# Coherent sheaves on projective schemes: Serre's theorems

*Written by GPT-6.1 Sol (OpenAI), in Codex, at Ultra effort, October 2026. Self-checked by the writing AI, GPT-6.1 Sol, at Ultra effort. Public domain (CC0).*

The cohomology of the individual twists on projective space controls every coherent sheaf there. One first expresses a sheaf as a quotient of finitely many twists. Its kernel is again coherent, and the long exact sequence moves the question one cohomological degree upward. Since the standard cover gives a finite upper bound, descending induction proves both finiteness and eventual vanishing.

We use [Cohomology of projective space](cohomology-of-projective-space.md) and the quasi-coherent foundations of the earlier lessons. The planned prerequisite lessons *Ample invertible sheaves* and *Projective morphisms and Chow's lemma*, in *Morphisms of schemes*, provide the geometric construction: over an affine base, a sufficiently high power of an ample line bundle on a scheme of finite type defines an immersion into a finite-dimensional projective space. If the scheme is proper, this immersion is closed. These are [Stacks, Tags 01VT and 01W6], whose open proofs are linked below. The cohomological consequences, including the converse ampleness criterion, are proved here.

## 1. Clearing denominators in a line bundle

For a line bundle \(L\) and a section \(s\in H^0(X,L)\), write \(X_s\) for the open where the section generates the line. Multiplication by \(s\) gives maps \(F\otimes L^a\to F\otimes L^{a+1}\).

**Lemma 1.1 (extension after a power).** If \(X\) is quasi-compact and quasi-separated, \(F\) is quasi-coherent, and \(s\) is as above, the natural map
\[
\underset{a\ge0}{\operatorname{colim}}\ H^0(X,F\otimes L^a)
\longrightarrow H^0(X_s,F),\qquad t\longmapsto t/s^a,
\]
is an isomorphism. In particular every section on \(X_s\) extends after multiplying by a sufficiently high power of \(s\).

**Proof.** Choose a finite affine cover \(U_i\) trivializing \(L\). On \(U_i\), write \(s=f_i\) in that frame and \(F=\widetilde M_i\). Then \(U_i\cap X_s=D(f_i)\), and sections there are elements of \((M_i)_{f_i}\). Each is a fraction with a finite denominator, so a section on \(X_s\) has local lifts in \(F\otimes L^a\) for a common exponent \(a\).

On each overlap the difference of two lifts becomes zero after inverting \(s\). The overlap is quasi-compact by quasi-separatedness. Cover it by finitely many affines trivializing \(L\); on each, an element that is zero in a localization is killed by a power of its local equation. A common power kills the difference on the whole overlap. There are finitely many pairs, so one further power makes all lifts agree. They glue to a global twisted section. This proves surjectivity. If a global twisted section restricts to zero, the same localization argument on the finite cover kills it by a common power of \(s\); it is therefore zero in the colimit. This proves injectivity. \(\square\)

The same argument works with a section of \(L^b\), using powers \(L^{ab}\). It does not require \(s\) to be a nonzerodivisor. The associated section-module recovery is [Stacks, Tag 0AG5]; here the denominator argument also explains its compatibility with the usual maps from homogeneous elements.

Now let \(R\) be Noetherian, \(S=R[T_0,\ldots,T_N]\), and \(P=\mathbf P_R^N\). The scheme \(P\) is Noetherian: each standard chart is a polynomial ring over \(R\). A coherent sheaf on a locally Noetherian scheme is a quasi-coherent sheaf with finite modules on affine opens. Kernels and cokernels of maps between such sheaves are coherent, because submodules of finite modules over Noetherian rings are finite.

**Proposition 1.2 (finite generation by twists).** For every coherent \(F\) on \(P\), there is a finite surjection
\[
\mathcal O_P(-a)^{\oplus m}\twoheadrightarrow F
\]
for some \(a\ge0\). Moreover \(F(d)\) is generated by finitely many global sections for every sufficiently large \(d\).

