Recall that a ring \(A\) is called graded if it admits a direct sum decomposition \(A=\oplus_{n=0}^{\infty}A_n\) as abelian groups, with the property that \(A_iA_j\subseteq A_{i+j}\) for all \(i,j\geq 0\). Prove that a graded commutative ring \(A=\oplus_{n=0}^{\infty} A_n\) is Noetherian if and only if \(A_0\) is Noetherian and \(A\) is finitely generated as an algebra over \(A_0\).
Solution
1. (\(\Leftarrow\)) If \(A_0\) is Noetherian and \(A\) is finitely generated as an \(A_0\)-algebra, then \(A\) is Noetherian.
1.1. \(A\) is a quotient of a polynomial ring \(A_0[x_1, \ldots, x_r]\) (over the finitely many algebra generators).
a finitely generated \(A_0\)-algebra is a quotient of a polynomial ring in finitely many variables.
1.2. \(A_0[x_1, \ldots, x_r]\) is Noetherian.
Hilbert’s basis theorem, applied inductively, since \(A_0\) is Noetherian.
1.3. A quotient of a Noetherian ring is Noetherian.
standard.
1.4. Hence \(A\) is Noetherian.
<2>1–<2>3.
2. (\(\Rightarrow\)) If \(A\) is Noetherian, then \(A_0\) is Noetherian and \(A\) is finitely generated as an \(A_0\)-algebra.
2.1. \(A_0\) is Noetherian.
\(A_0 = A/A_+\) where \(A_+ = \bigoplus_{n \ge 1} A_n\) is an ideal, so \(A_0\) is a quotient of the Noetherian ring \(A\), hence Noetherian.
2.2. The ideal \(A_+ = \bigoplus_{n \ge 1} A_n\) is finitely generated.
\(A\) is Noetherian, so every ideal is finitely generated.
2.3. Let \(A_+ = (x_1, \ldots, x_r)\) with each \(x_i\) homogeneous of positive degree.
\(A_+\) is generated by homogeneous elements (it is a homogeneous ideal).
2.4. \(A\) is generated as an \(A_0\)-algebra by \(x_1, \ldots, x_r\).
every homogeneous element of positive degree lies in \(A_+\), hence is an \(A_0\)-linear combination of the \(x_i\) (and their products); by induction on degree, every element of \(A\) is a polynomial in the \(x_i\) with coefficients in \(A_0\).
2.5. Hence \(A\) is finitely generated as an \(A_0\)-algebra.
4.
3. Q.E.D.
1 and 2.