Valuative and geometric characterizations of Cox sheaves
aa r X i v : . [ m a t h . AG ] D ec VALUATIVE AND GEOMETRIC CHARACTERIZATIONS OFCOX SHEAVES
BENJAMIN BECHTOLD
Abstract.
We give an intrinsic characterization of Cox sheaves on Krullschemes in terms of their valuative algebraic properties. We also provide ageometric characterization of their graded relative spectra in terms of goodquotients of graded schemes, extending the work of [3] on relative spectra ofCox sheaves on normal varieties. Moreover, we obtain an irredundant charac-terization of Cox rings which in turn produces a normality criterion for certaingraded rings.
Introduction
Cox sheaves on normal (pre-)varieties X currently are an active field of researchwith focus on questions of finite generation and explicit calculation of their ring R ( X ) of global sections (called the Cox ring ) and quotient constructions describing X in terms of a Cox ring and combinatorics [4, 7, 8, 10, 11, 17, 18, 19, 26]. Knownproperties of Cox rings include triviality of homogeneous units and graded factori-ality [2, 3, 6], i.e. factoriality of the monoid of non-zero homogeneous elements. Inthe case of a free class group Cl( X ) the latter is equivalent to genuine factorialityof the Cox ring R ( X ) by a result from [1].Our present purpose is to investigate and characterize Cox sheaves in the moregeneral setting of Krull schemes, i.e. schemes with a finite cover by spectra ofKrull rings, compare [22]. A Cox sheaf on X is a Cl( X )-graded O X -algebra R withhomogeneous components O X ( D ) for all [ D ] ∈ Cl( X ), equipped with a natural multiplication. This translates into the more formal requirement that there existsa graded morphism from the divisorial O X -algebra associated to WDiv( X ) π : O X (WDiv( X )) := M D ∈ WDiv( X ) O X ( D ) · χ D −→ R such that each restriction π : O X ( D ) → R [ D ] is an isomorphism, see Section 3for more motivation of this definition. All Cox sheaves on X are linked via themaps from O X (WDiv( X )) and thus they have the same graded invariants and theirmonoids of homogeneous elements are isomorphic modulo units, although the Coxsheaves themselves need not be isomorphic. Furthermore, one Cox sheaf has locallyor globally finitely generated sections if and only if all Cox sheaves on X do, seeProposition 3.5.Our main result is an intrinsic characterization of Cox sheaves in terms of theirvaluative algebraic properties. We start by illustrating our terminology. Recallthat on a Krull scheme X each prime divisor Y defines a discrete valuation ν Y,X : K ( X ) ∗ → Z with valuation ring K ( X ) ν Y,X = O X,Y and O X ( U ) is the intersectionover all O X,Y with Y ∈ U . In terms of sheaves each Y defines a discrete valuation ν Y : K ∗ → ı Y ( Z ) from the constant sheaf K ∗ onto the skyscraper sheaf at Y and O X is the intersection O X = \ Y prime K ν Y ⊆ K over the corresponding discrete valuation sheaves K ν Y . Thus, O X is a Krull sheaf .We will show that Cox sheaves admit a similar description and may even be char-acterized in such terms. Let K := Cl( X ) and let R be a Cox sheaf. The role as anambient sheaf of R is taken by the constant K -graded sheaf S assigning the stalk R ξ at the generic point. Every homogeneous component of S is of type K and hence S is K -simple , i.e every homogeneous section is invertible. Each prime divisor Y on X now defines a discrete K -valuation µ Y : S + → ı Y ( Z ) on the subsheaf S + ⊆ S of K -homogeneous non-zero elements and a corresponding discrete K -valuation sheaf S µ Y ⊆ S whose sections over U are generated by all K -homogeneous elements thatare valuated non-negatively by µ Y,U . R is then the intersection R = \ Y prime S µ Y ⊆ S and is thereby a K -Krull sheaf . In this terminology, whose precise definitions aregiven in Section 2, our result is the following. Theorem 1.