**Proof.** On \(D_+(T_i)\), choose finitely many module generators of \(F\). Lemma 1.1 extends each generator, after multiplication by a power of \(T_i\), to a global section of some \(F(a_{ij})\). Choose a common \(a\ge0\) at least as large as every \(a_{ij}\), and multiply these sections by \(T_i^{a-a_{ij}}\). On the \(i\)-th chart the multiplier is a unit in the chart's twist frame, so the resulting global sections of \(F(a)\) still generate there. All charts together give a finite global generating family, which is equivalent to the displayed surjection. For \(d\ge a\), tensor it with \(\mathcal O(d)\). The source is a sum of copies of \(\mathcal O(d-a)\), generated by its polynomial monomials; their images generate \(F(d)\). This also treats \(N=0\), where the scheme is affine and every twist is trivial. \(\square\)

Global generation means that the evaluation map from global sections tensored with \(\mathcal O_P\) is surjective. It does not mean that every local generator is itself the restriction of an untwisted global section. The power of the chart coordinate is what makes local information extend.

## 2. Descending induction proves Serre's theorems

**Theorem 2.1 (finiteness and vanishing on projective space).** For a coherent \(F\) on \(\mathbf P_R^N\), with \(R\) Noetherian:

- \(H^q(P,F)=0\) for \(q>N\).
- Every \(H^q(P,F)\) is a finite \(R\)-module.
- There exists \(d_0(F)\) such that \(H^q(P,F(d))=0\) for every \(q>0\) and every \(d\ge d_0(F)\).
- \(F(d)\) is globally generated for every sufficiently large \(d\).

**Proof.** The \(N+1\) standard charts and all their intersections are affine. Their ordered Čech complex computes quasi-coherent cohomology and has no terms above degree \(N\). This proves the first assertion, even without coherence.

Prove finiteness and eventual vanishing simultaneously, by descending induction on \(q\), for all coherent sheaves. The assertions hold in degrees greater than \(N\). Choose a finite quotient by twists, with coherent kernel:
\[
0\longrightarrow K\longrightarrow E\longrightarrow F\longrightarrow0,
\qquad E=\bigoplus_j\mathcal O(b_j).
\]
The exact cohomology segment
\[
H^q(P,E)\longrightarrow H^q(P,F)\longrightarrow H^{q+1}(P,K)
\]
has finite modules at both ends: the left one by the monomial computation, the right one by induction. Its middle term is an extension of a quotient of the left module by a submodule of the right module. The latter submodule is finite because \(R\) is Noetherian. Hence the middle term is finite. This includes \(q=0\).

For \(q>0\), twist the short exact sequence by \(\mathcal O(d)\), which is an exact operation because this sheaf is invertible. In the corresponding segment, \(H^q(E(d))\) is zero for sufficiently large \(d\) by the twist calculation; \(H^{q+1}(K(d))\) is zero for sufficiently large \(d\) by induction. Thus \(H^q(F(d))=0\) eventually. Only the finitely many degrees \(1,\ldots,N\) can occur, so their thresholds have a maximum. Proposition 1.2 proves the last assertion. \(\square\)

This is [Stacks, Tag 01YS]. The proof uses as many coherent kernels as the induction needs, rather than asserting that every coherent sheaf has a finite resolution by line bundles. Such a resolution would require further hypotheses and is unnecessary here.

For a closed immersion \(i:X\hookrightarrow P\), pushforward identifies coherent sheaves on \(X\) with coherent sheaves on \(P\) annihilated by the defining ideal. Affine-locally this is just restriction of scalars along a quotient of Noetherian rings. Pushforward is exact, and
\[
H^q(X,F)=H^q(P,i_*F),\qquad
i_*(F\otimes i^*\mathcal O(d))=(i_*F)(d).
\]
The cohomology equality follows either from the affine-morphism theorem or from the common affine-cover complex. The tensor equality can be checked in a local line-bundle frame. Theorem 2.1 therefore applies to every closed projective subscheme with its hyperplane bundle.

