§8. Blow-up preschemes; projecting cones; projective closure

8.1. Blow-up preschemes

(8.1.1)

Let $Y$ be a prescheme, and, for every integer $n\ge 0$, let ${\mathcal{I}}_n$ be a quasi-coherent sheaf of ideals of ${\mathcal{O}}_Y$; suppose that the following conditions are satisfied:

  ℐ₀ = 𝒪_Y,    ℐₙ ⊂ ℐₘ    for m ≤ n,                                       (8.1.1.1)

  ℐₘ · ℐₙ ⊂ ℐ_{m+n}     for any m, n.                                       (8.1.1.2)

Note that these hypotheses imply

$${\mathcal{I}}_1^n\subset {\mathcal{I}}_n.(\mathrm{8.1.1.3})$$

Set

$$\mathcal{S}={\oplus }_{n\ge 0}{\mathcal{I}}_n.(\mathrm{8.1.1.4})$$

It follows from (8.1.1.1) and (8.1.1.2) that $\mathcal{S}$ is a quasi-coherent graded ${\mathcal{O}}_Y$-algebra, and therefore defines a $Y$-scheme $X=\mathrm{Proj}(\mathcal{S})$. If $\mathcal{J}$ is an invertible sheaf of ideals of ${\mathcal{O}}_Y$, then ${\mathcal{I}}_n{\otimes }_{{\mathcal{O}}_Y}{\mathcal{J}}^{\otimes n}$ is canonically identified with ${\mathcal{I}}_n\cdot {\mathcal{J}}^n$. If we then replace the ${\mathcal{I}}_n$ by the ${\mathcal{I}}_n\cdot {\mathcal{J}}^n$, and in doing so replace $\mathcal{S}$ by a quasi-coherent ${\mathcal{O}}_Y$-algebra ${\mathcal{S}}_{(\mathcal{J})}$, then $X_{(\mathcal{J})}=\mathrm{Proj}({\mathcal{S}}_{(\mathcal{J})})$ is canonically isomorphic to $X$ (3.1.8).

(8.1.2)

Suppose $Y$ is locally integral, so that the sheaf $\mathcal{R}(Y)$ of rational functions is a quasi-coherent ${\mathcal{O}}_Y$-algebra (I, 7.3.7). We say that an ${\mathcal{O}}_Y$-submodule $\mathcal{I}$ of $\mathcal{R}(Y)$ is a fractional ideal of $\mathcal{R}(Y)$ if it is of finite type (0, 5.2.1). Suppose given, for every $n\ge 0$, a quasi-coherent fractional ideal ${\mathcal{I}}_n$ of $\mathcal{R}(Y)$ such that ${\mathcal{I}}_0={\mathcal{O}}_Y$ and such that condition (8.1.1.2) is satisfied (but not necessarily the second condition (8.1.1.1)); we can then again define, by formula (8.1.1.4), a quasi-coherent graded ${\mathcal{O}}_Y$-algebra and the corresponding $Y$-scheme $X=\mathrm{Proj}(\mathcal{S})$. We again have a canonical isomorphism from $X$ to $X_{(\mathcal{J})}$ for every invertible fractional ideal $\mathcal{J}$ of $\mathcal{R}(Y)$.

Definition.

Let $Y$ be a prescheme (resp. a locally integral prescheme), and let $\mathcal{I}$ be a quasi-coherent sheaf of ideals of ${\mathcal{O}}_Y$ (resp. a quasi-coherent fractional ideal of $\mathcal{R}(Y)$). We say that the $Y$-scheme $X=\mathrm{Proj}({\oplus }_{n\ge 0}{\mathcal{I}}^n)$ is obtained by blowing up the ideal $\mathcal{I}$, or is the blow-up prescheme of $Y$ relative to $\mathcal{I}$. When $\mathcal{I}$ is a quasi-coherent sheaf of ideals of ${\mathcal{O}}_Y$ and $Y^{\prime }$ the closed subprescheme of $Y$ defined by $\mathcal{I}$, we also say that $X$ is the $Y$-scheme obtained by blowing up $Y^{\prime }$.

By definition, $\mathcal{S}={\oplus }_{n\ge 0}{\mathcal{I}}^n$ is then generated by ${\mathcal{S}}_1=\mathcal{I}$; if $\mathcal{I}$ is an ${\mathcal{O}}_Y$-module of finite type, then $X$ is projective over $Y$ (5.5.2). Without any hypothesis on $\mathcal{I}$, the ${\mathcal{O}}_X$-module ${\mathcal{O}}_X(1)$ is invertible (3.2.5) and very ample, by (4.4.3) applied to the structure morphism $X\to Y$.

We note that, if $f:X\to Y$ is the structure morphism, then the restriction of $f$ to $f^{-1}(Y\setminus Y^{\prime })$ is an isomorphism onto $Y\setminus Y^{\prime }$ whenever $\mathcal{I}$ is an ideal of ${\mathcal{O}}_Y$ and $Y^{\prime }$ is the closed subprescheme it defines: indeed, since the question is local on $Y$, it suffices to assume that $\mathcal{I}={\mathcal{O}}_Y$, and our claim then follows from (3.1.7).

If we replace $\mathcal{I}$ by ${\mathcal{I}}^d$ ($d>0$), the blow-up $Y$-scheme $X$ is replaced by a canonically isomorphic $Y$-scheme $X^{\prime }$ (8.1.1); similarly, for every invertible ideal (resp. invertible fractional ideal) $\mathcal{J}$, the blow-up prescheme $X_{(\mathcal{J})}$ relative to the ideal $\mathcal{I}\mathcal{J}$ is canonically isomorphic to $X$ (8.1.1).

In particular, whenever $\mathcal{I}$ is an invertible ideal (resp. invertible fractional ideal), the $Y$-scheme obtained by blowing up $\mathcal{I}$ is isomorphic to $Y$ (3.1.7).

Proposition.

Let $Y$ be an integral prescheme.

1. For every sequence $({\mathcal{I}}_n)$ of quasi-coherent fractional ideals of $\mathcal{R}(Y)$ satisfying (8.1.1.2)

   and such that ${\mathcal{I}}_0={\mathcal{O}}_Y$, the $Y$-scheme $X=\mathrm{Proj}({\oplus }_{n\ge 0}{\mathcal{I}}_n)$ is integral and the structure morphism $f:X\to Y$ is dominant.

2. Let $\mathcal{I}$ be a quasi-coherent fractional ideal of $\mathcal{R}(Y)$, and let $X$ be the $Y$-scheme given by the blow-up of $Y$ relative to $\mathcal{I}$. If $\mathcal{I}\ne 0$, then the structure morphism $f:X\to Y$ is birational and surjective.

Proof.

(i) This follows from the fact that $\mathcal{S}={\oplus }_{n\ge 0}{\mathcal{I}}_n$ is an integral ${\mathcal{O}}_Y$-algebra (3.1.12 and 3.1.14), since for every $y\in Y$, ${\mathcal{O}}_y$ is an integral ring (I, 5.1.4).

(ii) By (i), $X$ is integral; if, furthermore, $x$ and $y$ are the generic points of $X$ and $Y$ (respectively), then $f(x)=y$, and it remains to show that $\kappa (x)$ is of rank 1 over $\kappa (y)$. But $x$ is also the generic point of the fibre $f^{-1}(y)$; if $\psi$ is the canonical morphism $Z\to Y$, where $Z=\mathrm{Spec}(\kappa (y))$, then the prescheme $f^{-1}(y)$ is identified with $\mathrm{Proj}({\mathcal{S}}^{\prime })$, where ${\mathcal{S}}^{\prime }=\psi \ast (\mathcal{S})$ (3.5.3). It is clear that ${\mathcal{S}}^{\prime }={\oplus }_{n\ge 0}({\mathcal{I}}_y{)}^n$, and since $\mathcal{I}$ is a quasi-coherent fractional ideal of $\mathcal{R}(Y)$ that is not zero, ${\mathcal{I}}_y\ne 0$ (I, 7.3.6), whence ${\mathcal{I}}_y=\kappa (y)$; then $\mathrm{Proj}({\mathcal{S}}^{\prime })$ is identified with $\mathrm{Spec}(\kappa (y))$ (3.1.7), whence the conclusion.

