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Proj of a finitely generated graded algebra is projective

Statement

Assume AC inherited from the Proj construction. Let $S = ⨁_{n \geq 0} S_{n}$ be a finitely generated graded commutative $C$ -algebra with $S_{0} = C$ and every $S_{n}$ finite-dimensional, and suppose $S$ is generated as a $C$ -algebra by homogeneous elements $f_{1}, \dots, f_{N}$ of positive degrees $d_{1}, \dots, d_{N}$ ; put $L = lcm (d_{1}, \dots, d_{N})$ (with $L = 1$ when $N = 0$ ) and $d = k L$ with $k = \max (1, N - 1)$ .

Then the Veronese subalgebra $S^{(d)} = ⨁_{j \geq 0} S_{d j}$ is generated in degree one by $S_{d}$ : every element of $S^{(d)}$ is a polynomial in elements of the finite-dimensional space $S_{d}$ . Consequently, if $S_{d} \neq 0$ , choosing a $C$ -basis $g_{0}, \dots, g_{M}$ of $S_{d}$ and the induced graded surjection $B = C [x_{0}, \dots, x_{M}] \to S^{(d)}$ , $x_{i} \mapsto g_{i}$ , the canonical isomorphism $Proj S ≅ Proj S^{(d)}$ exhibits $Proj S$ as a closed subscheme of the projective space $P_{C}^{M} = Proj B$ of finite type over $C$ , and $O (1)$ pulls back to $O_{S^{(d)}} (1)$ .

If $S_{+} = 0$ , then $S = C$ ; and if $S_{d} = 0$ , degree-one generation makes $S^{(d)} = C$ . In either case $Proj S = \emptyset$ is the closed subscheme $V_{+} (x_{0})$ of $P_{C}^{0}$ , so the conclusion holds vacuously in that case as well.

Facts & Assumptions

Given: The Axiom of Choice as inherited from the Proj construction; a finitely generated graded $C$ -algebra $S$ with $S_{0} = C$ and $\dim_{C} S_{n} < \infty$ for all $n$ ; homogeneous generators $f_{1}, \dots, f_{N}$ of positive degrees $d_{1}, \dots, d_{N}$ when $S_{+} \neq 0$ ; the numbers $L = lcm (d_{1}, \dots, d_{N})$ , $k = \max (1, N - 1)$ and $d = k L$ .

[F1]

The Axiom of Choice states that every family of nonempty sets has a choice function; it is inherited by this item from the Proj construction, whose localization and chart-gluing data use it. (The Axiom of Choice, Projective scheme of a homogeneous quotient and its standard affine charts)

[F2]

Finite homogeneous generation. $S$ is generated as a $C$ -algebra by finitely many homogeneous elements, and every $S_{n}$ is finite-dimensional. If $S_{+} \neq 0$ then the degree-zero part of a generating set may be omitted, so that $S$ is generated by finitely many homogeneous elements $f_{1}, \dots, f_{N}$ of positive degrees; every element of $S_{n}$ is then a $C$ -linear combination of monomials $f_{1}^{a_{1}} \dots f_{N}^{a_{N}}$ with $\sum a_{i} d_{i} = n$ . (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Nonnegatively graded rings and modules, homogeneous elements, and twists)

[F3]

Proj dictionary. For a graded ring $S$ with $S_{0} = C$ , the points of $Proj S$ are the homogeneous primes not containing $S_{+}$ , the standard opens are $D_{+} (f) = Spec S_{(f)}$ for homogeneous $f$ of positive degree, and these charts cover $Proj S$ . For the standard graded polynomial ring $B = C [x_{0}, \dots, x_{M}]$ one has $Proj B = P_{C}^{M}$ . (Projective scheme of a homogeneous quotient and its standard affine charts, Projective space is Proj of a polynomial ring)

[F4]

Closed subschemes of projective space. For a homogeneous ideal $I \subseteq B$ the canonical closed immersion $Proj (B / I) ↪ Proj B = P_{C}^{M}$ identifies $Proj (B / I)$ with $V_{+} (I)$ ; its chart rings are the quotients $B_{(g)} \to (B / I)_{(g)}$ . (Closed subschemes of projective space and saturated ideals)

[F5]

Veronese invariance. There is a canonical isomorphism $Proj S ≅ Proj S^{(d)}$ mapping $D_{+} (f)$ to $D_{+} (f^{d})$ with the same coordinate ring $S_{(f)} = S_{(f^{d})}^{(d)}$ , under which $O_{S^{(d)}} (1)$ corresponds to $O_{S} (d)$ . (Proj is invariant under Veronese regrading)

