Proposition 4

a) The ring $B$ is local and noetherian, with maximal ideal

One has $\dim(A) = \dim(B)$ and $[\kappa_B : \kappa_A] = 1$. The monomials $t_1^{\alpha(1)} ... t_r^{\alpha(r)}$, with $0 \leq \alpha(i) < d_i$ for $1 \leq i \leq r$, form a basis of the $A$-module $B$.

b) Let $\lambda$ be the canonical homomorphism of $m_A/m_A^2$ into $m_B/m_B^2$. Then the kernel of $\lambda$ is the $\kappa_A$-vector space generated by $\gamma_1, ..., \gamma_r$. The classes of the elements $t_1, ..., t_r$ form a basis over $\kappa_A$ of the cokernel of $\lambda$.

c) For $B$ to be regular, it is necessary and sufficient that $A$ be regular and that $\gamma_1, ..., \gamma_r$ be linearly independent in the $\kappa_A$-vector space $m_A/m_A^2$.

The $A$-algebra $B$ is isomorphic to the tensor product $B_1 \otimes_A \cdots \otimes_A B_r$ with $B_i = A[T]/(P_i)$ for $1 \leq i \leq r$. It follows that the monomials $t_1^{\alpha(1)} ... t_r^{\alpha(r)}$, with $0 \leq \alpha(i) < d_i$ for $1 \leq i \leq r$, form a basis of the $A$-module $B$. In particular, $B$ is integral over $A$, hence $A$ and $B$ have the same dimension by Th. 1 of § 2, No. 3.

According to Cor. 3 of Prop. 9 of IV, § 2, No. 5, the ring $B$ is noetherian, and every maximal ideal of $B$ contains $B.m_A$. Moreover, in view of the assumption made on $P_1, ..., P_r$ and the relation $P_i(t_i) = 0$, one has $t_i^{d_i} \in B.m_A$ for $1 \leq i \leq r$. Hence every maximal ideal of $B$ contains $t_1, ..., t_r$, and therefore also the ideal $q' = B.m_A + Bt_1 + \cdots + Bt_r$. Now one has $m_A = A \cap q'$ and $B = A + q'$, hence $B/q'$ is isomorphic to $A/m_A$ and $q'$ is a maximal ideal of $B$; consequently, $B$ is local and one has $[\kappa_B : \kappa_A] = 1$. This proves a).

Put $r = m_A^2 + \sum_{i=1}^r Ac_i$, and denote by $\varphi$ the canonical homomorphism of $(A/m_A^2)[T_1, ..., T_r]$ onto $B/m_B^2$. Since one has $m_B = B.m_A + \sum_{i=1}^r Bt_i$, the kernel $n$ of $\varphi$ is the ideal generated by the classes $\overline{P}_i(T_i)$ of the polynomials $P_i(T_i)$ modulo $m_A^2.A[T_1, ..., T_r]$ and the monomials $T_iT_j$ and $xT_i$ for $1 \leq i, j \leq r$ and $x$ in $m_A/m_A^2$. According to the assumption made on $P_i$, namely $P_i(T) \equiv T^{d_i} \mod m_A.A[T]$, one can replace $\overline{P}_i(T_i)$ by $\gamma_i$ in this description of $n$; consequently, the ring $B/m_B^2$ is isomorphic to the quotient of $(A/r)[T_1, ..., T_r]$ by the graded ideal generated by the monomials $T_i T_j$ and $x T_i$ for $x$ in $m_A / r$. Let $\tau_i$ denote the class of $t_i$ modulo $m_B^2$; one therefore has

whence

Assertion b) follows at once from this.

According to formula (6) and the relation $[ \kappa_B : \kappa_A ] = 1$, one has

Now, the $\kappa_A$-vector space $r / m_A^2$ is generated by $\gamma_1, ..., \gamma_r$ and one has

Assertion c) follows at once from formulas (7) and (8).