§3. Homogeneous spectrum of a sheaf of graded algebras

3.1. Homogeneous spectrum of a quasi-coherent graded ${\mathcal{O}}_Y$-algebra

(3.1.1)

Let $Y$ be a prescheme, $\mathcal{S}$ a graded ${\mathcal{O}}_Y$-algebra, and $\mathcal{M}$ a graded $\mathcal{S}$-module. If $\mathcal{S}$ is quasi-coherent, then each of its homogeneous components ${\mathcal{S}}_n$ is a quasi-coherent ${\mathcal{O}}_Y$-module, since they are the images of $\mathcal{S}$ under a homomorphism from $\mathcal{S}$ to itself (I, 1.3.8 and 1.3.9); likewise, if $\mathcal{M}$ is quasi-coherent as an ${\mathcal{O}}_Y$-module, then its homogeneous components ${\mathcal{M}}_n$ are also quasi-coherent, and conversely. For an integer $d>0$, we denote by ${\mathcal{S}}^{(d)}$ the direct sum of the ${\mathcal{O}}_Y$-modules ${\mathcal{S}}_{nd}$ (for $n\in \mathbb{Z}$), which is quasi-coherent if $\mathcal{S}$ is (I, 1.3.9); for every integer $k$ with $0\le k\le d-1$, we denote by ${\mathcal{M}}^{(d,k)}$ (or ${\mathcal{M}}^{(d)}$ for $k=0$) the direct sum of the ${\mathcal{M}}_{nd+k}$ (for $n\in \mathbb{Z}$), which is a graded ${\mathcal{S}}^{(d)}$-module, quasi-coherent if $\mathcal{S}$ and $\mathcal{M}$ are quasi-coherent (I, 9.6.1). We denote by $\mathcal{M}(n)$ the graded $\mathcal{S}$-module such that $(\mathcal{M}(n){)}_k={\mathcal{M}}_{n+k}$ for all $k\in \mathbb{Z}$; if $\mathcal{S}$ and $\mathcal{M}$ are quasi-coherent, then $\mathcal{M}(n)$ is a quasi-coherent graded $\mathcal{S}$-module (I, 9.6.1).

We say that $\mathcal{M}$ is a graded $\mathcal{S}$-module of finite type (resp. admits a finite presentation) if, for every $y\in Y$, there exists an open neighbourhood $U$ of $y$ and integers $n_i$ (resp. integers $m_i$ and $n_j$) such that there is a surjective degree-0 homomorphism

$${\oplus }_{i=1}^r(\mathcal{S}(n_i)|U)\to \mathcal{M}|U$$

(resp. such that $\mathcal{M}|U$ is isomorphic to the cokernel of a degree-0 homomorphism

  ⊕_{i=1}^r (𝒮(m_i)|U) → ⊕_{j=1}^s (𝒮(n_j)|U)).

Let $U$ be an affine open of $Y$, with ring $A=\Gamma (U,{\mathcal{O}}_Y)$; by hypothesis, the graded $({\mathcal{O}}_Y|U)$-algebra $\mathcal{S}|U$ is isomorphic to $\tilde{S}$, where $S=\Gamma (U,\mathcal{S})$ is a graded $A$-algebra (I, 1.4.3). Set $X_U=\mathrm{Proj}(\Gamma (U,\mathcal{S}))$. Let $U^{\prime }\subset U$ be a second affine open of $Y$, with ring $A^{\prime }$, and let $j:U^{\prime }\to U$ be the canonical injection, which corresponds to the restriction homomorphism $A\to A^{\prime }$; we have $\mathcal{S}|U^{\prime }=j\ast (\mathcal{S}|U)$, and so $S^{\prime }=\Gamma (U^{\prime },\mathcal{S})$ is canonically identified with $S{\otimes }_A{A}^{\prime }$ (I, 1.6.5). We

conclude (2.8.10) that $X_{U^{\prime }}$ is canonically identified with $X_U{\times }_U{U}^{\prime }$, and so also with $f_U^{-1}(U^{\prime })$, where $f_U$ denotes the structure morphism $X_U\to U$ (I, 4.4.1). We denote by ${\sigma }_{U^{\prime },U}$ the canonical isomorphism $f_U^{-1}(U^{\prime })\stackrel{\sim }{\to }{X}_{U^{\prime }}$ so defined, and by ${\rho }_{U^{\prime },U}$ the open immersion $X_{U^{\prime }}\to X_U$ obtained by composing ${\sigma }_{U^{\prime },U}^{-1}$ with the canonical injection $f_U^{-1}(U^{\prime })\to X_U$. It is immediate that, if $U^{\prime \prime }\subset U^{\prime }$ is a third affine open of $Y$, then ${\rho }_{U^{\prime \prime },U}={\rho }_{U^{\prime \prime },U^{\prime }}\circ {\rho }_{U^{\prime },U}$.

Proposition.

Let $Y$ be a prescheme. For every quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$, there exists a prescheme $X$ over $Y$, unique up to $Y$-isomorphism, with the following property: if $f:X\to Y$ is the structure morphism, then for every affine open $U$ of $Y$, there exists an isomorphism ${\eta }_U$ from the induced prescheme $f^{-1}(U)$ to $X_U=\mathrm{Proj}(\Gamma (U,\mathcal{S}))$ such that, if $V$ is a second affine open of $Y$ contained in $U$, then the diagram

              η_V
   f⁻¹(V) ─────────→ X_V                                                  (3.1.2.1)
     │                │
     ↓                ↓ ρ_{V, U}
   f⁻¹(U) ─────────→ X_U
              η_U

commutes.

Proof. For two affine opens $U$, $V$ of $Y$, let $X_{U,V}$ be the prescheme induced on $f_U^{-1}(U\cap V)$ by X_U; we will define a $Y$-isomorphism ${\theta }_{U,V}:X_{V,U}\stackrel{\sim }{\to }{X}_{U,V}$. For this, consider an affine open $W\subset U\cap V$: by composing the isomorphisms

                   σ_{W, U}            σ_{W, V}⁻¹
   f_U⁻¹(W) ─────────────→ X_W ─────────────→ f_V⁻¹(W),

we obtain an isomorphism ${\tau }_W$, and one checks at once that, if $W^{\prime }\subset W$ is an affine open, then ${\tau }_{W^{\prime }}$ is the restriction of ${\tau }_W$ to $f_U^{-1}(W^{\prime })$; the ${\tau }_W$ are therefore the restrictions of a $Y$-isomorphism ${\theta }_{V,U}$. Moreover, if $U$, $V$, $W$ are three affine opens of $Y$, and ${\theta }_{U,V}^{\prime }$, ${\theta }_{V,W}^{\prime }$, ${\theta }_{U,W}^{\prime }$ denote the restrictions of ${\theta }_{U,V}$, ${\theta }_{V,W}$, ${\theta }_{U,W}$ to the inverse images of $U\cap V\cap W$ in X_V, X_W, X_W respectively, it follows from the above definitions that ${\theta }_{U,V}^{\prime }\circ {\theta }_{V,W}^{\prime }={\theta }_{U,W}^{\prime }$. The existence of an $X$ satisfying the stated properties therefore follows from (I, 2.3.1); its uniqueness up to $Y$-isomorphism is trivial in view of (3.1.2.1).

(3.1.3)

We say that the prescheme $X$ defined in (3.1.2) is the homogeneous spectrum of the quasi-coherent graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$, and we denote it by $\mathrm{Proj}(\mathcal{S})$. It is immediate that $\mathrm{Proj}(\mathcal{S})$ is separated over $Y$ ((2.4.2) and (I, 5.5.5)); if $\mathcal{S}$ is an ${\mathcal{O}}_Y$-algebra of finite type (I, 9.6.2), then $\mathrm{Proj}(\mathcal{S})$ is of finite type over $Y$ ((2.7.1, (ii)) and (I, 6.3.1)).

If $f$ is the structure morphism $X\to Y$, then for every prescheme induced by $Y$ on an open subset $U$ of $Y$, $f^{-1}(U)$ is identified with the homogeneous spectrum $\mathrm{Proj}(\mathcal{S}|U)$.

Proposition.

Let $f\in \Gamma (Y,{\mathcal{S}}_d)$ for $d>0$. There exists an open subset $X_f$ of the underlying space of $X=\mathrm{Proj}(\mathcal{S})$ with the following property: for every affine open $U$ of $Y$, $X_f\cap {\varphi }^{-1}(U)=D_{+}(f|U)$ in ${\varphi }^{-1}(U)$ identified with $X_U=\mathrm{Proj}(\Gamma (U,\mathcal{S}))$ (where $\varphi$ denotes the structure morphism $X\to Y$).

