Exercise 11, § 5

We retain the notations of the preceding exercise; we suppose that the rings $A$ and $B$ are local and complete.

a) Now prove that $B$ is isomorphic to the quotient of an algebra of formal series $S = A[[X_1, \ldots, X_n]]$ by an ideal $a$ contained in $(X_1, \ldots, X_n)$; one can take $n = \dim_{\kappa_A}(\mathfrak{n}/\mathfrak{n}^2)$, where $\mathfrak{n}$ is the image of $m_B$ in $B/m_A B$.

b) Prove that the following conditions are equivalent: (i) $B$ is a finite and flat $A$-algebra and a complete intersection ring ; (ii) the ideal $a$ is generated by a sequence $(P_1, \ldots, P_n)$ completely transverse for $\kappa_A \otimes_A S$. (cf. exerc. 5.)

c) If these conditions are satisfied, prove that one has $\eta(B) = \gamma(B) = v(\varepsilon(\det(\frac{\partial P_i}{\partial X_j})))$ (use exerc. 9 and 10, d)).

d) Now prove that one can find a sequence $(f_1, \ldots, f_n)$ of elements of $a$ completely transverse for $\kappa_A \otimes_A S$ such that the $A$-algebra $B' = S/(f_1, \ldots, f_n)$ satisfies $\gamma(B') = \gamma(B)$.

(Deduce from the exact sequence $a/a^2 \to \Omega_A(S) \otimes_S B \to \Omega_A(B) \to 0$ (A, III, p. 137) an exact sequence of $A$-modules $a/a^2 \otimes_S A \to \Omega_A(S) \otimes_S A \to I/I^2 \to 0$ ; take elements $f_1, \ldots, f_n$ of $a$ whose images in $\Omega_A(S) \otimes_S A$ form a basis over $A$ of the image of $a/a^2 \otimes_S A$.)

e) Deduce using exerc. 10, e) that one has $\gamma(B) \geq \eta(B)$ and that equality holds if and only if $B$ is a complete intersection ring.