Let X be a Krull scheme with generic point ξ , fraction field K , es-sential valuations ν Y for the prime divisors Y ∈ X , and let K be an abelian groupand R a K -graded sheaf on X . Then R is isomorphic to a Cox sheaf if and only ifthe following hold: (i) the constant sheaf S assigning R ξ is K -simple with deg K ( S + ) = K and S = K , (ii) R is a K -Krull sheaf in S defined by discrete K -valuations { µ Y } Y whichrestrict to { ν Y } Y on K ∗ , in particular R = O X , (iii) div K := P Y µ Y is surjective and div K,X has kernel R ( X ) + , ∗ = R ( X ) ∗ .In this case, the required Cl( X ) -grading is provided by the natural isomorphism K → Cl( X ) , deg K ( f ) [div K,X ( f )] . If R is a Cox sheaf with the required Cl( X )-grading, then the above map is theidentity on Cl( X ). This intrinsic characterization of Cox sheaves underlines the factthat they form a natural class of graded sheaves. Properties (i) and (ii) are directgraded analoga of the properties of the structure sheaf, they occur in various graded O X -algebras e.g. in divisorial O X -algebras O X ( L ) of subgroups L ≤ WDiv( X ).Property (iii) ensures the correct Cl( X )-grading, the first part being equivalentto surjectivity, the second to injectivity of the canonical map K → Cl( X ). Thereference for the equation R ( X ) + , ∗ = O ( X ) ∗ is [3].In Theorem 2 below we give further details on Cox sheaves. We briefly explainthe occuring graded properties and invariants. A K -integral ring R (i.e. a ringwithout K -homogeneous zero divisors) is K -factorial if the monoid R + of non-zerohomogeneous elements is factorial. The homogeneous fraction ring Q + ( R ) is thelocalization of R by R + . A K -Krull ring is the graded analogon of a Krull ring.Its essential K -valuations form the minimal family of K -valuations defining R in Q + ( R ). They correspond bijectively to the K -prime divisors , i.e. the minimal non-zero K -prime ideals p of R . The K -valuation ring of an essential K -valuation ν p isthe graded localization R p . More detailed information on K -Krull rings is found inSection 1. The essential K -valuations of a K -Krull sheaf R are defined in terms ofthe K -Krull rings R ( U ) for all affine U , see Section 2. Theorem 2.
Let X be a Krull scheme. Then each Cox sheaf R on X is quasi-coherent and has the following properties: (i) For each open U , the ring R ( U ) is K -factorial and deg K ( R ( U ) + ) generates K . If U is affine, then S ( U ) = Q + ( R ( U )) and deg K ( R ( U ) + ) equals K . ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 3 (ii) { µ Y } Y are the essential K -valuations of R and the sections of their valu-ation sheaves S µ Y are S µ Y ( U ) = (cid:26) R Y Y ∈ U S ( X ) Y / ∈ U in particular R ( U ) = \ Y ∈ U R Y ⊆ S ( X )(iii) The stalk at x ∈ X is the K -local K -Krull ring R x = \ x ∈{ Y } S ( X ) µ Y,X = \ x ∈{ Y } R Y ⊆ S ( X ) The homogeneous elements of its K -maximal ideal a x resp. the homoge-neous units of R x are a x ∩ R + x = { f x ∈ R + x ; there ex. U ∋ x, f ∈ R ( U ) + s.th. x ∈ | div K,U ( f ) |}R + , ∗ x = { f x ∈ R + x ; there ex. U ∋ x, f ∈ R ( U ) + s.th. div K,U ( f ) = 0 } and deg K ( R + , ∗ x ) ⊆ Cl( X ) is the subgroup of classes [ D ] represented by adivisor D which is principal near x . (iv) For a prime divisor Y , each generator of the maximal ideal of O X,Y =( R Y ) also generates a Y , in particular, R Y has units in every degree. K -factoriality of the rings R ( U ) is due to surjectivity of div K,U which is essen-tially the argument from [2]. The first proof of K -factoriality of Cox rings is due to[6], it is valid for Cox sheaves of finite type on normal prevarieties. Cox sheaves onaffine Krull schemes also allow the following description. Corollary 3. A K -graded sheaf R on an affine Krull scheme X is a Cox sheaf ifand only if it is the sheaf R = e R associated to a K -graded O ( X ) -algebra R suchthat: (i) R is K -factorial (in particular, R is a K -Krull ring) and R + , ∗ = R ∗ , (ii) R = O ( X ) , Q + ( R ) = Q ( R ) and deg K ( Q + ( R )) = K , (iii) the essential K -valuations of R restrict bijectively on Q ( R ) to the essentialvaluations of R , The canonical choice for a geometric realization of a quasi-coherent K -graded O X -algebra F is its graded relative spectrum Spec
K,X ( F ) which is glued from the K -spectra (i.e. sets of K -prime ideals) of F ( U ) for all affine open U ⊆ X . This objectbelongs to the category of graded schemes (which contains the category of schemes,i.e. 0-graded schemes, as a full subcategory) wherein structure sheaves are gradedand morphisms between affine graded schemes are comorphisms of maps of gradedrings, see Section 4. From the perspective of [12] the category of graded schemesis situated between the categories of F -schemes and classical schemes within theircommon parent category of sesquiad schemes, see Remark 4.4. Graded algebraicproperties of F naturally correspond to geometric properties of Spec K,X ( F ). TheCl( X )-graded relative spectrum of a Cox sheaf R on X together with the canonicalmorphism q : Spec Cl( X ) ,X ( R ) → X is called its graded characteristic space . Sincea Cox sheaf is a K -Krull sheaf, its graded characteristic space is a K -Krull scheme which is the generalization of Krull schemes in the category of graded schemes.A K -Krull scheme b X comes with natural notions of K -Weil divisors as sums of K -prime divisors (i.e. points with one-codimensional closure), K -principal divisors for all non-zero homogeneous sections of the sheaf K K assigning the stalk O b X, b ξ at the generic point, and K -class groups . Our second main result is the followinggeometric characterization of graded characteristic spaces. B. BECHTOLD
Theorem 4.
Let q : b X → X be a morphism from a K -graded scheme to a scheme.Then X is a Krull scheme and q is a graded characteristic space if and only if thefollowing hold: (i) b X is a K -graded K -Krull scheme with deg K ( K K ( b X ) + ) = K , (ii) q is a good quotient and induces a commutative diagram of presheaves K ∗ div / / ∼ = q ∗ (cid:15) (cid:15) WDiv( q ∗ K K ) ∗ q ∗ div K / / q ∗ WDiv K b Y q ( b Y ) ∼ = O O (iii) Cl K ( b X ) = 0 , and O ( b X ) + , ∗ = O ( b X ) ∗ .If b X = Spec K,X ( R ) with a Cox sheaf R then with div K := P Y µ Y the followingcommutative diagram extends the diagram of (ii): S + div K / / / / ∼ = q ∗ (cid:15) (cid:15) WDiv q ∗ K + K q ∗ div K / / / / q ∗ WDiv K b Y q ( b Y ) ∼ = O O For each prime divisor Y the preimage q − ( Y ) consists of a single K -prime divisor b Y . If b x ∈ b X is the unique point contained in all closures of points mapped to x ∈ X ,then b x ∈ { b Y } if and only if x ∈ { Y } . In particular, O b X, b x = R x . This result extends the geometric characterization of relative spectra of Coxsheaves of finite type on normal prevarieties given in [3]; indeed, with respect tonormal prevarieties Theorem 4 allows a translation into terms of good quotients byquasi-torus actions, see Theorem 6.5. In the following theorem we also generalizetheir characterization of Cox rings.