Furthermore, the prescheme induced on $X_f$ is affine over $Y$ and is canonically isomorphic to $\mathrm{Spec}({\mathcal{S}}^{(d)}/(f-1){\mathcal{S}}^{(d)})$ (1.3.1).

Proof. We have $f|U\in \Gamma (U,{\mathcal{S}}_d)=(\Gamma (U,\mathcal{S}){)}_d$. If $U$, $U^{\prime }$ are affine opens of $Y$ with $U^{\prime }\subset U$, then $f|U^{\prime }$ is the image of $f|U$ under the restriction homomorphism

  Γ(U, 𝒮) → Γ(U', 𝒮),

so $D_{+}(f|U^{\prime })$ is equal (with the notation of (3.1.1)) to the prescheme induced on the inverse image ${\rho }_{U^{\prime },U}^{-1}(D_{+}(f|U))$ in $X_{U^{\prime }}$ (2.8.1); whence the first assertion. Furthermore, the prescheme induced on $D_{+}(f|U)$ by X_U is canonically identified with $\mathrm{Spec}((\Gamma (U,\mathcal{S}){)}_{(f|U)})$, these identifications being compatible with the restriction homomorphisms (2.8.1); the second assertion then follows from (2.2.5) and the commutativity of diagram (2.8.2.1).

We also say that $X_f$ (as an open subset of the underlying space $X$) is the set of $x\in X$ where $f$ does not vanish.

Corollary.

If $f\in \Gamma (Y,{\mathcal{S}}_d)$ and $g\in \Gamma (Y,{\mathcal{S}}_e)$, then

$$X_{fg}=X_f\cap X_g.(\mathrm{3.1.5.1})$$

Proof. It suffices to consider the intersection of both sides with a set ${\varphi }^{-1}(U)$, where $U$ is an affine open of $Y$, and to apply formula (2.3.3.2).

Corollary.

Let $(f_{\alpha })$ be a family of sections of $\mathcal{S}$ over $Y$ such that $f_{\alpha }\in \Gamma (Y,{\mathcal{S}}_{d_{\alpha }})$; if the sheaf of ideals of $\mathcal{S}$ generated by this family (0, 5.1.1) contains all the ${\mathcal{S}}_n$ from some rank on, then the underlying space $X$ is the union of the $X_{f_{\alpha }}$.

Proof. For every affine open $U$ of $Y$, ${\varphi }^{-1}(U)$ is the union of the $X_{f_{\alpha }}\cap {\varphi }^{-1}(U)$ (2.3.14).

Corollary.

Let $\mathcal{A}$ be a quasi-coherent ${\mathcal{O}}_Y$-algebra; set

  𝒮 = 𝒜[T] = 𝒜 ⊗_ℤ ℤ[T]

where $T$ is an indeterminate (and $\mathbb{Z}$, $\mathbb{Z}[T]$ are viewed as constant sheaves on $Y$). Then $X=\mathrm{Proj}(\mathcal{S})$ is canonically identified with $\mathrm{Spec}(\mathcal{A})$. In particular, $\mathrm{Proj}({\mathcal{O}}_Y[T])$ is identified with $Y$.

Proof. Applying (3.1.6) to the unique section $f\in \Gamma (Y,\mathcal{S})$ that equals $T$ at each point of $Y$, we see that $X_f=X$. Here $d=1$, and ${\mathcal{S}}^{(1)}/(f-1){\mathcal{S}}^{(1)}=\mathcal{S}/(f-1)\mathcal{S}$ is canonically isomorphic to $\mathcal{A}$; whence the corollary, by (1.2.2).

Let $g\in \Gamma (Y,{\mathcal{O}}_Y)$; taking $\mathcal{S}={\mathcal{O}}_Y[T]$, we have $g\in \Gamma (Y,{\mathcal{S}}_0)$; set

  h = gT ∈ Γ(Y, 𝒮_1).

If $X=\mathrm{Proj}(\mathcal{S})$, then the canonical identification defined in (3.1.7) identifies $X_h$ with the open subset $Y_g$ of $Y$ (in the sense of (0, 5.5.2)): indeed, we may reduce to the case $Y=\mathrm{Spec}(A)$ affine, and everything then reduces (taking (2.2.5) into account) to the fact that the ring of fractions $A_g$ is canonically identified with $A[T]/(gT-1)A[T]$ (0, 1.2.3).

Proposition.

Let $\mathcal{S}$ be a quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebra.

1. For every $d>0$, there is a canonical $Y$-isomorphism from $\mathrm{Proj}(\mathcal{S})$ to $\mathrm{Proj}({\mathcal{S}}^{(d)})$.

1. Let ${\mathcal{S}}^{\prime }$ be the graded ${\mathcal{O}}_Y$-algebra given by the direct sum of ${\mathcal{O}}_Y$ with the ${\mathcal{S}}_n$ ($n\ge 0$); then $\mathrm{Proj}({\mathcal{S}}^{\prime })$ and $\mathrm{Proj}(\mathcal{S})$ are canonically $Y$-isomorphic.
2. Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_Y$-module (0, 5.4.1), and let ${\mathcal{S}}_{(\mathcal{L})}$ be the graded ${\mathcal{O}}_Y$-algebra given by the direct sum of the ${\mathcal{S}}_d\otimes {\mathcal{L}}^{\otimes d}$ ($d\ge 0$); then $\mathrm{Proj}(\mathcal{S})$ and $\mathrm{Proj}({\mathcal{S}}_{(\mathcal{L})})$ are canonically $Y$-isomorphic.

Proof. In each of the three cases, it suffices to define the isomorphism locally on $Y$, since the verification of compatibility with restriction from one open to a smaller one is trivial. We may thus suppose $Y$ affine, and then (i) follows from (2.4.7, (i)) and (ii) from (2.4.8). As for (iii), if we further suppose $\mathcal{L}$ isomorphic to ${\mathcal{O}}_Y$ (allowed, the question being local on $Y$), the isomorphism between $\mathrm{Proj}(\mathcal{S})$ and $\mathrm{Proj}({\mathcal{S}}_{(\mathcal{L})})$ is evident; to define a canonical isomorphism, let $Y=\mathrm{Spec}(A)$ and $\mathcal{S}=S$, where $S$ is a graded $A$-algebra, and let $c$ be a generator of the free $A$-module $L$ such that $\mathcal{L}=L$; then, for every $n>0$, $x_n\mapsto x_n\otimes c^{\otimes n}$ is an $A$-isomorphism from $S_n$ to $S_n\otimes L^{\otimes n}$, and these $A$-isomorphisms define an $A$-isomorphism of graded algebras ${\varphi }_c:S\to S_{(L)}={\oplus }_{n\ge 0}{S}_n\otimes L^{\otimes n}$. Now let $f\in S_{+}$ be homogeneous of degree $d$; for every $x\in S_{nd}$, we have $(x\otimes c^{nd})/(f\otimes c^d{)}^n=(x\otimes (ϵc{)}^{nd})/(f\otimes (ϵc{)}^d{)}^n$ for every invertible $ϵ\in A$, which shows that the isomorphism $S_{(f)}\to (S_{(L)}{)}_{(f\otimes c^d)}$ induced from ${\varphi }_c$ is independent of the generator $c$ of $L$, completing the proof.

(3.1.9)

Recall (0, 4.1.3) and (I, 1.3.14) that, for the quasi-coherent graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$ to be generated by the ${\mathcal{O}}_Y$-module ${\mathcal{S}}_1$, it is necessary and sufficient that there exist an affine open covering $(U_{\alpha })$ of $Y$ such that the graded algebra $\Gamma (U_{\alpha },\mathcal{S})$ over $\Gamma (U_{\alpha },{\mathcal{S}}_0)$ is generated by the set $\Gamma (U_{\alpha },{\mathcal{S}}_1)$ of its homogeneous elements of degree 1. For every open $V$ of $Y$, $\mathcal{S}|V$ is then generated by the $({\mathcal{O}}_Y|V)$-module ${\mathcal{S}}_1|V$.

Proposition.

Suppose there exists a finite affine open covering $(U_i)$ of $Y$ such that each graded algebra $\Gamma (U_i,\mathcal{S})$ is of finite type over $\Gamma (U_i,{\mathcal{O}}_Y)$. Then there exists $d>0$ such that ${\mathcal{S}}^{(d)}$ is generated by ${\mathcal{S}}_d$, with ${\mathcal{S}}_d$ an ${\mathcal{O}}_Y$-module of finite type.

