## 27.8 Proj of a graded ring

In this section we construct Proj of a graded ring following [II, Section 2, EGA].

Let $S$ be a graded ring. Consider the topological space $\text{Proj}(S)$ associated to $S$, see Algebra, Section 10.57. We will endow this space with a sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ such that the resulting pair $(\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ will be a scheme.

Recall that $\text{Proj}(S)$ has a basis of open sets $D_{+}(f)$, $f \in S_ d$, $d \geq 1$ which we call *standard opens*, see Algebra, Section 10.57. This terminology will always imply that $f$ is homogeneous of positive degree even if we forget to mention it. In addition, the intersection of two standard opens is another: $D_{+}(f) \cap D_{+}(g) = D_{+}(fg)$, for $f, g \in S$ homogeneous of positive degree.

Lemma 27.8.1. Let $S$ be a graded ring. Let $f \in S$ homogeneous of positive degree.

If $g\in S$ homogeneous of positive degree and $D_{+}(g) \subset D_{+}(f)$, then

$f$ is invertible in $S_ g$, and $f^{\deg (g)}/g^{\deg (f)}$ is invertible in $S_{(g)}$,

$g^ e = af$ for some $e \geq 1$ and $a \in S$ homogeneous,

there is a canonical $S$-algebra map $S_ f \to S_ g$,

there is a canonical $S_0$-algebra map $S_{(f)} \to S_{(g)}$ compatible with the map $S_ f \to S_ g$,

the map $S_{(f)} \to S_{(g)}$ induces an isomorphism

\[ (S_{(f)})_{g^{\deg (f)}/f^{\deg (g)}} \cong S_{(g)}, \]

these maps induce a commutative diagram of topological spaces

\[ \xymatrix{ D_{+}(g) \ar[d] & \{ \mathbf{Z}\text{-graded primes of }S_ g\} \ar[l] \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(S_{(g)}) \ar[d] \\ D_{+}(f) & \{ \mathbf{Z}\text{-graded primes of }S_ f\} \ar[l] \ar[r] & \mathop{\mathrm{Spec}}(S_{(f)}) } \]

where the horizontal maps are homeomorphisms and the vertical maps are open immersions,

there are compatible canonical $S_ f$ and $S_{(f)}$-module maps $M_ f \to M_ g$ and $M_{(f)} \to M_{(g)}$ for any graded $S$-module $M$, and

the map $M_{(f)} \to M_{(g)}$ induces an isomorphism

\[ (M_{(f)})_{g^{\deg (f)}/f^{\deg (g)}} \cong M_{(g)}. \]

Any open covering of $D_{+}(f)$ can be refined to a finite open covering of the form $D_{+}(f) = \bigcup _{i = 1}^ n D_{+}(g_ i)$.

Let $g_1, \ldots , g_ n \in S$ be homogeneous of positive degree. Then $D_{+}(f) \subset \bigcup D_{+}(g_ i)$ if and only if $g_1^{\deg (f)}/f^{\deg (g_1)}, \ldots , g_ n^{\deg (f)}/f^{\deg (g_ n)}$ generate the unit ideal in $S_{(f)}$.

