Ample invertible sheaves

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

Very ampleness gives projective coordinates immediately. Ampleness allows us to take powers until sections provide enough affine neighbourhoods. This topological definition also controls sheaves: after a sufficiently large twist, every quasi-coherent sheaf of finite type is generated by global sections. For finite-type morphisms, those sections eventually supply projective embeddings in the immersion sense.

We use section ratios and the embedding constructions of Very ample sheaves. The foundational prerequisites are Quasi-coherent sheaves on schemes and Proj of a graded ring, the ninth and eleventh planned lessons of Sheaves and schemes. They supply affine module descriptions, tensor products, homogeneous localizations and the standard affine charts of Proj. Exact open affine providers are Stacks, Tag 01IA and the Proj construction. We will prove the section-extension mechanism needed here before applying those foundations.

1. Affine neighbourhoods from sections

For a section \(s\) of an invertible sheaf, \(X_s\) denotes its generation locus. An invertible sheaf \(\mathcal L\) on \(X\) is ample if \(X\) is quasi-compact and the affine opens \(X_s\), for sections \(s\in\Gamma(X,\mathcal L^{\otimes d})\) with \(d>0\), cover \(X\). Quasi-compactness is part of this definition, as in Stacks, Tag 01PS.

Lemma 1.1. If \(U\) is affine and \(s\) is a section of an invertible sheaf on \(U\), then \(U_s\) is affine.

Proof. Write \(U=\operatorname{Spec}A\), and let \(N\) be the invertible \(A\)-module of sections. Consider the algebra

If \(s\) is zero in \(N\otimes_A\kappa(\mathfrak p)\), this algebra has zero fibre over \(\mathfrak p\), since its relation becomes \(0=1\). Thus \(\operatorname{Spec}B\to U\) factors through \(U_s\). On any open where \(s\) is a basis of \(N\), the algebra is \(A'[T]/(T-1)\cong A'\). Those local identifications give an isomorphism \(\operatorname{Spec}B\cong U_s\). \(\square\)

This is Stacks, Tag 01PV. The section need not be a function on all of \(U\); the algebra (1.1) handles a nontrivial invertible module.

Proposition 1.2. For every integer \(r>0\), \(\mathcal L\) is ample if and only if \(\mathcal L^{\otimes r}\) is ample. An ample sheaf restricts to an ample sheaf on every closed subscheme. Moreover, a scheme admitting an ample sheaf is quasi-separated.

Proof. If \(s\) is a section of \(\mathcal L^{\otimes d}\), then \(X_{s^r}=X_s\), and \(s^r\) is a section of \((\mathcal L^{\otimes r})^{\otimes d}\). This proves one direction of the power assertion; the other follows by viewing sections of \(\mathcal L^{\otimes rd}\) as sections of powers of \(\mathcal L\).

For a closed subscheme \(Z\subset X\), quasi-compactness passes to \(Z\), and \(Z_{s|_Z}=Z\cap X_s\) is closed in the affine \(X_s\), hence affine. These opens cover \(Z\).

Finally choose the affine section opens as an affine cover of \(X\). The intersection of two of them is affine by Lemma 1.1 applied to the restriction of the second section on the first affine. In particular those intersections are quasi-compact, which is the affine-cover criterion for quasi-separatedness. \(\square\)

The power and closed-restriction assertions are Stacks, Tags 01PT and 01PU. We will strengthen quasi-separatedness to separatedness below.

2. Clearing denominators for sheaf sections

The following lemma is the engine of the equivalence theorem. It does not require Noetherian rings or coherent sheaves.

Lemma 2.1 (extension after a twist). Let \(X\) be quasi-compact and quasi-separated, \(\mathcal L\) invertible, \(s\in\Gamma(X,\mathcal L^{\otimes d})\), and \(\mathcal F\) quasi-coherent. Every section of \(\mathcal F\) on \(X_s\) can be written

A global numerator whose restriction vanishes on \(X_s\) is killed by some further power of \(s\). In particular, for the graded section ring

the natural map \(G_{(s)}\to\Gamma(X_s,\mathcal O_X)\) is an isomorphism.

