The Stacks project

37.51 Proj and Spec

In this section we clarify the relationship between the Proj and the spectrum of a graded ring.

Let $R$ be a ring. Let $A$ be a graded $R$-algebra, see Algebra, Section 10.56. For $m \geq 0$ we denote $A_{\geq m} = \bigoplus _{d \geq m} A_ d$. Consider the graded ring

\[ B = \bigoplus \nolimits _{d \geq 0} A_{\geq d} \]

For $d' \geq d$ and $a \in A_{d'}$ let us denote $a^{(d)} \in B$ the element in $B_ d$ corresponding to $a$. Let us denote $\sigma : A \to B$ and $\psi : A \to B$ the two obvious ring maps: if $a \in A_ d$, then $\sigma (a) = a^{(0)}$ and $\psi (a) = a^{(d)}$. Then $\psi $ is a graded ring map and $\sigma $ turns $B$ into a graded algebra over $A$. There is also a surjective graded ring map $\tau : B \to A$ which for $d' \geq d$ and $a \in A_{d'}$ sends $a^{(d)}$ to $0$ if $d' > d$ and to $a$ if $d' = d$.

Affine schemes and spectra. We set $X = \mathop{\mathrm{Spec}}(A)$. The irrelevant ideal $A_+$ cuts out a closed subscheme $Z = V(A_+) = \mathop{\mathrm{Spec}}(A/A_+) = \mathop{\mathrm{Spec}}(A_0)$. Set $U = X \setminus Z$.

\[ U \longrightarrow X \longrightarrow Z \]

Projective schemes and Proj. Set $P = \text{Proj}(A)$. We may and do view $P$ as a scheme over $\mathop{\mathrm{Spec}}(A_0) = Z$. Set $L = \text{Proj}(B)$. We may and do view $L$ as a scheme over $\mathop{\mathrm{Spec}}(B_0) = \mathop{\mathrm{Spec}}(A) = X$; observe that the identification of $B_0$ with $A$ is given by $\sigma $. The surjection $\tau $ defines a closed immersion $0 : P \to L$. Since $A \xrightarrow {\sigma } B \to A$ is equal to the map $A \to A_0 \to A$ we conclude that

\[ \xymatrix{ P \ar[d] \ar[r]_0 & L \ar[d] \\ Z \ar[r] & X } \]

is commutative.

We claim that $\psi $ defines a morphism $L \to P$. To see this, by Constructions, Lemma 27.11.1, it suffices to check $\psi (A_+) \not\subset \mathfrak p$ for every homogeneous prime ideal $\mathfrak p \subset B$ with $B_+ \not\subset \mathfrak p$. First, pick $g \in B_+$ homogeneous $g \not\in \mathfrak p$. Then we can write $g$ as a finite sum $g = \sum a_ i^{(d)}$ with $a_ i \in A_{d_ i}$ for some $d_ i \geq d$. We conclude that there exist $d' \geq d$ and $a \in A_{d'}$ such that $a^{(d)} \not\in \mathfrak p$. Then

\[ (a^{(d)})^{d'} = (a^{d'})^{(d'd)} = a^{(d)} (a^{d' - 1})^{(d(d' - 1))} = \psi (a) (a^{d' - 1})^{(d(d' - 1))} \]

(the notation leaves something to be desired) is not in $\mathfrak p$. Hence $\psi (a) \not\in \mathfrak p$, proving the claim. Thus we can extend our diagram above to a commutative diagram

\[ \xymatrix{ P \ar[d] \ar[r]_0 & L \ar[d] \ar[r]_\pi & P \ar[d] \\ Z \ar[r] & X \ar[r] & Z } \]

where $X \to Z$ is given by $A_0 \to A$. Since $\tau \circ \psi = \text{id}_ A$ we see $\pi \circ 0 = \text{id}_ P$.

Observe that $\pi $ is an affine morphism. This is clear from the construction in Constructions, Lemma 27.11.1. In fact, if $f \in A_ d$ for some $d > 0$, then setting $g = \psi (f)$ we have $\pi ^{-1}(D_+(f)) = D_+(g)$. In this case we have the following equality of homogeneous parts

\[ (B[1/g])_{m'} = \bigoplus \nolimits _{m \geq m'} (A[1/f])_ m \]

This isomorphism is compatible with further localization. Taking $m' = 0$ we see that $\pi _*\mathcal{O}_ L$ is the direct sum of $\mathcal{O}_ P(m)$ for $m \geq 0$1. We conclude $L$ is identified with the relative spectrum:

\[ L = \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \geq 0} \mathcal{O}_ P(m) \right) \]

In particular $L \to P$ is a cone2, see Constructions, Section 27.7. Moreover, it is clear that $0 : P \to L$ is the vertex of the cone.

