Let \(S\) be a graded ring, generated by \(S_1\) as an \(S_0\)-algebra, and put \(A=S_0\). For any integer \(d > 0\), let \(S^{(d)}\) be the graded ring \(\bigoplus_{n \geq 0} S^{(d)}_n\) where \(S^{(d)}_n = S_{nd}\). Let \(X = \Proj S\). Show that \(\Proj S^{(d)} \cong X\), and that the sheaf \(\OO(1)\) on \(\Proj S^{(d)}\) corresponds via this isomorphism to \(\OO_X(d)\).
This construction is related to the \(d\)-uple embedding as follows. If \(x_0, \ldots, x_r\) generate \(S_1\), corresponding to an embedding \(X \injects \PP^r_A\), then the monomials of degree \(d\) in the \(x_i\) generate \(S^{(d)}_1 = S_d\). These define a projective embedding of \(\Proj S^{(d)}\) which is the image of \(X\) under the \(d\)-uple embedding of \(\PP^r_A\).
Solution
Put \(R=S^{(d)}\) and \(Y=\Proj R\). For a positive-degree homogeneous element \(f\) of a graded ring \(B\), write \(B_{(f)}=(B_f)_0\). Use the associated-sheaf convention \(\OO_{\Proj B}(e)=\widetilde{B(e)}\), with \(B(e)_n=B_{n+e}\).
1. The opens \(D_X(s)\) for \(s\in S_1\) cover \(X\), and the opens \(D_Y(s^d)\) cover \(Y\).Here \(D_X\) and \(D_Y\) denote the homogeneous distinguished opens in the respective Proj schemes.
Since \(S\) is generated by \(S_1\), a homogeneous prime containing every element of \(S_1\) contains \(S_+\) and does not define a point of \(X\). This proves the first covering assertion.
Every monomial of degree \(nd\) in elements of \(S_1\) is a product of \(n\) monomials of degree \(d\). Thus \(R\) is generated as an \(A\)-algebra by \(R_1=S_d\). If a homogeneous prime of \(R\) contains \(s^d\) for every \(s\in S_1\), it contains every product \(s_1\cdots s_d\in S_d\): the \(d\)th power of that product is \(s_1^d\cdots s_d^d\). These products span \(S_d\) over \(A\), so the prime contains \(R_1\), hence \(R_+\). It does not define a point of \(Y\), proving the second covering assertion.
2. There is a canonical isomorphism \(\phi:X\xrightarrow{\cong}Y\).
For \(s\in S_1\), the inclusion \(R\subseteq S\) induces \[ R_{(s^d)}\longrightarrow S_{(s)},\qquad a/(s^d)^n\longmapsto a/s^{dn},\quad a\in S_{dn}. \] It is injective: if a numerator maps to zero, some power of \(s\) annihilates it in \(S\). Increasing the exponent to a multiple of \(d\) gives the same annihilating equality in the subring \(R\). It is surjective because, for \(a\in S_e\), a fraction \(a/s^e\) can be written \[ \frac{a}{s^e}=\frac{a s^{dn-e}}{(s^d)^n} \] for any \(n\) with \(dn\ge e\). The numerator on the right has degree \(dn\) in \(S\) and degree \(n\) in \(R\).
On the intersection with the chart for \(t\in S_1\), the function \(t^d/s^d\) corresponds to \((t/s)^d\). Inverting it is equivalent to inverting \(t/s\). Thus the affine isomorphisms identify the chart intersections and their restriction maps, all of which send a fraction to the same fraction after localization. They glue along the covers of step 1, as do their inverses, giving \(\phi\) [@Har10a, Chapter II, §2]. The map is canonical because every chart map is induced by \(R\subseteq S\).
3. The twisting sheaves correspond by\[ \boxed{\phi^*\OO_Y(1)\cong\OO_X(d)}. \]
On \(D_Y(s^d)\), the unit \(s^d\) in the localization has degree one for the \(R\)-grading and gives a frame of \(\OO_Y(1)\). On \(D_X(s)\), the section \(s^d\) gives a frame of \(\OO_X(d)\), since \(s\) has degree one in \(S\) [@Har10a, Proposition II.5.12]. Send the first frame to the second. On an overlap, the transition function for the first is \(t^d/s^d\), and for the second it is \((t/s)^d\). Step 2 identifies these functions, so the local module isomorphisms glue to the asserted isomorphism.
4. With finitely many degree-one generators \(x_0,\ldots,x_r\), the resulting projective embedding is the restriction of the \(d\)-uple embedding.
For a multiindex \(\alpha=(\alpha_0,\ldots,\alpha_r)\in\NN^{r+1}\), put \(|\alpha|=\sum_i\alpha_i\) and \(x^\alpha=x_0^{\alpha_0}\cdots x_r^{\alpha_r}\). Let \(e_i\) be the multiindex whose \(i\)th entry is one and whose other entries are zero. The monomials \(x^\alpha\) with \(|\alpha|=d\) span \(S_d\) over \(A\). By step 1, they generate \(R\) as an \(A\)-algebra. Hence, with one degree-one variable \(z_\alpha\) for each such monomial, there is a graded surjection \[ A[z_\alpha:|\alpha|=d]\longrightarrow R,\qquad z_\alpha\longmapsto x^\alpha. \] It induces a closed immersion \(Y\hookrightarrow\PP_A^N\), where \(N=\binom{r+d}{d}-1\) [@Har10a, Exercise II.3.12]. Under step 3 its coordinate sections pull back to the monomials \(x^\alpha\) in \(\Gamma(X,\OO_X(d))\).
Applying steps <1>1–<1>3 first to the polynomial ring \(P=A[X_0,\ldots,X_r]\) gives \(\Proj P^{(d)}\cong\PP_A^r\). The surjection \(A[z_\alpha]\to P^{(d)}\), \(z_\alpha\mapsto X^\alpha\), therefore gives the closed immersion defined by all degree-\(d\) monomials, namely the \(d\)-uple embedding. The graded quotient \(P\to S\) induces \(P^{(d)}\to R\), so the same construction restricted to \(X\) is the closed immersion just obtained. On the chart where \(x_i\ne0\), its coordinate map is explicitly \[ z_\alpha/z_{d e_i}\longmapsto x^\alpha/x_i^d. \] Thus the morphisms agree as scheme morphisms, including their maps on structure sheaves.
5. Q.E.D.
Step 2 gives the Proj isomorphism, step 3 identifies the twisting sheaf, and step 4 identifies the projective embedding.