Proof. Choose a finite affine cover \(U_i\) on which \(\mathcal L^{\otimes d}\) is trivial. Write \(s=f_i e_i\) in the chosen basis. On \(U_i\cap X_s=D(f_i)\), quasi-coherence turns the given section into a fraction with denominator \(f_i^{m_i}\). Increase the finitely many exponents to one \(m\). Multiplying the local fraction by \(s^m\) gives a section \(t_i\) of \(\mathcal F\otimes\mathcal L^{\otimes dm}\) on \(U_i\).

The sections \(t_i,t_j\) agree on \(U_i\cap U_j\cap X_s\), but can differ elsewhere on \(U_i\cap U_j\). That overlap is quasi-compact. Cover it by finitely many affine opens trivializing the line sheaf. On each, a section zero after localization at the coefficient of \(s\) is annihilated by some power of that coefficient. Taking a maximum on the cover, and then over all pairs, gives one power \(s^q\) annihilating every overlap difference. The sections \(t_i s^q\) now glue to a global numerator, proving (2.1).

The same argument on the original finite affine cover proves the annihilation assertion for a global section vanishing on \(X_s\); it uses quasi-compactness alone for this injectivity part. Apply the extension and annihilation assertions to \(\mathcal F=\mathcal O_X\). A numerator in degree \(dm\) divided by \(s^m\) is exactly a degree-zero element of \(G_s\), proving surjectivity and injectivity of the displayed ring map. \(\square\)

This is Stacks, Tag 01PW. There are two finite steps: clearing denominators on charts, then killing discrepancies on overlaps. Omitting the latter step loses the reason for quasi-separatedness.

3. Four equivalent descriptions

When the positive-degree section opens cover \(X\), their ratios define the canonical map

It is the graded-section construction of Stacks, Tags 01PZ and 01NK. On a chart its ring map is \(a/s^m\mapsto a/s^m\) in the trivialization supplied by \(s\). Arbitrary positive degrees are allowed; the section ring need not be generated in degree one.

Theorem 3.1. Let \(X\) be quasi-compact and \(\mathcal L\) invertible. The following are equivalent:

1. \(\mathcal L\) is ample.
2. The opens \(X_s\), for all positive-degree homogeneous sections, form a basis for the topology.
3. Those opens cover \(X\), and (3.1) is an open immersion.
4. \(X\) is quasi-separated and, for every finite-type quasi-coherent \(\mathcal F\), the sheaf \(\mathcal F\otimes\mathcal L^{\otimes n}\) is globally generated for all sufficiently large \(n\).

Every scheme satisfying these conditions is separated. Among the section opens, the affine ones also form a basis.

Proof of the geometric equivalences. Under the first condition, choose finitely many affine \(X_{s_i}\) covering \(X\). Proposition 1.2 gives quasi-separatedness, and Lemma 2.1 identifies their rings with \(G_{(s_i)}\). Thus (3.1) identifies each \(X_{s_i}\) with the whole standard open \(D_+(s_i)\). These identifications are restrictions of the same canonical map and agree on overlaps. They identify \(X\) with the open subscheme \(\bigcup_iD_+(s_i)\), proving the third condition.

For the third implying the second, standard homogeneous opens form a basis on Proj, and their inverse images are precisely the section opens. Restrict that basis to the open image. For the second implying the first, given \(x\) in an arbitrary small affine neighbourhood \(U\), choose \(x\in X_s\subset U\). Lemma 1.1 makes \(X_s=U\cap X_s\) affine. This also proves the affine-basis assertion.

Finally Proj of any graded ring is separated over its affine degree-zero base, by the first lesson, and that affine base is separated over \(\operatorname{Spec}\mathbf Z\). An open subscheme is separated. Hence the third condition proves separatedness of \(X\).