We will prove a converse of (8.1.4) in (III, 2.3.8).

(8.1.5)

We return to the setting and notation of (8.1.1). By definition, the injection homomorphisms ${\mathcal{I}}_{n+1}\to {\mathcal{I}}_n$ (8.1.1.1) define, for every $k\in \mathbb{Z}$, an injective homomorphism of degree zero of graded $\mathcal{S}$-modules

$$u_k:{\mathcal{S}}_{+}(k+1)\to \mathcal{S}(k);(\mathrm{8.1.5.1})$$

since ${\mathcal{S}}_{+}(k+1)$ and $\mathcal{S}(k+1)$ are canonically (TN)-isomorphic, they give a canonical correspondence between $u_k$ and an injective homomorphism of ${\mathcal{O}}_X$-modules (3.4.2):

$${\tilde{u}}_k:{\mathcal{O}}_X(k+1)\to {\mathcal{O}}_X(k).(\mathrm{8.1.5.2})$$

Recall (3.2.6) that we have defined canonical homomorphisms

  λ : 𝒪_X(h) ⊗_{𝒪_X} 𝒪_X(k) → 𝒪_X(h+k)                                      (8.1.5.3)

and since the diagram

   𝒮(h) ⊗_𝒮 𝒮(k) ⊗_𝒮 𝒮(l) ──→ 𝒮(h+k) ⊗_𝒮 𝒮(l)
           │                          │
           ↓                          ↓
   𝒮(h) ⊗_𝒮 𝒮(k+l) ─────────→ 𝒮(h+k+l)

commutes, it follows from the functoriality of the $\lambda$ (3.2.6) that the homomorphisms (8.1.5.3) define on

  𝒮_X = ⊕_{n ∈ ℤ} 𝒪_X(n)                                                    (8.1.5.4)

the structure of a quasi-coherent graded ${\mathcal{O}}_X$-algebra. Furthermore, the diagram

   𝒮(h) ⊗_𝒮 𝒮(k+1) ──────→ 𝒮(h+k+1)
        │                         │
   1⊗u_k│                         │u_{k+h}
        ↓                         ↓
   𝒮(h) ⊗_𝒮 𝒮(k) ────────→ 𝒮(h+k)

commutes; the functoriality of the $\lambda$ then implies that we have a commutative diagram

   𝒪_X(h) ⊗_{𝒪_X} 𝒪_X(k+1) ──λ──→ 𝒪_X(h+k+1)
            │                           │
      1⊗ũ_k │                           │ũ_{k+h}                            (8.1.5.5)
            ↓                           ↓
   𝒪_X(h) ⊗_{𝒪_X} 𝒪_X(k) ───λ──→ 𝒪_X(h+k)

where the horizontal arrows are the canonical homomorphisms. We can thus say that the ${\tilde{u}}_k$ define an injective homomorphism (of degree zero) of graded ${\mathcal{S}}_X$-modules

$$\tilde{u}:{\mathcal{S}}_X(1)\to {\mathcal{S}}_X.(\mathrm{8.1.5.6})$$

(8.1.6)

Keeping the notation of (8.1.5), we now note that, for $n\ge 0$, the composite homomorphism $v_n=u_{n-1}\circ u_{n-2}\circ \cdots \circ u_0$ is an injective homomorphism ${\mathcal{O}}_X(n)\to {\mathcal{O}}_X$; we denote by ${\mathcal{I}}_{n,X}$ its image, which is thus a quasi-coherent sheaf of ideals of ${\mathcal{O}}_X$, isomorphic to ${\mathcal{O}}_X(n)$. Furthermore, the diagram

   𝒪_X(m) ⊗_{𝒪_X} 𝒪_X(n) ──λ──→ 𝒪_X(m+n)
            │                          │
     ṽ_m⊗ṽ_n│                          │ṽ_{m+n}
            ↓                          ↓
   𝒪_X ⊗_{𝒪_X} 𝒪_X ────id────→ 𝒪_X

commutes for $m\ge 0$, $n\ge 0$. From this we deduce the following inclusions:

  ℐ_{0,X} = 𝒪_X,    ℐ_{n,X} ⊂ ℐ_{m,X}      for 0 ≤ m ≤ n,                   (8.1.6.1)

  ℐ_{m,X} · ℐ_{n,X} ⊂ ℐ_{m+n,X}            for m ≥ 0, n ≥ 0.                (8.1.6.2)

Proposition.

Let $Y$ be a prescheme, $\mathcal{I}$ a quasi-coherent sheaf of ideals of ${\mathcal{O}}_Y$, and $X=\mathrm{Proj}({\oplus }_{n\ge 0}{\mathcal{I}}^n)$ the $Y$-scheme obtained by blowing up $\mathcal{I}$. Then, for every $n>0$, there is a canonical isomorphism

  𝒪_X(n) ⥲ ℐⁿ · 𝒪_X = ℐ_{n,X}                                                (8.1.7.1)

(cf. (0, 4.3.5)), and consequently ${\mathcal{I}}^n\cdot {\mathcal{O}}_X$ is a very ample invertible ${\mathcal{O}}_X$-module if $n>0$.

Proof. The last assertion is immediate, since ${\mathcal{O}}_X(1)$ is invertible (3.2.5) and very ample for $Y$ by definition (4.4.3 and 4.4.9). On the other hand, by definition, the image of $v_n$ is precisely ${\mathcal{I}}^n\mathcal{S}$, and (8.1.7.1) thus follows from the exactness of the functor $\tilde{M}$ (3.2.4) and from formula (3.2.4.1).

Corollary.

Under the hypotheses of (8.1.7), if $f:X\to Y$ is the structure morphism and $Y^{\prime }$ is the closed subprescheme of $Y$ defined by $\mathcal{I}$, then the closed subprescheme $X^{\prime }=f^{-1}(Y^{\prime })$ of $X$ is defined by $\mathcal{I}\cdot {\mathcal{O}}_X$ (which is canonically isomorphic to ${\mathcal{O}}_X(1)$), whence a canonical short exact sequence

$$0\to {\mathcal{O}}_X(1)\to {\mathcal{O}}_X\to {\mathcal{O}}_{X^{\prime }}\to 0.(\mathrm{8.1.8.1})$$

Proof. This follows from (8.1.7.1) and from (I, 4.4.5).

(8.1.9)

Under the hypotheses of (8.1.7), we can be more precise about the structure of the ${\mathcal{I}}_{n,X}$. Note that the homomorphism

$${\tilde{u}}_{-1}:{\mathcal{O}}_X\to {\mathcal{O}}_X(-1)$$

canonically corresponds to a section $s$ of ${\mathcal{O}}_X(-1)$ over $X$, which we call the canonical section (relative to $\mathcal{I}$) (0, 5.1.1). On the other hand, in the diagram (8.1.5.5) the horizontal arrows are isomorphisms (3.2.7); by replacing in this diagram $h$ by $k$ and $k$ by $-1$, we obtain $u_k=1_k\otimes u_{-1}$ (where $1_k$ denotes the identity on ${\mathcal{O}}_X(k)$); in other words, the homomorphism ${\tilde{u}}_k$ is nothing other than tensor multiplication by the canonical section $s$ (for every $k\in \mathbb{Z}$). The homomorphism ũ (8.1.5.6) is thus interpreted in the same way.

Consequently, for every $n\ge 0$, the homomorphism ${\tilde{v}}_n:{\mathcal{O}}_X(n)\to {\mathcal{O}}_X$ is nothing other than tensor multiplication by $s^{\otimes n}$; we deduce:

Corollary.

With the notation of (8.1.8), the underlying space of $X^{\prime }$ is the set of $x\in X$ such that $s(x)=0$, where $s$ denotes the canonical section of ${\mathcal{O}}_X(-1)$.

Proof. Indeed, if $c_x$ is a generator of the fibre $({\mathcal{O}}_X(1){)}_x$ at a point $x$, then $s_x\otimes c_x$ is canonically identified with a generator of the fibre of ${\mathcal{I}}_{1,X}$ at the point $x$, and is therefore invertible if and only if $s_x\notin {\mathfrak{m}}_x({\mathcal{O}}_X(-1){)}_x$, or, equivalently, if and only if $s(x)\ne 0$.