Let $f \in A_ d$ for some $d > 0$ and $g = \psi (f) \in B_ d$ as in the previous paragraph. Looking at the structure of the ring maps

\[ \xymatrix{ A_0 \ar[r] \ar[d] & A \ar[d]^\sigma \ar[r] & A_0 \ar[d] \\ (A[1/f])_0 \ar[r]^-\psi & (B[1/g])_0 = \bigoplus \nolimits _{m \geq 0} (A[1/f])_ m \ar[r]^-\tau & (A[1/f])_0 } \]

some compuations3 in graded rings will show that

  1. $\sigma (A_+)(B[1/g])_0 \subset \mathop{\mathrm{Ker}}(\tau : (B[1/g])_0 \to (A[1/f])_0)$,

  2. $\sigma (f) \in (B[1/g])_0$ is a nonzerodivisor,

  3. $\sigma (f) (B[1/g])_0 = \sigma (A_ d) (B[1/g])_0$ as ideals,

  4. $\sigma (f) (B[1/g])_0$ and $\mathop{\mathrm{Ker}}(\tau : (B[1/g])_0 \to (A[1/f])_0)$ have the same radical,

  5. if $d = 1$, then $\sigma (f) (B[1/g])_0 = \mathop{\mathrm{Ker}}(\tau : (B[1/g])_0 \to (A[1/f])_0)$.

We see in particular that

\[ 0(D_+(f)) = V(\sigma (f)) \subset D_+(g) = \mathop{\mathrm{Spec}}((B[1/g])_0) \]

set theoretically. In other words, the ideal generated by $\sigma (A_ d)$ cuts out an effective Cartier divisor on $D_+(g)$ which is set theoretically equal to the image of the closed immersion $0 : P \to L$.

We claim that $L \to X$ is an isomorphism over $U$. Namely, if $f \in A_ d$ for some $d > 0$, then

\[ \mathop{\mathrm{Spec}}(A_ f) \times _ X L = \text{Proj}(A_ f \otimes _ A B) = \text{Proj}(B_{\sigma (f)}) \]

For each $e$ we have $(B_{\sigma (f)})_ e = A_ f \otimes _ B B_ e = A_ f \otimes _ A A_{\geq e} = A_ f$, the final equality induced by the injection $A_{\geq e} \subset A$. Hence $B_{\sigma (f)} \cong A_ f[T]$ with $T$ in degree $1$. This proves the claim as $\text{Proj}(A_ f[T]) \to \mathop{\mathrm{Spec}}(A_ f)$ is an isomorphism. From now on we identify $U$ with the corresponding open of $L$.

The identification made in the previous paragraph lets us consider the restriction $\pi |_ U : U \to P$. Pick $f \in A_ d$ for some $d > 0$ and $g = \psi (f) \in B_ d$ as we have done above several times. Then

\[ U \cap \pi ^{-1}(D_+(f)) = U \cap D_+(g) \]

is the complement of the zero locus of $\sigma (f) \in (B[1/g])_0$ via the identification of $D_+(g)$ with the spectrum of $(B[1/g])_0$. This is assertion (4) above. Therefore $U \cap D_+(g)$ is affine and

\[ \mathcal{O}_ L(U \cap D_+(g)) = (B[1/g])_0[1/\sigma (f)] = \bigoplus \nolimits _{m \in \mathbf{Z}} (A[1/f])_ m \]

where the last equal sign is the natural extension of the identification $(B[1/g])_0 = \bigoplus _{m \geq 0} (A[1/f])_ m$ made above. Exactly as we did before with $\pi : L \to P$ we conclude that $\pi |_ U : U \to P$ is affine and

\[ U = \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \in \mathbf{Z}} \mathcal{O}_ P(m) \right) \]

as schemes over $P$.

Summarising the above, our constructions produce a commutative diagram
\begin{equation} \label{more-morphisms-equation-proj-and-spec} \vcenter { \xymatrix{ \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \in \mathbf{Z}} \mathcal{O}_ P(m) \right) \ar[r] \ar@{=}[d] & L = \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \geq 0} \mathcal{O}_ P(m) \right) \ar[d]^\sigma \ar[r]_-\pi & P \ar[d] \\ U \ar[r] & X \ar[r] & Z } } \end{equation}

of schemes where $\pi $ is a cone whose zero section $0 : P \to L$ maps set theoretically onto the inverse image of $Z$ in $L$.

Let $W \subset P$ be the largest open such that $\mathcal{O}_ P(1)|_ W$ is invertible and the natural maps induce isomorphisms $\mathcal{O}_ P(m)|_ W \cong \mathcal{O}_ P(1)^{\otimes m}|_ W$ for all $m \in \mathbf{Z}$, i.e., the open of Constructions, Lemma 27.10.4 for $d = 1$. Then we see that $L|_ W = \pi ^{-1}(W) \to W$ is a vector bundle (Constructions, Section 27.6) of rank $1$, namely,

\[ L|_ W = \mathbf{V}(\mathcal{O}_ P(1)|_ W) \]

in Grothendieckian notation. This is immediate from the above showing that $L|_ W$ is equal to the relative spectrum of the symmetric algebra over $\mathcal{O}_ W$ on $\mathcal{O}_ P(1)|_ W$. Then clearly the morphism $0|_ W : W \to L|_ W$ is the zero section of this vector bundle. In particular $0(W)$ is an effective Cartier divisor on $L|_ W$. Moreover, the open $U|_ W = (\pi |_ U)^{-1}(W)$ is the complement of the zero section.