Proof of eventual generation. Start with the affine cover \(X_{s_i}\) of the ample condition. Choose a positive common multiple \(D\) of the degrees \(d_i\), and put \(\mathcal H=\mathcal L^{\otimes D}\), \(h_i=s_i^{D/d_i}\). The sections \(h_i\) globally generate \(\mathcal H\), and their affine opens are the same covering opens.

For a finite-type \(\mathcal F\), choose a finite list of module generators on each affine \(X_{h_i}\). By Lemma 2.1 each generator is a numerator in \(\Gamma(X,\mathcal F\otimes\mathcal H^{\otimes m_{ij}})\) divided by \(h_i^{m_{ij}}\). Increase all exponents to a common \(M\), multiplying the respective numerators by powers of \(h_i\). The resulting global sections generate \(\mathcal F\otimes\mathcal H^{\otimes M}\) on each \(X_{h_i}\), hence everywhere. Since \(\mathcal H\) is globally generated, tensoring these generators with generators of \(\mathcal H^{\otimes r}\) proves generation for every \(M+r\).

Apply this conclusion to the finitely many sheaves \(\mathcal F\otimes\mathcal L^{\otimes j}\), \(0\leq j<D\). Taking the maximum of their thresholds proves generation of \(\mathcal F\otimes\mathcal L^{\otimes n}\) for every sufficiently large integer \(n\), including every residue class modulo \(D\). This proves the fourth condition.

Proof of the converse from generation. Fix \(x\in U\subset X\), where \(U\) is affine. Let \(\mathcal I\) be the quasi-coherent ideal defining the reduced closed complement \(X\setminus U\). Its stalk at \(x\) is \(\mathcal O_{X,x}\). On a qcqs scheme, every quasi-coherent sheaf is the directed union of its finite-type quasi-coherent subsheaves. The exact full open proof of this ingredient is Stacks, Tag 01PG; its finite extension construction is what permits an arbitrary, possibly infinitely generated ideal here. Consequently there is a finite-type quasi-coherent subideal \(\mathcal J\subset\mathcal I\) with \(\mathcal J_x=\mathcal O_{X,x}\).

For large positive \(n\), the fourth condition generates \(\mathcal J\otimes\mathcal L^{\otimes n}\) by global sections. Some such section has a unit coefficient at \(x\), because its stalk there is a rank-one module over a local ring. Regard it, by the inclusion of ideals, as \(s\in\Gamma(X,\mathcal L^{\otimes n})\). It vanishes on \(X\setminus U\), so \(x\in X_s\subset U\). Lemma 1.1 makes \(X_s\) affine. These affine section opens cover \(X\), proving ampleness. \(\square\)

This is Stacks, Tag 01Q3. Quasi-separatedness in condition 4 is an explicit hypothesis for its converse. The threshold depends on \(\mathcal F\); the theorem does not claim one threshold for every sheaf at once.

It also does not assert that every section open is affine. On \(\mathbf A^2_k\setminus\{0\}\), the structure sheaf is ample, as shown below, but the generation locus of its section \(1\) is the whole nonaffine scheme. The affine section opens form a basis within the larger family.

4. Relative ampleness and very ample powers

An invertible sheaf \(\mathcal L\) is \(f\)-ample if \(f:X\to S\) is quasi-compact and \(\mathcal L|_{f^{-1}(V)}\) is ample for every affine open \(V\subset S\). This is Stacks, Tag 01VH. Such a morphism is separated: over each affine \(V\), the preceding theorem makes the inverse image separated absolutely, hence separated over \(V\), and separatedness is target-local. Relative ampleness does not force finite type: the structure sheaf on \(\operatorname{Spec}k[x_1,x_2,\ldots]\) is ample, since the section \(1\) has the entire affine scheme as its generation locus, but its morphism to \(\operatorname{Spec}k\) is not of finite type.