Proof. By (2.1.6, (v)), for each $i$ there exists an integer $m_i$ such that $\Gamma (U_i,{\mathcal{S}}_{n{m}_i})=(\Gamma (U_i,{\mathcal{S}}_{m_i}){)}^n$ for every $n>0$; it suffices to take $d$ to be a common multiple of the $m_i$, in view of (2.1.6, (i)).

Corollary.

Under the hypotheses of (3.1.10), $\mathrm{Proj}(\mathcal{S})$ is $Y$-isomorphic to a homogeneous spectrum $\mathrm{Proj}({\mathcal{S}}^{\prime })$, where ${\mathcal{S}}^{\prime }$ is a graded ${\mathcal{O}}_Y$-algebra generated by ${\mathcal{S}}_1^{\prime }$, with ${\mathcal{S}}_1^{\prime }$ an ${\mathcal{O}}_Y$-module of finite type.

Proof. It suffices to take ${\mathcal{S}}^{\prime }={\mathcal{S}}^{(d)}$, with $d$ determined by the property of (3.1.10), and to apply (3.1.8, (i)).

(3.1.12)

If $\mathcal{S}$ is a quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebra, we know (I, 5.1.1) that its nilradical $\mathcal{N}$ is a quasi-coherent ${\mathcal{O}}_Y$-module; we say that ${\mathcal{N}}_{+}=\mathcal{N}\cap {\mathcal{S}}_{+}$ is the nilradical of ${\mathcal{S}}_{+}$; this is a quasi-coherent graded ${\mathcal{S}}_0$-module, since we reduce at once to the case $Y$ affine, and the proposition then follows from (2.1.10). For every $y\in Y$, $({\mathcal{N}}_{+}{)}_y$ is then the nilradical of $({\mathcal{S}}_{+}{)}_y=({\mathcal{S}}_y{)}_{+}$ (I, 5.1.1). We say that the graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$ is essentially reduced if ${\mathcal{N}}_{+}=0$, which is equivalent

to saying that ${\mathcal{S}}_y$ is an essentially reduced graded ${\mathcal{O}}_y$-algebra for every $y\in Y$. For every graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$, $\mathcal{S}/{\mathcal{N}}_{+}$ is essentially reduced.

We say that $\mathcal{S}$ is integral if, for every $y\in Y$, ${\mathcal{S}}_y$ is an integral ring and moreover $({\mathcal{S}}_y{)}_{+}=({\mathcal{S}}_{+}{)}_y\ne 0$.

Proposition.

Let $\mathcal{S}$ be a positively-graded ${\mathcal{O}}_Y$-algebra. If $X=\mathrm{Proj}(\mathcal{S})$, then the $Y$-scheme $X_{red}$ is canonically isomorphic to $\mathrm{Proj}(\mathcal{S}/{\mathcal{N}}_{+})$; in particular, if $\mathcal{S}$ is essentially reduced, then $X$ is reduced.

Proof. That $X^{\prime }=\mathrm{Proj}(\mathcal{S}/{\mathcal{N}}_{+})$ is reduced follows immediately from (2.4.4, (i)), the property being local; moreover, for every affine open $U\subset Y$, ${\varphi }^{\prime -1}(U)=({\varphi }^{-1}(U){)}_{red}$ (where $\varphi$ and ${\varphi }^{\prime }$ denote the structure morphisms $X\to Y$ and $X^{\prime }\to Y$); one checks at once that the canonical $U$-morphisms ${\varphi }^{\prime -1}(U)\to {\varphi }^{-1}(U)$ are compatible with restriction and so define a closed immersion $X^{\prime }\to X$ that is a homeomorphism of underlying spaces; whence the conclusion (I, 5.1.2).

Proposition.

Let $Y$ be an integral prescheme, and $\mathcal{S}$ a quasi-coherent graded ${\mathcal{O}}_Y$-algebra such that ${\mathcal{S}}_0={\mathcal{O}}_Y$.

1. If $\mathcal{S}$ is integral (3.1.12), then $X=\mathrm{Proj}(\mathcal{S})$ is integral and the structure morphism $\varphi :X\to Y$ is dominant.
2. Suppose further that $\mathcal{S}$ is essentially reduced. Then conversely, if $X$ is integral and $\varphi$ is dominant, then $\mathcal{S}$ is integral.

Proof.

(i) If $(U_{\alpha })$ is a base of $Y$ consisting of non-empty affine opens, it suffices to prove the proposition with $Y$ replaced by one of the $U_{\alpha }$ and $\mathcal{S}$ by $\mathcal{S}|U_{\alpha }$: indeed, on the one hand the underlying spaces ${\varphi }^{-1}(U_{\alpha })$ will be irreducible (hence non-empty) opens of $X$ such that ${\varphi }^{-1}(U_{\alpha })\cap {\varphi }^{-1}(U_{\beta })\ne \varnothing$ for every pair of indices (since $U_{\alpha }\cap U_{\beta }$ contains some $U_{\gamma }$), so $X$ will be irreducible (0, 2.1.4); on the other hand, $X$ will be reduced, since this is a local property, and so $X$ will be integral and $\varphi (X)$ dense in $Y$.

Suppose then that $Y=\mathrm{Spec}(A)$, $A$ integral (I, 5.1.4), and $\mathcal{S}=S$, with $S$ a graded $A$-algebra; the hypothesis is that for every $y\in Y$, $S_y=S_y$ is an integral graded ring with $(S_y{)}_{+}\ne 0$. It suffices to show $S$ is an integral ring, since then $S_{+}\ne 0$ and we may apply (2.4.4, (ii)). Let $f,g\ne 0$ in $S$ and suppose $fg=0$; for every $y\in Y$ we have $(f/1)(g/1)=0$ in $S_y$, so $f/1=0$ or $g/1=0$ by hypothesis. Suppose, say, $f/1=0$ in $S_y$; this means there exists $a\in A$ with $a\notin {\mathfrak{j}}_y$ and $af=0$; then for every $z\in Y$, $(a/1)(f/1)=0$ in the integral ring $S_z$, and since $a/1\ne 0$ (since $A$ is integral), $f/1=0$, which forces $f=0$.

(ii) The question being local on $Y$, we may again assume $Y=\mathrm{Spec}(A)$, $A$ integral, and $\mathcal{S}=\tilde{S}$. By hypothesis, for every $y\in Y$, $(S_y{)}_{+}$ contains no non-zero nilpotent element, and the same holds for $(S_0{)}_y=A_y$ by hypothesis; so $S_y$ is a reduced ring for every $y\in Y$, whence $S$ itself is reduced (I, 5.1.1). The hypothesis that $X$ is integral implies that $S$ is essentially integral (2.4.4, (ii)), and everything reduces to showing that the annihilator $\mathfrak{J}$ of $S_{+}$ in $A=S_0$ is reduced to 0 (2.1.11). Otherwise we would have

$(S_h{)}_{+}=0$ for some $h\ne 0$ in $\mathfrak{J}$, hence (3.1.1) ${\varphi }^{-1}(D(h))=\varnothing$, and $\varphi (X)$ would not be dense in $Y$ contrary to hypothesis (since $D(h)\ne \varnothing$, $h$ being non-nilpotent).

3.2. Sheaf on $\mathrm{Proj}(\mathcal{S})$ associated to a graded $\mathcal{S}$-module

(3.2.1)

Let $Y$ be a prescheme, $\mathcal{S}$ a quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebra, and $\mathcal{M}$ a quasi-coherent graded $\mathcal{S}$-module (on $(Y,{\mathcal{O}}_Y)$, or equivalently (I, 9.6.1) on the ringed space $(Y,\mathcal{S})$). With the notation of (3.1.1), we denote by ${\mathcal{M}}_U$ the quasi-coherent ${\mathcal{O}}_{X_U}$-module $\Gamma (U,\mathcal{M})$; for $U^{\prime }\subset U$, $\Gamma (U^{\prime },\mathcal{M})$ is canonically identified with $\Gamma (U,\mathcal{M}){\otimes }_A{A}^{\prime }$ (I, 1.6.4); thus ${\mathcal{M}}_{U^{\prime }}={\rho }_{U^{\prime },U}\ast ({\mathcal{M}}_U)$ (2.8.11).

Proposition.

