Non-commutative $\PP^1$-bundles over commutative schemes

In this paper we develop the theory of non-commutative $\PP^1$-bundles over commutative (smooth) schemes. Such non-commutative $\PP^1$-bundles occur in the theory of $D$-modules but our definition is more general. We can show that every non-commutative deformation of a Hirzebruch surface is given by a non-commutative $\PP^1$-bundle over $\PP^1$ in our sense.


Introduction
In this paper we develop the theory of non-commutative P 1 -bundles over commutative (smooth) schemes. Such non-commutative P 1 -bundles occur in the theory of D-modules (see [5]) but our definition is more general. The extra generality is needed to cover basic examples in non-commutative algebraic geometry [30]. As an indication that our definition is the "right one" we present a proof that every noncommutative deformation of a Hirzebruch surface is given by a non-commutative P 1 -bundle over P 1 (see below).
Let us explain our definition. Assume that X is a scheme of finite type over a field k. Following [30] and later [23,24] we define a shbimod(X −X) as the category of coherent O X×X modules whose support is finite over X on the left and right. We call the elements of shbimod(X − X) "sheaf-bimodules" to distinguish them from the somewhat more general bimodules which were introduced in [31]. The category of coherent sheaves on X may be identified with the objects in shbimod(X − X) supported on the diagonal.
Convolution makes shbimod(X − X) into a monoidal category so we may define a "Z-graded sheaf-algebra" on X to be a graded algebra object in shbimod(X − X). If A is a graded sheaf-algebra then we may define a category Gr(A) of graded Amodules. Following [1] we define QGr(A) as Gr(A) divided by the modules which are direct limits of right bounded ones.
A first approximative approach to non-commutative P 1 -bundles on X, advocated in [23,24,30], is to consider abelian categories of the form QGr(A) where A is a graded sheaf-algebra on X which resembles the symmetric algebra of a locally free sheaf of rank two on X.
In order to explain this definition we need a notion of locally free sheaf in shbimod(X − X). We say that E ∈ shbimod(X − X) is locally free (of rank n) if pr 1 * E and pr 2 * E are locally free (of rank n). If E ∈ shbimod(X − X) then we may define the tensor algebra T X E in the obvious way. If E is locally free of rank two then in [23,24,30] a non-commutative symmetric algebra of rank two associated to E is defined as a graded sheaf-algebra of the form T X E/(Q) where Q ⊂ E ⊗ E is Q is locally free of rank one. While this is a reasonable definition there are some problems with it.
• It is not so easy to find suitable Q inside E ⊗ E (see the complicated computations in [30]). • The dependence of QGr(T X E/(Q)) on Q has not been made clear. In this paper we solve these problems by showing that Q is actually superfluous (!) if X is smooth. In other words the theory can be set up in a manner which does not depend on an additional choice of Q.
We need the concept of a sheaf-Z-algebra on X . This is a sheaf algebra version of a usual Z-algebra [7,27]. Thus a sheaf Z-algebra on X is defined by giving for i, j ∈ Z an object A ij in shbimod(X − X) together with "multiplication maps" A ij ⊗ A jk → A ik and "identity maps" O X → A ii satisfying the usual axioms. As in the graded case we may define abelian categories Gr(A) and QGr(A).
Let E be locally free of rank n. Then it is easy to show that − ⊗ OX E has a right adjoint − ⊗ OX E * , where E * ∈ shbimod(X − X) is also locally free of rank n (this depends on X being smooth). Repeating this construction, we may define E * 2 = E * * by requiring that − ⊗ OX E * * is the right adjoint of − ⊗ OX E * . By induction we define E * 0 = E, E * m+1 = (E * m ) * for m ≥ 0 and by considering left adjoints we may define E * m for m < 0.
We now define S(E) as the Z-algebra which satisfies (a) S(E) mm = O X ; (b) S(E) m,m+1 = E * m ; (c) S(E) is freely generated by the S(E) m,m+1 , subject to the relations given by the images of i m .
Definition 1.1. The non-commutative P 1 -bundle P(E) on X associated to the locally free sheaf bimodule of rank two E on X is the category QGr(S(E)).
It is easy to see that if E is an ordinary commutative vector bundle of rank 2 on X then Gr(S X (E)) ∼ = Gr(S(E)). Thus the notion of a non-commutative P 1 -bundle if a generalization of the commutative one. This is no longer true in higher rank but even then the algebra S(E) could be interesting in its own right.
We will show (see §5.2) that if E ∈ shbimod(X − X) is locally free of rank n and Q ⊂ E ⊗ E is of rank one and satisfies a suitable non-degeneracy condition then Gr(T X E/(Q) = Gr(S(E)). This shows that the current definition of P 1 -bundles is indeed a generalization of the earlier one.
Let us now give a more detailed description of the content of this paper. Our first main result is the following. Theorem 1.2. If E is locally free of rank two then S(E) is a noetherian sheaf-Zalgebra in the sense that Gr(S(E)) is a locally noetherian Grothendieck category.
To prove this we follow a standard approach (see [3]) which consists in defining a suitable quotient D of A = S(E) through the functor of point modules. The sheaf Z-algebra D will be noetherian by construction and we will show that there is an invertible ideal J ⊂ A ≥2 such that D = A/J . Then we may conclude by invoking a suitable variant of the Hilbert basis theorem.
Point modules over sheaf-(Z)-algebras have been defined in Adam Nyman's PhDthesis [20] and he has shown that the corresponding functor is representable (under suitable hypotheses). In particular it follows from his results that the point functor of S(E) is representable by P X×X (E). We reproduce the proof of this fact since we need the exact nature of the bijections involved.
Our second main result is the following. Theorem 1.3. Assume that Z is a Hirzebruch surface. Then every deformation of Z is a non-commutative P 1 -bundle over P 1 .
For a precise definition of the notion of deformation we use we refer to §8.2 (which is based on [29]). The proof of Theorem 1.3 is based on the observation that on Z there are canonical exceptional line bundles which may be lifted to any deformation. Imitating some standard constructions in commutative algebraic geometry using the resulting objects yields the desired result.
After this paper was put on the arXiv the theory of non-commutative P 1 -bundles has been further developed. In [21,22] it was proved that they are Ext-finite and satisfy a classical form of Serre duality. These papers use Theorem 7.1.2 below. In return the current proof of Theorem 1.3 uses some results from [21,22].
In [16] it was shown that non-commutative P 1 -bundles share a number of geometric properties with their commutative counterparts. These results are stated in the language of non-commutative algebraic geometry (where Grothendieck categories play the role of spaces, see e.g. [26,31]). In this setting one may define a structure map f : P(E) → X and Izuru Mori shows that the fibers do not intersect. He also defines a certain "quasi-section" for f and computes its self-intersection. In [17] Izuru Mori computes the derived category of non-commutative P 1 -bundles.
In [8] the authors attack the reverse question. They generalize a standard characterization of ruled surfaces [12] to the non-commutative case. Due to some new non-commutative phenomena that have to be dealt with they do not yet obtain a full analogue but nonetheless non-commutative P 1 -bundles appear as a basic example. Along the way the authors prove that non-commutative P 1 -bundles satisfy the Bondal-Kapranov strengthening of Serre duality [6] and are "strongly noetherian" (which is important for the construction of Hilbert schemes in this generality [2]).

Acknowledgement
The author wishes to thank Daniel Chan, Colin Ingalls, Izuru Mori, Adman Nyman and Paul Smith for useful discussions. In addition he thanks the anonymous referee for his careful reading of the manuscript.

Notations and conventions
Unless otherwise specified all schemes below will be of finite type over a field k.