Lemma 4.1. If \(S\) is affine, \(f\) is quasi-compact, and \(\mathcal L\) is \(f\)-very ample in the projective-bundle immersion convention, then \(\mathcal L\) is ample on \(X\).

Proof. Factor the given immersion as a closed immersion into an open \(W\subset\mathbf P(E)\), with \(E\) a module over the affine base ring. At a point of its image, choose a standard homogeneous open \(D_+(h)\subset W\). Its inverse image is closed in that affine chart, hence affine, and is \(X_s\) for the pulled-back degree-\(d\) section \(s\) of \(\mathcal L^{\otimes d}\). Such opens cover \(X\). Since \(X\) is quasi-compact, this is ampleness. \(\square\)

Theorem 4.2. If \(S\) is quasi-compact and \(f:X\to S\) is of finite type, the following are equivalent:

When \(S\) is affine, the powers can give immersions into finite-dimensional projective spaces. If \(f\) is also proper, those immersions are closed.

Proof over an affine base. Assume \(\mathcal L\) ample. Choose finitely many affine section opens \(X_{s_i}\) covering \(X\), and replace their sections by powers of a common degree. Write them as \(h_i\) for \(\mathcal H=\mathcal L^{\otimes D}\). Each affine ring \(B_i=\Gamma(X_{h_i},\mathcal O_X)\) is a finite-type algebra over the base ring \(A\). Choose finite algebra generators \(b_{ij}\). Lemma 2.1 writes them as \(t_{ij}/h_i^{e_{ij}}\). Choose \(m\) at least every \(e_{ij}\). The sections

generate the sheaf, since the first family already does. Their map to finite projective space has affine inverse image \(X_{h_i}\) on the chart of \(h_i^m\), and its ratios include every \(b_{ij}\). Thus it is closed on those target charts, which cover its image, and is an immersion into the full projective space. This proves very ampleness of one power.

By Theorem 3.1, all sufficiently large powers of \(\mathcal L\) are globally generated. Tensor them with the power just embedded. The very-ample tensor theorem of the preceding lesson proves H-very ampleness of every sufficiently large power. Conversely, a very ample power on this quasi-compact source is ample by Lemma 4.1, and Proposition 1.2 recovers ampleness of \(\mathcal L\).

Passage to a quasi-compact base. Choose a finite affine cover \(V_i\) of \(S\). If \(\mathcal L\) is \(f\)-ample, the affine argument gives immersions for each sufficiently large power on every \(X_i=f^{-1}(V_i)\). Take the maximum of the finitely many thresholds. We explain why these local very-ample descriptions give a global one for that same power.

The morphism \(f\) is separated and quasi-compact. Hence \(\mathcal E=f_*\mathcal L^{\otimes d}\) is quasi-coherent. This familiar pushforward fact can be seen on an affine base by a finite affine source cover and finite affine covers of its overlaps: sections are the kernel of the resulting map between finite products of modules. Localization of the base commutes with that kernel and those finite products, so the direct image has the affine quasi-coherence test. This is the exact proof of Stacks, Tag 01LC.

The evaluation \(f^*\mathcal E\to\mathcal L^{\otimes d}\) is surjective, because locally the sections furnishing each immersion are included in it. It gives \(r:X\to\mathbf P(\mathcal E)\). On \(V_i\), let a finite local tuple supply the known immersion \(i_i:X_i\to\mathbf P^{N_i}_{V_i}\). Inside \(\mathbf P(\mathcal E|_{V_i})\), the open where this tuple generates the tautological quotient has a projection \(\rho_i\) to \(\mathbf P^{N_i}_{V_i}\). It is affine on each target coordinate chart: its inverse image is the corresponding standard affine chart of \(\mathbf P(\mathcal E|_{V_i})\). In particular \(\rho_i\) is separated. The map \(r|_{X_i}\) lands in that open and its composite with \(\rho_i\) is \(i_i\), as the section ratios agree.