[F6]

Twisting sheaves. For a graded ring $A$ with $A_{0} = C$ , the sheaf $O_{Proj A} (1)$ is the associated sheaf of the shifted graded module $A (1)$ , with sections $A (1)_{(g)}$ on the chart $D_{+} (g)$ ; a degree-zero homomorphism $M \to N$ of graded $A$ -modules induces a morphism $\tilde{M} \to \tilde{N}$ of associated sheaves, computed on charts by the corresponding localization maps. (Twisting sheaf on Proj, Associated sheaf of a graded module on Proj)

Proof

technique · direct

1.1F1F2given

If $S_{+} = 0$ then $S_{n} = 0$ for all $n \geq 1$ and hence $S = S_{0} = C$ ; since every homogeneous prime contains $S_{+} = 0$ , the set $Proj S$ is empty, and $\emptyset$ is the closed subscheme $V_{+} (x_{0}) \subseteq P_{C}^{0}$ , so the assertions about $Proj S$ hold vacuously. Assume henceforth $S_{+} \neq 0$ and fix homogeneous generators $f_{1}, \dots, f_{N}$ of $S$ of positive degrees $d_{1}, \dots, d_{N}$ , with $L = lcm (d_{1}, \dots, d_{N})$ , $k = \max (1, N - 1)$ and $d = k L$ .

1.2F2algebra

The generation claim. We show that every monomial $f_{1}^{a_{1}} \dots f_{N}^{a_{N}}$ of degree $m d$ with $m \geq 2$ is a product of $m$ monomials of degree $d$ . Suppose not, and choose a counterexample with $m \geq 2$ minimal; call it $M = f_{1}^{a_{1}} \dots f_{N}^{a_{N}}$ , so that $a = (a_{1}, \dots, a_{N})$ has no decomposition $a = b_{1} + \dots + b_{m}$ with $b_{j} \in Z_{\geq 0}^{N}$ and $\sum_{i} b_{j i} d_{i} = d$ . Then $a$ has no sub-vector of degree $d = k L$ : if $c \leq a$ had $\sum_{i} c_{i} d_{i} = k L$ , then $a - c$ would have degree $(m - 1) k L$ and would, by minimality of $m$ , decompose into $m - 1$ vectors of degree $k L$ , so $a$ would decompose into $m$ of them.

2.1choosealgebrastep 1.2

Maximal $L$ -blocks. Call $v \in Z_{\geq 0}^{N}$ an $L$ -block of $a$ if $v \leq a$ and $\sum_{i} v_{i} d_{i} = L$ , and choose a maximal-length family $v_{1}, \dots, v_{ℓ}$ of $L$ -blocks of $a$ with $\sum_{j \leq ℓ} v_{j} \leq a$ ; such a maximum exists because every $L$ -block has degree $L > 0$ . Then $ℓ \leq k - 1$ , since otherwise $v_{1} + \dots + v_{k}$ would be a sub-vector of $a$ of degree $k L$ , contradicting step 1.2. Put $u : = a - \sum_{j \leq ℓ} v_{j} \in Z_{\geq 0}^{N}$ . The vector $u$ contains no $L$ -block: any $L$ -block $v \leq u$ would be disjoint from all $v_{j}$ , contradicting maximality of $ℓ$ . Moreover $\sum_{i} u_{i} d_{i} = m d - ℓ L \geq (m k - k + 1) L = ((m - 1) k + 1) L \geq (k + 1) L \geq N L$ , using $m \geq 2$ and $k \geq N - 1$ .

3.1F2algebrastep 2.1

The box bound. For each $i$ , since $L / d_{i}$ is a positive integer and $(L / d_{i}) e_{i}$ has degree $L$ , the absence of an $L$ -block in $u$ forces $u_{i} < L / d_{i}$ , i.e. $u_{i} \leq L / d_{i} - 1$ . Hence $\sum_{i} u_{i} d_{i} \leq \sum_{i} (L / d_{i} - 1) d_{i} = N L - \sum_{i} d_{i} < N L$ , contradicting the lower bound $\geq N L$ from step 2.1. Therefore every monomial of degree $m d$ , $m \geq 2$ , is a product of $m$ monomials of degree $d$ ; since $S_{m d}$ is spanned by such monomials and $S_{0} = C$ , the Veronese algebra satisfies $S^{(d)} = C [S_{d}]$ with $S_{d}$ finite-dimensional.