**Proof.**
Recall that $D_{+}(g) = \mathop{\mathrm{Spec}}(S_{(g)})$ with identification given by the ring maps $S \to S_ g \leftarrow S_{(g)}$, see Algebra, Lemma 10.57.3. Thus $f^{\deg (g)}/g^{\deg (f)}$ is an element of $S_{(g)}$ which is not contained in any prime ideal, and hence invertible, see Algebra, Lemma 10.17.2. We conclude that (a) holds. Write the inverse of $f$ in $S_ g$ as $a/g^ d$. We may replace $a$ by its homogeneous part of degree $d\deg (g) - \deg (f)$. This means $g^ d - af$ is annihilated by a power of $g$, whence $g^ e = af$ for some $a \in S$ homogeneous of degree $e\deg (g) - \deg (f)$. This proves (b). For (c), the map $S_ f \to S_ g$ exists by (a) from the universal property of localization, or we can define it by mapping $b/f^ n$ to $a^ nb/g^{ne}$. This clearly induces a map of the subrings $S_{(f)} \to S_{(g)}$ of degree zero elements as well. We can similarly define $M_ f \to M_ g$ and $M_{(f)} \to M_{(g)}$ by mapping $x/f^ n$ to $a^ nx/g^{ne}$. The statements writing $S_{(g)}$ resp. $M_{(g)}$ as principal localizations of $S_{(f)}$ resp. $M_{(f)}$ are clear from the formulas above. The maps in the commutative diagram of topological spaces correspond to the ring maps given above. The horizontal arrows are homeomorphisms by Algebra, Lemma 10.57.3. The vertical arrows are open immersions since the left one is the inclusion of an open subset.

The open $D_{+}(f)$ is quasi-compact because it is homeomorphic to $\mathop{\mathrm{Spec}}(S_{(f)})$, see Algebra, Lemma 10.17.10. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology.

The third statement follows directly from Algebra, Lemma 10.17.2.
$\square$

In Sheaves, Section 6.30 we defined the notion of a sheaf on a basis, and we showed that it is essentially equivalent to the notion of a sheaf on the space, see Sheaves, Lemmas 6.30.6 and 6.30.9. Moreover, we showed in Sheaves, Lemma 6.30.4 that it is sufficient to check the sheaf condition on a cofinal system of open coverings for each standard open. By the lemma above it suffices to check on the finite coverings by standard opens.

Definition 27.8.2. Let $S$ be a graded ring. Suppose that $D_{+}(f) \subset \text{Proj}(S)$ is a standard open. A *standard open covering* of $D_{+}(f)$ is a covering $D_{+}(f) = \bigcup _{i = 1}^ n D_{+}(g_ i)$, where $g_1, \ldots , g_ n \in S$ are homogeneous of positive degree.

Let $S$ be a graded ring. Let $M$ be a graded $S$-module. We will define a presheaf $\widetilde M$ on the basis of standard opens. Suppose that $U \subset \text{Proj}(S)$ is a standard open. If $f, g \in S$ are homogeneous of positive degree such that $D_{+}(f) = D_{+}(g)$, then by Lemma 27.8.1 above there are canonical maps $M_{(f)} \to M_{(g)}$ and $M_{(g)} \to M_{(f)}$ which are mutually inverse. Hence we may choose any $f$ such that $U = D_{+}(f)$ and define

\[ \widetilde M(U) = M_{(f)}. \]

Note that if $D_{+}(g) \subset D_{+}(f)$, then by Lemma 27.8.1 above we have a canonical map

\[ \widetilde M(D_{+}(f)) = M_{(f)} \longrightarrow M_{(g)} = \widetilde M(D_{+}(g)). \]

Clearly, this defines a presheaf of abelian groups on the basis of standard opens. If $M = S$, then $\widetilde S$ is a presheaf of rings on the basis of standard opens. And for general $M$ we see that $\widetilde M$ is a presheaf of $\widetilde S$-modules on the basis of standard opens.

Let us compute the stalk of $\widetilde M$ at a point $x \in \text{Proj}(S)$. Suppose that $x$ corresponds to the homogeneous prime ideal $\mathfrak p \subset S$. By definition of the stalk we see that

\[ \widetilde M_ x = \mathop{\mathrm{colim}}\nolimits _{f\in S_ d, d > 0, f\not\in \mathfrak p} M_{(f)} \]