Proposition.

Let $Y$ be an integral prescheme, $\mathcal{I}$ a quasi-coherent fractional ideal of $\mathcal{R}(Y)$, and $X$ the $Y$-scheme obtained by blowing up $\mathcal{I}$. Then $\mathcal{I}\cdot {\mathcal{O}}_X$ is an invertible ${\mathcal{O}}_X$-module that is very ample for $Y$.

Proof. Since the question is local on $Y$ (4.4.5), we may restrict to the case where $Y=\mathrm{Spec}(A)$, with $A$ an integral ring with field of fractions $K$, and $\mathcal{I}=\tilde{\mathfrak{J}}$, where $\mathfrak{J}$ is a fractional ideal of $K$; there then exists an element $a\ne 0$ of $A$ such that $a\mathfrak{J}\subset A$. Set $S={\oplus }_{n\ge 0}{\mathfrak{J}}^n$; the map $x\mapsto ax$ is an $A$-isomorphism from ${\mathfrak{J}}^{n+1}=(S(1){)}_n$ onto $a{\mathfrak{J}}^{n+1}=a\mathfrak{J}\cdot S_n\subset {\mathfrak{J}}^n=S_n$,

and thus defines a (TN)-isomorphism of degree zero of graded $S$-modules $S_{+}(1)\to a\mathfrak{J}\cdot S$. On the other hand, $x\mapsto a^{-1}x$ is an isomorphism of degree zero of graded $S$-modules $a\mathfrak{J}\cdot S\stackrel{\sim }{\to }\mathfrak{J}\cdot S$. By composition (3.2.4), we thus obtain an isomorphism of ${\mathcal{O}}_X$-modules ${\mathcal{O}}_X(1)\stackrel{\sim }{\to }\mathcal{I}\cdot {\mathcal{O}}_X$, and since $S$ is generated by $S_1=\mathfrak{J}$, ${\mathcal{O}}_X(1)$ is invertible (3.2.5) and very ample (4.4.3 and 4.4.9), whence our assertion.

8.2. Preliminary results on localisation in graded rings

(8.2.1)

Let $S$ be a graded ring, where for the moment we do not assume the degrees to be positive. We set

  S^≥ = ⊕_{n≥0} S_n,    S^≤ = ⊕_{n≤0} S_n                                   (8.2.1.1)

which are graded subrings of $S$, with degrees respectively all positive and all negative. If $f$ is a homogeneous element of degree $d$ (positive or negative) of $S$, then the ring of fractions $S_f=S^{\prime }$ is again endowed with the structure of a graded ring, by taking for $S_n^{\prime }$ ($n\in \mathbb{Z}$) the set of $x/f^k$ with $x\in S_{n+kd}$ ($k\ge 0$); we set $S_{(f)}=S_0^{\prime }$, and we will write $S_f^{\ge }$ and $S_f^{\le }$ for $S^{\prime \ge }$ and $S^{\prime \le }$ respectively. If $d>0$, then

$$(S^{\ge }{)}_f=S_f(\mathrm{8.2.1.2})$$

since, if $x\in S_{n+kd}$ with $n+kd<0$, we can write $x/f^k=x{f}^h/f^{h+k}$ and $n+(h+k)d>0$ for $h$ sufficiently large and > 0. We conclude by definition that

$$(S^{\ge }{)}_{(f)}=(S_f^{\ge }{)}_0=S_{(f)}.(\mathrm{8.2.1.3})$$

If $M$ is a graded $S$-module, we similarly set

  M^≥ = ⊕_{n≥0} M_n,    M^≤ = ⊕_{n≤0} M_n                                   (8.2.1.4)

which are respectively a graded $S^{\ge }$-module and a graded $S^{\le }$-module, and whose intersection is the S_0-module M_0. If $f\in S_d$, we again define $M_f$ as the graded $S_f$-module whose elements of degree $n$ are the $z/f^k$ for $z\in M_{n+kd}$ ($k\ge 0$); we denote by $M_{(f)}$ the set of elements of degree zero of $M_f$, which is an $S_{(f)}$-module, and we will write $M_f^{\ge }$ and $M_f^{\le }$ for $(M_f{)}^{\ge }$ and $(M_f{)}^{\le }$ respectively. If $d>0$, we see as above that

$$(M^{\ge }{)}_f=M_f(\mathrm{8.2.1.5})$$

and

$$(M^{\ge }{)}_{(f)}=(M_f^{\ge }{)}_0=M_{(f)}.(\mathrm{8.2.1.6})$$

(8.2.2)

Let $\mathbf{z}$ be an indeterminate, which we shall call the homogenisation variable. If $S$ is a graded ring (in positive or negative degrees), then the polynomial algebra [¹]

$$\hat{S}=S[\mathbf{z}](\mathrm{8.2.2.1})$$

[¹] There can be no confusion here with the use of the notation Ŝ to denote the separated completion of a ring.

is a graded $S$-algebra, when we take for the degree of $f{\mathbf{z}}^n$ ($n\ge 0$), with $f$ homogeneous,

  deg(f𝐳ⁿ) = n + deg f.                                                      (8.2.2.2)

Lemma (8.2.3).

1. We have canonical isomorphisms of (non-graded) rings

     Ŝ_{(𝐳)} ⥲ Ŝ/(𝐳 − 1)Ŝ ⥲ S.                                            (8.2.3.1)
   

2. We have a canonical isomorphism of (non-graded) rings

     Ŝ_{(f)} ⥲ S_f^≤                                                       (8.2.3.2)
   

   for every $f\in S_d$, with $d>0$.

Proof. The first of the isomorphisms in (8.2.3.1) was defined in (2.2.5) and the second is trivial; the isomorphism ${\hat{S}}_{(\mathbf{z})}\stackrel{\sim }{\to }S$ thus defined makes correspond to $x{\mathbf{z}}^n/{\mathbf{z}}^{n+k}$ (where $deg(x)=k$, $k\ge -n$) the element $x$. The homomorphism (8.2.3.2) makes correspond to $x{\mathbf{z}}^n/f^k$ (where $deg(x)=kd-n$) the element $x/f^k$, of degree $-n$ in $S_f^{\le }$, and it is again clear that we have an isomorphism here.

(8.2.4)

Let $M$ be a graded $S$-module. It is clear that the $S$-module

  M̂ = M ⊗_S Ŝ = M ⊗_S S[𝐳]                                                 (8.2.4.1)

is the direct sum of the $S$-modules $M\otimes S{\mathbf{z}}^n$, and thus of the abelian groups $M_k\otimes S{\mathbf{z}}^n$ ($k\in \mathbb{Z}$, $n\ge 0$); we define on $\hat{M}$ the structure of a graded Ŝ-module by setting

  deg(x ⊗ 𝐳ⁿ) = n + deg x                                                    (8.2.4.2)

for every homogeneous $x$ in $M$. We leave to the reader the task of proving the analogue of (8.2.3):

Lemma (8.2.5).

1. There is a canonical di-isomorphism of (non-graded) modules

     M̂_{(𝐳)} ⥲ M.                                                          (8.2.5.1)
   

2. For every $f\in S_d$ ($d>0$), there is a di-isomorphism of (non-graded) modules

     M̂_{(f)} ⥲ M_f^≤.                                                      (8.2.5.2)
   

(8.2.6)

Let $S$ be a positively-graded ring, and consider the decreasing sequence of graded ideals of $S$

  S_{[n]} = ⊕_{m≥n} S_m       (n ≥ 0)                                       (8.2.6.1)

(so in particular $S_{[0]}=S$ and $S_{[1]}=S_{+}$). Since $S_{[m]}{S}_{[n]}\subset S_{[m+n]}$ is clear, we can define a graded ring $S^{♮}$ by setting

  S^♮ = ⊕_{n≥0} S_n^♮     with   S_n^♮ = S_{[n]}.                            (8.2.6.2)

$S_0^{♮}$ is then the ring $S$ considered as a non-graded ring, and $S^{♮}$ is consequently an $S_0^{♮}$-algebra. For every homogeneous element $f\in S_d$ ($d>0$), we denote by $f^{♮}$ the element $f$ considered as belonging to $S_{[d]}=S_d^{♮}$. With this notation:

Lemma (8.2.7).