4.1F3F4F5step 1.1step 3.1

If $S_{d} = 0$ , step 3.1 gives $S^{(d)} = C$ , so its Proj is empty and is a closed subscheme of $P^{0}$ as in step 1.1; [F5] gives the same conclusion for $Proj S$ . Otherwise choose a basis $g_{0}, \dots, g_{M}$ of the nonzero finite-dimensional space $S_{d}$ . The graded map $B = C [x_{0}, \dots, x_{M}] \to S^{(d)}$ sending $x_{i}$ to $g_{i}$ is surjective by step 3.1. Its homogeneous kernel $I$ identifies $S^{(d)}$ with $B / I$ , and [F4] exhibits its Proj as $V_{+} (I) \subseteq P^{M}$ . The remaining chart arguments concern this nonempty-basis case.

5.1F6algebrastep 4.1

The twisting sheaves. On the covering charts $D_{+} (x_{i})$ of $Proj B$ , the degree-zero module $B (1)_{(x_{i})}$ is free with frame $x_{i}$ : every fraction of shifted degree zero is $x_{i}$ times a degree-zero fraction. Its pullback to $D_{+} (g_{i})$ is therefore free with the corresponding frame $g_{i}$ , which likewise generates $S^{(d)} (1)_{(g_{i})}$ . On overlaps the transition ratios $x_{j} / x_{i}$ pull back to $g_{j} / g_{i}$ . Thus these frame identifications glue to identify the pullback of $O_{Proj B} (1)$ with $O_{S^{(d)}} (1)$ by [F6].

5.2F3F4step 3.1step 4.1

Finite type. The charts $D_{+} (g_{i})$ have rings obtained by quotienting the polynomial ring in the degree-zero ratios $x_{j} / x_{i}$ , hence are finitely generated $C$ -algebras; finitely many charts suffice because $S^{(d)}$ is generated over $C$ by the finite set $g_{0}, \dots, g_{M}$ by step 3.1, so $D_{+} (g_{0}), \dots, D_{+} (g_{M})$ cover $Proj S^{(d)}$ . Thus $Proj S^{(d)}$ is a $C$ -scheme of finite type.

6.1F5step 1.1step 4.1step 5.1step 5.2∎

Finally [F5] gives the canonical isomorphism $Proj S ≅ Proj S^{(d)}$ mapping $D_{+} (f)$ to $D_{+} (f^{d})$ . Composed with step 4.1 it exhibits $Proj S$ as a closed subscheme of $P_{C}^{M}$ , of finite type over $C$ by step 5.2, and by step 5.1 the twisting sheaf $O (1)$ pulls back to $O_{S^{(d)}} (1)$ , as claimed; the degenerate case $S = C$ was settled in step 1.1.

Remarks

The correcting range of $d$ . The repaired statement singles out the common multiple $d = k L$ with $k = \max (1, N - 1)$ ; the original scaffold claimed the conclusion for every common multiple of the degrees of a generating set, which is false. With $S = C [x_{1}, x_{2}, x_{3}, x_{4}]$ graded by $\deg x_{1} = 2$ , $\deg x_{2} = 12$ , $\deg x_{3} = 15$ , $\deg x_{4} = 20$ , one has $L = 60$ , and the monomial $x_{1} x_{2}^{4} x_{3}^{2} x_{4}^{2}$ has degree $120 = 2 L$ although it is not a product of two monomials of degree $L$ : no sub-multiset of the degrees ${2, 12, 12, 12, 12, 15, 15, 20, 20}$ sums to $60$ . Hence $S^{(L)}$ is not generated in degree one, while the lemma supplies $k = N - 1 = 3$ and then $S^{(3 L)}$ is. (This is Deligne's classical phenomenon, quoted in the weighted-projective-space literature; the proof above is self-contained.)
Multiples of $d$ . If $A$ is a standard graded $C$ -algebra, then so is every Veronese $A^{(l)}$ : $A_{l m} = (A_{l})^{m}$ . Applying this to $A = S^{(d)}$ shows that $S^{(d^{'})}$ is generated in degree one for every multiple $d^{'}$ of $d$ , Moreover, the maximal-block and box-bound argument of steps 1.2–3.1 works for every integer $k \geq \max (1, N - 1)$ : its only bound on $k$ is $k \geq N - 1$ . Thus the conclusion holds for all sufficiently large multiples $k L$ of $L$ .
No Hilbert--Mumford input. The proof uses only the monomial combinatorics of the degrees $d_{i}$ and the definitions of $Proj$ and its twisting sheaves; no criterion for (semi)stability is involved.

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Sources

Victoria Hoskins, Moduli Problems and Geometric Invariant Theory, FU Berlin lecture notes (2015/16) (standard reference, not scraped)
I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series 296, Cambridge University Press, 2003 (standard reference, not scraped)