**Theorem 2.2 (proper schemes with an ample line bundle).** Let \(X\to\operatorname{Spec}R\) be proper, with \(R\) Noetherian, and let \(L\) be ample. For coherent \(F\), all \(H^q(X,F)\) are finite over \(R\); the groups \(H^q(X,F\otimes L^d)\), \(q>0\), vanish for all sufficiently large \(d\); and \(F\otimes L^d\) is globally generated for all sufficiently large \(d\).

**Proof.** The geometric prerequisite gives a closed immersion \(i:X\hookrightarrow\mathbf P_R^N\) and an integer \(b>0\) with \(L^b=i^*\mathcal O(1)\). Apply Theorem 2.1 to each of the finitely many coherent sheaves \(i_*(F\otimes L^j)\), \(0\le j<b\). If \(d=bm+j\), its twist by \(\mathcal O(m)\) is \(i_*(F\otimes L^d)\). The theorem gives vanishing and global generation for all sufficiently large \(m\) in each residue class. Taking their finitely many thresholds proves the assertion for every large \(d\). Finiteness follows already from \(i_*F\). \(\square\)

This residue-class step is essential: an embedding given by \(L^b\) initially controls only multiples of \(b\). The result is [Stacks, Tag 0B5T].

**Corollary 2.3 (relative finiteness and vanishing).** If \(S\) is locally Noetherian, \(f:X\to S\) is locally projective, and \(F\) is coherent, every \(R^qf_*F\) is coherent. If \(S\) is Noetherian, \(f\) is proper, and \(L\) is \(f\)-ample, then there is a single \(d_0\) such that
\[
R^qf_*(F\otimes L^d)=0\qquad(q>0,\ d\ge d_0).
\]

**Proof.** The earlier higher-direct-image theorem makes these sheaves quasi-coherent and identifies them, over a base affine \(\operatorname{Spec}R\), with the modules \(H^q(X_R,F)\). For a locally projective morphism, base-local closed embeddings into projective space give their finiteness by Theorem 2.1. Finite modules over Noetherian rings define coherent sheaves, proving the first assertion. For the second, choose a finite affine cover of \(S\); relative ampleness means that \(L\) is ample on each inverse image. Theorem 2.2 supplies thresholds there, and their maximum works on \(S\). These are [Stacks, Tags 02O4 and 02O1]. Over a base that is only locally Noetherian, this proof gives thresholds locally; it does not supply a uniform threshold without a finite cover. \(\square\)

## 3. Graded modules and their invisible tails

Let \(B\) be a nonnegatively graded ring, with \(B_0=R\) Noetherian, generated by finitely many degree-one elements. Set \(X=\operatorname{Proj}B\). A graded surjection from a polynomial ring over \(R\) embeds \(X\) as a closed subscheme of projective space and identifies its \(\mathcal O_X(1)\) with the hyperplane bundle. Thus the preceding results apply.

For coherent \(F\), define the nonnegative section module
\[
G(F)=\bigoplus_{d\ge0}H^0(X,F(d)).
\]
Multiplication by elements of \(B\) supplies its graded \(B\)-module structure.

**Lemma 3.1 (finite section module).** The module \(G(F)\) is finitely generated over \(B\).

**Proof.** Present the pushed-forward sheaf on projective space as a quotient of a finite sum \(E\) of twists, with coherent kernel \(K\). For every large \(d\), vanishing of \(H^1(K(d))\) makes \(H^0(E(d))\to H^0(F(d))\) surjective. The high-degree section tail of \(E\) is a finite module over the polynomial ring: it is a truncation of a finite sum of shifted polynomial modules, except for finitely many low degrees. This also holds on \(\mathbf P_R^0\), where each twist's sections form a shifted copy of the one-variable polynomial module in sufficiently high degrees. A truncation is a submodule of a finite module over a Noetherian ring, hence is finite. Its quotient, the high-degree tail of \(G(F)\), is finite. The finitely many remaining homogeneous pieces are finite \(R\)-modules by Theorem 2.1, so adding generators for them proves finiteness of all \(G(F)\). The polynomial-ring action factors through \(B\), which proves the claimed \(B\)-finiteness. \(\square\)