Sheaf-bimodules
4.1. Generalities. In the current and the next section we recapitulate the definition of sheaf-bimodules from [30] and we give additional properties. Since we will need to work with certain families of objects it will be convenient to develop the material over a base-scheme S. In the applications we will assume S = Spec k.
Below S is a scheme and α : X → S, β : Y → S, γ : Z → S will be Sschemes. An S-central coherent X − Y -sheaf-bimodule E is by definition a coherent O X×SY -module such that the support of E is finite over both X and Y . We denote the corresponding abelian category by shbimod S (X − Y ). More generally an Scentral X − Y -sheaf-bimodule will be a quasi-coherent sheaf on X × S Y which is a filtered direct limit of objects in shbimod S (X − Y ). We denote the corresponding category by ShBimod then the tensor product E ⊗ OY F is defined as pr 13 * (pr * 12 E ⊗ OX×Y ×Z pr * 23 F ). It is easy to show that this definition yields all the expected properties (see [30]). Now assume that we have finite S-maps u : Any bimodule E can be presented in this form since we may take W to be the scheme-theoretic support of E. From the definition it is easy to check that It is useful to know that the functor − ⊗ OX E actually determines E. Let us define Bimod(X − Y ) as the category of right exact functors Qch(X) → Qch(Y ) commuting with direct sums (this is equivalent to the definition in [31]). Then we have a functor F : which sends E to the functor − ⊗ OX E. We have the following result. Proof. We have to show how to reconstruct E from the functor − ⊗ OX E.
Choose an affine open covering X = i U i and let u i : U i → X, u ij : U i ∩U j → X be the inclusion maps.
Assume that H : Qch(X) → Qch(Y ) is a right exact functor commuting with direct sums. Then H(u i * O Ui ) will be a quasi-coherent sheaf on Y with an O X (U i ) structure. There is a corresponding quasi-coherent sheaf In a similar way we find quasi-coherent sheaves H ij on ( It would be interesting to give a more precise characterization of the essential image of the functor F . One useful observation is that if E ∈ ShBimod S (X − Y ) then − ⊗ OX E preserves exactness of short exact sequence of vector bundles. This leads to the following example.
If we compute F as in the proof of Lemma 4.1.1 then we find F = 0 which gives another reason why H is not in the essential image of F .
A partial result in this context has been obtained by Nyman in [19].
The following lemma shows that tensor products of locally free bimodules behave as they should.
Thus we have to show that if u : W → X is a finite map and U, V ′ are coherent O W -modules such that V ′ is locally free and u * U is locally free then u * (U ⊗ OW V ′ ) is locally free. Since the question is local on X we may reduce to the case that X is affine. Then W is affine as well and hence V ′ is a direct summand of a free O W -module. So we reduce to the case V ′ = O W which is obvious.
It is sufficient to prove the assertion on the rank for all pullbacks Spec l → S for l algebraically closed. Hence we may assume that S = Spec l with k algebraically closed. Now let m, n be respectively the left rank of E and F . We have to show that length(O x ⊗ OX E ⊗ OY F ) = mn for all closed points x ∈ X. Since O x ⊗ OX E is an extension of m objects of the form O yi for some y i ∈ Y , this is clear.
In the sequel we will use the following lemma to show that certain sheaves are locally free. Lemma 4.1.5. Assume that ψ : R → S is a local ring homomorphism between noetherian commutative local rings with maximal ideals m, n. Let u : M → N be a morphism between finitely generated S modules where N is in addition flat over R. Assume that u ⊗ R R/m is injective with S/mS-free cokernel. Then u is also injective with S-free cokernel.
Proof. Let C be the cokernel of u. By hypotheses C/mC is free over S/mS. Choose an isomorphism (S/mS) k → C/mC and lift this to a map θ : S k → C. Let T its cokernel. Tensoring with R/m yields T /mT = 0. Since ψ(m) ⊂ n we obtain T = 0 by Nakayama's lemma. Now factor θ through a map θ ′ : S k → N and let K be the pullback of θ ′ and u. Thus we have an exact sequence: Since N is flat over R this sequence remains exact if we tensor with R/mR. Since (S/mS) k is isomorphic to coker u ⊗ R R/m we deduce that K/mK = 0. By Nakayama we obtain K = 0. This clearly implies what we want.
If α is smooth then we will say that α is equidimensional if the fibers of α are equidimensional and if furthermore they all have the same dimension. We will say that α is of relative dimension n if it is equidimensional and if all fibers have dimension n.
The following result will be very convenient: Proof. Assume that E is locally free on the left. We will show that it is also locally free on the right. First consider the case that S = Spec k. Then X and Y are regular of the same dimension. As above we may assume that E = δ U ǫ for finite maps δ : W → X, ǫ : W → Y . We then have the following chain of implications: The last implication follows from the fact that Y is regular. Now consider the case where S is general. From the hypotheses that δ * U is locally free over X we obtain that U is flat over S and hence ǫ * U is also flat over S (since ǫ is finite).
Thus ǫ * U is flat over S. Since ǫ is finite the formation of ǫ * U commutes with base change. By the above discussion we know that for every s ∈ S we have that ǫ * (U s ) is locally free over Y s . Then lemma 4.1.5 with M = 0 shows that ǫ * U itself is locally free.
Below we assume that α : X → S, β : Y → S, γ : Z → S are smooth and equidimensional of the same relative dimension. Now assume that E is an object in shbimod S (X − Y ) which is locally free on the left (and hence on the right). We will define/construct the right and left duals E * , * E to E. For brevity we restrict the discussion below to the right dual. Everything has obvious analogues for the left dual.
We want E * ∈ shbimod S (Y − X) and in addition we should have According to lemma 4.1.1 this property defines E * up to unique isomorphism, if it exists. We now describe − ⊗ OY E * more precisely. With the same notations as before we assume E = u U v where U ∈ coh(W ). Let us denote with v ! the right adjoint to v * . Then it is easy to verify that one has Thus the left structure of E * is given by the dual of the right structure of E. Let Rv ! be the right derived functor to v ! (note that this is somewhat at variance with the usual definitions). Then it is clear that we also have Furthermore if ω X/S denotes the relative dualizing complex then we have where * E is defined as E * but using left adjoints.
The author learned this beautiful formula from notes by Kontsevich [13] where it is shown that it holds more generally in the setting of derived categories. In our current setting it follows trivially from (4.3). Proof. According to (4.1) the left structure of E * is given by the ordinary vector bundle dual of the right structure of E. Thus the right rank of E equals the left rank of E * . In the same way we find that the right rank of E * equals the left rank of E * * . Now from lemma 4.1.7 we easily obtain that the left rank of E * * equals the left rank of E which finishes the proof.
The following lemma will be used many times. Lemma 4.1.9. The formation of (−) * is compatible with base change for locally free coherent sheaf-bimodules.
Proof. If E is a locally free coherent sheaf-bimodule on X and we have a base extension T → S then using the formula (4.3) we see that there is at least a map of sheaf-bimodules (E * ) T → (E T ) * . Then by looking at the left or right structure we see that this map is an isomorphism.
Using standard properties of adjoint functors together with lemma 4.1.1 we obtain canonical maps in ShBimod S (X − Y ) In the sequel we will need some properties of these maps. (1) i is injective and its cokernel is locally free. (2) j is surjective (and hence its kernel is trivially locally free).
Proof. We only consider (1). (2) is similar. With a similar method as the one that was used in the proof of Proposition 4.1.6 it suffices to prove this in the case that S = Spec k. If we restrict to this case then it is sufficient to prove that for all closed points x ∈ X the map O x → O x ⊗ OX E ⊗ OY E * is non-zero. Now this map is obtained by adjointness from the identity map Since this map is obviously non-zero we are done.
Below it will be convenient to have a slight generalization of the relationship that exists between members of a pair (E, E * ).
Therefore we make the following definition.
Using the results in [4] or [1] one obtains that Q ∈ shbimod S (X − Z) is invertible if and only if Q ∼ = id X (L) β where L ∈ Pic(X) and β is an S isomorphism between X and Z.
Definition 4.1.12. Let E, F be locally free objects respectively in shbimod(X −Y ) and shbimod(Y − Z). Assume that Q is an invertible object in shbimod(X − Z) and assume furthermore that Q is contained in E ⊗ OY F . We say that Q is nondegenerate if the following composition Clearly if Q is non-degenerate in E ⊗ OY F then we have Sheaf algebras and sheaf Z-algebras. In this section the notations will be as in the previous section. It is clear that ShBimod S (X − X) is a monoidal category so we can routinely define algebras and I-algebras in this category (see [7] for the definition of ordinary Z-algebras. If we replace the indexing set Z by an arbitrary set I then we obtain the notion of an I-algebra). We will call these (S-central) sheaf-algebras and (S-central) sheaf-I-algebras. For example a sheafalgebra on X is an object A in ShBimod S (X − X) together with a multiplication map A ⊗ OX A → A and a unit map O X → A having the usual properties. If A is a sheaf-algebra on X then we define Mod(A) as the category consisting of objects in Qch(X) together with a multiplication map M ⊗ OX A → M, again satisfying the usual properties. In the same way we may define ShBimod(A − A). This and similar notions will be used routinely in the sequel. We leave the obvious definitions to the reader. The previous paragraph makes clear what we mean by a sheaf-I-algebra on X. However in the sequel we will use this notion in somewhat greater generality. So we will discuss this next.
Assume that Ξ is a family of S schemes α i : X i → S indexed by i ∈ I. A sheaf I-algebra on Ξ is defined by giving for i, j ∈ I an object A ij in ShBimod S (X i − X j ) together with "multiplication maps" A ij ⊗ OX j A jk → A ik and an "identity map" O Xi → A ii satisfying the usual axioms.
If A is a sheaf-Ξ-algebra then an A-module is a formal direct sum ⊕ i∈I M i where M i ∈ Qch(X i ) together with multiplication maps M i ⊗ OX i A ij → M j , again satisfying the usual axioms. We denote the category of A-modules by Gr(A). It is easy to see that Gr(A) is a Grothendieck category.
Unless otherwise specified we will now assume that I = Z even though some (but not all) notions below make sense more generally. We will say that A is noetherian if Gr(A) is a locally noetherian abelian category. In the case that A is noetherian we borrow a number of definitions from [1]. Let M ∈ Gr(A). We say that M is is left, resp. right bounded if M i = 0 for i ≪ 0 resp. i ≫ 0. We say that M is bounded if M is both left and right bounded. We say M is torsion if it is a direct limit of right bounded objects. We denote the corresponding category by Tors(A). Following [1] we also put QGr(A) = Gr(A)/ Tors(A). Furthermore we define the following functors. τ : Gr(A) → Tors(A) is the torsion functor associated to Tors(A); π : Gr(A) → QGr(A) is the quotient functor; ω : QGr(A) → Gr(A) is the right adjoint to π and finally (−) = ωπ.
In these notes we will use the convention that if Xyz is an abelian category then xyz denotes the full subcategory of Xyz whose objects are given by the noetherian objects. Following this convention we introduce qgr(A) and tors(A). Note that if M ∈ tors(A) then M is right bounded, just as in the ordinary graded case. It is also easy to see that qgr(A) is equal to gr(A)/ tors(A). We put A ≥l = ⊕ j−i≥l A ij and similarly A ≤l = ⊕ j−i≤l A ij . A ≥0 and A ≤0 are both sheaf-Z-subalgebras of A and A ≥l and A ≤l are sheaf-bimodules over respectively A ≥0 and A ≤0 .
We say that A is positive if A = A ≥0 .