Let $S$ be a positively-graded ring, $f$ a homogeneous element of $S_d$ ($d>0$). We have canonical ring isomorphisms

  S_f ⥲ ⊕_{n ∈ ℤ} S(n)_{(f)}                                                 (8.2.7.1)

$$(S_f^{\ge }{)}_{f/1}\stackrel{\sim }{\to }{S}_f(\mathrm{8.2.7.2})$$

$$S_{(f^{♮})}^{♮}\stackrel{\sim }{\to }{S}_f^{\ge }(\mathrm{8.2.7.3})$$

where the first two are isomorphisms of graded rings.

Proof. It is immediate, by definition, that $(S_f{)}_n=(S(n{)}_f{)}_0$, whence the isomorphism (8.2.7.1), which is in fact the identity. Next, since $f/1$ is invertible in $S_f$, there is a canonical isomorphism $S_f\stackrel{\sim }{\to }(S_f^{\ge }{)}_{f/1}=(S_f{)}_{f/1}$, by (8.2.1.2) applied to $S_f$; the inverse isomorphism is, by definition, the isomorphism (8.2.7.2). Finally, if $x={\sum }_{m\ge n}{y}_m$ is an element of $S_{[n]}$ with $n=kd$, then the element $x/(f^{♮}{)}^k$ corresponds to the element ${\sum }_m{y}_m/f^k$ of $S_f^{\ge }$, and we quickly verify that this defines an isomorphism (8.2.7.3).

(8.2.8)

If $M$ is a graded $S$-module, we similarly define, for every $n\in \mathbb{Z}$,

$$M_{[n]}={\oplus }_{m\ge n}{M}_m(\mathrm{8.2.8.1})$$

and, since $S_{[m]}{M}_{[n]}\subset M_{[m+n]}$ ($m\ge 0$), we can define a graded $S^{♮}$-module $M^{♮}$ by setting

  M^♮ = ⊕_{n ∈ ℤ} M_n^♮      with   M_n^♮ = M_{[n]}.                         (8.2.8.2)

We leave to the reader the proof of:

Lemma (8.2.9).

With the notation of (8.2.7) and (8.2.8), there are canonical di-isomorphisms of modules

  M_f ⥲ ⊕_{n ∈ ℤ} M(n)_{(f)}                                                 (8.2.9.1)

$$(M_f^{\ge }{)}_{f/1}\stackrel{\sim }{\to }{M}_f(\mathrm{8.2.9.2})$$

$$M_{(f^{♮})}^{♮}\stackrel{\sim }{\to }{M}_f^{\ge }(\mathrm{8.2.9.3})$$

where the first two are di-isomorphisms of graded modules.

Lemma (8.2.10).

Let $S$ be a positively-graded ring.

1. For $S^{♮}$ to be an $S_0^{♮}$-algebra of finite type (resp. a Noetherian $S_0^{♮}$-algebra), it is necessary and sufficient that $S$ be an $S_0^{♮}$-algebra of finite type (resp. a Noetherian $S_0^{♮}$-algebra).
2. For $S_{n+1}^{♮}=S_1^{♮}{S}_n^{♮}$ ($n\ge n_0$), it is necessary and sufficient that $S_{n+1}=S_1{S}_n$ ($n\ge n_0$).
3. For $S_n^{♮}=(S_1^{♮}{)}^n$ ($n\ge n_0$), it is necessary and sufficient that $S_n=S_1^n$ ($n\ge n_0$).
4. If $(f_{\alpha })$ is a set of homogeneous elements of $S_{+}$ such that $S_{+}$ is the radical in $S_{+}$ of the ideal of $S_{+}$ generated by the $f_{\alpha }$, then $S_{+}^{♮}$ is the radical in $S_{+}^{♮}$ of the ideal of $S_{+}^{♮}$ generated by the $f_{\alpha }^{♮}$.

Proof.

(i) If $S^{♮}$ is an $S_0^{♮}$-algebra of finite type, then $S_{+}=S_1^{♮}$ is a module of finite type over $S=S_0^{♮}$, by (2.1.6, i), and so $S$ is an S_0-algebra of finite type (2.1.4); if $S^{♮}$ is a Noetherian ring, then so too is $S_0^{♮}=S$ (2.1.5). Conversely, if $S$ is an S_0-algebra

of finite type, then we know (2.1.6, ii) that there exist $h>0$ and $m_0>0$ such that $S_{n+h}=S_h{S}_n$ for $n\ge m_0$; we can clearly assume $m_0\ge h$. Furthermore, the $S_m$ are S_0-modules of finite type (2.1.6, i). So, if $n\ge m_0+h$, then $S_n^{♮}=S_h{S}_{n-h}^{♮}=S_h^{♮}{S}_{n-h}^{♮}$; and if $m<m_0+h$ then, letting $E=S_{m_0}+\cdots +S_{m_0+h-1}$, we have

  S_m^♮ = S_m + ⋯ + S_{m_0+h−1} + S_h E + S_h² E + ⋯.

For $1\le m\le m_0$, let $G_m$ be the union of finite systems of generators of the S_0-modules $S_i$ for $m\le i\le m_0+h-1$, considered as a subset of $S_{[m]}$. For $m_0+1\le m\le m_0+h-1$, let $G_m$ be the union of finite systems of generators of the S_0-modules $S_i$ for $m\le i\le m_0+h-1$ and of $S_{h}E$, considered as a subset of $S_{[m]}$. It is clear that $S_m^{♮}=S_0^{♮}{G}_m$ for $1\le m\le m_0+h-1$, and so the union $G$ of the $G_m$ for $1\le m\le m_0+h-1$ is a system of generators of the $S_0^{♮}$-algebra $S^{♮}$. We conclude that, if $S=S_0^{♮}$ is a Noetherian ring, then so too is $S^{♮}$.

(ii) It is clear that, if $S_{n+1}=S_1{S}_n$ for $n\ge n_0$, then $S_{n+1}^{♮}=S_1{S}_n^{♮}$, and a fortiori $S_{n+1}^{♮}=S_1^{♮}{S}_n^{♮}$ for $n\ge n_0$. Conversely, this last equality can be written

  S_{n+1} + S_{n+2} + ⋯ = (S_1 + S_2 + ⋯)(S_n + S_{n+1} + ⋯)

and comparing terms of degree $n+1$ (in $S$) on both sides gives $S_{n+1}=S_1{S}_n$.

(iii) If $S_n=S_1^n$ for $n\ge n_0$, then $S_n^{♮}=S_1^n+S_1^{n+1}+\cdots$; since $S_1^{♮}$ contains $S_1+S_1^2+\cdots$, we have $S_n^{♮}\subset (S_1^{♮}{)}^n$, hence $S_n^{♮}=(S_1^{♮}{)}^n$ for $n\ge n_0$. Conversely, the only terms of $(S_1^{♮}{)}^n=(S_1+S_2+\cdots {)}^n$ that are of degree $n$ in $S$ are those of $S_1^n$; the equality $S_n^{♮}=(S_1^{♮}{)}^n$ thus implies $S_n=S_1^n$.

(iv) It suffices to show that, if an element $g\in S_{k+h}$ is considered as an element of $S_k^{♮}$ ($k>0$, $h\ge 0$), then there exists an integer $n>0$ such that $g^n$ is a linear combination (in $S_{kn}^{♮}$) of the $f_{\alpha }^{♮}$ with coefficients in $S^{♮}$. By hypothesis, there exists an integer $m_0$ such that, for $m\ge m_0$, we have, in $S$, $g^m={\sum }_{\alpha }{c}_{\alpha m}{f}_{\alpha }$, where the indices $\alpha$ are independent of $m$; furthermore, we can clearly assume that the $c_{\alpha m}$ are homogeneous, with

  deg(c_{αm}) = m(k + h) − deg f_α

in $S$. So take $m_0$ sufficiently large to ensure $k{m}_0>deg{f}_{\alpha }$ for all $f_{\alpha }$ appearing in $g^{m_0}$; for all $\alpha$, let $c_{\alpha m}^{\prime }$ be the element $c_{\alpha m}$ considered as having degree $km-deg{f}_{\alpha }$ in $S^{♮}$; we then have, in $S^{♮}$, $g^m={\sum }_{\alpha }{c}_{\alpha m}^{\prime }{f}_{\alpha }^{♮}$, which completes the proof.

