Exercise 10, § 5

Let $A$ be a discrete valuation ring, $v$ its normalized valuation, $K$ its field of fractions. Let us consider a noetherian $A$-algebra $B$ endowed with a homomorphism of $A$-algebras $\varepsilon : B \to A$. We denote by $I$ the kernel of $\varepsilon$, and by $J$ its annihilator in $B$.

(a) Prove that the following conditions are equivalent: (i) the $A$-module $I/I^2$ is of finite length ; (ii) the local ring of $K \otimes_A B$ at the maximal ideal $K \otimes_A I$ is isomorphic (as a $K$-algebra) to $K$ ; (iii) one has $K \otimes_A B = (K \otimes_A I) \oplus (K \otimes_A J)$.

We now suppose that these conditions are satisfied ; we denote by $\gamma(B, \varepsilon)$, or simply $\gamma(B)$, the length of the $A$-module $I/I^2$, and by $\eta(B, \varepsilon)$, or simply $\eta(B)$, that of the $A$-module $A/\varepsilon(J)$.

(b) Let $b$ be an ideal of $B$ contained in $I$ ; we consider the $A$-algebra $B' = B/b$ and the homomorphism $\varepsilon' : B' \to A$ deduced from $\varepsilon$. Prove that $B'$ also satisfies the conditions of (a), and that one has $\gamma(B', \varepsilon') \leq \gamma(B, \varepsilon)$ and $\eta(B', \varepsilon') \leq \eta(B, \varepsilon)$.

(c) Suppose that the $A$-module $B$ is free of finite type. Prove that $J$ is a free $A$-module of rank one, a direct factor in $B$, and that one has $\mathrm{Tr}_{B/A}(x) = \varepsilon(x)$ for all $x \in J$.

(d) Suppose moreover that $B$ is a Gorenstein ring ; denote by $\ell$ a basis of the $B$-module $B^*$ (§ 3, exerc. 5), and by $\delta$ a generator of the different ideal of $B$ over $A$ (exerc. 7, b)). Prove the equality $\eta(B) = v(\varepsilon(\delta))$ (prove that $\ell(J)$ is equal to $A$, and use (c)).

e) We place ourselves in the situation of b), with $b \neq 0$ ; we suppose that the $A$-algebras $B$ and $B'$ are finite and flat, and that $B$ is a Gorenstein ring. Prove that one has $\eta(B') < \eta(B)$ (observe that the image of $J$ in $B/m_A B$ is the socle of $B/m_A B$, hence is contained in $b/m_A b$).