Lemma 4.2.1. [18]
A is noetherian if and only if A ≥0 and A ≤0 are noetherian.
We will use the following generalization of the Hilbert basis-theorem.

Lemma 4.2.2. Assume that A is positive and let I ⊂ A ≥1 be an invertible ideal in A (that is an invertible object in ShBimod(A − A) which is contained in A). If A/I is noetherian then so is A.
A is said to be strongly graded if the canonical map A ij ⊗ OX j A jk → Ascr ik is surjective for all i, j, k. We have [18]  An interesting fact about sheaf-Z-algebras is that they admit a useful form of twisting. Let A be a sheaf-Z-algebra over Ξ and let Ξ ′ = (X ′ i ) i∈Z be another family of S-schemes.
It is easy to see that the functor

4.3.
Ampleness. If Ξ = (α i : X i → S) i∈Z and Ω = (β i : Y i → S) i∈Z are collections of S-schemes then a map γ : Ω → Ξ is a collection of maps (γ i : Y i → X i ) i∈Z such that α i γ i = β i . Assume now that the following condition holds for γ: (C) Let i, j ∈ Z be arbitrary and let Z be an arbitrary closed subset of Y i × S Y j which is finite over both factors. Then the image of Z in X i × S X j is also finite over both factors.
Here is an example why this condition is not vacuous even if Y i → X i is proper. Let S = Spec k and let (E, +) be an elliptic curve over k.
and hence is not finite over both factors.
If B is a sheaf-Z-algebra on Ω and γ satisfies (C) then we may define sheaf Z-algebra This functor factors through a functorγ * : QGr(B) → QGr(γ * B). In the sequel we will study the properties of this functor in some special cases.
Let us now assume that B is a positive sheaf-Z-algebra on Ω such that all B ij are coherent. Assume furthermore that all γ i are proper. Examining [4,30] leads to the following notion. (2) For every i ∈ Z and M ∈ coh(Y i ) we have that M ⊗ OY i B ij is relatively generated by global sections for the map γ j for j ≫ 0.
Generalizing [1,4,30] we then obtain: Assume that condition (C) holds and that all γ i are proper. Assume furthermore that B is ample for γ. Thenγ * is an equivalence of categories. In addition γ * (B) is noetherian and the functor γ * preserves noetherian objects. [20]. We reproduce his definition below. We first introduce another notion of local freeness. If α : X → S is an S-scheme and P ∈ coh(X) then we say that P is coherent over S if the support of P is finite over S.

Point modules. Point modules over sheaf-(Z-)algebras have been introduced by Adam Nyman in his PhD-thesis
We say that P is locally free (of rank n) over S if P is coherent over S and α * P is locally free (of rank n). If P is locally free of rank one over S then it is of the form ζ * Q for a unique section ζ : S → X of α and Q a line bundle on S. Using a slight abuse of notation we write P −1 for ζ * (Q −1 ). If α : X → S and β : Y → S are S-schemes and if P 1 ∈ coh(X), P 2 ∈ coh(Y ) are locally free of rank one over S then so is P 1 ⊠ S P 2 = pr * 1 (P 1 ) ⊗ OX× S Y pr * 2 (P 2 ) Note that if P 1 = ζ 1 * (Q 1 ) and P 2 = ζ 2 * (Q 2 ) then We will need the following result.
Lemma 4.4.1. Assume that α : X → S and β : Y → S are S-schemes and let E ∈ ShBimod S (X − Y ). Let P 0 ∈ coh(X), P 1 ∈ coh(Y ) be locally free of rank one over S. Then we have canonical isomorphisms:

Furthermore under this isomorphism, epimorphisms correspond to each other.
Proof. This is a direct computation. Let P 0 = ζ 0 * (Q 0 ), P 1 = ζ 1 * (Q 1 ) where ζ 1 : S → X, ζ 2 : S → Y are sections of α and β respectively. We have If we look at the following pullback diagram: We now compute To prove the claim about preservation of epimorphisms one simply checks that epimorphisms are preserved in each individual step.
Assume now that A is a positively graded sheaf-Z-algebra on Ξ. Just as in the case of ordinary algebras one may define a concept of point modules in Gr(A). 2. An m-shifted point module over A is an A-module P generated in degree m such that for n ≥ m we have that P n is locally free of rank one over S. A 0-shifted point module will be simply called a point-module. An extended point module over A is an A-module P such that for all m, P ≥m is an m-shifted point module.
To study point modules it will be convenient to introduce the notion of a trun- We define a [m : n]-truncated A point module P as an A [m:n] -module generated in degree m such that for n ≥ i ≥ m we have that P i is locally free of rank one.
It is natural to declare two (truncated, extended, shifted) point modules P, Q to be equivalent if there exists a line bundle L on S such that Q n = α * n L ⊗ OX n P n The main feature of (extended) point modules is that they define certain sheaf-Z-algebras which may be used to study A. Let P be an extended point module over A. Thus for every i we have that P i is locally free of rank one over S and is a strongly graded sheaf-Z-algebra on S. Let Ω = (S) i∈Z be the trivial constant system of S-schemes and let ζ : Ω → Ξ be defined by (ζ i ) i . Then the right A-module structure of P yields us through lemma 4.4.1 a surjective map A m,n → ζ * B m,n (P ) and a straightforward verification shows that this map is compatible with multiplication. Hence we obtain a surjective map of sheaf-Z-algebras A → ζ * B(P ).
In the sequel we will need families of the concepts that were introduced above. If θ : W → S is an S-scheme then we can consider the base extended algebra A W which is just ⊕ m,n (θ, θ) * (A m,n ) where we have denoted the base extension of θ to a map X n,W → X n also by θ. We define a family of point modules over A parametrized by W to be a point module on A W . Families of extended and truncated point modules are defined in a similar way.
Assume that P is a family of extended point modules parametrized by W . Then B(P ) is a W -central sheaf-Z-algebra on W . As above we have Proof. By the definition of a point module we have a surjective map Let Ω = (W ) i∈Z be the constant system associated to W and let µ : Ω → Ξ be given by (µ i ) i . Then µ satisfies (C) and the map A W → B(P ) gives by adjointness rise to a map A → µ * B(P ).
The first map is a section and so it is a closed immersion. In particular it is proper. The second map is also proper since it is the base extension of a proper map. Thus µ i is also proper. Now we can verify (C). Since (µ i , µ j ) is proper it is sufficient to verify that the image of (µ i , µ j ) is finite on the left and right. This is clear since by the previous lemma this image is contained in the support of A ij and A ij was coherent by hypotheses.
Equivalences among families of point modules are defined in the same way as for ordinary point modules (see above). For use in the sequel we introduce the following (somewhat adhoc) notations. Generalities. We will consider the following particular case of a sheaf-Zalgebra. Let α : X → S, β : Y → S be smooth equidimensional maps of the same relative dimension and let E ∈ shbimod S (X − Y ) be locally free. Define In a similar way we define We define E * n as in the introduction. I.e.
We then define S(E) as the sheaf-Z-sheaf-algebra generated by the E * n subject to the relations i(O Xn ). More precisely We say that S(E) is a non-commutative symmetric algebra in standard form.
In the sequel it will sometimes be convenient to define more general symmetric algebras. We will do so now and then we will show that these more general symmetric algebras are equivalent to those in standard form.
Let α n : X n → S be arbitrary smooth equidimensional maps of the same relative dimension. Assume that (E n ) n , (Q n ) n are respectively a series of locally free objects in shbimod(X n − X n+1 ) and invertible objects in shbimod(X n − X n+2 ) which are non-degenerate subobjects of E n ⊗ OX n+1 E n+1 . We then define A to be the (X n ) nsheaf-Z-algebra generated by the E n subject to the relations Q n . Thus A nn = O Xn , A n,n+1 = E n and A n,n+2 = E n ⊗ E n+1 /Q n , etc. . . . We will call an algebra of the form A a non-commutative symmetric algebra. We expect a non-commutative symmetric algebra to have good homological properties but this has only been proved in the rank two case (see below). Now let X = X 0 , α = α 0 , Y = X 1 , β = α 1 and define X ′ n , α ′ n in the same way as X n , α n in (5.1)(5.2). Thus Using (4.5) we find: The inclusion becomes an inclusion and it is easy to see that this inclusion is derived from the canonical inclusion . Thus we have shown that every non-commutative symmetric algebra is obtained from one in standard form by twisting (see §4.2).
We will say that A is a non-commutative symmetric algebra of rank r if E 0 has rank r on both sides. From lemma 4.1.8 together with (5.3) we then obtain that all E n have rank r on both sides. [30,24,23]. Let X be a scheme and let E ⊂ shbimod S (X − X) be locally free. Let Q ∈ E ⊗ OX E be a non-degenerate invertible subobject and let H = T X (E)/(Q). The following lemma makes the connection between H and S(E). Proof. If A is a sheaf-Z-graded algebra on X then we define the Z-graded sheaf algebraǍ by

Relation with the definition from
It is clear that we have Gr(Ǎ) = Gr(A). Furthermore it is also clear thatǍ is a non-commutative symmetric algebra with E i = E and Q i = Q for all i. Since such a non-commutative symmetric algebra is obtained by twisting from S(E) we are done.
5.3. Point modules over non-commutative symmetric algebras or rank two. We let the notations be as in the previous sections but we assume in addition that A has rank two. We start with the following result. Proof. Both claims are similar so we only consider the second one. Since we may shift A we may without loss of generality assume that m = 0. In that case P is described by a triple (P 0 , P 1 , φ) where P 0 ∈ coh(X 0 ), P 1 ∈ coh(X 1 ) are locally free of rank one over S and φ : P 0 ⊗ OX E 0 → P 1 is a surjective map. We have to extend this triple to a quintuple (P 0 , P 1 , P 2 , φ, ψ) where P 2 ∈ coh(X 2 ) is also locally free of rank one over S and ψ : P 1 ⊗ OX 1 E 1 → P 2 is another surjective map. The entries in such a quintuple are not arbitrary since the relation Q 0 has to be satisfied. To clarify this restriction we note that point modules and truncated point modules are preserved under twisting (see §4.2). Hence we may without loss of generality assume that A is in standard form, i.e. A = S(E) for some sheaf-bimodule E which is locally free of rank two on both sides.
In order for (P 0 , P 1 , P 2 , φ, ψ) to define an object in Gr(A [0:2] ) module we need that the composition P 0 → P 0 ⊗ OX 0 E ⊗ OX 1 E * φ⊗E * −−−→ P 1 ⊗ OX 1 E * ψ − → P 2 is equal to zero since this composition represents the action of Q 0 . From lemma 5.3.2 below it follows that this composition may be described in the following alternative way: where φ * is obtained from φ by adjointness. Thus the pair (ψ, P 2 ) is a quotient of coker φ * . If we now show that coker φ * is itself locally free of rank one then we are done. This last fact follows from lemma 5.3.4 below. Proof. This is standard. Proof. This is a direct consequence of Lemma 4.1.4 if we view F as an S − Xbimodule.
Lemma 5.3.4. Let α : X → S, β : Y → S be smooth equidimensional maps of the same relative dimension. Let E ∈ shbimod S (X − Y ) be locally free of rank two on both sides. Assume that we have objects P 0 ∈ coh(X), P 1 ∈ coh(Y ) which are locally free of rank one over S, together with a surjective map φ : P 0 ⊗ OX E → P 1 . Then the adjoint map φ * : P 0 → P 1 ⊗ OY E * is injective and has a cokernel which is locally free of rank one over S.
Proof. Using lemma 4.1.5 it suffices to prove this in the case that S = Spec k. But then it is sufficient to show that φ * is not zero (as P 1 ⊗ OY E * has rank two by Lemma 5.3.3). Since φ is not zero this is clear.
Using the bijections exhibited in Proposition 5.3.1 together with the fact that the relations in A have degree two we now easily obtain: It follows from the proof of Proposition (5.3.1) (see (5.5)) that if P is an extended point module over A then there are exact sequences on X j+2 In fact this was only shown if A is in standard form, but the general case follows by twisting. Now write P j in the usual form ζ j * (Q j ) where ζ j is a section of α j and Q j ∈ Pic(S). Then applying α j+2 * to (5.6) we obtain an exact sequence on S Tensoring the previous exact sequence on the left with Q −1 i yields an exact sequence → 0 By dualizing (5.6), tensoring on the left with Q j , applying a suitable variant of (4.5), applying α j * , tensoring with Q k and finally changing indices we obtain the following analogous exact sequence Projective bundles associated to quasi-coherent sheaves. If Z is a scheme and U is a coherent sheaf on Z then we define If W is an arbitrary scheme and χ is a W -point of P Z (U) then χ defines a pair (χ ′ , L) where χ ′ is the composition W χ − → P Z (U) → Z and L ∈ Pic(W ) is given by χ * (O(1)). Clearly L is a quotient of χ ′ * (U). It is standard that conversely every pair (χ ′ , L) where χ ′ is a map W → Z and L ∈ Pic(W ) is a quotient of χ ′ * (U) corresponds to a unique χ : W → P Z (U).
We will use the following result in the following sections.
In particular it is equal to some P n k(x) . Here is a somewhat more specialized result.
Proposition 5.4.2. Assume that β : Z → X is a map of schemes and assume that E ∈ coh(Z) is coherent over X. Then the obvious map o : P Z (E) → P X (β * E) is a closed immersion. If X is a smooth connected curve over k and E is locally free of rank two over X then o is either surjective or else its image is a divisor.
Proof. All claims are local on X so we may and we will assume that X = Spec R is affine. In addition we may replace Z by the scheme-theoretic support of E, i.e. we may assume that β is finite. It follows that Z is also affine, say Z = Spec T . Therefore E is obtained from a finitely generated T module E and P Z (E) = Proj S T (E), P X (E) = Proj S R (E). The map o is obtained from the obvious map To prove that o is a closed immersion we simply remark that S T (E) → S R (E) is surjective in degree ≥ 1. Now we make the additional hypotheses on our data, i.e. X is a smooth connected curve over k and E is locally free of rank two over X. To prove our claim we may now make the additional simplifying assumption that X = Spec R where R is a discrete valuation ring.
The fact that E is Cohen-Macaulay implies that T has no embedded components. So T is free of rank one or two over R and R embeds in T .
If T is free of rank one then T = R and hence o is an isomorphism. So assume that T has rank two. Thus T = R[z] where z satisfies a monic quadratic equation over R.
We now have to show that the kernel K of S R (E) → S T (E) is generated by one element. Let E = Rx+Ry. Then K is generated by (z ·x)x−x(z ·x), (z ·y)x−y(z ·x) and (z · y)y − y(z · y). Write z · x = ax + by, z · y = cx + dy with a, b, c, d ∈ R. Then Thus K is indeed generated by a single quadratic element.
Remark 5.4.3. The preceding result is false if X is not a curve.
Consider the following example : Counting dimensions of fibers we see that P X×Y (E) has dimension 2.
Clearly P X×Y (E) contains two closed subsets respectively given by P X×Y (O ∆ ) = ∆ and P X×Y (O Γ ) = Γ which must be irreducible components since they also have dimension 2. Furthermore outside the point (o, o) ∈ X × Y the map ∆ Γ → P X×Y (E) is an isomorphism. However the fiber F of (o, o) in P X×Y (E) is P 1 whereas ∆ Γ gives us at most two points.
Thus F must be contained in an additional irreducible component. If this irreducible component is not F itself then it must contain some points of P X×Y (E) not above (o, o). But then F must be equal to ∆ or Γ which is a contradiction. It follows that P X×Y (E) is not equidimensional and in particular it cannot be a divisor in P X (pr 1 E).
The problem with this example is that the support ∆ ∪ Γ of E is not Cohen-Macaulay.