Here the set $\{ f \in S_ d, d > 0, f \not\in \mathfrak p\} $ is preordered by the rule $f \geq f' \Leftrightarrow D_{+}(f) \subset D_{+}(f')$. If $f_1, f_2 \in S \setminus \mathfrak p$ are homogeneous of positive degree, then we have $f_1f_2 \geq f_1$ in this ordering. In Algebra, Section 10.57 we defined $M_{(\mathfrak p)}$ as the module whose elements are fractions $x/f$ with $x, f$ homogeneous, $\deg (x) = \deg (f)$, $f \not\in \mathfrak p$. Since $\mathfrak p \in \text{Proj}(S)$ there exists at least one $f_0 \in S$ homogeneous of positive degree with $f_0 \not\in \mathfrak p$. Hence $x/f = f_0x/ff_0$ and we see that we may always assume the denominator of an element in $M_{(\mathfrak p)}$ has positive degree. From these remarks it follows easily that

\[ \widetilde M_ x = M_{(\mathfrak p)}. \]

Next, we check the sheaf condition for the standard open coverings. If $D_{+}(f) = \bigcup _{i = 1}^ n D_{+}(g_ i)$, then the sheaf condition for this covering is equivalent with the exactness of the sequence

\[ 0 \to M_{(f)} \to \bigoplus M_{(g_ i)} \to \bigoplus M_{(g_ ig_ j)}. \]

Note that $D_{+}(g_ i) = D_{+}(fg_ i)$, and hence we can rewrite this sequence as the sequence

\[ 0 \to M_{(f)} \to \bigoplus M_{(fg_ i)} \to \bigoplus M_{(fg_ ig_ j)}. \]

By Lemma 27.8.1 we see that $g_1^{\deg (f)}/f^{\deg (g_1)}, \ldots , g_ n^{\deg (f)}/f^{\deg (g_ n)}$ generate the unit ideal in $S_{(f)}$, and that the modules $M_{(fg_ i)}$, $M_{(fg_ ig_ j)}$ are the principal localizations of the $S_{(f)}$-module $M_{(f)}$ at these elements and their products. Thus we may apply Algebra, Lemma 10.24.1 to the module $M_{(f)}$ over $S_{(f)}$ and the elements $g_1^{\deg (f)}/f^{\deg (g_1)}, \ldots , g_ n^{\deg (f)}/f^{\deg (g_ n)}$. We conclude that the sequence is exact. By the remarks made above, we see that $\widetilde M$ is a sheaf on the basis of standard opens.

Thus we conclude from the material in Sheaves, Section 6.30 that there exists a unique sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ which agrees with $\widetilde S$ on the standard opens. Note that by our computation of stalks above and Algebra, Lemma 10.57.5 the stalks of this sheaf of rings are all local rings.

Similarly, for any graded $S$-module $M$ there exists a unique sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules $\mathcal{F}$ which agrees with $\widetilde M$ on the standard opens, see Sheaves, Lemma 6.30.12.

Definition 27.8.3. Let $S$ be a graded ring.

The *structure sheaf $\mathcal{O}_{\text{Proj}(S)}$ of the homogeneous spectrum of $S$* is the unique sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ which agrees with $\widetilde S$ on the basis of standard opens.

The locally ringed space $(\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ is called the *homogeneous spectrum* of $S$ and denoted $\text{Proj}(S)$.

The sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules extending $\widetilde M$ to all opens of $\text{Proj}(S)$ is called the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$. This sheaf is denoted $\widetilde M$ as well.

We summarize the results obtained so far.

Lemma 27.8.4. Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$.

For every $f \in S$ homogeneous of positive degree we have

\[ \Gamma (D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}. \]

For every $f\in S$ homogeneous of positive degree we have $\Gamma (D_{+}(f), \widetilde M) = M_{(f)}$ as an $S_{(f)}$-module.

Whenever $D_{+}(g) \subset D_{+}(f)$ the restriction mappings on $\mathcal{O}_{\text{Proj}(S)}$ and $\widetilde M$ are the maps $S_{(f)} \to S_{(g)}$ and $M_{(f)} \to M_{(g)}$ from Lemma 27.8.1.

Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}$.

Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{F}_ x = M_{(\mathfrak p)}$ as an $S_{(\mathfrak p)}$-module.