There exists on $\mathrm{Proj}(\mathcal{S})=X$ a unique quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{M}$ such that, for every affine open $U$ of $Y$, ${\eta }_U\ast (\Gamma (U,\mathcal{M}))=\tilde{\mathcal{M}}|f^{-1}(U)$ (where ${\eta }_U$ denotes the isomorphism $f^{-1}(U)\stackrel{\sim }{\to }\mathrm{Proj}(\Gamma (U,\mathcal{S}))$ and $f$ is the structure morphism $X\to Y$).

Proof. Since ${\rho }_{U^{\prime },U}$ is identified with the injection morphism $f^{-1}(U^{\prime })\to f^{-1}(U)$ (3.1.2.1), the proposition follows at once from the relation ${\mathcal{M}}_{U^{\prime }}={\rho }_{U^{\prime },U}\ast ({\mathcal{M}}_U)$ and the gluing principle for sheaves (0, 3.3.1).

We say that $\tilde{\mathcal{M}}$ is the ${\mathcal{O}}_X$-module associated to the quasi-coherent graded $\mathcal{S}$-module $\mathcal{M}$.

Proposition.

Let $\mathcal{M}$ be a quasi-coherent graded $\mathcal{S}$-module, and let $f\in \Gamma (Y,{\mathcal{S}}_d)$ ($d>0$). If ${\xi }_f$ is the canonical isomorphism from $X_f$ to the $Y$-prescheme $Z_f=\mathrm{Spec}({\mathcal{S}}^{(d)}/(f-1){\mathcal{S}}^{(d)})$ (3.1.4), then $({\xi }_f{)}_{\ast }(\mathcal{M}|X_f)$ is the ${\mathcal{O}}_{Z_f}$-module ${\mathcal{M}}^{(d)}/(f-1){\mathcal{M}}^{(d)}$ (1.4.3).

Proof. The question being local on $Y$, we reduce immediately to (2.2.5), using the commutativity of diagram (2.8.12.1).

Proposition.

The ${\mathcal{O}}_X$-module $\tilde{\mathcal{M}}$ is a covariant additive exact functor in $\mathcal{M}$ from the category of quasi-coherent graded $\mathcal{S}$-modules to the category of quasi-coherent ${\mathcal{O}}_X$-modules, which commutes with inductive limits and direct sums.

Proof. The question being local on $Y$, we reduce to (I, 1.3.11), (I, 1.3.9), and (2.5.4).

In particular, if $\mathcal{N}$ is a quasi-coherent graded sub-$\mathcal{S}$-module of $\mathcal{M}$, then $\mathcal{N}$ is canonically identified with a quasi-coherent sub-${\mathcal{O}}_X$-module of $\mathcal{M}$; more specifically, for every quasi-coherent graded sheaf of ideals $\mathcal{J}$ of $\mathcal{S}$, $\tilde{\mathcal{J}}$ is a quasi-coherent sheaf of ideals of ${\mathcal{O}}_X$.

If $\mathcal{M}$ is a quasi-coherent graded $\mathcal{S}$-module and $\mathcal{I}$ a quasi-coherent sheaf of ideals of ${\mathcal{O}}_Y$, then $\mathcal{I}\mathcal{M}$ is a quasi-coherent graded sub-$\mathcal{S}$-module of $\mathcal{M}$, and

$$\mathcal{I}\mathcal{M}=\mathcal{I}\cdot \mathcal{M}(\mathrm{3.2.4.1})$$

(the right-hand side in the sense of (0, 4.3.5)). It suffices to verify this in the case $Y=\mathrm{Spec}(A)$ affine, $\mathcal{S}=S$ with $S$ a graded $A$-algebra, $\mathcal{M}=M$

with $M$ a graded $S$-module, and $\mathcal{I}=\mathfrak{J}$ with $\mathfrak{J}$ an ideal of $A$. For every homogeneous element $f$ of $S_{+}$, the restriction to $D_{+}(f)=\mathrm{Spec}(S_{(f)})$ of the left-hand side of (3.2.4.1) is associated to $(\mathfrak{J}M{)}_{(f)}=\mathfrak{J}\cdot M_{(f)}$, and the same holds for the restriction of the right-hand side, by (I, 1.3.13) and (I, 1.6.9).

Proposition.

Let $f\in \Gamma (Y,{\mathcal{S}}_d)$ ($d>0$). On the open subset $X_f$, the $({\mathcal{O}}_X|X_f)$-module $\mathcal{S}(nd)|X_f$ is canonically isomorphic to ${\mathcal{O}}_X|X_f$ for every $n\in \mathbb{Z}$. In particular, if the ${\mathcal{O}}_Y$-algebra $\mathcal{S}$ is generated by ${\mathcal{S}}_1$ (3.1.9), then the ${\mathcal{O}}_X$-modules $\mathcal{S}(n)$ are invertible for every $n\in \mathbb{Z}$.

Proof. Indeed, for every affine open $U$ of $Y$, we defined in (2.5.7) a canonical isomorphism from $\widetilde{\mathcal{S}(nd)}|(X_f\cap {\varphi }^{-1}(U))$ to ${\mathcal{O}}_X|(X_f\cap {\varphi }^{-1}(U))$, in view of (3.1.4) (where $\varphi$ is the structure morphism $X\to Y$); one checks at once that these isomorphisms are compatible with restriction from $U$ to an affine open $U^{\prime }\subset U$, whence the first assertion. For the second, note that if $\mathcal{S}$ is generated by ${\mathcal{S}}_1$, there is an affine open covering $(U_{\alpha })$ of $Y$ such that $\Gamma (U_{\alpha },\mathcal{S})$ is generated by $\Gamma (U_{\alpha },\mathcal{S}{)}_1=\Gamma (U_{\alpha },{\mathcal{S}}_1)$; we then apply (2.5.9), the property of being invertible being local.

We also set, for every $n\in \mathbb{Z}$,

$${\mathcal{O}}_X(n)=\widetilde{\mathcal{S}(n)}(\mathrm{3.2.5.1})$$

and, for every ${\mathcal{O}}_X$-module $\mathcal{F}$,

  ℱ(n) = ℱ ⊗_{𝒪_X} 𝒪_X(n).                                                (3.2.5.2)

It follows at once from these definitions that, for every open $U$ of $Y$,

$$\widetilde{(\mathcal{S}|U)(n)}={\mathcal{O}}_X(n)|f^{-1}(U)$$

where $f$ is the structure morphism $X\to Y$.

Proposition.

Let $\mathcal{M}$ and $\mathcal{N}$ be quasi-coherent graded $\mathcal{S}$-modules. There is a canonical homomorphism, functorial in $\mathcal{M}$ and $\mathcal{N}$,

  λ : ℳ̃ ⊗_{𝒪_X} 𝒩̃ → (ℳ ⊗_𝒮 𝒩)̃                                            (3.2.6.1)

and a canonical homomorphism, functorial in $\mathcal{M}$ and $\mathcal{N}$,

  μ : (𝓗𝓸𝓶_𝒮(ℳ, 𝒩))̃ → 𝓗𝓸𝓶_{𝒪_X}(ℳ̃, 𝒩̃).                                   (3.2.6.2)

Furthermore, if $\mathcal{S}$ is generated by ${\mathcal{S}}_1$ (3.1.9), then $\lambda$ is an isomorphism; if in addition $\mathcal{M}$ admits a finite presentation (3.1.1), then $\mu$ is an isomorphism.

Proof. The isomorphisms $\lambda$ and $\mu$ were defined in (2.5.11.2) and (2.5.12.2) when $Y$ is affine; these definitions being local, they extend at once to the general case considered here, in view of (2.8.14).

Corollary.

If $\mathcal{S}$ is generated by ${\mathcal{S}}_1$, then for any $m,n\in \mathbb{Z}$,

  𝒪_X(m) ⊗_{𝒪_X} 𝒪_X(n) = 𝒪_X(m + n)                                      (3.2.7.1)

  𝒪_X(n) = (𝒪_X(1))^{⊗ n}                                                 (3.2.7.2)

up to canonical isomorphism.

Corollary.

If $\mathcal{S}$ is generated by ${\mathcal{S}}_1$, then for every graded $\mathcal{S}$-module $\mathcal{M}$ and every $n\in \mathbb{Z}$,

$$\mathcal{M}(n)=\mathcal{M}(n)(\mathrm{3.2.8.1})$$

up to canonical isomorphism.

Proof. This follows from the corresponding properties for $Y$ affine ((2.5.14) and (2.5.15)) together with (2.8.11).

Remarks.