(8.2.11)

Consider the graded S_0-algebra

  S^♮ ⊗_S S_0 = S^♮/S_+ S^♮ = ⊕_{n≥0} S_{[n]}/S_+ S_{[n]}.                  (8.2.11.1)

Since $S_n$ is a quotient S_0-module of $S_{[n]}/S_{+}{S}_{[n]}$, there is a canonical homomorphism of graded S_0-algebras

  S^♮ ⊗_S S_0 → S                                                          (8.2.11.2)

which is clearly surjective, and consequently corresponds (2.9.2) to a canonical closed immersion

  Proj(S) → Proj(S^♮ ⊗_S S_0).                                              (8.2.11.3)

Proposition.

The canonical morphism (8.2.11.3) is bijective. For the homomorphism (8.2.11.2) to be (TN)-bijective, it is necessary and sufficient that there exist some $n_0$ such that $S_{n+1}=S_1{S}_n$ for $n\ge n_0$. If this latter condition is satisfied, then (8.2.11.3) is an isomorphism; the converse is true whenever $S$ is Noetherian.

Proof. To prove the first claim, it suffices (2.8.3) to show that the kernel $\mathfrak{J}$ of the homomorphism (8.2.11.2) consists of nilpotent elements. But if $f\in S_{[n]}$ is an element whose class modulo $S_{+}{S}_{[n]}$ belongs to this kernel, then this implies $f\in S_{[n+1]}$; then $f^{n+1}$, considered as an element of $S_{[n(n+1)]}$, is also an element of $S_{+}{S}_{[n(n+1)]}$, since it can be written as $f\cdot f^n$; so the class of $f^{n+1}$ modulo $S_{+}{S}_{[n(n+1)]}$ is zero, which proves our claim. Since the hypothesis that $S_{n+1}=S_1{S}_n$ for $n\ge n_0$ is equivalent to $S_{n+1}^{♮}=S_1^{♮}{S}_n^{♮}$ for $n\ge n_0$ (8.2.10, ii), this hypothesis is equivalent, by definition, to (8.2.11.2) being (TN)-injective, and thus (TN)-bijective, and so (8.2.11.3) is an isomorphism, by (2.9.1). Conversely, if (8.2.11.3) is an isomorphism, then the sheaf $\tilde{\mathfrak{J}}$ on $\mathrm{Proj}(S^{♮}{\otimes }_S{S}_0)$ is zero (2.9.2, i); since $S^{♮}{\otimes }_S{S}_0$ is Noetherian, as a quotient of $S^{♮}$ (8.2.10, i), we conclude from (2.7.3) that $\mathfrak{J}$ satisfies condition (TN), and so $S_{n+1}^{♮}=S_1^{♮}{S}_n^{♮}$ for $n\ge n_0$, which finishes the proof, by (8.2.10, ii).

(8.2.13)

Consider now the canonical injections $(S_{+}{)}^n\to S_{[n]}$, which define an injective homomorphism of degree zero of graded rings

$${\oplus }_{n\ge 0}(S_{+}{)}^n\to S^{♮}.(\mathrm{8.2.13.1})$$

Proposition.

For the homomorphism (8.2.13.1) to be a (TN)-isomorphism, it is necessary and sufficient that there exist some $n_0$ such that $S_n=S_1^n$ for all $n\ge n_0$. Whenever this is the case, the morphism corresponding to (8.2.13.1) is everywhere defined and is also an isomorphism

$$\mathrm{Proj}(S^{♮})\stackrel{\sim }{\to }\mathrm{Proj}({\oplus }_{n\ge 0}(S_{+}{)}^n);$$

the converse is true whenever $S$ is Noetherian.

Proof. The first two claims are evident, given (8.2.10, iii) and (2.9.1). The third will follow from (8.2.10, i and iii) and the following lemma:

Lemma (8.2.14.1).

Let $T$ be a positively-graded ring that is also a T_0-algebra of finite type. If the morphism corresponding to the injective homomorphism ${\oplus }_{n\ge 0}{T}_1^n\to T$ is everywhere defined and is also an isomorphism $\mathrm{Proj}(T)\to \mathrm{Proj}({\oplus }_{n\ge 0}{T}_1^n)$, then there exists some $n_0$ such that $T_n=T_1^n$ for $n\ge n_0$.

Let $g_i$ ($1\le i\le r$) be generators of the T_0-module T_1. The hypothesis implies first that the $D_{+}(g_i)$ cover $\mathrm{Proj}(T)$ (2.8.1). Let $(h_j{)}_{1\le j\le s}$ be a system of homogeneous elements of $T_{+}$, with $deg(h_j)=n_j$, that form, together with the $g_i$, a system of generators of the ideal $T_{+}$, or, equivalently (2.1.3), a system of generators of $T$ as a T_0-algebra; if we set $T^{\prime }={\oplus }_{n\ge 0}{T}_1^n$, then the element $h_j/g_i^{n_j}$ of the ring $T_{(g_i)}$ must, by hypothesis, belong to the subring $T_{(g_i)}^{\prime }$, and so there exists some integer $k$ such that $T_1^k{h}_j\subset T_1^{k+n_j}$ for all $j$. We thus conclude, by induction on $r$, that $T_1^k{h}_j^r\subset T^{\prime }$ for all $r\ge 1$, and, by definition of the $h_j$, we thus have $T_1^{k}T\subset T^{\prime }$. Also, there exists, for all $j$, an integer $m_j$ such that $h_j^{m_j}$ belongs to the ideal of $T$ generated by the $g_i$ (2.3.14), so $h_j^{m_j}\in T_{1}T$, and

$h_j^{m_{j}k}\in T_1^{k}T\subset T^{\prime }$. There is thus an integer $m_0\ge k$ such that $h_j^m\in T_1^{m{n}_j}$ for $m\ge m_0$. So, if $q$ is the largest of the integers $n_j$, then $n_0=qs{m}_0+k$ is the required number. Indeed, an element of $S_n$, for $n\ge n_0$, is the sum of monomials belonging to $T_1^{\alpha }u$, where $u$ is a product of powers of the $h_j$; if $\alpha \ge k$, then it follows from the above that $T_1^{\alpha }u\subset T_1^n$; in the other case, one of the exponents of the $h_j$ is $\ge m_0$, so $u\in T_1^{\beta }v$ where $\beta \ge k$ and $v$ is again a product of powers of the $h_j$; we can then reduce to the previous case, and so we conclude that $T_1^{\alpha }u\subset T_1^n$ in all cases.

Remark.

The condition $S_n=S_1^n$ for $n\ge n_0$ clearly implies that $S_{n+1}=S_1{S}_n$ for $n\ge n_0$, but the converse is not necessarily true, even if we assume that $S$ is Noetherian. For example, let $K$ be a field, $A=K[\mathbf{x}]$, and $B=K[\mathbf{y}]/{\mathbf{y}}^{2}K[\mathbf{y}]$, where $\mathbf{x}$ and $\mathbf{y}$ are indeterminates, with $\mathbf{x}$ of degree 1 and $\mathbf{y}$ of degree 2, and let $S=A{\otimes }_{K}B$, so that $S$ is a graded algebra over $K$ with a basis given by the elements 1, ${\mathbf{x}}^n$ ($n\ge 1$), and ${\mathbf{x}}^n\mathbf{y}$ ($n\ge 0$). It is immediate that $S_{n+1}=S_1{S}_n$ for $n\ge 2$, but $S_1^n=K{\mathbf{x}}^n$ while $S_n=K{\mathbf{x}}^n+K{\mathbf{x}}^n\mathbf{y}$ for $n\ge 2$.