Theorem 5. If X is a Krull scheme with class group K and Cox ring R , then thefollowing hold: (i) R is K -factorial, (ii) R + , ∗ = R ∗ , (iii) deg K ( R + ) generates K ,If X has a cover by affine complements of divisors (e.g. X is separated or of affineintersection), then (iv) each localization R p at a K -prime divisor has units in every degree.Conversely, if K is finitely generated and R satisfies (i) - (iv), then there exists aKrull scheme X of affine intersection with class group K and Cox ring R ( X ) = R . Here, property (iv) implies property (iii). The additional assumption is neededin order to translate the fact that the stalks R Y have units in every degree intoa property of the global ring R ( X ). In the case that R is finitely generated overan algebraically closed base field K , property (iv) translates into freeness of theaction of H = Spec max ( K [ K ]) on a big open subset of Spec max ( R ) - see Remark 6.4,which is the property featured in the characterization of finitely generated Coxrings of normal prevarieties of affine intersection given by [3]. Our characterizationof Cox rings of Krull schemes with cover by affine divisor complements and finitelygenerated class groups by conditions (i), (ii) and (iv) is irredundant, see Remark 5.2Together with normality of Cox rings of normal prevarieties over K [3, Thm. I.5.1.1]we obtain: ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 5
Corollary 6.
Let K be a finitely generated abelian group, K an algebraically closedfield and R a K -graded affine K -algebra satisfying the above properties (i)- (iv).Then R is normal. The paper is organized as follows: Section 1 lays the algebraic foundations forthe later parts, introducing K -Krull rings and providing the key preparation forthe calculation of the essential K -valuations of a Cox sheaf R and hence the K -prime divisors of Spec K,X ( R ) in Theorem 1.10. In Section 2 we introduce K -Krullsheaves and discuss the example of divisorial O X -algebras O X ( L ) associated tosubgroups L ≤ WDiv( X ). In Section 3 we give background on the definition ofCox sheaves and prove Theorem 1. Section 4 offers a first general introduction tograded schemes and good quotients thereof as well as K -Krull schemes and their K -Weil divisors and class groups. The example of graded spectra of monoid algebras(Example 4.3) relates graded schemes to F -schemes, i.e. schemes over the fieldwith one element as treated in [13]. We also indicate how graded schemes fit intothe more general framework of sesquiad schemes from [12], see Remark 4.4. InSection 5 we prove Theorems 4 and 5 using Theorem 1. In Section 6 we pointout some aspects of the connection between graded schemes of finite type anddiagonalizable actions on prevarieties, in particular, we reformulate Theorem 4 inthis more familiar setting. This requires the concept of invariant structure sheaveswhose stalks naturally encode generic isotropy groups of the action, see Remark 6.4.Furthermore, we provide details on the connection between orbit closures, gradedschemes and combinatorics (Remark 6.3), and go on to show that the toric gradedscheme corresponding to a toric variety is canonically identified with the definingpolyhedral fan (Remark 6.6).The author is grateful for helpful discussions with J¨urgen Hausen.1. K -Krull rings We start by recalling some generalities and notations from graded Algebra. Allrings are taken to be commutative with unit. All abelian groups used to graderings are written additively. A K -graded ring is a ring with a decomposition R = L w ∈ K R w into abelian groups such that R w R w ′ ⊆ R w + w ′ . The sets of K -homogeneous elements with and without zero, and the group of K -homogeneousunits are denoted R + , , R + and R + , ∗ , respectively. A morphism of graded ringsis a map φ : R → R ′ with accompanying group homomorphism ψ : K → K ′ suchthat φ restricts to group homomorphisms R w → R ′ ψ ( w ) . A morphism