Exercise 10, § 6

Soit $\rho : A \to B$ un homomorphisme d’anneaux. Pour tout $\mathfrak{p} \in \mathrm{Spec}(B)$, on pose

We propose to prove in this exercise that, when B is an A-algebra of finite type, the mapping $\mathfrak{p} \mapsto d(\mathfrak{p})$ of $\mathrm{Spec}(B)$ into $\mathbf{R}$ is upper semi-continuous (“Chevalley’s theorem”).

Let $\delta \subset B \otimes_A B$ be the kernel of the canonical A-homomorphism $B \otimes_A B \to B$. For every integer $r$ put

The $B \otimes_A B$-algebra $P_{B/A}^r$ is called the algebra of principal parts of order $r$. We consider $B \otimes_A B$ as a B-algebra by the homomorphism $b \mapsto b \otimes 1$, so that $P_{B/A}^r$ is, for every $r$, a B-algebra and the canonical homomorphisms of passing to the quotient $P_{B/A}^{r+1} \to P_{B/A}^r$ define a projective system of B-algebras.

$^1$ This is the dimension as a vector space over the field $\kappa(\mathfrak{p})$.

a) Let $p : A \to A'$ be a ring homomorphism. Put $B' = B \otimes_A A'$ and denote by $q : B \to B'$ the canonical homomorphism. Let $r$ be an integer. Show that the unique homomorphism of $B'$-algebras

which associates to the class of $(b_1 \otimes b_2) \otimes 1$ ($b_i \in B$), the class of $q(b_1) \otimes q(b_2)$, is an isomorphism.

b) Let $S$ be a multiplicative subset of $B$. Show that the canonical homomorphism

is an isomorphism.

c) Suppose that $B$ is an A-algebra of finite presentation, that is, isomorphic to a quotient of $A[T_1, ..., T_n]$ by an ideal of finite type. Show that for every $r \in \mathbf{N}$, $P_{B/A}'$ is a B-module of finite presentation. For every $p \in \mathrm{Spec}(B)$, put $P_{B/A}'[p] = \lim P_{B/A}' \otimes_B \kappa(p)$.

Show that $P_{B/A}'[p]$ is a complete local $\kappa(p)$-algebra with residue field $\kappa(p)$. Let $k \in \mathbf{N}$. Show that the set of $p \in \mathrm{Spec}(B)$ such that $\dim(P_{B/A}'[p]) \geq k$ is a closed subset of $\mathrm{Spec}(B)$. (For every $r \in \mathbf{N}$, let $F_{r,k}$ be the set of $p$ such that $\dim_{\kappa(p)}(P_{B/A}' \otimes \kappa(p)) \geq \binom{r+k-1}{r-1}$.)

It will be shown that $F_{r,k}$ is closed (exerc. 9) and that consequently $\Phi_k = \bigcap_r F_{r,k}$ is closed. We shall then apply § 6, No. 3, Cor. 2 to Prop. 6.)

d) Let $m$ be an ideal of $B$ and let $\mu \subset (B/m) \otimes_A B$ be the kernel of the canonical mapping $(B/m) \otimes_A B \to B/m$. Show that there is a canonical isomorphism

Assume that $A$ is a field, that $B$ is an $A$-algebra of finite type, and that $m$ is a maximal ideal of $B$. Show that $P_{B/A}'[m]$ is isomorphic to the completed localization of $(B/m) \otimes_A B$ at the maximal ideal $\mu$. Deduce that one has $\dim(P_{B/A}'[m]) = \dim_m(B)$. (Apply the theorem of zeros and Theorem 1 of § 2, No. 3. One will notice that the extension $B/m$ of $A$ is contained in a finite galois extension of a finite radicial extension of $A$.)

e) Assume that $B$ is an algebra of finite type over $A$. Let $p \in \mathrm{Spec}(B)$. Show that one has

(First remark that for every $r \in \mathbf{N}$, one has $P_{B/A}' \otimes_B \kappa(p) = (P_{B/A}' \otimes_B B') \otimes_{B'} \kappa(p)$ by putting $B' = B \otimes_A \kappa(\rho^{-1}(p))$, and that consequently, by virtue of a) , one has $P_{B/A}'[p] = P_{B'/\kappa(\rho^{-1}(p))}'[pB']$. One will thus reduce to proving, in the case where $A$ is a field, the equality

Remark then that in the case where $p$ is maximal, this equality follows from d). In the general case, let $q_1, ..., q_l$ be the minimal prime ideals of $B$ contained in $p$. Show using the theorem of zeros that there exists a dense open set $U$ of $V(p)$ such that for every maximal ideal $m$ belonging to $U$, the minimal prime ideals contained in $m$ are contained in $U$. Using c), show that one can, by decreasing $U$, assume in addition that, for every $p' \in U$, one has $\dim(P_{B/A}'[p]) = \dim(P_{B/A}'[p'])$. Conclude.)

f) Assume that $B$ is an $A$-algebra of finite type. Show that the function $p \mapsto d(p) = \dim_p(B \otimes_A \kappa(\rho^{-1}(p)))$ is upper semi-continuous. (When $B$ is of finite presentation, use e) and c). In the general case, $B$ is of the form $A[T_1, ..., T_n]/J$. Let $J_\alpha$ be the family of ideals of finite type of $A[T_1, ..., T_n]$ contained in $J$, ordered by inclusion, and put $B_\alpha = A[T_1, ..., T_n]/J_\alpha$. For every $\alpha$, one has $\mathrm{Spec}(B) \subset \mathrm{Spec}(B_\alpha) \subset \mathrm{Spec}(A[T_1, ..., T_n])$, and $\mathrm{Spec}(B) = \bigcap_\alpha \mathrm{Spec}(B_\alpha)$. For every $p \in \mathrm{Spec}(B)$, put

Show that $d_\alpha(p) \geq d(p)$ and that $d(p) = \inf_\alpha d_\alpha(p)$ by virtue of the noetherian character of $\mathrm{Spec}(\kappa(\rho^{-1}(p))[T_1, ..., T_n])$. Deduce that $p \mapsto d(p)$ is the greatest lower bound of a decreasing filtered family of upper semi-continuous functions.)

EXERCISES

AC VIII .103