8.3. Projecting cones

(8.3.1)

Let $Y$ be a prescheme; in all of this section, we consider only $Y$-preschemes and $Y$-morphisms. Let $\mathcal{S}$ be a quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebra; we further assume that ${\mathcal{S}}_0={\mathcal{O}}_Y$. Following the notation introduced in (8.2.2), we let

  𝒮̂ = 𝒮[𝐳] = 𝒮 ⊗_{𝒪_Y} 𝒪_Y[𝐳]                                              (8.3.1.1)

which we consider as a positively-graded ${\mathcal{O}}_Y$-algebra by defining the degrees as in (8.2.2.2), so that, for every affine open $U$ of $Y$,

  Γ(U, 𝒮̂) = Γ(U, 𝒮)[𝐳].

In what follows, we write

  X = Proj(𝒮),    C = Spec(𝒮),    Ĉ = Proj(𝒮̂)                              (8.3.1.2)

(where, in the definition of $C$, we consider $\mathcal{S}$ as a non-graded ${\mathcal{O}}_Y$-algebra), and we say that $C$ (resp. Ĉ) is the affine cone (resp. projective cone) defined by $\mathcal{S}$; we will sometimes say "cone" instead of "affine cone". By abuse of language, we also say that $C$ (resp. Ĉ) is the affine projecting cone (resp. projective projecting cone) of $X$, with the implicit understanding that the prescheme $X$ is given in the form $\mathrm{Proj}(\mathcal{S})$; finally, we say that Ĉ is the projective closure of $C$ (with the datum of $\mathcal{S}$ being implicit in the structure of $C$).

Proposition.

There exist canonical $Y$-morphisms

  Y ──ε──→ C ──i──→ Ĉ                                                       (8.3.2.1)

  X ──j──→ Ĉ                                                                (8.3.2.2)

such that $ϵ$ and $j$ are closed immersions, and $i$ is an affine morphism which is a dominant open immersion, for which

$$i(C)=\hat{C}\setminus j(X);(\mathrm{8.3.2.3})$$

furthermore, Ĉ is the smallest closed subprescheme of Ĉ containing $i(C)$.

Proof. To define $i$, consider the open subset of Ĉ given by

$${\hat{C}}_{\mathbf{z}}=\mathrm{Spec}(\hat{\mathcal{S}}/(\mathbf{z}-1)\hat{\mathcal{S}})(\mathrm{8.3.2.4})$$

(3.1.4), where $\mathbf{z}$ is canonically identified with a section of $\hat{\mathcal{S}}$ over $Y$. The isomorphism $i:C\stackrel{\sim }{\to }{\hat{C}}_{\mathbf{z}}$ then corresponds to the canonical isomorphism (8.2.3.1)

$$\hat{\mathcal{S}}/(\mathbf{z}-1)\hat{\mathcal{S}}\stackrel{\sim }{\to }\mathcal{S}.$$

The morphism $ϵ$ corresponds to the augmentation homomorphism $\mathcal{S}\to {\mathcal{S}}_0={\mathcal{O}}_Y$, which has kernel ${\mathcal{S}}_{+}$ (1.2.7), and, since the latter is surjective, $ϵ$ is a closed immersion (1.4.10). Finally, $j$ corresponds (3.5.1) to the surjective homomorphism of degree zero $\hat{\mathcal{S}}\to \mathcal{S}$, which restricts to the identity on $\mathcal{S}$ and is zero on $\mathbf{z}\hat{\mathcal{S}}$, the kernel; $j$ is everywhere defined and is a closed immersion, by (3.6.2).

To prove the other claims of (8.3.2), we may clearly restrict to the case where $Y=\mathrm{Spec}(A)$ is affine, and $\mathcal{S}=S$, with $S$ a graded $A$-algebra, whence $\hat{\mathcal{S}}=\hat{S}$; the homogeneous elements $f$ of $S_{+}$ can then be identified with sections of $\hat{\mathcal{S}}$ over $Y$, and the open subset of Ĉ, denoted $D_{+}(f)$ in (2.3.3), can be written as ${\hat{C}}_f$ (3.1.4); similarly, the open subset of $C$ denoted $D(f)$ in (I, 1.1.1) can be written as $C_f$ (0, 5.5.2). With this in mind, it follows from (2.3.14) and from the definition of Ŝ that, in this case, the open subsets ${\hat{C}}_{\mathbf{z}}=i(C)$ and ${\hat{C}}_f$ (with $f$ homogeneous in $S_{+}$) form a cover of Ĉ. Furthermore, with this notation,

$$i^{-1}({\hat{C}}_f)=C_f;(\mathrm{8.3.2.5})$$

indeed, ${\hat{C}}_f\cap i(C)={\hat{C}}_f\cap {\hat{C}}_{\mathbf{z}}={\hat{C}}_{f\mathbf{z}}=\mathrm{Spec}({\hat{S}}_{(f\mathbf{z})})$. But, if $d=deg(f)$, then ${\hat{S}}_{(f\mathbf{z})}$ is canonically isomorphic to $({\hat{S}}_{(\mathbf{z})}{)}_{f/{\mathbf{z}}^d}$ (2.2.2), and it follows from the definition of the isomorphism (8.2.3.1) that the image of $({\hat{S}}_{(\mathbf{z})}{)}_{f/{\mathbf{z}}^d}$ under the corresponding isomorphism of rings of fractions is exactly $S_f$. Since $C_f=\mathrm{Spec}(S_f)$, this proves (8.3.2.5) and shows at the same time that the morphism $i$ is affine; furthermore, the restriction of $i$ to $C_f$, considered as a morphism to ${\hat{C}}_f$, corresponds (I, 1.7.3) to the canonical homomorphism ${\hat{S}}_{(f)}\to {\hat{S}}_{(f\mathbf{z})}$, and, by the above and (8.2.3.2), we may claim the following result:

(8.3.2.6)

If $Y=\mathrm{Spec}(A)$ is affine and $\mathcal{S}=\tilde{S}$, then, for every homogeneous $f$ in $S_{+}$, ${\hat{C}}_f$ is canonically identified with $\mathrm{Spec}(S_f^{\le })$, and the morphism $C_f\to {\hat{C}}_f$ given by restricting $i$ then corresponds to the canonical injection $S_f^{\le }\to S_f$.

Now note that (for $Y$ affine) the complement of ${\hat{C}}_{\mathbf{z}}$ in $\hat{C}=\mathrm{Proj}(\hat{S})$

is, by definition, the set of graded prime ideals of Ŝ containing $\mathbf{z}$, which is exactly $j(X)$, by definition of $j$, which proves (8.3.2.3).

Finally, to prove the last claim of (8.3.2), we may assume $Y$ affine. With the above notation, note that in the ring Ŝ, $\mathbf{z}$ is not a zero divisor; since $i(C)={\hat{C}}_{\mathbf{z}}$, it suffices to prove the following lemma:

Lemma (8.3.2.7).

Let $T$ be a positively-graded ring, $Z=\mathrm{Proj}(T)$, and $g$ a homogeneous element of $T$ of degree $d>0$. If $g$ is not a zero divisor in $T$, then $Z$ is the smallest closed subprescheme of $Z$ containing $Z_g=D_{+}(g)$.

By (I, 4.1.9), the question is local on $Z$; for every homogeneous element $h\in T_e$ ($e>0$), it thus suffices to prove that $Z_h$ is the smallest closed subprescheme of $Z_h$ containing $Z_{gh}$; it follows from the definitions and from (I, 4.3.2) that this condition is equivalent to asking for the canonical homomorphism $T_{(h)}\to T_{(gh)}$ to be injective. But this homomorphism can be identified with the canonical homomorphism $T_{(h)}\to (T_{(h)}{)}_{g^e/h^d}$ (2.2.3). But since $g^e$ is not a zero divisor in $T$, $g^e/h^d$ is not a zero divisor in $T_h$ (nor a fortiori in $T_{(h)}$), since the fact that $(g^e/h^d)(t/h^m)=0$ (for $t\in T$ and $m>0$) implies the existence of some $n>0$ such that $h^n{g}^{e}t=0$, whence $h^{n}t=0$, and thus $t/h^m=0$ in $T_h$. This finishes the proof (0, 1.2.2).