Sheafification of graded modules is exact: on \(D_+(f)\), it is homogeneous degree zero after localization. Lemma 1.1 gives the canonical recovery
\[
\widetilde{G(F)}\simeq F.
\]
Indeed on a degree-one chart every local section is a fraction \(t/f^a\) with \(t\in H^0(X,F(a))\). The lemma identifies these fractions and their equality precisely with \(G(F)_{(f)}\). This is an isomorphism on all chart modules and hence on sheaves. Truncating the full integer-indexed section module at zero loses no fractions, since one may multiply numerator and denominator by additional powers of \(f\).

**Lemma 3.2 (a sheaf sees the high-degree tail).** For a finite graded \(B\)-module \(M\), the natural map
\[
M_d\longrightarrow H^0(X,\widetilde M(d))
\]
is an isomorphism for all sufficiently large \(d\). A finite graded module has zero associated sheaf exactly when it is zero in all sufficiently large degrees.

**Proof.** First prove the second assertion directly. Let \(f_1,\ldots,f_t\in B_1\) generate the positive-degree algebra, and choose homogeneous generators \(m_j\) of \(M\). If \(\widetilde M=0\), then \(m_j/f_i^{\deg m_j}\) is zero in the degree-zero localization; a negative exponent in this expression means multiplication by a positive power. Since \(f_i\) is invertible there, this implies \(f_i^{a_{ij}}m_j=0\) for some \(a_{ij}\ge0\). A monomial in the \(f_i\) of sufficiently large total degree has an exponent at least one of these killing bounds, so it kills \(m_j\). Every high-degree element of \(M\) is a sum of such monomial multiples of its generators. Thus \(M_d=0\) for large \(d\). Conversely, eventual zero makes every degree-zero localized fraction zero after raising its denominator, so the associated sheaf is zero.

Apply this to the canonical graded map
\[
M_{\ge0}\longrightarrow G(\widetilde M).
\]
Both modules are finite: the first by Noetherianity, the second by Lemma 3.1. Its sheafification is an isomorphism. On a chart the composite with the recovery map sends a localized homogeneous element to that very section, so it is the identity on the module defining \(\widetilde M\). Its kernel and cokernel are consequently finite modules with zero associated sheaf. They vanish in high degree by the assertion just proved, giving the first claim. \(\square\)

In this setting, call a finite graded module **irrelevant torsion** if each element is killed by some power of \(B_+=\bigoplus_{d>0}B_d\). Finite generation and degree-one generation make this equivalent to eventual zero of its graded pieces: a uniform power kills the finitely many generators, and conversely a high enough degree kills every generator. This usage of torsion concerns the irrelevant ideal, not ordinary torsion over a domain.

**Theorem 3.3 (Serre's graded equivalence).** Sheafification gives an equivalence
\[
\frac{\{\text{finite graded }B\text{-modules}\}}
{\{\text{irrelevant torsion modules}\}}
\simeq\operatorname{Coh}(X),
\]
with quasi-inverse \(F\mapsto G(F)\) in the quotient category.