(i) If $\mathcal{S}=\mathcal{A}[T]$ with $\mathcal{A}$ a quasi-coherent ${\mathcal{O}}_Y$-algebra (3.1.7), one checks at once that all the invertible ${\mathcal{O}}_X$-modules ${\mathcal{O}}_X(n)$ are canonically isomorphic to ${\mathcal{O}}_X$.

Furthermore, let $\mathcal{N}$ be a quasi-coherent $\mathcal{A}$-module, and set $\mathcal{M}=\mathcal{N}{\otimes }_{\mathcal{A}}\mathcal{A}[T]$. It then follows from (3.2.3) and (3.1.7) that, under the canonical identification of $X=\mathrm{Proj}(\mathcal{A}[T])$ with $X^{\prime }=\mathrm{Spec}(\mathcal{A})$, the ${\mathcal{O}}_X$-module $\mathcal{M}$ is identified with the ${\mathcal{O}}_{X^{\prime }}$-module $\mathcal{N}$ associated to $\mathcal{N}$ (in the sense of (1.4.3)).

(ii) Let $\mathcal{S}$ be an arbitrary graded ${\mathcal{O}}_Y$-algebra, and let ${\mathcal{S}}^{\prime }$ be the graded ${\mathcal{O}}_Y$-algebra such that ${\mathcal{S}}_0^{\prime }={\mathcal{O}}_Y$ and ${\mathcal{S}}_n^{\prime }={\mathcal{S}}_n$ for every $n>0$; the canonical isomorphism from $X=\mathrm{Proj}(\mathcal{S})$ to $X^{\prime }=\mathrm{Proj}({\mathcal{S}}^{\prime })$ (3.1.8, (ii)) identifies ${\mathcal{O}}_X(n)$ with ${\mathcal{O}}_{X^{\prime }}(n)$ for every $n\in \mathbb{Z}$: this follows from the same proposition in the affine case (2.5.16) and from the fact that, on the affine opens of $Y$, these identifications commute with restriction. Similarly, let $X^{(d)}=\mathrm{Proj}({\mathcal{S}}^{(d)})$; the canonical isomorphism from $X$ to $X^{(d)}$ (3.1.8, (i)) identifies ${\mathcal{O}}_X(nd)$ with ${\mathcal{O}}_{X^{(d)}}(n)$ for every $n\in \mathbb{Z}$.

Proposition.

Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_Y$-module, and let $g$ be the canonical isomorphism $X_{(\mathcal{L})}=\mathrm{Proj}({\mathcal{S}}_{(\mathcal{L})})\stackrel{\sim }{\to }X=\mathrm{Proj}(\mathcal{S})$ (3.1.8, (iii)). For every $n\in \mathbb{Z}$, $g_{\ast }({\mathcal{O}}_{X_{(\mathcal{L})}}(n))$ is canonically isomorphic to ${\mathcal{O}}_X(n){\otimes }_Y{\mathcal{L}}^{\otimes n}$.

Proof. Suppose first that $Y$ is affine with ring $A$ and $\mathcal{L}=\tilde{L}$, with $L$ a free $A$-module of rank 1. With the notation of the proof of (3.1.8, (iii)), we define, for $f\in S_d$, an isomorphism from $(S(n){)}_{(f)}{\otimes }_A{L}^{\otimes n}$ to $(S_{(L)}(n){)}_{(f\otimes c^d)}$ by sending $(x/f^k)\otimes c^n$, with $x\in S_{kd+n}$, to $(x\otimes c^{n+kd})/(f\otimes c^d{)}^k$; one checks at once that this isomorphism is independent of the chosen generator $c$ of $L$; furthermore, the isomorphisms so defined for each $f\in S_{+}$ are compatible with the restriction operators $D_{+}(f)\to D_{+}(fg)$. Finally, in the general case, one sees easily, going back to the definitions (3.1.1), that the isomorphisms so defined for each affine open $U$ of $Y$ are compatible with passage from $U$ to an affine open $U^{\prime }\subset U$.

3.3. Graded $\mathcal{S}$-module associated to a sheaf on $\mathrm{Proj}(\mathcal{S})$

Throughout this entire section we suppose that the graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$ is generated by ${\mathcal{S}}_1$ (3.1.9). Recall that, by (3.1.8, (i)), this restriction is not essential, given the finiteness conditions (3.1.10).

(3.3.1)

Let $p$ be the structure morphism $X=\mathrm{Proj}(\mathcal{S})\to Y$. For every ${\mathcal{O}}_X$-module $\mathcal{F}$, set

  Γ_*(ℱ) = ⊕_{n ∈ ℤ} p_*(ℱ(n))                                            (3.3.1.1)

and in particular

  Γ_*(𝒪_X) = ⊕_{n ∈ ℤ} p_*(𝒪_X(n)).                                       (3.3.1.2)

We know (0, 4.2.2) that there is a canonical homomorphism

  p_*(ℱ) ⊗_{𝒪_Y} p_*(𝒢) → p_*(ℱ ⊗_{𝒪_X} 𝒢)

for two ${\mathcal{O}}_X$-modules $\mathcal{F}$ and $\mathcal{G}$; we therefore deduce from (3.2.7.1) that ${\Gamma }_{\ast }({\mathcal{O}}_X)$ is endowed with the structure of a graded ${\mathcal{O}}_Y$-algebra, and (3.2.5.2) similarly defines on ${\Gamma }_{\ast }(\mathcal{F})$ the structure of a graded ${\Gamma }_{\ast }({\mathcal{O}}_X)$-module.

By (3.2.5) and the left-exactness of $p_{\ast }$ (0, 4.2.1), ${\Gamma }_{\ast }(\mathcal{F})$ is a covariant additive left-exact functor in $\mathcal{F}$ from the category of ${\mathcal{O}}_X$-modules to the category of graded ${\mathcal{O}}_Y$-modules (with morphisms of degree 0). In particular, if $\mathcal{J}$ is a sheaf of ideals of ${\mathcal{O}}_X$, then ${\Gamma }_{\ast }(\mathcal{J})$ is identified with a graded sheaf of ideals of ${\Gamma }_{\ast }({\mathcal{O}}_X)$.

(3.3.2)

Let $\mathcal{M}$ be a quasi-coherent graded $\mathcal{S}$-module. For every affine open $U$ of $Y$, we defined in (2.6.2) a homomorphism of abelian groups

  α_{0, U} : Γ(U, ℳ_0) → Γ(p⁻¹(U), ℳ̃).

It is immediate that these homomorphisms commute with restriction (2.8.13.1) and so define (without using the hypothesis that $\mathcal{S}$ is generated by ${\mathcal{S}}_1$) a homomorphism of sheaves of abelian groups

$${\alpha }_0:{\mathcal{M}}_0\to p_{\ast }(\tilde{\mathcal{M}}).(\mathrm{3.3.2.1})$$

Applying this to each ${\mathcal{M}}_n=(\mathcal{M}(n){)}_0$ and using (3.2.8.1), we define a homomorphism of sheaves of abelian groups

$${\alpha }_n:{\mathcal{M}}_n\to p_{\ast }(\tilde{\mathcal{M}}(n))(\mathrm{3.3.2.2})$$

for every $n\in \mathbb{Z}$, whence a functorial homomorphism (of degree 0) of graded sheaves of abelian groups

$$\alpha :\mathcal{M}\to {\Gamma }_{\ast }(\tilde{\mathcal{M}})(\mathrm{3.3.2.3})$$

(also denoted ${\alpha }_{\mathcal{M}}$).

Taking $\mathcal{M}=\mathcal{S}$ in particular, one checks that $\alpha :\mathcal{S}\to {\Gamma }_{\ast }({\mathcal{O}}_X)$ is a homomorphism of graded ${\mathcal{O}}_Y$-algebras, and that (3.3.2.3) is a di-homomorphism of graded modules relative to this homomorphism of graded algebras.

We also note that to each ${\alpha }_n$ corresponds (0, 4.4.3) a canonical homomorphism of ${\mathcal{O}}_X$-modules

$${\alpha }_n♯:p\ast ({\mathcal{M}}_n)\to \tilde{\mathcal{M}}(n).(\mathrm{3.3.2.4})$$

One checks without difficulty that this homomorphism is precisely the one which corresponds functorially (3.2.4) to the canonical homomorphism (of degree 0) of graded ${\mathcal{O}}_Y$-modules

  ℳ_n ⊗_{𝒪_Y} 𝒮 → ℳ(n)                                                   (3.3.2.5)

where the grading on the right-hand side comes naturally from that of $\mathcal{S}$. We may restrict to the case $Y=\mathrm{Spec}(A)$ affine, $\mathcal{M}=M$, and $\mathcal{S}=S$, with the graded $A$-algebra $S$ generated by S_1, so that as $f$ runs over S_1 the $D_{+}(f)$ form a covering of $X$. Returning to the definitions (2.6.2) and using (I, 1.6.7), the restriction to $D_{+}(f)$ of the homomorphism (3.3.2.4) corresponds (I, 1.3.8) to the homomorphism of $S_{(f)}$-modules $M_n{\otimes }_A{S}_{(f)}\to (S(n){)}_{(f)}$ sending $x\otimes 1$ (with $x\in M_n$) to $x/1$; this proves the claim.

