(graded ring)
A \(\ZZ\)-graded ring is a commutative ring \(S_\bullet=\bigoplus_{n\in\ZZ}S_n\) with \(1\in S_0\) whose multiplication restricts to maps \(S_m\times S_n\to S_{m+n}\). Its base ring is \(S_0\); each \(S_n\) is an \(S_0\)-module. The grading is nonnegative if \(S_n=0\) for all \(n<0\). An element of \(S_n\) is a homogeneous element of degree \(n\).
The ring is finitely generated over \(S_0\) if a finite set of homogeneous elements generates it as an \(S_0\)-algebra. It is generated in degree one if \(S_1\) generates it as an \(S_0\)-algebra.
(Irrelevant ideal)
For a nonnegatively graded ring \(S_\bullet\), the irrelevant ideal is \[ S_+\coloneqq\bigoplus_{n\ge1}S_n. \]
A nonnegatively graded ring \(S_\bullet\) is finitely generated as an \(S_0\)-algebra if and only if \(S_+\) is finitely generated as an ideal of \(S_\bullet\).
Finite homogeneous algebra generators of positive degree generate \(S_+\) as an ideal, by writing each positive-degree monomial as a multiple of one of those generators. Conversely, take finite homogeneous ideal generators \(a_1,\ldots,a_r\) of \(S_+\); such generators can be obtained by replacing finite ideal generators by their homogeneous components. For homogeneous \(s\in S_n\) with \(n>0\), write \(s=\sum_i b_ia_i\) with each nonzero \(b_i\) homogeneous of degree \(n-\deg a_i<n\). Induction on \(n\) expresses all the \(b_i\), and hence \(s\), as polynomials in the \(a_i\) with coefficients in \(S_0\).
(Positive-degree terms in an integer grading)
For a field \(k\), give \(k[t,t^{-1}]\) the grading \(\deg t=1\). Its positive-degree sum is not an ideal: it contains \(t\) but not \(t^{-1}t=1\).
(Homogeneous ideal)
An ideal \(I \subseteq S_\bullet\) is homogeneous if it is generated by homogeneous elements, equivalently if \(I = \bigoplus_n (I \cap S_n)\).
(Operations on homogeneous ideals)
Sums, products, intersections and radicals of homogeneous ideals are homogeneous. A homogeneous ideal \(\mfa \neq S_\bullet\) is prime if and only if for all homogeneous \(f, g \in S_\bullet\), \(fg \in \mfa\) implies \(f \in \mfa\) or \(g \in \mfa\). [@Har10a, §I.2]
(Monomials of a fixed degree)
Let \(k\) be a field and let \(n,d\ge0\) be integers. In \(S=k[x_0,\ldots,x_n]\) with the usual grading, the degree-\(d\) part \(S_d\) has the monomials of total degree \(d\) as a \(k\)-basis, and \[ \dim_k S_d = \binom{d+n}{n} . \] The monomial \(x_0^{e_0} \cdots x_n^{e_n}\) corresponds to the string of \(e_0\) stars, a bar, \(e_1\) stars, a bar, \(\ldots\), a bar, \(e_n\) stars: \(d\) stars and \(n\) bars in some order. For three variables (\(n = 2\)), written \(x_1, x_2, x_3\), and \(d = 3\), the \(\binom{5}{2} = 10\) monomials correspond to the strings:
| monomial | string | monomial | string |
|---|---|---|---|
| \(x_1^3\) | \(\star\star\star\mid\mid\) | \(x_1x_3^2\) | \(\star\mid\mid\star\star\) |
| \(x_1^2x_2\) | \(\star\star\mid\star\mid\) | \(x_2^3\) | \(\mid\star\star\star\mid\) |
| \(x_1^2x_3\) | \(\star\star\mid\mid\star\) | \(x_2^2x_3\) | \(\mid\star\star\mid\star\) |
| \(x_1x_2^2\) | \(\star\mid\star\star\mid\) | \(x_2x_3^2\) | \(\mid\star\mid\star\star\) |
| \(x_1x_2x_3\) | \(\star\mid\star\mid\star\) | \(x_3^3\) | \(\mid\mid\star\star\star\) |
With \(n\ge1\) variables \(x_1,\ldots,x_n\) instead, the count is \(\binom{d+n-1}{n-1}\).
For a homogeneous polynomial \(f\in k[x_0,\ldots,x_n]\) of degree \(d\) and \(\lambda\in k^\times\), \[ f(\lambda x_0,\ldots,\lambda x_n)=\lambda^d f(x_0,\ldots,x_n). \] Thus its vanishing is invariant under changing homogeneous coordinates by a common nonzero scalar.
Let \(S\) be generated by finitely many elements \(s_0,\ldots,s_r\in S_1\) as an \(S_0\)-algebra. The surjection \(S_0[x_0,\ldots,x_r]\to S\) sending \(x_i\) to \(s_i\) induces a closed immersion \(X=\Proj S\hookrightarrow\PP^r_{S_0}\). The sheaf \(\OO_X(1)\) is invertible and trivial on each \(D_+(s_i)\), with frame \(s_i\) [@Har10a, Chapter II, §2 and Proposition II.5.12]. If \(S_0\) is noetherian and \(\mcf\) is quasi-coherent, the canonical evaluation map \(\widetilde{\Gamma_*(\mcf)}\to\mcf\) for the graded section module is an isomorphism [@Har10a, Proposition II.5.15].
(Infinitely many degree-one generators)
For a field \(k\), let \(S=k[x_0,x_1,\ldots]\) with every variable of degree one. The affine chart \(D_+(x_0)\) of \(\Proj S\) has coordinate ring \(k[x_1/x_0,x_2/x_0,\ldots]\). This ring is not noetherian, since the ideals generated by the first \(j\) variables form a strictly increasing chain. Consequently \(\Proj S\) is not a closed subscheme of any finite-dimensional projective space over \(k\), whose closed subschemes are noetherian.
For a nonnegatively graded ring \(S\), the quotient \(S/S_+\) is canonically \(S_0\). Thus the closed subscheme \(\Spec(S/S_+)\subseteq\Spec S\) is \(\Spec S_0\); for \(S=k[x_0,\ldots,x_n]\) it is the affine origin. The points of \(\Proj S\) are homogeneous prime ideals not containing \(S_+\). The module \(S/S_+\) has zero associated sheaf because localizing it at any positive-degree element gives zero.
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