There is a canonical ring map $ S_0 \longrightarrow \Gamma (\text{Proj}(S), \widetilde S) $ and a canonical $S_0$-module map $ M_0 \longrightarrow \Gamma (\text{Proj}(S), \widetilde M) $ compatible with the descriptions of sections over standard opens and stalks above.

Moreover, all these identifications are functorial in the graded $S$-module $M$. In particular, the functor $M \mapsto \widetilde M$ is an exact functor from the category of graded $S$-modules to the category of $\mathcal{O}_{\text{Proj}(S)}$-modules.

**Proof.**
Assertions (1) - (5) are clear from the discussion above. We see (6) since there are canonical maps $M_0 \to M_{(f)}$, $x \mapsto x/1$ compatible with the restriction maps described in (3). The exactness of the functor $M \mapsto \widetilde M$ follows from the fact that the functor $M \mapsto M_{(\mathfrak p)}$ is exact (see Algebra, Lemma 10.57.5) and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 17.3.1.
$\square$

Lemma 27.8.6. Let $S$ be a graded ring. Let $f \in S$ be homogeneous of positive degree. Suppose that $D(g) \subset \mathop{\mathrm{Spec}}(S_{(f)})$ is a standard open. Then there exists an $h \in S$ homogeneous of positive degree such that $D(g)$ corresponds to $D_{+}(h) \subset D_{+}(f)$ via the homeomorphism of Algebra, Lemma 10.57.3. In fact we can take $h$ such that $g = h/f^ n$ for some $n$.

**Proof.**
Write $g = h/f^ n$ for some $h$ homogeneous of positive degree and some $n \geq 1$. If $D_{+}(h)$ is not contained in $D_{+}(f)$ then we replace $h$ by $hf$ and $n$ by $n + 1$. Then $h$ has the required shape and $D_{+}(h) \subset D_{+}(f)$ corresponds to $D(g) \subset \mathop{\mathrm{Spec}}(S_{(f)})$.
$\square$

Lemma 27.8.7. Let $S$ be a graded ring. The locally ringed space $\text{Proj}(S)$ is a scheme. The standard opens $D_{+}(f)$ are affine opens. For any graded $S$-module $M$ the sheaf $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules.

**Proof.**
Consider a standard open $D_{+}(f) \subset \text{Proj}(S)$. By Lemmas 27.8.1 and 27.8.4 we have $\Gamma (D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}$, and we have a homeomorphism $\varphi : D_{+}(f) \to \mathop{\mathrm{Spec}}(S_{(f)})$. For any standard open $D(g) \subset \mathop{\mathrm{Spec}}(S_{(f)})$ we may pick an $h \in S_{+}$ as in Lemma 27.8.6. Then $\varphi ^{-1}(D(g)) = D_{+}(h)$, and by Lemmas 27.8.4 and 27.8.1 we see

\[ \Gamma (D_{+}(h), \mathcal{O}_{\text{Proj}(S)}) = S_{(h)} = (S_{(f)})_{h^{\deg (f)}/f^{\deg (h)}} = (S_{(f)})_ g = \Gamma (D(g), \mathcal{O}_{\mathop{\mathrm{Spec}}(S_{(f)})}). \]

Thus the restriction of $\mathcal{O}_{\text{Proj}(S)}$ to $D_{+}(f)$ corresponds via the homeomorphism $\varphi $ exactly to the sheaf $\mathcal{O}_{\mathop{\mathrm{Spec}}(S_{(f)})}$ as defined in Schemes, Section 26.5. We conclude that $D_{+}(f)$ is an affine scheme isomorphic to $\mathop{\mathrm{Spec}}(S_{(f)})$ via $\varphi $ and hence that $\text{Proj}(S)$ is a scheme.

In exactly the same way we show that $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules. Namely, the argument above will show that

\[ \widetilde M|_{D_{+}(f)} \cong \varphi ^*\left(\widetilde{M_{(f)}}\right) \]

which shows that $\widetilde M$ is quasi-coherent.
$\square$

Lemma 27.8.8. Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is separated.