5.5.
Representability of the point functor. The following result has been proved by Adam Nyman [20]. We reproduce the proof since we need the exact nature of the isomorphisms involved.
Proof. In view of the above discussion it is clearly sufficient to prove this for Points 0,1,A . We will start by giving an alternative description of Points 0,1,A (S). Without loss of generality we may assume that A = S(E).
If we apply this the discussion before the statement of the theorem with Z = X × S Y , U = E, W = S then we find Since this bijection is obviously compatible with base extension we find that the functor Points A,0,1 is represented by P X×SY (E). This finishes the proof.

Properties of the universal point algebra
From now on we assume that our base scheme S is Spec k and therefore we will omit S from the notations. Otherwise the notations will be as in the previous section.

A vanishing result.
Theorem 6.1.1. Let s : E →Ē be a projective map of relative dimension one. Let L ∈ coh(E) and assume that the restriction to every fiber of L is generated by global sections and has vanishing higher cohomology. Then R i s * L = 0 for i > 0 and the canonical map s * s * L → L is a surjective.
Proof. This is not an immediate consequence of semi-continuity since we are not assuming that L is flat overĒ.
We use the theorem on formal functions. For y ∈Ē let E n = E ×Ē Spec OĒ ,y /m n y where m y is the maximal ideal corresponding to y. In addition let L n be the restriction of L to E n . Then one has [10, Thm III.11.1] Thus in order to show that R i s * (L) = 0 for i > 0 it is sufficient to show that (H1 n ) H i (E n , L n ) = 0 for all y and all n.
Similarly it is easy to see that for s * s * L → L to be surjective it is sufficient that the condition (H2 n ) Γ(E n , L n ) ⊗ k O En → L n is surjective holds for all y and all n.
Our proof will be by induction on n. It follows from the hypotheses that (H1 1 ) and (H2 1 ) are satisfied.
Assume now that (H1 n ) and (H2 n ) are satisfied. We have an exact sequence Thus F = ker(L n+1 → L n ) is the quotient of a sheaf with vanishing higher cohomology, and since we are in dimension one it follows that F itself has vanishing higher cohomology. Thus it follows that is exact, and furthermore the induction hypotheses imply that H i (E n+1 , L n+1 ) = 0 for i > 0. So this proves (H2 n+1 ). In order to prove (H1 n+1 ) we use the following commutative diagram with exact rows: Since the outermost vertical maps are surjective the same holds for the middle one. This proves (H1 n+1 ).
6.2. The case of non-commutative symmetric algebras. In this section the notations are as before. In particular A is a non-commutative symmetric algebra of rank two over Ξ = (X i ) i∈Z (see §5). As usual we put E i = A i,i+1 . By definition E i has rank two on both sides. Put E j = P Xj ×Xj+1 (E j ). Since E j represents Points A , there is a universal extended point P j over A E j . We now let B j = B(P j ), be the associated sheaf-Z-algebras and we aim to study these in more detail. As above let ζ j i : E j → X i,E j be the support of P j i . We may write ζ j i as a pair (µ j i , id E j ) where µ j i is a map from E j to X i . Again as above we write P j i = ζ j i, * (Q j i ) for Q j i ∈ Pic(E j ). We will also write α j i : X i,E j → E j for the map obtained by base extension from α i : X i → Spec k.
Our first observation is that since the E j all represent the same functor there must exist isomorphisms θ j : E j+1 → E j and objects L j ∈ Pic(E j ) such that This may be rewritten as µ j+1 In the sequel we will define θ jl : E j → E l as the composition θ l θ l+1 · · · θ j−1 if j ≥ l and by a similar formula if j < l. Thus we find µ j i = µ l i θ jl and B j mn = θ jl * B l mn From the proof that E m represents Points A it follows that B m m,m+1 = O E m (1) and (µ m m , µ m m+1 ) is the projection map E m = P Xm×Xm+1 (E m ) → X m × X m+1 . This allows us to describe B i mn in terms of the O E j (1) and the isomorphisms θ pq .
To understand these factorizations let us first consider µ j j and µ j j+1 which together represent the canonical map E j → X j × X j+1 . As an intermediate step consider the Stein factorization E j → G j → X j × X j+1 of this last map. By construction [10, Cor. III.11.5] G j is finite over the scheme theoretic image Z j of E j in X j ×X j+1 . Since Z j is finite over both X j and X j+1 we obtain from the construction ofĒ j [10, Cor. III.11.5] that E j → G j → X j and E j → G j → X j+1 are the Stein factorizations of respectively µ j j : E j → X j and µ j j+1 : E j → X j+1 . In particular we obtainĒ j j =Ē j j+1 and s j j = s j j+1 . Now using the fact that Stein factorizations are (obviously) compatible with isomorphisms we obtain from this by applying suitable θ pq thatĒ p j =Ē p j+1 and s p j = s p j+1 for all p. ThusĒ p j and s p j are independent of j and we may writē E p j =Ē p , s p j = s p . Thus the result of this discussion is that we have commutative diagrams: X i X i Now we investigate the scheme-theoretic closed fibers of s j . By lemma 5.4.1 the scheme-theoretic fibers of E j → Z j are either points or P 1 's and hence in particular they are connected. The fibers of E j →Ē j are also connected by the properties of the Stein factorization. Hence it follows that the mapĒ j → Z j is settheoretically a bijection. In particular s j and E j → Z j have the same closed fibers. We conclude that the fibers if s j are either points or P 1 's. Now let Ω be the constant system of schemes (E 0 ) i∈Z and let µ 0 = (µ 0 i ) i . From Corollary 4.4.4 it follows that µ 0 satisfies condition (C). We can now prove the following technical result which will be used below in the proof that a non-commutative symmetric algebra is noetherian (see §7.3 below).
We compute andμ 0 j is finite, it is sufficient to prove the analogues of 2. and 3. in Theorem . According to Theorem 6.1.1 we have to show that M ⊗ O E 0 B 0 ij when restricted to the P 1 fibers of s 0 becomes eventually generated by global sections. This follows from the fact that according to (6.1) the P 1 -fibers are preserved under the θ's and the fact that O E m (1) when restricted to a P 1 -fiber of s m is equal to O P 1 (1).

