PID (in fact, it is a DVR).

Now, let $u={b}_{0}+{b}_{1}t+{b}_{2}{t}^{2}+\mathrm{\cdots}$. Now, $u\notin B$ because it uses all of the ${b}_{i}$ and thus the coefficients^{} do not define a finite extension of $(\mathbb{Z}/p\mathbb{Z})({b}_{1}^{p},{b}_{2}^{p},\mathrm{\dots})$. However, $u$ is integral over $B$: ${b}_{i}^{p}\in B\Rightarrow {u}^{p}\in B$ which implies that the degree of $u$ over the field of fractions^{} is $p$. Hence a basis for $K(u)/K$ is $\{1,u,{u}^{2},\mathrm{\dots},{u}^{p-1}\}$. There are other elements integral over $B$:

$${b}_{1}+{b}_{2}t+{b}_{3}{t}^{2}+\mathrm{\cdots}=\frac{u-{b}_{0}}{t}=\frac{-{b}_{0}}{t}+\frac{1}{t}u$$ | ||

$${b}_{2}+{b}_{3}t+{b}_{4}{t}^{2}+\mathrm{\cdots}=\frac{u-{b}_{0}-{b}_{1}t}{{t}^{2}}=\frac{-{b}_{0}-{b}_{1}t}{{t}^{2}}+\frac{1}{{t}^{2}}u$$ | ||

$$\mathrm{\vdots}$$ |

Clearly the denominators are getting bigger, so the integral closure^{} of $B$ cannot be finitely generated^{} as a $B$-module.

Title | separability is required for integral closures to be finitely generated |
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Canonical name | SeparabilityIsRequiredForIntegralClosuresToBeFinitelyGenerated |

Date of creation | 2013-03-22 17:02:15 |

Last modified on | 2013-03-22 17:02:15 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 6 |

Author | rm50 (10146) |

Entry type | Example |

Classification | msc 13B21 |

Classification | msc 12F05 |