**Proof.**
We have to show that the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated. We will use Schemes, Lemma 26.21.7. Thus it suffices to show given any pair of standard opens $D_{+}(f)$ and $D_{+}(g)$ that $D_{+}(f) \cap D_{+}(g) = D_{+}(fg)$ is affine (clear) and that the ring map

\[ S_{(f)} \otimes _{\mathbf{Z}} S_{(g)} \longrightarrow S_{(fg)} \]

is surjective. Any element $s$ in $S_{(fg)}$ is of the form $s = h/(f^ ng^ m)$ with $h \in S$ homogeneous of degree $n\deg (f) + m\deg (g)$. We may multiply $h$ by a suitable monomial $f^ ig^ j$ and assume that $n = n' \deg (g)$, and $m = m' \deg (f)$. Then we can rewrite $s$ as $s = h/f^{(n' + m')\deg (g)} \cdot f^{m'\deg (g)}/g^{m'\deg (f)}$. So $s$ is indeed in the image of the displayed arrow.
$\square$

Lemma 27.8.9. Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is quasi-compact if and only if there exist finitely many homogeneous elements $f_1, \ldots , f_ n \in S_{+}$ such that $S_{+} \subset \sqrt{(f_1, \ldots , f_ n)}$. In this case $\text{Proj}(S) = D_+(f_1) \cup \ldots \cup D_+(f_ n)$.

**Proof.**
Given such a collection of elements the standard affine opens $D_{+}(f_ i)$ cover $\text{Proj}(S)$ by Algebra, Lemma 10.57.3. Conversely, if $\text{Proj}(S)$ is quasi-compact, then we may cover it by finitely many standard opens $D_{+}(f_ i)$, $i = 1, \ldots , n$ and we see that $S_{+} \subset \sqrt{(f_1, \ldots , f_ n)}$ by the lemma referenced above.
$\square$

Lemma 27.8.10. Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ has a canonical morphism towards the affine scheme $\mathop{\mathrm{Spec}}(S_0)$, agreeing with the map on topological spaces coming from Algebra, Definition 10.57.1.

**Proof.**
We saw above that our construction of $\widetilde S$, resp. $\widetilde M$ gives a sheaf of $S_0$-algebras, resp. $S_0$-modules. Hence we get a morphism by Schemes, Lemma 26.6.4. This morphism, when restricted to $D_{+}(f)$ comes from the canonical ring map $S_0 \to S_{(f)}$. The maps $S \to S_ f$, $S_{(f)} \to S_ f$ are $S_0$-algebra maps, see Lemma 27.8.1. Hence if the homogeneous prime $\mathfrak p \subset S$ corresponds to the $\mathbf{Z}$-graded prime $\mathfrak p' \subset S_ f$ and the (usual) prime $\mathfrak p'' \subset S_{(f)}$, then each of these has the same inverse image in $S_0$.
$\square$

Lemma 27.8.11. Let $S$ be a graded ring. If $S$ is finitely generated as an algebra over $S_0$, then the morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(S_0)$ satisfies the existence and uniqueness parts of the valuative criterion, see Schemes, Definition 26.20.3.

**Proof.**
The uniqueness part follows from the fact that $\text{Proj}(S)$ is separated (Lemma 27.8.8 and Schemes, Lemma 26.22.1). Choose $x_ i \in S_{+}$ homogeneous, $i = 1, \ldots , n$ which generate $S$ over $S_0$. Let $d_ i = \deg (x_ i)$ and set $d = \text{lcm}\{ d_ i\} $. Suppose we are given a diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \text{Proj}(S) \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & \mathop{\mathrm{Spec}}(S_0) } \]

as in Schemes, Definition 26.20.3. Denote $v : K^* \to \Gamma $ the valuation of $A$, see Algebra, Definition 10.50.13. We may choose an $f \in S_{+}$ homogeneous such that $\mathop{\mathrm{Spec}}(K)$ maps into $D_{+}(f)$. Then we get a commutative diagram of ring maps

\[ \xymatrix{ K & S_{(f)} \ar[l]^{\varphi } \\ A \ar[u] & S_0 \ar[l] \ar[u] } \]