On the structure of non-commutative symmetric algebras of rank two
In this section the notations are the same as in the previous ones.
7.1. Ranks and exact sequences. Let e i ∈ Γ(X n , A nn ) = Γ(X n , O Xn ) be the section corresponding to 1. The structure of the relations in A implies that there is an exact sequence of O Xm − A sheaf-bimodules given by We will show below that this exact sequence is exact on the left. The following proposition is proved in the same way as Proposition 5.3.1 and Theorem 5.3.5. From the fact that a point module is uniquely determined by its restriction to A [0:1] one obtains that if k is algebraically closed then for every x ∈ X there is at least one point module P such that P 0 = O x . Now we will consider line-modules. For x a rational point in X m we define L m,x = O x ⊗ OX m e m A. For simplicity we write L x for L 0,x .
If P is a point module then we have Thus it follows that if k is algebraically closed then every L x maps onto at least one point-module. In the same way one sees that L m,x maps to an m-shifted point module.
Let L x → P be a surjective map to a point module and let K be its kernel. Since length(L x ) 1 = 2 and length P 1 = 1 we deduce that K 1 ∼ = O y for some y ∈ X. Thus there is a non-zero map L 1,y → K 1 . Since coker(L 1,y → L x ) has the same truncation to A [0:1] as P it follows from Proposition 7.1.1 that we have an exact sequence We will call this a standard exact sequence. A similar standard exact sequence exists for L m,x : where P is now an m-shifted point module.
We can now prove the following result. Proof. Without loss of generality we may assume that k is algebraically closed. As far as (1) is concerned we will only consider the left structure of A. The statement about the right structure follows by symmetry. Assume that we have shown that A m,n is locally free on the left of rank n− m+ 1 for n − m ≤ t. We tensor (7 This yields On the other hand we have from (7.3) length(L m,x ) m+t+1 ≤ 1 + length(L m+1,y ) m+t+1 = t + 2 Combining these two inequalities yields length(L m,x ) m+t+1 = t + 2 for all m, x. Since (L m,x ) m+t+1 = O x ⊗ OX A m,m+t+1 this yields that A m,m+t+1 is locally free of rank t + 2 on the left. By induction we obtain the corresponding statement for all m, n. From this we easily obtain that (7.1) and (7.3) are exact on the left. 7.2. Two different types. We need the following notation. Let X = X ′ X ′′ and Y = Y ′ Y ′′ be disjoint unions of schemes and let p ′ , p ′′ : We use a similar construction for sheaf-Z-algebras. We leave the obvious definitions to the reader.
We will now analyze the F ∈ shbimod(X − Y ) which are locally free of rank two on both sides. As usual we assume that X, Y are smooth of the same dimension and equidimensional.
Let Z be the scheme theoretic support of F . Since F is Cohen-Macaulay, all components of Z have the same dimension and there are no embedded components.
Assume that Z has an irreducible component Z ′ on which the restriction of F has rank two (generically). Z ′ lies over connected components X ′ and Y ′ of X and Y . Let X ′′ and Y ′′ be the union of the other connected components of X and Y . Counting ranks we see that there can be no other irreducible components of Z lying above X ′ and Y ′ and hence F = F ′ ⊞ F ′′ where F ′ ∈ shbimod(X ′ − Y ′ ) and F ′′ ∈ shbimod(X ′′ − Y ′′ ).
Let us return to F ′ . Since Z ′ is integral and has degree one over X ′ and Y ′ and since X ′ and Y ′ are furthermore integrally closed we obtain that Z is the graph of an isomorphism σ : X → Y and F is a vector bundle of rank two on Z ′ .
It is clear that S(F ) = S(F ′ ) ⊞ S(F ′′ ). A similar decomposition then holds for every non-commutative symmetric algebra by twisting. Furthermore we leave it to the reader to check that Gr(S(F ′ )) is equivalent to Gr(S Z ′ (F ′ )) and hence corresponds to a commutative P 1 -bundle.
To formalize this let us make the following definition.
Definition 7.2.1. Let A be a non-commutative symmetric algebra of rank two and let E = A 01 . We say that A is of Type I if E is a rank two bundle over the graph of an automorphism and we say that A is of Type II if the restrictions of E to the irreducible components of its support all have rank one generically.
Thus we have obtained the following result.

