Exercise 18, § 2

Let $A$ be a local integrally closed domain, $K$ its field of fractions, $m$ its maximal ideal and $k = A/m$ its residue field. Let $K'$ be a finite algebraic extension of $K$ of degree $n$, $A'$ a subring of $K$ containing $A$, integral over $A$ and with $K'$ as field of fractions and $m_i$ ($1 \leq i \leq r$) the maximal ideals of $A$. An ideal $m$ is called unramified in $A'$ if there exists a basis $(w_i)_{1 \leq i \leq r}$ of $K'$ over $K$ consisting of elements of $A'$ and whose discriminant $D_{K'/K}(w_1, \ldots, w_r)$ (Algebra, Chapter IX, § 2) belongs to $A - m$.

(a) Show that, if $m$ is unramified in $A'$, $K'$ is a separable extension of $K$, $A'$ is the integral closure of $A$ in $K'$ and is a free $A$-module every basis of which over $A$ is a basis of $K'$ over $K$; $mA'$ is the Jacobson radical of $A'$ and $A'/mA'$ is a semi-simple $k$-algebra of rank $n$ over $k$ which is separable over $k$ and the direct composition of the fields $A'/m_i'$.

(b) Let $d_i$ be the separable factor of the degree of $A'/m_i'$ over $A/m$. Show that, if $k$ is infinite, $\sum_i d_i \leq n$ (use Exercise 17 (c)); moreover, the following conditions are equivalent: (α) $m$ is unramified in $A'$. (β) $A'$ is a finitely generated $A$-module and $A'/mA'$ is a semi-simple $k$-algebra and separable over $k$. (γ) $\sum_{i=1}^r d_i \leq n$. (To see that (β) implies (γ), note that there is a system of generators of the $A$-module $A'$ whose classes mod. $mA'$ form a basis of $A'/mA'$ over $k$, using Chapter II, § 3, no. 2, Corollary 2 to Proposition 4; deduce that

To see that (γ) implies (α), use Exercise 17 (c) to show that there exists a basis of $A'/mA'$ over $k$ consisting of the powers $\xi^i$ of an element $\xi$ ($0 \leq i \leq n-1$); if $x \in A'$ is an element of the class $\xi$, deduce that $D_{K'/K}(1, x, \ldots, x^{n-1})$ belongs to $A - m$.) Give an example where $K'$ is separable over $K$, $A'/mA' = A/m$ and $A'$ is not a finitely generated $A$-module (cf. Exercise 9 (c)).

(c) Extend the results of (b) to the case where $k$ is finite (in the notation of Exercise 16, show that, if $n$ is unramified in $B'$, then $m$ is unramified in $A'$; taking account of the fact that $B/n$ is infinite, use Exercise 16 (c), thus obtaining an element $t \in B'$ such that

show that $z$ may be assumed to be a polynomial in $A'[X]$ and, by expressing the above discriminant as a polynomial in $X$, obtain the existence of a system of $n$ elements $w_i$ ($1 \leq i \leq n$) of $A'$ such that $D_{K'/K}(w_1, \ldots, w_r) \in A - m$).