After renumbering we may assume that $\varphi (x_ i^{\deg (f)}/f^{d_ i})$ is nonzero for $i = 1, \ldots , r$ and zero for $i = r + 1, \ldots , n$. Since the open sets $D_{+}(x_ i)$ cover $\text{Proj}(S)$ we see that $r \geq 1$. Let $i_0 \in \{ 1, \ldots , r\} $ be an index minimizing $\gamma _ i = (d/d_ i)v(\varphi (x_ i^{\deg (f)}/f^{d_ i}))$ in $\Gamma $. For convenience set $x_0 = x_{i_0}$ and $d_0 = d_{i_0}$. The ring map $\varphi $ factors though a map $\varphi ' : S_{(fx_0)} \to K$ which gives a ring map $S_{(x_0)} \to S_{(fx_0)} \to K$. The algebra $S_{(x_0)}$ is generated over $S_0$ by the elements $x_1^{e_1} \ldots x_ n^{e_ n}/x_0^{e_0}$, where $\sum e_ i d_ i = e_0 d_0$. If $e_ i > 0$ for some $i > r$, then $\varphi '(x_1^{e_1} \ldots x_ n^{e_ n}/x_0^{e_0}) = 0$. If $e_ i = 0$ for $i > r$, then we have

\begin{align*} \deg (f) v(\varphi '(x_1^{e_1} \ldots x_ r^{e_ r}/x_0^{e_0})) & = v(\varphi '(x_1^{e_1 \deg (f)} \ldots x_ r^{e_ r \deg (f)}/x_0^{e_0 \deg (f)})) \\ & = \sum e_ i v(\varphi '(x_ i^{\deg (f)}/f^{d_ i})) - e_0 v(\varphi '(x_0^{\deg (f)}/f^{d_0})) \\ & = \sum e_ i d_ i \gamma _ i - e_0 d_0 \gamma _0 \\ & \geq \sum e_ i d_ i \gamma _0 - e_0 d_0 \gamma _0 = 0 \end{align*}

because $\gamma _0$ is minimal among the $\gamma _ i$. This implies that $S_{(x_0)}$ maps into $A$ via $\varphi '$. The corresponding morphism of schemes $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(S_{(x_0)}) = D_{+}(x_0) \subset \text{Proj}(S)$ provides the morphism fitting into the first commutative diagram of this proof.
$\square$

We saw in the proof of Lemma 27.8.11 that, under the hypotheses of that lemma, the morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(S_0)$ is quasi-compact as well. Hence (by Schemes, Proposition 26.20.6) we see that $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(S_0)$ is universally closed in the situation of the lemma. We give several examples showing these results do not hold without some assumption on the graded ring $S$.

Example 27.8.12. Let $\mathbf{C}[X_1, X_2, X_3, \ldots ]$ be the graded $\mathbf{C}$-algebra with each $X_ i$ in degree $1$. Consider the ring map

\[ \mathbf{C}[X_1, X_2, X_3, \ldots ] \longrightarrow \mathbf{C}[t^\alpha ; \alpha \in \mathbf{Q}_{\geq 0}] \]

which maps $X_ i$ to $t^{1/i}$. The right hand side becomes a valuation ring $A$ upon localization at the ideal $\mathfrak m = (t^\alpha ; \alpha > 0)$. Let $K$ be the fraction field of $A$. The above gives a morphism $\mathop{\mathrm{Spec}}(K) \to \text{Proj}(\mathbf{C}[X_1, X_2, X_3, \ldots ])$ which does not extend to a morphism defined on all of $\mathop{\mathrm{Spec}}(A)$. The reason is that the image of $\mathop{\mathrm{Spec}}(A)$ would be contained in one of the $D_{+}(X_ i)$ but then $X_{i + 1}/X_ i$ would map to an element of $A$ which it doesn't since it maps to $t^{1/(i + 1) - 1/i}$.