**Proof.** The quotient is the Serre quotient: it makes every map with irrelevant torsion kernel and cokernel invertible. The torsion subcategory is closed under subobjects, quotients and extensions, immediately from eventual zero, so this quotient is defined. Exact sheafification kills precisely these modules by Lemma 3.2 and therefore factors through it. Lemma 3.1 defines the proposed quasi-inverse. The recovery map identifies the composite on sheaves with the identity. On modules, pass through \(M_{\ge0}\): its inclusion into \(M\) has torsion cokernel, and its map to \(G(\widetilde M)\) has torsion kernel and cokernel by Lemma 3.2. Both become isomorphisms in the quotient. These natural maps give the two inverse functors. \(\square\)

The results are [Stacks, Tags 0AG6, 0AG7 and 0BXD]; the categorical construction used here is the openly proved Serre quotient of [Stacks, Tag 02MS]. The equivalence permits many graded presentations of the same sheaf; finite changes in the low-degree tail are invisible. Degree-one generation is part of the hypotheses. With arbitrary positive generator degrees, twists need not all be invertible, and the corresponding general statement uses irrelevant modules rather than this simplified tail argument. If \(B_+=0\), its Proj is empty and every finite graded \(B_0\)-module has only finitely many nonzero pieces; both categories in the equivalence are zero.

## 4. Vanishing detects ampleness

We use the affine-open definition: on a quasi-compact scheme, an invertible sheaf \(L\) is ample when affine opens \(X_s\), for sections of positive powers of \(L\), cover the scheme. The equivalence with eventual global generation is part of the prerequisite *Ample invertible sheaves*, with open proof [Stacks, Tag 01Q3].

**Theorem 4.1 (cohomological criterion).** Let \(X\) be proper over a Noetherian ring \(R\), and let \(L\) be invertible. The following conditions are equivalent:

1. \(L\) is ample.
2. For every coherent \(F\), the groups \(H^q(X,F\otimes L^d)\) are zero for all \(q>0\) and all sufficiently large \(d\).
3. For every coherent ideal \(I\subset\mathcal O_X\), there is some \(d\ge1\) with \(H^1(X,I\otimes L^d)=0\).

The exponent in the second condition may depend on \(F\); the third condition requires only one suitable positive exponent for each ideal.

**Proof.** Theorem 2.2 proves \(1\Rightarrow2\), and \(2\Rightarrow3\) is immediate. For the converse, \(X\) is Noetherian because it is of finite type over \(R\). Fix a closed point \(x\), and choose an affine neighborhood \(U\) on which \(L\) is trivial. Give \(Z=X\setminus U\) and \(Z\cup\{x\}\) their reduced closed subscheme structures, with coherent ideals \(I\) and \(I'\). Then \(I'\subset I\), and
\[
I/I'\simeq i_{x*}\kappa(x).
\]
This equality is local: on \(U\) it is the quotient by the maximal ideal of \(x\), while on \(X\setminus\{x\}\) the two ideals agree. Trivializing \(L\) on \(U\) identifies the twist of this quotient with the same residue-field skyscraper.

Choose \(d\ge1\) with \(H^1(X,I'\otimes L^d)=0\). The exact sequence of the two twisted ideals gives a surjection
\[
H^0(X,I\otimes L^d)\twoheadrightarrow\kappa(x).
\]
Lift the residue class one to a section \(s\). Viewed in \(H^0(X,L^d)\), this section is nonzero at \(x\) and vanishes at every point of \(Z\). Hence
\[
x\in X_s\subset U.
\]
In the chosen trivialization \(s|_U\) is a function \(g\), so \(X_s=D(g)\) inside the affine \(U\), and is affine. We have found an affine section open around every closed point.

The union \(W\) of all affine positive-power section opens is open. If \(X\setminus W\) were nonempty, this closed subset of a Noetherian space would contain a point closed in \(X\): choose a minimal nonempty closed subset; a scheme's underlying space is \(T_0\), so such a subset is a singleton. That contradicts the construction for closed points. Thus \(W=X\), proving ampleness. \(\square\)

This is [Stacks, Tag 0B5U], using the affine-neighborhood construction of [Stacks, Tag 0B5P]. The converse only used Noetherianity and the ideal vanishing condition; properness was used to obtain the forward implication. The proof does not assume that all points are closed, or that the residue fields equal the base field.