(8.3.3)

We will often identify the affine cone $C$ with the subprescheme induced by the projective cone Ĉ on the open subset $i(C)$ by means of the open immersion $i$. The closed subprescheme of $C$ associated to the closed immersion $ϵ$ is called the apex prescheme of $C$; we also say that $ϵ$, which is a $Y$-section of $C$, is the apex section, or zero section, of $C$; we may identify $Y$ with the apex prescheme of $C$ by means of $ϵ$. Also, $i\circ ϵ$ is a $Y$-section of Ĉ, and thus also a closed immersion (I, 5.4.6), corresponding to the canonical surjective homomorphism of degree zero $\hat{\mathcal{S}}=\mathcal{S}[\mathbf{z}]\to {\mathcal{O}}_Y[\mathbf{z}]$ (3.1.7), whose kernel is ${\mathcal{S}}_{+}[\mathbf{z}]={\mathcal{S}}_{+}\hat{\mathcal{S}}$; the subprescheme of Ĉ associated to this closed immersion is also called the apex prescheme of Ĉ, and $i\circ ϵ$ the apex section of Ĉ; it may be identified with $Y$ by means of $i\circ ϵ$. Finally, the closed subprescheme of Ĉ associated to $j$ is called the locus at infinity of Ĉ, and may be identified with $X$ by means of $j$.

(8.3.4)

The subpreschemes of $C$ (resp. Ĉ) induced on the open subsets

  E = C ∖ ε(Y),     Ê = Ĉ ∖ i(ε(Y))                                         (8.3.4.1)

are called (by an abuse of language) the punctured affine cone and the punctured projective cone (respectively) defined by $\mathcal{S}$; we note that, despite this nomenclature, $E$ is not necessarily affine over $Y$, nor Ê projective over $Y$ (8.4.3). When we identify $C$ with $i(C)$, we thus have the underlying spaces

  C ∪ Ê = Ĉ,        C ∩ Ê = E                                               (8.3.4.2)

so that Ĉ may be regarded as being obtained by gluing the open subpreschemes $C$ and Ê; furthermore, by (8.3.2.3),

$$E=\hat{E}\setminus j(X).(\mathrm{8.3.4.3})$$

If $Y=\mathrm{Spec}(A)$ is affine, then, with the notation of (8.3.2),

  E = ⋃ C_f,    Ê = ⋃ Ĉ_f,    C_f = C ∩ Ĉ_f                                 (8.3.4.4)

where $f$ runs over the set of homogeneous elements of $S_{+}$ (or only a subset $M$ of this set, with $M$ generating an ideal of $S_{+}$ whose radical in $S_{+}$ is $S_{+}$ itself, or, equivalently, such that the $X_f$ for $f\in M$ cover $X$ (2.3.14)). The gluing of $C$ and ${\hat{C}}_f$ along $C_f$ is thus determined by the injection morphisms $C_f\to C$ and $C_f\to {\hat{C}}_f$, which, as we have seen (8.3.2.6), correspond (respectively) to the canonical homomorphisms $S\to S_f$ and $S_f^{\le }\to S_f$.

Proposition.

With the notation of (8.3.1) and (8.3.4), the morphism associated (3.5.1) to the canonical injection $\varphi :\mathcal{S}\to \hat{\mathcal{S}}=\mathcal{S}[\mathbf{z}]$ is a surjective affine morphism (called the canonical retraction)

$$p:\hat{E}\to X(\mathrm{8.3.5.1})$$

such that

$$p\circ j=1_X.(\mathrm{8.3.5.2})$$

Proof. To prove the proposition, we may restrict to the case where $Y$ is affine. Taking into account the expression (8.3.4.4) for Ê, the fact that the domain of definition $G(\varphi )$ of $p$ is equal to Ê will follow from the first of the following claims:

(8.3.5.3)

If $Y=\mathrm{Spec}(A)$ is affine and $\mathcal{S}=\tilde{S}$, then, for every homogeneous $f\in S_{+}$,

$$p^{-1}(X_f)={\hat{C}}_f(\mathrm{8.3.5.4})$$

and the restriction of $p$ to ${\hat{C}}_f=\mathrm{Spec}(S_f^{\le })$, considered as a morphism from ${\hat{C}}_f$ to $X_f$, corresponds to the canonical injection $S_{(f)}\to S_f^{\le }$. If moreover $f\in S_1$, then ${\hat{C}}_f$ is isomorphic to $X_f{\otimes }_{\mathbb{Z}}\mathbb{Z}[T]$ (where $T$ is an indeterminate).

Indeed, equation (8.3.5.4) is exactly a particular case of (2.8.1.1), and the second claim is exactly the definition of $\mathrm{Proj}(\varphi )$ when $Y$ is affine (2.8.1). Then equation (8.3.5.2) and the fact that $p$ is surjective show that the composition $\mathcal{S}\to \hat{\mathcal{S}}\to \mathcal{S}$ of the canonical homomorphisms is the identity on $\mathcal{S}$. Finally, the last claim of (8.3.5.3) follows from the fact that $S_f^{\le }$ is isomorphic to $S_{(f)}[T]$ whenever $f\in S_1$ (2.2.1).

Corollary.

The restriction

$$\pi :E\to X(\mathrm{8.3.6.1})$$

of $p$ to $E$ is a surjective affine morphism. If $Y$ is affine and $f$ is homogeneous in $S_{+}$, then

$${\pi }^{-1}(X_f)=C_f(\mathrm{8.3.6.2})$$

and the restriction of $\pi$ to $C_f$ corresponds to the canonical injection $S_{(f)}\to S_f$. If moreover $f\in S_1$, then $C_f$ is isomorphic to $X_f{\otimes }_{\mathbb{Z}}\mathbb{Z}[T,T^{-1}]$ (where $T$ is an indeterminate).

Proof. Equation (8.3.6.2) follows immediately from (8.3.5.3) and (8.3.2.5), and shows the surjectivity of $\pi$; we have already seen that the immersion $i$, restricted to $C_f$, corresponds

to the injection $S_f^{\le }\to S_f$ (8.3.2). Finally, the last claim is a consequence of the fact that, for $f\in S_1$, $S_f$ is isomorphic to $S_{(f)}[T,T^{-1}]$ (2.2.1).

Remark.

When $Y$ is affine, the elements of the underlying space of $E$ are the (not-necessarily-graded) prime ideals $\mathfrak{p}$ of $S$ not containing $S_{+}$, by definition of the immersion $ϵ$ (8.3.2). For such an ideal $\mathfrak{p}$, the $\mathfrak{p}\cap S_n$ clearly satisfy the conditions of (2.1.9), and so there exists exactly one graded prime ideal $\mathfrak{q}$ of $S$ such that $\mathfrak{q}\cap S_n=\mathfrak{p}\cap S_n$ for all $n$; the map $\pi :E\to X$ of underlying spaces can then be understood via the equation

$$\pi (\mathfrak{p})=\mathfrak{q}.(\mathrm{8.3.7.1})$$

Indeed, to prove this equation, it suffices to consider some homogeneous $f$ in $S_{+}$ such that $\mathfrak{p}\in D(f)$, and to note that ${\mathfrak{q}}_{(f)}$ is the inverse image of ${\mathfrak{p}}_f$ under the injection $S_{(f)}\to S_f$.

Corollary.

If $\mathcal{S}$ is generated by ${\mathcal{S}}_1$, then the morphisms $p$ and $\pi$ are of finite type; for every $x\in X$, the fibre $p^{-1}(x)$ is isomorphic to $\mathrm{Spec}(\kappa (x)[T])$, and the fibre ${\pi }^{-1}(x)$ is isomorphic to $\mathrm{Spec}(\kappa (x)[T,T^{-1}])$.

Proof. This follows immediately from (8.3.5) and (8.3.6) by noting that, when $Y$ is affine and $S$ is generated by S_1, the $X_f$ for $f\in S_1$ form a cover of $X$ (2.3.14).

Remark.