Example 27.8.13. Let $R = \mathbf{C}[t]$ and

\[ S = R[X_1, X_2, X_3, \ldots ]/(X_ i^2 - tX_{i + 1}). \]

The grading is such that $R = S_0$ and $\deg (X_ i) = 2^{i - 1}$. Note that if $\mathfrak p \in \text{Proj}(S)$ then $t \not\in \mathfrak p$ (otherwise $\mathfrak p$ has to contain all of the $X_ i$ which is not allowed for an element of the homogeneous spectrum). Thus we see that $D_{+}(X_ i) = D_{+}(X_{i + 1})$ for all $i$. Hence $\text{Proj}(S)$ is quasi-compact; in fact it is affine since it is equal to $D_{+}(X_1)$. It is easy to see that the image of $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(R)$ is $D(t)$. Hence the morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(R)$ is not closed. Thus the valuative criterion cannot apply because it would imply that the morphism is closed (see Schemes, Proposition 26.20.6 ).

Example 27.8.14. Let $A$ be a ring. Let $S = A[T]$ as a graded $A$ algebra with $T$ in degree $1$. Then the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(A)$ (see Lemma 27.8.10) is an isomorphism.

Example 27.8.15. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme, and let $U \subset X$ be an open subscheme. Grade $A[T]$ by setting $\deg T = 1$. Define $S$ to be the subring of $A[T]$ generated by $A$ and all $fT^ i$, where $i \ge 0$ and where $f \in A$ is such that $D(f) \subset U$. We claim that $S$ is a graded ring with $S_0 = A$ such that $\text{Proj}(S) \cong U$, and this isomorphism identifies the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(A)$ of Lemma 27.8.10 with the inclusion $U \subset X$.

Suppose $\mathfrak p \in \text{Proj}(S)$ is such that every $fT \in S_1$ is in $\mathfrak p$. Then every generator $fT^ i$ with $i \ge 1$ is in $\mathfrak p$ because $(fT^ i)^2 = (fT)(fT^{2i-1}) \in \mathfrak p$ and $\mathfrak p$ is radical. But then $\mathfrak p \supset S_+$, which is impossible. Consequently $\text{Proj}(S)$ is covered by the standard open affine subsets $\{ D_+(fT)\} _{fT \in S_1}$.

Observe that, if $fT \in S_1$, then the inclusion $S \subset A[T]$ induces a graded isomorphism of $S[(fT)^{-1}]$ with $A[T, T^{-1}, f^{-1}]$. Hence the standard open subset $D_+(fT) \cong \mathop{\mathrm{Spec}}(S_{(fT)})$ is isomorphic to $\mathop{\mathrm{Spec}}(A[T, T^{-1}, f^{-1}]_0) = \mathop{\mathrm{Spec}}(A[f^{-1}])$. It is clear that this isomorphism is a restriction of the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(A)$. If in addition $gT \in S_1$, then $S[(fT)^{-1}, (gT)^{-1}] \cong A[T, T^{-1}, f^{-1}, g^{-1}]$ as graded rings, so $D_+(fT) \cap D_+(gT) \cong \mathop{\mathrm{Spec}}(A[f^{-1}, g^{-1}])$. Therefore $\text{Proj}(S)$ is the union of open subschemes $D_+(fT)$ which are isomorphic to the open subschemes $D(f) \subset X$ under the canonical morphism, and these open subschemes intersect in $\text{Proj}(S)$ in the same way they do in $X$. We conclude that the canonical morphism is an isomorphism of $\text{Proj}(S)$ with the union of all $D(f) \subset U$, which is $U$.

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