A morphism whose composite with a separated morphism is an immersion is itself an immersion: factor through its closed graph and the base change of the given immersion. Thus \(r|_{X_i}\) is an immersion, and the target-local immersion property makes \(r\) an immersion globally. Its tautological pullback is \(\mathcal L^{\otimes d}\), proving the last condition in (4.1). The last implies the middle trivially. The middle implies the first by restricting to every affine base open and applying Lemma 4.1 and the power assertion.

Finally a map from a proper \(S\)-scheme to separated projective space is proper. Its immersion image is closed by the proper-map theorem, so the finite-space embeddings over an affine base are closed when \(f\) is proper. \(\square\)

These are Stacks, Tags 01VT and 01VU. The quasi-compactness of the base supplies a single bound across its affine cover. For a general base, the global module \(\mathcal E\) above need not be a finite free module; the theorem asserts the projective-bundle notion of very ampleness.

A quasi-projective morphism means a finite-type morphism admitting a relatively ample invertible sheaf. An H-quasi-projective morphism admits a quasi-compact immersion into \(\mathbf P^N_S\). These are Stacks, Tag 01VW. Over an affine base the two notions agree by Theorem 4.2 and Lemma 4.1. Local embeddings over different base opens alone do not supply a single specified global invertible sheaf; that distinction matters over a general base.

5. The structure sheaf and quasi-affineness

Here quasi-affine means isomorphic to a quasi-compact open subscheme of an affine scheme.

Theorem 5.1. \(\mathcal O_X\) is ample if and only if \(X\) is quasi-affine. In that case its canonical map to \(\operatorname{Spec}\Gamma(X,\mathcal O_X)\) is a quasi-compact open immersion.

Proof. Put \(A=\Gamma(X,\mathcal O_X)\). The section ring for \(\mathcal O_X\) is \(A[T]\), with \(T\) of degree one. Its Proj has the single chart \(D_+(T)=\operatorname{Spec}A\), so \(\operatorname{Proj}A[T]\cong\operatorname{Spec}A\). If \(\mathcal O_X\) is ample, Theorem 3.1 identifies its canonical map with an open immersion into this affine scheme. It is quasi-compact because the source is quasi-compact and the target is separated: affine target opens meet a finite affine cover of the source in quasi-compact opens. Thus \(X\) is quasi-affine.

Conversely, let \(X\subset\operatorname{Spec}R\) be a quasi-compact open. For each \(x\in X\), choose a distinguished affine \(D(r)\) containing \(x\) and contained in \(X\). The restriction of \(r\) is a global function on \(X\), and its generation locus there is exactly \(D(r)\). These affine section opens cover \(X\), proving ampleness of \(\mathcal O_X\). \(\square\)

This is Stacks, Tag 01QE. For \(X=\mathbf A^2_k\setminus\{0\}\), the global functions are \(k[x,y]_x\cap k[x,y]_y=k[x,y]\), so the canonical map is the given punctured-plane open immersion. The affine section opens \(X_x=D(x)\) and \(X_y=D(y)\) cover it. The scheme is not affine: were it affine, its canonical map to the spectrum of its global functions would be an isomorphism, contradicting the missing origin.

6. Exercises with solutions

Exercise 6.1 (easy). For a field \(k\) and \(n\geq1\), show that \(\mathcal O(-1)\) on \(\mathbf P^n_k\) is not ample. Show that \(\mathcal O(1)\) is ample. Explain the case \(n=0\).

Solution. Restrict to a coordinate projective line. The chart compatibility calculation for \(\mathbf P^1\) gives no nonzero sections of \(\mathcal O(-d)\) for any \(d>0\). Hence this sheaf cannot have a nonempty generation locus covering the line and is not ample. Closed restriction of ampleness would contradict that fact, so \(\mathcal O(-1)\) on \(\mathbf P^n\) is not ample. For \(\mathcal O(1)\), the standard coordinate sections have affine generation loci and cover projective space, giving ampleness directly. When \(n=0\), projective space is \(\operatorname{Spec}k\), and all twists are trivial and ample; the dimension condition is necessary.