The punctured affine cone corresponding to the graded ${\mathcal{O}}_Y$-algebra ${\mathcal{O}}_Y[T]$ (where $T$ is an indeterminate) may be identified with ${\mathbb{G}}_m=\mathrm{Spec}({\mathcal{O}}_Y[T,T^{-1}])$, since it is exactly C_T, as we have seen in (8.3.2) (see (8.4.4) for a more general result). This prescheme is canonically endowed with the structure of a "$Y$-scheme in commutative groups". This idea will be explained in detail later, but, for now, may be quickly summarised as follows. A $Y$-scheme in groups is a $Y$-scheme $G$ endowed with two $Y$-morphisms, $p:G{\times }_{Y}G\to G$ and $s:G\to G$, satisfying conditions formally analogous to the axioms of the composition law and the symmetry law of a group: the diagram

   G × G × G ──p×1──→ G × G
        │              │
    1×p │              │ p
        ↓              ↓
     G × G ────p────→ G

should commute (associativity), and there should be a condition corresponding to the fact that, for groups, the maps

  (x, y) ↦ (x, x⁻¹, y) ↦ (x, x⁻¹y) ↦ x(x⁻¹y)

and

  (x, y) ↦ (x, x⁻¹, y) ↦ (x, y x⁻¹) ↦ (y x⁻¹) x

should both reduce to $(x,y)\mapsto y$; the sequence of morphisms corresponding, for example, to the first composite map is

  G × G ──(1,s)×1──→ G × G × G ──1×p──→ G × G ──p──→ G

and the reader should write down the second sequence.

It is immediate (I, 3.4.3) that the datum of a structure of a $Y$-scheme in groups on a $Y$-scheme $G$ is equivalent to the datum, for every $Y$-prescheme $Z$, of a group structure on the set ${\mathrm{Hom}}_Y(Z,G)$, with these structures being such that, for every $Y$-morphism $Z\to Z^{\prime }$, the corresponding map ${\mathrm{Hom}}_Y(Z^{\prime },G)\to {\mathrm{Hom}}_Y(Z,G)$ is a group homomorphism. In the particular case of ${\mathbb{G}}_m$ considered here, ${\mathrm{Hom}}_Y(Z,{\mathbb{G}}_m)$ may be identified with the set of $Z$-sections of $Z{\times }_Y{\mathbb{G}}_m$ (I, 3.3.14), and thus with the set of $Z$-sections of $\mathrm{Spec}({\mathcal{O}}_Z[T,T^{-1}])$; finally, the same reasoning as in (I, 3.3.15) shows that this set is canonically identified with the set of invertible elements of the ring $\Gamma (Z,{\mathcal{O}}_Z)$, and the group structure on this set is the structure coming from the multiplication in the ring $\Gamma (Z,{\mathcal{O}}_Z)$. The reader may verify that the morphisms $p$ and $s$ above are obtained as follows: they correspond, by (1.2.7) and (1.4.6), to the homomorphisms of ${\mathcal{O}}_Y$-algebras

  π : 𝒪_Y[T, T⁻¹] → 𝒪_Y[T, T⁻¹, T′, T′⁻¹]
  σ : 𝒪_Y[T, T⁻¹] → 𝒪_Y[T, T⁻¹]

and are entirely defined by the data $\pi (T)=T{T}^{\prime }$ and $\sigma (T)=T^{-1}$.

With this in mind, ${\mathbb{G}}_m$ can be considered as a "universal domain of operators" for every affine cone $C=\mathrm{Spec}(\mathcal{S})$, where $\mathcal{S}$ is a quasi-coherent positively-graded ${\mathcal{O}}_Y$-algebra. This means that we may canonically define a $Y$-morphism ${\mathbb{G}}_m{\times }_{Y}C\to C$ having the formal properties of an external law of a set endowed with a group of operators; or, again, as above for schemes in groups, we may give, for every $Y$-prescheme $Z$, an external law on ${\mathrm{Hom}}_Y(Z,C)$, having the group ${\mathrm{Hom}}_Y(Z,{\mathbb{G}}_m)$ as its set of operators, with the usual axioms of sets endowed with a group of operators, and a compatibility condition with respect to the $Y$-morphisms $Z\to Z^{\prime }$. In the current case, the morphism ${\mathbb{G}}_m{\times }_{Y}C\to C$ is defined by the datum of a homomorphism of ${\mathcal{O}}_Y$-algebras $\mathcal{S}\to \mathcal{S}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_Y[T,T^{-1}]=\mathcal{S}[T,T^{-1}]$, which associates, to each section $s_n\in \Gamma (U,{\mathcal{S}}_n)$ (where $U$ is an open subset of $Y$), the section $s_n{T}^n\in \Gamma (U,\mathcal{S}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_Y[T,T^{-1}])$.

Conversely, suppose that we are given a quasi-coherent, a priori non-graded, ${\mathcal{O}}_Y$-algebra, and, on $C=\mathrm{Spec}(\mathcal{S})$, a structure of a "$Y$-scheme in sets endowed with a group of operators" having the $Y$-scheme in groups ${\mathbb{G}}_m$ as its domain of operators; then we canonically obtain a grading of ${\mathcal{O}}_Y$-algebras on $\mathcal{S}$. Indeed, the datum of a $Y$-morphism ${\mathbb{G}}_m{\times }_{Y}C\to C$ is equivalent to that of a homomorphism of ${\mathcal{O}}_Y$-algebras $\psi :\mathcal{S}\to \mathcal{S}[T,T^{-1}]$, which may be written $\psi ={\sum }_{n\in \mathbb{Z}}{\psi }_n{T}^n$, where the ${\psi }_n:\mathcal{S}\to \mathcal{S}$ are homomorphisms of ${\mathcal{O}}_Y$-modules (with ${\psi }_n(s)=0$ except for finitely many $n$ for every section $s\in \Gamma (U,\mathcal{S})$, for any open subset $U$ of $Y$). One can then prove that the axioms of sets endowed with a group of operators imply that the ${\psi }_n(\mathcal{S})={\mathcal{S}}_n$ define a grading (in positive or negative degree) of ${\mathcal{O}}_Y$-algebras on $\mathcal{S}$, with the ${\psi }_n$ being the corresponding projectors. We also have the notion of a structure of an "affine cone" on every affine $Y$-scheme, defined in a "geometric" way without any reference to any prior grading.

We will not develop this point of view further here, and we leave the work of precisely formulating the definitions and results corresponding to the information given above to the reader.

8.4. Projective closure of a vector bundle

(8.4.1)

Let $Y$ be a prescheme and $\mathcal{E}$ a quasi-coherent ${\mathcal{O}}_Y$-module. If we take $\mathcal{S}$ to be the graded ${\mathcal{O}}_Y$-algebra ${\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E})$, then Definition (8.3.1.1) shows that $\hat{\mathcal{S}}$ may be identified with ${\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}\oplus {\mathcal{O}}_Y)$. With the affine cone $\mathrm{Spec}(\mathcal{S})$ defined by $\mathcal{S}$ being, by definition, $\mathbb{V}(\mathcal{E})$, and $\mathrm{Proj}(\mathcal{S})$ being, by definition, $\mathbb{P}(\mathcal{E})$, we see that:

Proposition.

The projective closure of a vector bundle $\mathbb{V}(\mathcal{E})$ over $Y$ is canonically isomorphic to $\mathbb{P}(\mathcal{E}\oplus {\mathcal{O}}_Y)$, and the locus at infinity of the latter is canonically isomorphic to $\mathbb{P}(\mathcal{E})$.

Remark.

Take, for example, $\mathcal{E}={\mathcal{O}}_Y^r$ with $r\ge 2$; then the punctured cones $E$ and Ê defined by $\mathcal{S}$ are neither affine nor projective over $Y$ if $Y\ne \varnothing$. The second claim is immediate, because $\hat{C}=\mathbb{P}({\mathcal{O}}_Y^{r+1})$ is projective over $Y$, and the underlying spaces of $E$ and Ê are non-closed open subsets of Ĉ, and so the canonical immersions $E\to \hat{C}$ and $\hat{E}\to \hat{C}$ are not projective (5.5.3), and we conclude by appealing to (5.5.5, v). Now, supposing for example that $Y=\mathrm{Spec}(A)$ is affine and $r=2$, then $C=\mathrm{Spec}(A[T_1,T_2])$, and $E$ is the prescheme induced by $C$ on the open subset $D(T_1)\cup D(T_2)$; but we have already seen that the latter is not affine (I, 5.5.11); a fortiori Ê cannot be affine, since $E$ is the open subset where the section $\mathbf{z}$ over Ê does not vanish (8.3.2).