of gradedrings is called degree-preserving if the accompanying map is the identity. For anyfixed K the category of graded rings has a subcategory of K -graded rings withdegree-preserving morphisms, and this subcategory has direct and inverse limits.A K -graded module M over a K -graded ring R is a R -module with a decomposi-tion M = L w ∈ K M w into abelian groups such that R w M w ′ ⊆ M w + w ′ , where theelements of S w ∈ K M w are called homogeneous elements. A K -graded submodule of M is a submodule of the form N = L w ∈ K N ∩ M w , i.e. a submodule generated byhomogeneous elements. Remark 1.1. If B → R is a morphism of graded rings, then R is also calleda graded algebra over B . The graded algebras over B form a category with theobvious morphisms. This category has coproducts: If φ R : B → R and φ S : B → S are morphisms accompanied by ψ L : K → L and ψ M : K → M then R ⊗ B S isnaturally graded by L × M/ im( ψ L × − ψ M ) and the canonical maps R → R ⊗ B S and S → R ⊗ B S are morphisms of graded algebras over B . This statement is usedto define fiber products of graded schemes. B. BECHTOLD
Classical algebraic properties of rings and their elements and ideals have gradedanaloga which are obtained by restricting the defining axioms to homogeneous ele-ments resp. graded ideals. In particular, there are natural concepts of graded divisi-bility theory, i.e. K -integrality , K -prime and K -irreducible elements, K -factoriality ,as well as K -prime and K -maximal ideals, K -locality , K -noetherianity etc. Severalauthors have studied such properties: [20] treats K -prime ideals and invariants ofgraded modules over K -noetherian rings. Graded divisibility theory was introducedand interpreted geometrically by [17, 2] who showed that K -factoriality is a naturalproperty of Cox rings of normal prevarieties. Graded integral closures and theirbehaviour under coarsening have been studied in [25]. The localization of R by a K -prime ideal p is denoted R p := ( R + \ p ) − R . By localizing a K -integral ring R by R + we obtain Q + ( R ), the K -homogeneous fraction ring in which every homo-geneous element is invertible, making it K -simple . In general, a K -graded ring R is K -simple if and only if R is a field and deg K ( R + ) is a group, and in that case rR = R deg K ( r ) holds for every r ∈ R + . A K -integral ring R is K -normal , if eachhomogeneous fraction that is integral over R (i.e. over R + , ) is an element of R .In the following major part of this section we take a slightly more detailed look at K -Krull rings, the graded equivalent of Krull rings. Proofs of their basic propertiesmay be obtained from the proofs of the respective properties of Krull rings (founde.g. in [15, 21]) by restricting the arguments to homogeneous elements resp. gradedideals. In Example 1.9 we treat a canonical class of K -Krull rings which form thealgebraic analogon of divisorial O X -algebras O X ( L ) of subgrops of Weil divisors.Theorem 1.10 gives details on their K -divisors and essential K -valuations and thusprovides the key for the calculation of the essential K -valuations of Cox sheaves inSection 3 and the K -Weil divisors of their graded characteristic spaces in Section 5. Definition 1.2.
Let S be a K -simple ring. A discrete K -valuation on S is a groupepimorphism ν : S + → Z with ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } for all w ∈ K , a, b ∈ S w \ a + b = 0. Its discrete K -valuation ring is the subring R ν ⊆ S generated bythe preimage of Z ≥ under ν .A K -Krull ring is an intersection R of discrete K -valuation rings R ν j ⊆ S , j ∈ J such that for each a ∈ R + only finitely many ν j ( a ) are non-zero.Let R be a K -integral ring. For K -graded R -submodules a , b of Q + ( R ), product ab and quotient [ a : b ] = { f ∈ Q + ( R ); f b ⊆ a } are again K -graded. A K -fractionalideal is a K -graded proper R -submodule a ≤ R Q + ( R ) with [ R : a ] = 0. Construction 1.3.