On \(\mathbf P_k^1\), a line bundle \(\mathcal O(a)\) has this property exactly when \(a>0\). For positive \(a\), the powers of the standard coordinate sections give affine charts, so it is ample. If \(a=0\), take \(F=\mathcal O(-2)\); its first cohomology stays nonzero under all twists by \(L\). If \(a<0\), already \(F=\mathcal O\) has nonzero first cohomology for every sufficiently large \(d\), since \(ad\le-2\). Thus testing just the structure sheaf would fail to rule out the trivial bundle: the criterion quantifies over all coherent sheaves.

## 5. Equations, sections and point ideals

Let \(k\) be a field and \(Z=V_+(f)\subset\mathbf P_k^N\), where \(f\) is a nonzero homogeneous polynomial of degree \(e>0\). Multiplication by \(f\) is injective in the polynomial ring, hence also in its chart localizations. Sheafifying gives
\[
0\longrightarrow\mathcal O(-e)\xrightarrow{\ f\ }\mathcal O
\longrightarrow i_*\mathcal O_Z\longrightarrow0.
\]
For \(N\ge2\), the first cohomology of every ambient twist is zero. Consequently, for every integer \(d\),
\[
H^0(Z,\mathcal O_Z(d))=k[T_0,\ldots,T_N]_d/f\,k[T_0,\ldots,T_N]_{d-e}.
\]
The identification respects multiplication, so the entire graded section module is the homogeneous coordinate quotient, with zero negative pieces. For example the plane conic \(X^2+YZ=0\) has \(\dim H^0(\mathcal O_Z(d))=2d+1\) for \(d\ge0\), by subtracting \(\binom d2\) from \(\binom{d+2}2\), with the low-degree cases computed directly.

The ambient-dimension hypothesis matters. On \(\mathbf P_k^1\), the long exact sequence contains an additional contribution from \(H^1(\mathcal O(d-e))\). The graded coordinate module still recovers the section module in high degree by Lemma 3.2, but need not recover its low-degree pieces.

For a point-ideal example, take the two rational points
\[
P=[1:0:0],\qquad Q=[0:1:0]
\]
in the plane with coordinates \(X,Y,Z\). Their homogeneous ideal is
\[
(Y,Z)\cap(X,Z)=(Z,XY).
\]
Modulo \(Z\), this is the identity \((Y)\cap(X)=(XY)\) in \(k[X,Y]\), verified by divisibility of monomials. Thus the ideal sheaf \(I\) is generated in degrees one and two. Its twist \(I(2)\) is generated by \(XZ,YZ,Z^2,XY\), and every \(I(d)\), \(d\ge2\), is globally generated. In contrast, \(H^0(I(1))\) consists only of multiples of \(Z\). Near \(P\), with \(y=Y/X\) and \(z=Z/X\), the ideal is \((y,z)\); the section \(Z\) generates only \((z)\). Hence \(I(1)\) is not globally generated. Vanishing and global generation have different thresholds, as the next exercises make explicit.

## 6. Exercises with solutions

**Exercise 6.1 (easy: the cohomological bound).** Prove \(H^q(\mathbf P_R^N,F)=0\) for \(q>N\), for any quasi-coherent \(F\), and identify which hypotheses of Theorem 2.1 this proof does not use.

**Solution.** Intersections of the \(N+1\) standard affine charts are affine. Positive cohomology of \(F\) on them vanishes by the affine theorem. Acyclic-cover comparison therefore identifies derived cohomology with the ordered Čech complex, whose last term has degree \(N\). All cohomology above it is zero. Neither Noetherianity of \(R\) nor coherence or finite generation of \(F\) enters this argument. These hypotheses enter the later finiteness and coherent-kernel induction, rather than the cover bound.

**Exercise 6.2 (easy: functions on a projective scheme).** Let \(X\) be projective over a Noetherian ring \(R\). Prove that \(H^0(X,\mathcal O_X)\) is a finite \(R\)-algebra.