Proposition.

For every section $f\in \Gamma (Y,{\mathcal{S}}_d)$ ($d>0$), $X_f$ coincides with the set of points of $X$ where ${\alpha }_d(f)$ (considered as a section of ${\mathcal{O}}_X(d)$) does not vanish (0, 5.5.2).

Proof. (Note that ${\alpha }_d(f)$ is a section of $p_{\ast }({\mathcal{O}}_X(d))$ over $Y$, but by definition such a section is also a section of ${\mathcal{O}}_X(d)$ over $X$ (0, 4.2.1).) The definition of $X_f$ (3.1.4) reduces us to the case $Y$ affine, which was handled in (2.6.3).

(3.3.4)

From now on, in addition to the hypothesis at the start of this section, we suppose that for every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the $p_{\ast }(\mathcal{F}(n))$ are quasi-coherent on $Y$, so that ${\Gamma }_{\ast }(\mathcal{F})={\oplus }_{n\in \mathbb{Z}}{p}_{\ast }(\mathcal{F}(n))$ is also a quasi-coherent ${\mathcal{O}}_Y$-module ((I, 1.4.1) and (I, 1.3.9)); this will always be the case in particular if $X$ is of finite type over $Y$ (I, 9.2.2). We thus conclude that $\widetilde{{\Gamma }_{\ast }(\mathcal{F})}$ is defined and is a quasi-coherent ${\mathcal{O}}_X$-module. For every affine open $U$ of $Y$, we have ((I, 1.3.9) and (2.5.4))

  (Γ(U, ⊕_{n ∈ ℤ} p_*(ℱ(n))))̃
    = ⊕_{n ∈ ℤ} (Γ(U, p_*(ℱ(n))))̃
    = ⊕_{n ∈ ℤ} (Γ(p⁻¹(U), ℱ(n)))̃
    = (⊕_{n ∈ ℤ} Γ(p⁻¹(U), ℱ(n)))̃
    = (Γ_*(ℱ|p⁻¹(U)))̃

and so (2.6.4) we have a canonical homomorphism

  β_U : (Γ(U, ⊕_{n ∈ ℤ} p_*(ℱ(n))))̃ → ℱ|p⁻¹(U).

Furthermore, the commutativity of (2.8.13.2) shows that these homomorphisms commute with restriction on $Y$; we thus obtain a canonical functorial homomorphism

$$\beta :\widetilde{{\Gamma }_{\ast }(\mathcal{F})}\to \mathcal{F}(\mathrm{3.3.4.1})$$

(also denoted ${\beta }_{\mathcal{F}}$) for quasi-coherent ${\mathcal{O}}_X$-modules.

Proposition.

Let $\mathcal{M}$ be a quasi-coherent graded $\mathcal{S}$-module, and $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-module; the composite homomorphisms

                α̃                    β
  ℳ̃ ─────→ (Γ_*(ℳ̃))̃ ─────→ ℳ̃                                            (3.3.5.1)

                  α                          Γ_*(β)
  Γ_*(ℱ) ─────→ Γ_*((Γ_*(ℱ))̃) ─────→ Γ_*(ℱ)                              (3.3.5.2)

are the identity isomorphisms.

Proof. The question is local on $Y$, so we reduce to (2.6.5).

3.4. Finiteness conditions

Proposition.

Let $Y$ be a prescheme, $\mathcal{S}$ a quasi-coherent ${\mathcal{O}}_Y$-algebra generated by ${\mathcal{S}}_1$ (3.1.9); suppose further that ${\mathcal{S}}_1$ is of finite type. Then $X=\mathrm{Proj}(\mathcal{S})$ is of finite type over $Y$.

Proof. The question being local on $Y$, we may suppose $Y$ affine with ring $A$; then $\mathcal{S}=\tilde{S}$ with $S=\Gamma (Y,\mathcal{S})$, and by hypothesis $S$ is an $A$-algebra generated by $S_1=\Gamma (Y,{\mathcal{S}}_1)$, where we may further assume that S_1 is an $A$-module of finite type ((I, 1.3.9) and (I, 1.3.12)). Then $S$ is a graded $A$-algebra of finite type, and we reduce to (2.7.1, (ii)).

(3.4.2)

Let $\mathcal{S}$ be a quasi-coherent graded ${\mathcal{O}}_Y$-algebra; for a quasi-coherent graded $\mathcal{S}$-module $\mathcal{M}$, we consider the following finiteness conditions:

• (TF) There exists an integer $n$ such that the $\mathcal{S}$-module ${\oplus }_{k\ge n}{\mathcal{M}}_k$ is of finite type.
• (TN) There exists an integer $n$ such that ${\mathcal{M}}_k=0$ for $k\ge n$.

If $\mathcal{M}$ satisfies (TN), then $\tilde{\mathcal{M}}=0$, since this is a local property on $Y$ (2.7.2).

Let $\mathcal{M}$, $\mathcal{N}$ be quasi-coherent graded $\mathcal{S}$-modules; we say that a homomorphism $u:\mathcal{M}\to \mathcal{N}$ of degree 0 is (TN)-injective (resp. (TN)-surjective, (TN)-bijective) if there exists an integer $n$ such that $u_k:{\mathcal{M}}_k\to {\mathcal{N}}_k$ is injective (resp. surjective, bijective) for $k\ge n$; then $u:\mathcal{M}\to \tilde{\mathcal{N}}$ is injective (resp. surjective, bijective) by (2.7.2), the question being local on $Y$, and in view of (I, 1.3.9); when $u$ is (TN)-bijective, we also say that $u$ is a (TN)-isomorphism.

Proposition.

Let $Y$ be a prescheme, $\mathcal{S}$ a quasi-coherent graded ${\mathcal{O}}_Y$-algebra generated by ${\mathcal{S}}_1$, with ${\mathcal{S}}_1$ assumed of finite type. Let $\mathcal{M}$ be a quasi-coherent graded $\mathcal{S}$-module.

1. If $\mathcal{M}$ satisfies (TF), then $\tilde{\mathcal{M}}$ is of finite type.
2. Suppose $\mathcal{M}$ satisfies (TF); for $\tilde{\mathcal{M}}=0$, it is necessary and sufficient that $\mathcal{M}$ satisfy (TN).

Proof. The questions being local on $Y$, we reduce to the case $Y$ affine with ring $A$, $\mathcal{S}=S$ with $S$ a graded $A$-algebra such that the ideal $S_{+}$ is of finite type, and $\mathcal{M}=M$ with $M$ a graded $S$-module; the proposition then follows from (2.7.3).

Theorem.

Let $Y$ be a prescheme, $\mathcal{S}$ a quasi-coherent graded ${\mathcal{O}}_Y$-algebra generated by ${\mathcal{S}}_1$, with ${\mathcal{S}}_1$ assumed of finite type; let $X=\mathrm{Proj}(\mathcal{S})$. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the canonical homomorphism $\beta$ (3.3.4) is an isomorphism.

Proof. First, $\beta$ is defined by virtue of (3.4.1). To see that $\beta$ is an isomorphism, we reduce to the case $Y$ affine with ring $A$, $\mathcal{S}=\tilde{S}$ with $S$ a graded $A$-algebra generated by S_1, and S_1 an $A$-module of finite type. It then suffices to apply (2.7.5).

Corollary.

Under the hypotheses of (3.4.4), every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ is isomorphic to an ${\mathcal{O}}_X$-module of the form $\tilde{\mathcal{M}}$, where $\mathcal{M}$ is a quasi-coherent graded $\mathcal{S}$-module. If furthermore $\mathcal{F}$ is of finite type, and if we assume that $Y$ is a quasi-compact scheme or that the underlying space of $Y$ is Noetherian, then we may take $\mathcal{M}$ to be of finite type.