Proposition 7.2.2. Let A be a non-commutative symmetric algebra of rank two. Then
is equivalent to the category of graded modules over the symmetric algebra of a rank two vector bundle over a smooth scheme.
Example 7.2.3. The most basic example of type II symmetric algebra is obtained by embedding a smooth elliptic curve C as a divisor of degree (2, 2) in P 1 × P 1 and letting E = u L v where L is a line bundle on C and (u, v) : C → P 1 × P 1 denotes the embedding. Such non-commutative symmetric algebras appeared naturally in [30] and provided one of the motivations for writing the current paper.
7.3. Non-commutative symmetric algebras of rank two are noetherian.
Since to prove A is noetherian we may treat the cases of Type I and Type II individually, and since the Type I case is easy we assume throughout that A is of Type II. From Theorems 6.2.1 and 4.3.3 we obtain that µ * B 0 ≥0 is noetherian. Furthermore by construction there is a map A → µ 0 * B 0 ≥0 . We would like to use this map in order to analyze A. However the analysis is complicated by the fact that E 0 may have components of different dimensions if dim X n > 1 (see Remark 5.4.3).
Therefore we will use the following trick. We will let F j be the union of all components in E j which are of maximal dimension and we let t j : F j → E j be the inclusion map. It is clear that with θ jl restricts to a map F j → F l which we will also denote by θ jl .
Let C mn = t * (B 0 mn ). Then C = ⊕ m≤n C mn is a Z-algebra on F 0 . Put λ j i = µ j i t i and λ = (λ 0 i ) i . From the fact that B ≥0 is ample for µ (Theorem 6.2.1) we easily obtain that C is ample for λ. We will now analyze the map A → λ * C.
Step 1. The map A ii → (λ * C) ii is monic. If we denote its cokernel by S ii then S ii is locally free of rank one on both sides.
To see this we will show that is monic and its cokernel is locally free of rank one. The corresponding statement for the right structure is similar.
We have O Xi = pr 1 * (A ii ) and pr 1 * (C ii ) = pr 1 * (λ 0 is monic and that its cokernel is locally free of rank one. Put B = P Xi (pr 1 * (E i )) and let O B (n) = O PX i (pr 1 * (Ei)) (n). Denote the projection map B → X i by p. By Proposition 5.4.2 the map E i → B is a closed immersion. So the composition F i → E i → B is a closed immersion as well. We denote this composition by v. Now since A is of Type II it easy to see that dim F i = dim X i . Hence F is a divisor in B. Generically E 0 will be invertible over its support and hence generically F will have degree two over X i . Since according to [10, II. Ex. 7.9] one has Pic(B) = Pic(X i )×Z x where x is the number of connected components of X and the factor Z x corresponds to the degrees over the generic fibers it follows We now apply Rp * to the exact sequence Using the known properties of the map p : B → X i [10, Ex. III.8.4] we extract from the long exact sequence for Rp * a short exact sequence We obtain in addition that R h λ i i * (O F i ) = 0 for h > 0. This may be rephrazed as the next step. Step Arguing as in Step 1 we reduce the problem to showing that the canonical map (1)) is an isomorphism. Tensoring (7.5) by O B (1) and applying Rp * we obtain what we want and in addition we obtain R h λ i i * (O F i (1)) = 0 for h > 0. This then yields the next step.
Indeed the image of (λ 0 i , λ 0 i+1 ) is finite over X i . Thus it is sufficient to prove pr 1 * R h (λ 0 i , λ 0 i+1 ) * (C i,i+1 ) = 0. By the Leray spectral sequence this then follows from R h λ 0 i, * (C i,i+1 ) = 0 which is restatement of Step 5. Now we translate the exact sequence (5.7) to our current situation. It becomes.
Step 2 and 4 one obtains by induction that the following sequence is exact Step 6. The map A ii+2 → (λ * C) ii+2 is an epimorphism. If we denote its kernel by T ii+2 then T ii+2 = S ii ⊗ OX i Q i . In particular T ii+2 is locally free of rank one on both sides.
To prove these statements we consider the following commutative diagram with exact rows.
(the second row is the dual version of (7.1)). Applying the snake lemma to (7.7) together with Step 1 yields what we want.
Step 7. Assume j ≥ i − 1. Then the complex We prove this by induction on j. The cases j = i − 1, i were covered by the previous steps. Assume now j ≥ i + 1. We consider the following commutative diagram with exact rows.
By induction we may assume that the first two columns are exact. Hence so is the third column.
Step 8. The canonical maps A ij ⊗ OX j T jj+2 → A ij+2 and T ii+2 ⊗ Xi+2 A i+2,j+2 → A ij+2 are monomorphism, and furthermore they define an isomorphism To see this note that by the previous step we already know that the first map is a monomorphism. A similar proof involving (5.8) shows that the second map is also a monomorphism.
Since by definition T ii+2 goes to zero under the map A → λ * C we also have that T ii+2 ⊗ Xi+2 A i+2,j+2 goes to zero. Thus the image of T ii+2 ⊗ Xi+2 A i+2,j+2 in A i,j+2 lies in the image of A ij ⊗ OX j T jj+2 . By symmetry the opposite inclusion will also hold and hence we are done.
Step 9. A is noetherian.
By the previous steps we have an invertible ideal J ⊂ A ≥2 given by From the fact that C is noetherian and the fact that all C ij are coherent we easily obtain that D is noetherian. We may now conclude by invoking lemma 4.2.2. A strongly ample sequence (O(n)) n∈Z in E is ample in the sense of [25]. Hence using the methods of [1] or [25] one obtains It would be interesting to know if a noncommutative P 1 -bundle always has an ample sequence. The next lemma is very weak but it is sufficient for us below.

Proof. We have maps of gr(A)-objects induced by the multiplication in
which are surjective in degree ≥ i + 1. Since A i,i+1 is generated by global sections on the right these may be turned into maps for certain t i which are still surjective in degree ≥ i + 1. Let M = πM with M ∈ gr(A) noetherian. Then there is some N such that M ≥N is generated in degree one. Hence there is some N ′ , which we will take ≥ N , such that there is an epimorphism 8.2. Deformations of abelian categories. For the convenience of the reader we will repeat the main statements from [29]. We first recall briefly some notions from [14]. Throughout R will be a commutative noetherian ring and mod(R) is its category of finitely generated modules.
Let C be an R-linear abelian category. Then we have bifunctors − ⊗ R − : C × mod(R) → C, Hom R (−, −) : mod(R) × C → C defined in the usual way. These functors may be derived in their mod(R)-argument to yield bi-delta-functors is an exact functor, or equivalently if Tor R i (M, −) = 0 for i > 0. By definition (see [14, §3]) C is R-flat if Tor R i or equivalently Ext i R is effaceable in its C-argument for i > 0. This implies that Tor R i and Ext i R are universal ∂-functors in both arguments.
If f : R → S is a morphism of commutative noetherian rings such that S/R is finitely generated and C is an R-linear abelian category then C S denotes the (abelian) category of objects in C equipped with an S-action. If f is surjective then C S identifies with the full subcategory of C given by the objects annihilated by ker f . The inclusion functor C S → C has right and left adjoints given respectively by Hom R (S, −) and − ⊗ R S. Now assume that J is an ideal in R and let R be the J-adic completion of R.
Recall that an abelian category D is said to be noetherian if it is essentially small and all objects are noetherian. Let D be an R-linear noetherian category and let Pro(D) be its category of pro-objects. We define D as the full subcategory of Pro(D) consisting of objects M such that M/M J n ∈ D for all n and such that in addition the canonical map M → proj lim n M/M J n is an isomorphism. The category D is R-linear. The following is basically a reformulation of Jouanolou's results [11]. In general, to simplify the notations, we will pretend that the equivalence D R/J ∼ = C is just the identify.
Thus below we consider the case that D is complete and formally flat and C = D R/J . The following definition turns out to be natural.
The results below allow one to lift properties from C to D.  The following result is a version of "Grothendieck's existence theorem". (2) D is flat (instead of just formally flat); (3) D is Ext-finite as R-linear category.