**Solution.** Over the affine base, the projective construction gives a closed embedding into some \(\mathbf P_R^N\). Indeed a finite-type module defining a projective bundle has a finite generating set over \(R\), and the induced surjection of symmetric algebras embeds that bundle into finite-dimensional projective space; compose with the defining closed embedding of \(X\). The sheaf \(i_*\mathcal O_X\) is coherent, so Theorem 2.1 makes its zeroth cohomology a finite \(R\)-module. This is the same module as \(H^0(X,\mathcal O_X)\). Its multiplication is the multiplication of global functions, so it is an \(R\)-algebra finite as a module. The assertion is stronger than finite generation merely as an algebra.

**Exercise 6.3 (medium: generation after twisting).** Starting with a surjection \(\bigoplus_j\mathcal O(b_j)\to F\), give an explicit sufficient bound for global generation of \(F(d)\). Apply it to the two-point ideal above.

**Solution.** Every summand \(\mathcal O(b_j+d)\) is globally generated once \(b_j+d\ge0\). Thus \(d\ge\max_j(-b_j)\) suffices, and the images of the polynomial monomial sections generate the quotient. For \(I\), the generators \(Z,XY\) give a sheaf surjection \(\mathcal O(-1)\oplus\mathcal O(-2)\to I\), so \(d\ge2\) suffices. At \(d=2\), the four sections are precisely \(XZ,YZ,Z^2,XY\). The local calculation \((y,z)\ne(z)\) at \(P\) shows this bound is attained: degree one does not work. This establishes generation directly, rather than attempting to deduce it from the vanishing of \(H^1(I(d))\) alone.

**Exercise 6.4 (medium: coordinate rings and low-degree sections).** Compute the nonnegative section module of a degree-\(e\) hypersurface in \(\mathbf P_k^N\), \(N\ge2\). Compare with \(Z=V_+(T_0T_1)\subset\mathbf P_k^1\).

**Solution.** Twist the hypersurface sequence. Its global-section segment is
\[
0\to S_{d-e}\xrightarrow{\ f\ }S_d\to H^0(Z,\mathcal O_Z(d))
\to H^1(\mathbf P_k^N,\mathcal O(d-e)).
\]
The last term is zero for \(N\ge2\), so assembling all \(d\ge0\) gives \(S/(f)\). If \(0\le d<e\), the source \(S_{d-e}\) is zero and the section space is \(S_d\); if \(d\ge e\), its dimension is \(\binom{d+N}N-\binom{d-e+N}N\).

On the line, the specified \(Z\) is the disjoint union of the two reduced rational points \([1:0]\) and \([0:1]\). Every twist restricts to a one-dimensional space at each point, so its section space has dimension two for every integer \(d\). The coordinate quotient has dimension one in degree zero and two in each positive degree, with basis \(T_0^d,T_1^d\). The degree-zero map is the diagonal \(k\to k^2\), and is not onto; in every positive degree it is an isomorphism. This is exactly a low-degree discrepancy that vanishes in the Serre quotient. The extra degree-zero section comes from \(H^1(\mathcal O(-2))=k\) in the exact sequence.

**Exercise 6.5 (challenging: prove the converse criterion with few tests).** Suppose \(X\) is Noetherian and \(L\) is invertible. Assume that for every coherent ideal \(J\), at least one positive power satisfies \(H^1(X,J\otimes L^d)=0\). Prove ampleness, without assuming eventual vanishing or properness.