Proof. The first assertion follows at once from (3.4.4) by taking $\mathcal{M}={\Gamma }_{\ast }(\mathcal{F})$. To establish the second, it suffices to show that $\mathcal{M}$ is the inductive limit of its graded sub-$\mathcal{S}$-modules of finite type ${\mathcal{N}}_{\lambda }$: indeed, it will follow that $\mathcal{M}$ is the inductive limit of the ${\mathcal{N}}_{\lambda }$ (3.2.4), hence $\mathcal{F}$ is the inductive limit of the $\beta ({\mathcal{N}}_{\lambda })$; since $X$ is quasi-compact ((3.4.1) and (I, 6.3.1)) and $\mathcal{F}$ is of finite type, $\mathcal{F}$ will necessarily equal one of the $\beta ({\mathcal{N}}_{\lambda })$ (0, 5.2.3).

To define the ${\mathcal{N}}_{\lambda }$ having $\mathcal{M}$ as inductive limit, it suffices to consider, for each $n\in \mathbb{Z}$, the quasi-coherent ${\mathcal{O}}_Y$-module ${\mathcal{M}}_n$, which is the inductive limit of its sub-${\mathcal{O}}_Y$-modules ${\mathcal{M}}_n^{({\mu }_n)}$ of finite type, by the hypothesis on $Y$ (I, 9.4.9); one sees at once that ${\mathcal{P}}_{{\mu }_n}=\mathcal{S}\cdot {\mathcal{M}}_n^{({\mu }_n)}$ is a graded $\mathcal{S}$-module of finite type, and that taking the ${\mathcal{N}}_{\lambda }$ to be the finite sums of $\mathcal{S}$-modules of the form ${\mathcal{P}}_{{\mu }_n}$ gives the desired objects.

Corollary.

Suppose the hypotheses of (3.4.4) are satisfied and, moreover, that the underlying space of $Y$ is quasi-compact; let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module of finite type. Then there exists $n_0$ such that for every $n\ge n_0$ the canonical homomorphism $\sigma :p\ast (p_{\ast }(\mathcal{F}(n)))\to \mathcal{F}(n)$ (0, 4.4.3) is surjective.

Proof. For every $y\in Y$, let $U$ be an affine open neighbourhood of $y$ in $Y$. There exists an integer $n_0(U)$ such that, for $n\ge n_0(U)$, $\mathcal{F}(n)|p^{-1}(U)$ is generated by finitely many of its sections over $p^{-1}(U)$ (2.7.9); but these sections are the canonical images of sections of $p\ast (p_{\ast }(\mathcal{F}(n)))$ over $p^{-1}(U)$ ((0, 3.7.1) and (0, 4.4.3)), so $\mathcal{F}(n)|p^{-1}(U)$ equals the canonical image of $p\ast (p_{\ast }(\mathcal{F}(n)))|p^{-1}(U)$. Finally, since $Y$ is quasi-compact, there is a finite cover of $Y$ by affine opens $U_i$, and taking $n_0$ to be the largest of the $n_0(U_i)$ finishes the proof.

Remarks.

If $p=(\psi ,\theta ):X\to Y$ is a morphism of ringed spaces and $\mathcal{F}$ an ${\mathcal{O}}_X$-module, then the fact that the canonical homomorphism $\sigma :p\ast (p_{\ast }(\mathcal{F}))\to \mathcal{F}$ is surjective can be made explicit as follows (0, 4.4.1): for every $x\in X$ and every section $s$ of $\mathcal{F}$ over an open neighbourhood $V$ of $x$, there exist an open neighbourhood $U$ of $p(x)$ in $Y$, finitely many sections $t_i$ ($1\le i\le m$) of $\mathcal{F}$ over $p^{-1}(U)$, a neighbourhood $W\subset V\cap p^{-1}(U)$ of $x$, and sections $a_i$ ($1\le i\le m$) of ${\mathcal{O}}_X$ over $W$ such that

  s|W = ∑_{i=1}^m a_i · (t_i|W).

When $Y$ is an affine scheme and $p_{\ast }(\mathcal{F})$ is quasi-coherent, this condition is equivalent to $\mathcal{F}$ being generated by its sections over $X$ (0, 5.5.1): indeed, if $Y=\mathrm{Spec}(A)$, we may suppose $U=D(f)$ with $f\in A$; then there exist an integer $n>0$ and sections $s_i$ of $\mathcal{F}$ over $X$ such that $t_i$ is the restriction to $p^{-1}(U)$ of $s_i{g}^n$, with $g=\theta (f)$ (by applying (I, 1.4.1) to $p_{\ast }(\mathcal{F})$); since $g$ is invertible over $p^{-1}(U)$, we have

  s|W = ∑_i b_i · (s_i|W)

with $b_i=a_i(g|W{)}^{-n}$, whence the claim. When $Y$ is affine, corollary (3.4.6) thus recovers (2.7.9).

We thus conclude that, when $Y$ is an arbitrary prescheme, the following three conditions on a quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ such that $p_{\ast }(\mathcal{F})$ is

quasi-coherent are equivalent:

• a) The canonical homomorphism $\sigma :p\ast (p_{\ast }(\mathcal{F}))\to \mathcal{F}$ is surjective.
• b) There exists a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{G}$ and a surjective homomorphism $p\ast (\mathcal{G})\to \mathcal{F}$.
• c) For every affine open $U$ of $Y$, $\mathcal{F}|p^{-1}(U)$ is generated by its sections over $p^{-1}(U)$.

We have just shown the equivalence of a) and c). On the other hand, it is clear that a) implies b), $p_{\ast }(\mathcal{F})$ being quasi-coherent by hypothesis. Conversely, every homomorphism $u:p\ast (\mathcal{G})\to \mathcal{F}$ factors as $p\ast (\mathcal{G})\to p\ast (p_{\ast }(\mathcal{F}))\stackrel{\sigma }{\to }\mathcal{F}$ (0, 3.5.4.4), so if $u$ is surjective so is $\sigma$, which shows that b) implies a).

Corollary.

Suppose the hypotheses of (3.4.4) are satisfied, and further that $Y$ is a quasi-compact scheme or that the underlying space of $Y$ is Noetherian. Let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module of finite type; then there exists an integer $n_0$ such that for every $n\ge n_0$, $\mathcal{F}$ is isomorphic to a quotient of an ${\mathcal{O}}_X$-module of the form $(p\ast (\mathcal{G}))(-n)$, where $\mathcal{G}$ is a quasi-coherent ${\mathcal{O}}_Y$-module of finite type (depending on $n$).

Proof. Since the structure morphism $X\to Y$ is separated and of finite type, $p_{\ast }(\mathcal{F}(n))$ is quasi-coherent ((I, 9.2.2, b)), hence the inductive limit of its quasi-coherent sub-${\mathcal{O}}_Y$-modules of finite type, by the hypothesis on $Y$ (I, 9.4.9). We thus deduce from (3.4.6), (0, 4.3.2), and (0, 5.2.3) that $\mathcal{F}(n)$ is the canonical image of an ${\mathcal{O}}_X$-module of the form $p\ast (\mathcal{G})$, where $\mathcal{G}$ is a quasi-coherent sub-${\mathcal{O}}_Y$-module of $p_{\ast }(\mathcal{F}(n))$ of finite type; the corollary follows from (3.2.5.2) and (3.2.7.1).

3.5. Functorial behaviour

(3.5.1)

Let $Y$ be a prescheme, $\mathcal{S}$, ${\mathcal{S}}^{\prime }$ two quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebras; set $X=\mathrm{Proj}(\mathcal{S})$, $X^{\prime }=\mathrm{Proj}({\mathcal{S}}^{\prime })$, and let $p$, $p^{\prime }$ be the structure morphisms of $X$ and $X^{\prime }$ into $Y$. Let $\varphi :{\mathcal{S}}^{\prime }\to \mathcal{S}$ be an ${\mathcal{O}}_Y$-homomorphism of graded algebras. For every affine open $U$ of $Y$, set $S_U=\Gamma (U,\mathcal{S})$, $S_U^{\prime }=\Gamma (U,{\mathcal{S}}^{\prime })$; the homomorphism $\varphi$ defines a homomorphism ${\varphi }_U:S_U^{\prime }\to S_U$ of graded A_U-algebras, where $A_U=\Gamma (U,{\mathcal{O}}_Y)$. There corresponds in $p^{-1}(U)$ an open subset $G({\varphi }_U)$ and a morphism ${\Phi }_U:G({\varphi }_U)\to p^{\prime -1}(U)$ (2.8.1). Furthermore, if $V\subset U$ is an affine open, the diagram

              φ_U
   S'_U ───────────→ S_U                                                  (3.5.1.1)
     │                │
     ↓                ↓
   S'_V ───────────→ S_V
              φ_V

commutes, and one checks at once from the definitions (2.8.1) that

$$G({\varphi }_V)=G({\varphi }_U)\cap p^{-1}(V)$$

and that ${\Phi }_V$ is the restriction of ${\Phi }_U$ to $G({\varphi }_V)$. We have thus defined an open subset $G(\varphi )$ of $X$ such that $G(\varphi )\cap p^{-1}(U)=G({\varphi }_U)$ for every affine open $U\subset Y$, and an affine $Y$-morphism $\Phi :G(\varphi )\to X^{\prime }$, which we say is

associated to $\varphi$ and which we denote by $\mathrm{Proj}(\varphi )$. When, for every $y\in Y$, there is an affine neighbourhood $U$ of $y$ such that the $\Gamma (U,{\mathcal{O}}_Y)$-module $\Gamma (U,{\mathcal{S}}_{+})$ is generated by $\varphi (\Gamma (U,{\mathcal{S}}_{+}^{\prime }))$, then $G({\varphi }_U)=p^{-1}(U)$, and so $G(\varphi )=X$.