8.3.
Deformations of Hirzebruch surfaces. Below (R, m) is a complete commutative local noetherian ring with residue field k = R/m. Everything will now either be over k or over R. Although in the main part of this paper we have set up the theory over a base scheme of finite type over a k it is not difficult to see that the results remain valid over Spec R. We will use this without further comment. When we say that something is "compatible with base change" we mean compatible with the passage from R to k. We usually abbreviate − ⊗ R k by (−) k . We also use a subscript k to indicate that something is defined over k.
We let X k be the Hirzebruch surface P(E k ) with E k = O P 1 k ⊕ O P 1 k (h), h ≥ 0 and we let D be an R-deformation of C = coh(X k ) in the sense of §8.2. The rest of this section will be devoted to proving the following result. Theorem 8.3.1. There exists a sheaf-bimodule E over P 1 R such that D is equivalent to qgr(S(E)).
Let t : X k → P 1 k be the projection map. Then we have standard line bundles ) we deduce that in particular H i (X k , O X k (m, n)) = 0 for i > 0 and m, n ≥ 0.
Since the O k (m, n) are exceptional in C they lift to objects O(m, n) in D using Proposition 8.2.5. Furthermore from the ampleness criterion in [10, Cor. V.2.18] together with Theorem 8.2.10(1) it follows that (O(n, n)) n is a strongly ample sequence in D. By item (3) of the same theorem we obtain that D is Ext-finite.
We now define some R-linear Z-algebras as well as C m − C n -bimodules for n ≥ m: From Proposition 8.2.7 it follows that C n and A mn are R-flat and compatible with base change. Hence We can now look for some properties of C n,k that lift to C n (see [28, §8.3] for a more elaborate example of how this is done). (P1) Then C n is generated by the (V n,i ) i . (P3) Put K n,i = ker(V n,i ⊗ V n,i+1 → C n,i,i+2 ). Then the relations between the V n,i in C n are generated by the K n,i . (P4) Rank counting reveals that rk K n,i = 1. The R-module K n,i is generated by a non-degenerate tensor r n,i in V n,i ⊗ R V n,i+1 . Using these properties it is now easy to describe C n . After choosing suitable bases x i , y i in V n,i we may assume that r i = y i x i+1 − x i y i+1 . Thus all C n are in fact isomorphic toŠ (see (5.4)) where S is the graded algebra R[x, y]. In particular qgr(C n ) ∼ = coh(P 1 R ) for all n. It also follows that after suitable reindexing A mn becomes in a natural way a bigraded S ⊗ R S-module which we denote by A ′ mn . We think of A ′ mn as an S-Sbimodule with independent left and right grading. The required reindexing is given by A ′ mn;ij = A mn;−i,j Here x, y act as x i−1 , y i−1 on the left and as x j , y j on the right.
The following diagram is commutative Here by (−) 0,− we mean taking the part of degree zero for the left grading.
A mn is locally free on the left and right of rank n − m + 1. In addition A mn,k ∼ = δ * S n−m E k where δ : P 1 k → P 1 k × P 1 k is the diagonal embedding. Proof. We first observe that A mn is in fact coherent. To this end it is sufficient to show that the diagonal submodule i A ′ mn;ii is a finitely generated i S i ⊗ R S imodule. This may be verified after tensoring with k.
From (8.5) one obtains The righthand side of (8.7) is the gradedi S i,k ⊗ k S i,k -module associated to the coherent O P 1 k ×P 1 k -module δ * S n−m E k (for the ample line bundle given by O k (1, 1)). Hence this graded module is finitely generated.
From the computation in the previous paragraph we also learn that A m,n ⊗ R k is indeed given by the sheaf S n−m E k supported on the diagonal.
We claim that the support of A mn is finite over both factors of P 1 R × P 1 R . Again it is clearly sufficient to check this over k but then it follows from the explicit form of A m,n ⊗ R k given above.
As indicated above A mn is flat over R. Hence the same is true for A mn . Since A mn ⊗ R k is locally free over both factors it follows from Lemma 4.1.5 that A m,n is locally free on the left and on the right. By tensoring with k we deduce that the left and right rank of A m,n are equal to n − m + 1. Proof. It is sufficient to prove that for every i we have that e i A mn lies in gr(C n ). Since e i A mn is a finitely generated R-module in every degree we may prove this after specialization.
We compute e i A mn,k = j≥i Γ(P 1 k , S n−m E k (j − i)) Thus e i A mn,k is up to finite length modules the graded S-module associated to the coherent P 1 -module S n−m E k (−i). Hence it is finitely generated. Proof. We first have to construct a natural transformation gr(C m ) −⊗C m Amn / / π gr(C n ) π w w w w w w w w w w w w w w w w w w w w w coh(P 1 R ) −⊗ P 1 R Amn / / coh(P 1 R ) Taking into account the equivalences gr(C m ) = gr(S) this diagram may be rewritten as Here ω 1 is ω applied to the left grading and similarly for π. The natural transformation is now obtained by functoriality from the canonical map M ⊗ S A ′ mn → π 1 ω 1 (M ⊗ S A ′ mn ) We claim this natural transformation is an isomorphism. Both branches of the diagram (8.8) represent right exact functors so it is sufficient to consider the value on the projective generators S k (i) of gr(S k ). This verification may be done after specialization.
We find S k (i) ⊗ S k A ′ mn,k = pq A ′ mn;p+i,q,k = p+q+i≥0 Γ(P 1 k , S n−m E k (q + p + i)) where have used (8.6). An easy verification shows that π 1 ω 1 (S k (i) ⊗ S k A ′ mn,k ) = p,q Γ(P 1 k , S n−m E k (q + p + i)) from which we deduce that (π 1 ω 1 (S k (i) ⊗ S k A ′ mn,k )/(S k (i) ⊗ S k A ′ mn,k )) l,− is finite dimensional for any l. This implies that the natural transformation in (8.9) is in fact a natural isomorphism.
The natural morphism A mn ⊗ Cn A nt → A mt induces via diagram (8.8) a natural transformation of functors − ⊗ P 1 R (A mn ⊗ P 1 R A nt ) → − ⊗ P 1 R A mt Using Lemma 4.1.1 one obtains from this a morphism of bimodules (8.10) A mn ⊗ P 1 R A nt → A mt Using a similar argument one shows that this morphism of bimodules satisfies the associativity axiom and hence produces a sheaf-Z-algebra on P 1 R given by A = n≥m A mn From the fact that A mm = C m on easily obtains A mm = O P 1 R . Explicitating the proof of Lemma 4.1.1 one obtains that over k (8.10) is given by the canonical maps S n−m E k ⊗ P 1 k S t−n E k → S t−m E k Therefore by a suitable version of Nakayama's lemma we deduce that (8.10) is an epimorphism and hence A is generated by E n def = A n,n+1 . Let Q n be the kernel of A n,n+1 ⊗ P 1 R A n+1,n+2 → A n,n+2 . We claim that this kernel is non-degenerate in A n,n+1 ⊗ P 1 R A n+1,n+2 . In Lemma 4.1.9 we have shown that dualizing of bimodules is compatible with base change. From this it easily follows that it is sufficient to check the nondegenerateness of Q n over k where it is obvious. Now let A ′ be the Z-algebra generated by the E n,n+1 subject to the relations given by the Q n . By construction there is a surjective map A ′ → A. Since A ′ and A are locally free in each degree and have the same rank it follows that this surjective map must actually be an isomorphism.
So summarizing we have shown the following: which is defined as follows. Let M ∈ Gr(C). Then M n def = M −,n is a right C nmodule. Furthermore the right action of C on M induces maps (8.11) M m ⊗ Cm A mn → M n Put M n = π(M n ) ∈ Qch(P 1 R ). Thanks to Lemma 8.3.4 the maps (8.11) become maps M m ⊗ P 1 R A mn → M n and one checks that ΣM def = n M n defines an object in Gr(A). Put σM = πΣM ∈ QGr(A) (where here π is the quotient functor Gr(A) → QGr(A)).
We claim that Σ sends finitely generated objects in Gr(C) to objects in gr(A). It suffices to prove this for the projective generators e im C.
We have for n ≥ m Hence we have to prove that the righthand side is a finitely generated C n -module.
Since the summands Hom D (O(−j, −n), O(−i, −m)) are all finitely generated Rmodules we may do this after specialization. We get which is indeed finitely generated. For reference below we note that from this computation we also get π(e im C −,n ⊗ R k) = S n−m E k (−i) (where here π is the quotient functor Gr(C n,k ) → QGr(C n,k ) ∼ = Qch(P 1 k )) and thus Σ(e im C ⊗ R k) = n≥m S n−m E k (−i) so that finally we get The left hand side is R-flat and commutes with base change as indicated above. We claim that this true for the right hand side as well.
Lemma 8.3.6. qgr(A) is a deformation of qgr(A) k = qgr(A k ) = qgr(SE k ) = coh(P 1 k ). Proof. According to [21] qgr(A) is Ext-finite. Therefore, according to Proposition 8.2.9 it is sufficient to prove that qgr(A) has a strongly ample sequence. To this end we verify the conditions for Lemma 8.1.1. It is standard that these conditions lift from k to R and hence we may check them over k. Over k they follow from the explicit description of A mn,k given in Lemma 8.3.2. constructed above is an isomorphism.
Proof. We first discuss the first statement. Given Lemma 8.3.6 it is sufficient to check that σ(e jn C) ⊗ R k satisfies the conditions of Proposition 8.2.7. It is easy to see that σ(e jn C) is compatible with base change and is R-flat. One may then invoke the explicit description of σ(e jn C ⊗ R k) given in (8.12).
To prove the last statement we note that is is true over k by (8.12). We may then invoke Nakayama's lemma for R (given that everything is compatible with base change as we have shown above).
Proof of Theorem 8.3.1. Given our preparatory work it is sufficient to prove D ∼ = qgr(A). By Theorem 8.2.10 we obtain that (O(n, n)) n is an ample sequence in D. Given (8.13) and the Z-algebra version of the Artin-Zhang theorem [1] it is sufficient to prove that (σ(e −n,−n C)) n forms a strongly ample sequence in qgr(A). Using Lemma 8.3.6 together with Theorem 8.2.10 this may be checked over k. Then we invoke again the explicit description of σ(e −n,−n C ⊗ R k) given in (8.12).