If $A$ is generated by $f_1, \ldots , f_ r \in A_1$ over $A_0$, then $(f_1, \ldots , f_ r)^ m = A_{\geq m}$ for all $m \geq 0$ and hence our $B$ above is the Rees algebra for $A_+ = (f_1, \ldots , f_ r)$. Thus in this case $L \to X$ is the blowup of $Z$ and $W = P$ where $W$ is as in the preceding paragraph.

If $P$ is quasi-compact, then for $d$ sufficiently divisible, the closed subscheme $D \subset L$ cut out by $\sigma (A_ d)\mathcal{O}_ L$ is an effective Cartier divisor, $0 : P \to L$ factors through $D$, and $0(P) = D$ set theoretically. This follows from Constructions, Lemma 27.8.9 and (1), (2), (3), and (4) proved above. (Take any $d$ divisible by the lcm of the degrees of the elements found in the lemma.)

We continue to assume $P$ is quasi-compact. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ P$-module. Let us set $\mathcal{F}_ U = \pi ^*\mathcal{F}|_ U$. Then we have
\begin{equation} \label{more-morphisms-equation-cohomology-torsor} R\Gamma (U, \mathcal{F}_ U) = \bigoplus \nolimits _{m \in \mathbf{Z}} R\Gamma (P, \mathcal{F} \otimes _{\mathcal{O}_ P} \mathcal{O}_ P(m)) \end{equation}

Moreover, this direct sum decomposition is functorial in $\mathcal{F}$ and the induced $A$-module structure on the right is the same as the $A$-module structure on the left coming from $U \subset X$. To prove the formula, since $\pi |_ U$ is affine and $(\pi |_ U)_*\mathcal{O}_ U = \bigoplus _{m \in \mathbf{Z}} \mathcal{O}_ P(m)$ we get

\begin{align*} R(\pi |_ U)_*\mathcal{F}_ U & = (\pi |_ U)_*\mathcal{F}_ U \\ & = (\pi |_ U)_*(\pi |_ U)^*\mathcal{F} \\ & = \mathcal{F} \otimes _{\mathcal{O}_ P} \bigoplus \nolimits _{m \in \mathbf{Z}} \mathcal{O}_ P(m) \\ & = \bigoplus \nolimits _{m \in \mathbf{Z}} \mathcal{F} \otimes _{\mathcal{O}_ P} \mathcal{O}_ P(m) \end{align*}

By Leray we find that $R\Gamma (U, \mathcal{F}_ U) = R\Gamma (P, R(\pi |_ U)_*\mathcal{F}_ U)$, see Cohomology, Lemma 20.13.6. The proof is finished because taking cohomology commutes with direct sums in this case, see Derived Categories of Schemes, Lemma 36.4.5. This is where we use that $P$ is quasi-compact; $P$ is separated by Constructions, Lemma 27.8.8.

Lemma 37.51.1. Let $R$ be a ring. Let $P$ be a proper scheme over $R$ and let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ P$-module. Set $A = \bigoplus _{m \geq 0} \Gamma (P, \mathcal{L}^{\otimes m})$. Then $P = \text{Proj}(A)$ and diagram ( becomes the diagram

\[ \xymatrix{ \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \in \mathbf{Z}} \mathcal{L}^{\otimes m} \right) \ar[r] \ar@{=}[d] & L = \underline{\mathop{\mathrm{Spec}}}_ P \left( \bigoplus \nolimits _{m \geq 0} \mathcal{L}^{\otimes m} \right) \ar[d]^\sigma \ar[r]_-\pi & P \ar[d] \\ U \ar[r] & X \ar[r] & Z } \]

having the properties explained above.

Proof. We have $P = \text{Proj}(A)$ by Morphisms, Lemma 29.43.17. Moreover, by Properties, Lemma 28.28.2 via this identification we have $\mathcal{O}_ P(m) = \mathcal{L}^{\otimes m}$ for all $m \in \mathbf{Z}$. $\square$

[1] It similarly follows that $\pi _*\mathcal{O}_ L(i) = \bigoplus _{m \geq -i} \mathcal{O}_ P(m)$.
[2] Often $L$ is a line bundle over $P$, see below.
[3] Parts (1) and (2) are clear. To see (3), note that if $a \in A_ d$, then $\sigma (a) = \sigma (f) \psi (a/f)$. For (4) note that $b/g^ m$ is in the kernel of $\tau $ if and only if $b \in A_{\geq md}$ maps to zero in $A_{md}$. Thus it suffices to show if $m' > md$ and $a \in A_{m'}$, then some power of $a^{(md)}/g^ m$ is in the ideal generated by $\sigma (f)$. Take $e$ such that $em' - emd \geq d$. Then
\[ (a^{(md)}/g^ m)^ e = (a^ e)^{(emd)}/g^{em} = (fa^ e)^{(emd + d)}/g^{em + 1} = \sigma (f) \cdot (a^ e)^{(emd + d)}/g^{em + 1} \]
as desired (apologies for the terrible notation). To see (5) argue as before and note that $a^{(md)}/g^ m = \sigma (f) \cdot a^{(md + 1)}/g^{m + 1}$ if $d = 1$.

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