Exercise 6.2 (medium). On \(\mathbf P^1_k\times_k\mathbf P^1_k\), prove that \(\mathcal O(a,b)\) is ample exactly when \(a,b>0\).

Solution. If both are positive, the two Veronese embeddings followed by Segre give a closed immersion with tautological pullback \(\mathcal O(a,b)\). The source is quasi-compact, so Lemma 4.1 gives ampleness. Conversely restrict an ample sheaf to fibres of the two projections, which are closed projective lines over rational points of the other factor. The restrictions are \(\mathcal O(a)\) and \(\mathcal O(b)\). For a negative exponent every positive power has no sections; for exponent zero all section opens are empty or the entire nonaffine projective line. Neither sheaf is ample. Thus each exponent must be positive.

Exercise 6.3 (medium). Show that a scheme admitting an ample invertible sheaf is separated, without assuming separatedness at the start of the global-section argument.

Solution. First use affine section opens to obtain quasi-separatedness: two such opens meet affinely by Lemma 1.1. This justifies the denominator extension of Lemma 2.1. Its ring isomorphisms identify the finite affine section cover with standard opens in the homogeneous spectrum of the section ring, so the canonical map is an open immersion. That Proj is separated over its affine degree-zero base, and the base is separated absolutely. The open subscheme is therefore separated. This order avoids using separatedness to prove the localization fact from which separatedness follows.

Exercise 6.4 (medium). Prove that \(\mathcal O_X\) is ample exactly when \(X\) is quasi-affine, and show why this does not force affineness.

Solution. Its section ring is \(A[T]\), \(A=\Gamma(X,\mathcal O_X)\), and \(\operatorname{Proj}A[T]=\operatorname{Spec}A\). The ampleness characterization therefore gives the canonical quasi-compact open immersion. Conversely, distinguished opens contained in a quasi-compact open of an affine scheme are generation loci of restricted global functions, so they give an affine section cover. The punctured plane is the counterexample to affineness: it has this cover and global function ring \(k[x,y]\), yet its canonical map omits the origin. Ampleness makes that map an open immersion, not necessarily a surjection.

Exercise 6.5 (medium). Let \(\mathcal L\) be ample and \(\mathcal M\) globally generated. Prove that \(\mathcal L\otimes\mathcal M\) is ample directly from section opens.

Solution. At \(x\), choose \(s\in\Gamma(X,\mathcal L^{\otimes d})\) with \(x\in X_s\) affine. Choose a global section \(t\) of \(\mathcal M\) generating it at \(x\). The product \(st^d\) is a section of \((\mathcal L\otimes\mathcal M)^{\otimes d}\), and its generation locus is \(X_s\cap X_t\). Lemma 1.1 makes this intersection affine, since it is a section locus inside the affine \(X_s\). These opens cover the already quasi-compact \(X\), proving ampleness. More generally it suffices that positive powers of \(\mathcal M\) have generation loci covering \(X\): if \(t\) has degree \(e\), use \(s^e t^d\). This is Stacks, Tag 0890.

References and proof providers

The Stacks project, read in the AI Integrated Stacks Project edition, gives the tagged ampleness, localization, relative-power and quasi-affine treatments cited above. The precise open ingredient in the converse generation proof is the qcqs finite-type submodule theorem at Tag 01PG, whose full proof and extension dependency are available at the linked locator. The canonical graded-section map uses the named foundational Proj construction. These works retain GNU FDL 1.2; no source text is reproduced here.

Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, draft of 27 July 2024, §16.2, was consulted for the complementary geometric and global-generation viewpoints. Its distinction between the proper and absolute settings informs the conventions stated here. All assigned characterizations and applications, and all five exercises, have been proved with the necessary quasi-compactness, quasi-separatedness and finite-type conditions explicit.