Proposition.

1. If $\mathcal{M}$ is a quasi-coherent graded $\mathcal{S}$-module, there is a canonical functorial isomorphism from the ${\mathcal{O}}_{X^{\prime }}$-module ${\mathcal{M}}_{[\varphi ]}$ to the ${\mathcal{O}}_{X^{\prime }}$-module ${\Phi }_{\ast }(\mathcal{M}|G(\varphi ))$.
2. If ${\mathcal{M}}^{\prime }$ is a quasi-coherent graded ${\mathcal{S}}^{\prime }$-module, there is a canonical functorial homomorphism $\nu$ from the $({\mathcal{O}}_X|G(\varphi ))$-module $\Phi \ast ({\mathcal{M}}^{\prime })$ to the $({\mathcal{O}}_X|G(\varphi ))$-module ${\mathcal{M}}^{\prime }{\otimes }_{{\mathcal{S}}^{\prime }}\mathcal{S}|G(\varphi )$. If ${\mathcal{S}}^{\prime }$ is generated by ${\mathcal{S}}_1^{\prime }$, then $\nu$ is an isomorphism.

Proof. The homomorphisms considered have already been defined when $Y$ is affine ((2.8.7) and (2.8.8)); in the general case, it suffices to check that they are compatible with restriction from an affine open of $Y$ to a smaller one, which follows at once from the commutativity of (3.5.1.1).

In particular, for every $n\in \mathbb{Z}$, we have a canonical homomorphism

$$\Phi \ast ({\mathcal{O}}_{X^{\prime }}(n))\to {\mathcal{O}}_X(n)|G(\varphi ).(\mathrm{3.5.2.1})$$

Proposition.

Let $Y$, $Y^{\prime }$ be two preschemes, $\psi :Y^{\prime }\to Y$ a morphism, and $\mathcal{S}$ a quasi-coherent graded ${\mathcal{O}}_Y$-algebra; set ${\mathcal{S}}^{\prime }=\psi \ast (\mathcal{S})$. Then the $Y^{\prime }$-scheme $X^{\prime }=\mathrm{Proj}({\mathcal{S}}^{\prime })$ is canonically identified with $\mathrm{Proj}(\mathcal{S}){\times }_Y{Y}^{\prime }$. Furthermore, if $\mathcal{M}$ is a quasi-coherent graded $\mathcal{S}$-module, then the ${\mathcal{O}}_{X^{\prime }}$-module $\psi \ast (\mathcal{M})$ is identified with $\mathcal{M}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$.

Proof. First note that $\psi \ast (\mathcal{S})$ and $\psi \ast (\mathcal{M})$ are quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-modules, as are their homogeneous components (0, 5.1.4). Let $U$ be an affine open of $Y$, $U^{\prime }\subset {\psi }^{-1}(U)$ an affine open of $Y^{\prime }$, $A$, $A^{\prime }$ the rings of $U$, $U^{\prime }$; then $\mathcal{S}|U=S$, with $S$ a graded $A$-algebra, and ${\mathcal{S}}^{\prime }|U^{\prime }$ is identified with $S{\otimes }_A{A}^{\prime }$ (I, 1.6.5); the first assertion then follows from (2.8.10) and (I, 3.2.6.2), since one verifies at once that the projection $\mathrm{Proj}({\mathcal{S}}^{\prime }|U^{\prime })\to \mathrm{Proj}(\mathcal{S}|U)$ defined by the above identification is compatible with restriction on $U$ and $U^{\prime }$ and so defines a morphism $\mathrm{Proj}({\mathcal{S}}^{\prime })\to \mathrm{Proj}(\mathcal{S})$. Now let

  q  : Proj(𝒮) → Y,    q' : Proj(𝒮') → Y'

be the structure morphisms; $q^{\prime -1}(U^{\prime })$ is then identified with $q^{-1}(U){\times }_U{U}^{\prime }$, and the two sheaves $\psi \ast (\mathcal{M})|q^{\prime -1}(U^{\prime })$ and $(\mathcal{M}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }})|q^{\prime -1}(U^{\prime })$ are then both canonically identified with $\widetilde{M{\otimes }_A{A}^{\prime }}$, where $M=\Gamma (U,\mathcal{M})$, by (2.8.10) and (I, 1.6.5); whence the second assertion, the compatibility of the above identifications with restriction again being immediate.

Corollary.

With the notation of (3.5.3), ${\mathcal{O}}_{X^{\prime }}(n)$ is canonically identified with ${\mathcal{O}}_X(n){\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$ for every $n\in \mathbb{Z}$ (with $X=\mathrm{Proj}(\mathcal{S})$).

Proof. Indeed, with the notation of (3.5.3), it is clear that $\psi \ast (\mathcal{S}(n))={\mathcal{S}}^{\prime }(n)$ for every $n\in \mathbb{Z}$.

(3.5.5)

Keeping the previous notation, denote by $\Psi$ the canonical projection $X^{\prime }\to X$, and set ${\mathcal{M}}^{\prime }=\psi \ast (\mathcal{M})$; we further suppose that $\mathcal{S}$ is generated by ${\mathcal{S}}_1$

and that $X$ is of finite type over $Y$; it then follows that ${\mathcal{S}}^{\prime }$ is generated by ${\mathcal{S}}_1^{\prime }$ (as one sees by reducing to the case $Y$ and $Y^{\prime }$ affine) and that $X^{\prime }$ is of finite type over $Y^{\prime }$ (I, 6.3.4). Let $\mathcal{F}$ be an ${\mathcal{O}}_X$-module and set ${\mathcal{F}}^{\prime }=\Psi \ast (\mathcal{F})$; it then follows from (3.5.4) and (0, 4.3.3) that ${\mathcal{F}}^{\prime }(n)=\Psi \ast (\mathcal{F}(n))$ for every $n\in \mathbb{Z}$. We further define a canonical $\Psi$-homomorphism ${\theta }_n:q_{\ast }(\mathcal{F}(n))\to q_{\ast }^{\prime }({\mathcal{F}}^{\prime }(n))$ as follows: in view of the commutativity of the diagram

                  Ψ
   X ←───────── X'
   │             │
 q │             │ q'
   ↓             ↓
   Y ←───────── Y'
                  ψ

it suffices to define a homomorphism $q_{\ast }(\mathcal{F}(n))\to {\psi }_{\ast }(q_{\ast }^{\prime }(\Psi \ast (\mathcal{F}(n))))=q_{\ast }({\Psi }_{\ast }(\Psi \ast (\mathcal{F}(n))))$, and we take the homomorphism ${\theta }_n=q_{\ast }({\rho }_n)$, with ${\rho }_n$ the canonical homomorphism $\mathcal{F}(n)\to {\Psi }_{\ast }(\Psi \ast (\mathcal{F}(n)))$ (0, 4.4.3). It is immediate that for every affine open $U$ of $Y$ and every affine open $U^{\prime }$ of $Y^{\prime }$ with $U^{\prime }\subset {\psi }^{-1}(U)$, the homomorphism ${\theta }_n$ gives, on sections, the canonical homomorphism (0, 3.7.2) $\Gamma (q^{-1}(U),\mathcal{F}(n))\to \Gamma (q^{\prime -1}(U^{\prime }),{\mathcal{F}}^{\prime }(n))$.