Exercise 4, § 6

Let $A$ be a commutative ring and $a$ an ideal of $A$. The Rees algebra of the ideal $a$ is the subalgebra $R(a)$ of the polynomial algebra $A[T]$ in one indeterminate, consisting of the polynomials $c_0 + c_1 T + \cdots + c_n T^n$, where $c_0 \in A$ and $c_k \in a^k$ for every integer $k \geq 1$.

(a) Show that there exists one and only one surjective $A$-algebra homomorphism $r : S(a) \to R(a)$ such that the restriction of $r$ to $a$ is the $A$-linear mapping $x \mapsto xT$.

(b) Suppose that the ring $A$ is an integral domain. Let $a_1, \ldots, a_n$ be elements of $A$ with $a_n \neq 0$; consider the $A$-algebra homomorphism

such that $u(X_i) = a_i T$. The kernel $r = \mathrm{Ker}(u)$ is a graded ideal whose homogeneous component $r_m$ of degree $m$ consists of the homogeneous polynomials $f$ of degree $m$ such that $f(a_1, \ldots, a_n) = 0$. If $r' \subset r$ is the ideal of $A[X_1, \ldots, X_n]$ generated by the homogeneous component $r_1$ of $r$, show that the $A$-module $r/r'$ is a torsion $A$-module. (To verify that, for all $f \in r_m$, there exists $c \neq 0$ in $A$ such that $cf \in r'$, argue by induction on $m$, by writing

where the $f_j$ are homogeneous of degree $m - 1$; consider the polynomial observe that $g \in r_1 \subset r'$ and form $a_n^{m-1} f - X_n^{m-1} g.$)

(c) Deduce from (b) that when $A$ is an integral domain, the kernel of the homomorphism $r$ of (a) is the torsion module of $S(a)$. Deduce that, for $S(a)$ to be an integral domain, it is necessary and sufficient that $S(a)$ be a torsion-free $A$-module. In particular, if the ideal $a$ is a projective $A$-module, $S(a)$ is an integral domain and isomorphic to $R(a)$.