**Solution.** Fix a closed point \(x\), a trivializing affine \(U\), and the reduced ideals \(I\) of \(X\setminus U\) and \(I'\) of \((X\setminus U)\cup\{x\}\). They are coherent because \(X\) is Noetherian. Their quotient is the residue-field skyscraper at \(x\). Apply the hypothesis to \(J=I'\), not to \(I\), to obtain an exponent \(d>0\) making the connecting obstruction to lifting a residue class zero. The class one lifts to \(s\in H^0(X,I\otimes L^d)\). This section is nonvanishing at \(x\), vanishes off \(U\), and defines the affine open \(X_s=D(s|_U)\). Repeat for every closed point. The complement of the union of these opens is closed; if nonempty, Noetherianity and the \(T_0\) property provide a closed point in it, a contradiction. The affine section opens cover, which is ampleness by definition. Only the existence of one exponent for each chosen ideal was used; it may vary with the point.

**Exercise 6.6 (medium: two thresholds).** For the two-point ideal \(I\subset\mathcal O_{\mathbf P_k^2}\), compute \(H^1(I(d))\) for every integer \(d\), and compare its vanishing threshold with global generation.

**Solution.** The exact sequence \(0\to I(d)\to\mathcal O(d)\to\mathcal O_{P\sqcup Q}(d)\to0\) and \(H^1(\mathcal O(d))=0\) identify the first group with the cokernel of
\[
H^0(\mathbf P_k^2,\mathcal O(d))\longrightarrow k^2.
\]
Choose the frames at the two points given by \(X^d\) and \(Y^d\), respectively, with dual frames for negative \(d\). For \(d\ge1\), the polynomials \(X^d,Y^d\) map to the two coordinate vectors, so the cokernel is zero. For \(d=0\), constants map diagonally, giving a one-dimensional cokernel. For \(d<0\), there are no ambient sections, so the cokernel is \(k^2\). Thus first-cohomology vanishing starts at degree one, while global generation starts at degree two. A vanishing theorem does not specify an optimal generation bound unless one also controls the presentation and its kernel.

## References and geometric prerequisites

- **[Stacks]** The Stacks project authors, *The Stacks project*, read in its AI Integrated Stacks Project edition. Serre's projective-space theorems: [Tag 01YS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-coherent-projective); degree-one Proj: [Tag 0AG6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-coherent-on-proj); eventual module recovery: [Tag 0AG7](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-recover-tail-graded-module); the equivalence: [Tag 0BXD](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-proposition-coherent-modules-on-proj). The section recovery used in Section 3 is [Tag 0AG5](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/properties.html#properties-lemma-proj-quasi-coherent), with its degree-one compatibility checked explicitly above. The general Serre quotient construction and its universal property are supplied by the open proof of [Tag 02MS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/homology.html#homology-lemma-serre-subcategory-is-kernel).
- Proper schemes with an ample bundle: [Tag 0B5T](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-coherent-proper-ample); relative vanishing: [Tag 02O1](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-kill-by-twisting); coherence of locally projective pushforwards: [Tag 02O4](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-locally-projective-pushforward); the criterion: [Tag 0B5U](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-vanshing-gives-ample), using [Tag 0B5P](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-quasi-compact-h1-zero-invertible).
- The planned prerequisite *Ample invertible sheaves*, in *Morphisms of schemes*, provides the definition and characterizations of ampleness, including [Tag 01Q3](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/properties.html#properties-proposition-characterize-ample) and the power defining a projective immersion in [Tag 01VT](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-finite-type-over-affine-ample-very-ample). The planned *Projective morphisms and Chow's lemma* provides the properness relations; the graph argument making a morphism from a proper scheme to a separated target proper is [Tag 01W6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-image-proper-scheme-closed). These are geometric prerequisites, rather than appeals to a later cohomological theorem.
- J.-P. Serre, *Faisceaux algébriques cohérents*, Annals of Mathematics (2) **61** (1955), 197–278, Chapter III, is the original source of the named projective-cohomology results. The proofs, teaching organization, examples and solutions here are independently written; the linked Stacks treatments and categorical prerequisite retain the GNU Free Documentation License 1.2. AI Integrated Stacks Project contains AI-proposed corrections and additions and is not reviewed by maintainers of the [official Stacks project](https://stacks.math.columbia.edu/).