Let R be a K -integral ring. A K -fractional ideal a is called a K -divisor of R if a = [ R : [ R : a ]]. The set Div K ( R ) of K -divisors of R equippedwith the operation sending a and b to a + b := [ R : [ R : ab ]]and the partial order defined by a ≤ b : ⇔ a ⊇ b is a partially ordered semi groupwith neutral element R in which each two elements have infimum and supremum.There is a canonical homomorphismdiv K : Q + ( R ) + → Div K ( R ) , f Rf with kernel R + , ∗ whose image P Div K ( R ) is called the group of K -principal divisors .The cokernel Cl K ( R ) is called the K -class semi-group of R .The semi-group of K -divisors characterizes R as follows: Theorem 1.4.
Let R a K -integral ring. (i) R is a K -Krull ring if and only if Div K ( R ) is a group and every non-emptyset of positive elements in Div K ( R ) has a minimal element. (ii) R is K -factorial if and only if R is a K -Krull ring with Cl K ( R ) = 0 . ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 7
Remark 1.5.
Let { ν j } j ∈ J be a defining family of the K -Krull ring R . For a K -fractional ideal a and j ∈ J we set ν ( a j ) := max {− ν j ( f ); f ∈ [ R : a ] ∩ Q + ( R ) + } ∈ Z . This notion is well-defined and satisfies ν j ( a ) = ν ([ R : [ R : a ]]). Furthermore, thereis a monomorphism of ordered groups Div K ( R ) −→ M j ∈ J Z , a ν j ( a ) } j ∈ J . Proposition 1.6.
Let R be a K -Krull ring. Then the following hold: (i) The minimal positive K -divisors are those that are K -prime as ideals in R , and these are the minimal non-zero K -prime ideals of R ; they are calledthe K -prime divisors of R and form a Z -basis of Div K ( R ) , (ii) For each K -prime divisor p the map ν p assigning to a ∈ Q + ( R ) the co-efficient with which p occurs in div K ( a ) is a K -valuation on Q + ( R ) . Its K -valuation ring is R p and we have p ∩ R + = ν − p ( Z > ) ∩ R + . The coeffi-cient of a K -divisor a at p is ν p ( a ) = min { ν p ( a ); a ∈ a ∩ Q + ( R ) + } . The family { ν p } p is minimal among all families defining R in Q + ( R ) , itis called the family of essential K -valuations of R . Remark 1.7.
Assertion (i) is in particular an existence statement: A K -Krull ringhas K -prime divisors if and only if it is not K -simple. A general K -integral ringneed not have K -prime divisors (i.e. minimal non-zero K -prime ideals), even if itis not K -simple. Proposition 1.8. A K -noetherian K -integral ring is a K -Krull ring if and only ifit is K -normal. Next, we treat the algebraic construction that lies beneath divisorial O X -algebrasof subgroups L of Weil divisors. Example 1.9.
Let A be a Krull ring with essential valuations { ν p } p and let φ : K → Div ( A ) be a homomorphism of abelian groups. The group algebra S := Q ( A )[ K ]is K -simple and µ p : S + −→ Z aχ w ν p ( a ) + ν p ( φ ( w )) = ν p (div( a ) + φ ( w ))defines a K -valuation on S for every prime divisor p . The ring R = T p S µ p is a K -Krull ring with homogeneous components R w = { a ∈ Q ( A ); a = 0 or div( a ) + φ ( w ) ≥ } · χ w for w ∈ K. Theorem 1.10.
In the above notation, the ring R has the following properties: (i) R = A and Q + ( R ) is canonically isomorphic to Q ( A )[ K ] , (ii) { µ p } p are the essential K -valuations of R and there is an isomorphism Div K ( R ) −→ Div ( A ) α : b X p µ p ( b ) p [ R : [ R : R a ]] ←− [ a : β which restricts to a bijection { K − prime divisors of R } −→ { prime divisors of A } q q ∩ A h µ − p ( Z > ) ∩ R i ←− [ p B. BECHTOLD and induces an isomorphism Cl K ( R ) ∼ = Cl( A ) / im( φ ) , (iii) The localization R q by a K -prime divisor q of R has units in every degreewith ( R q ) = A p where p = q ∩ A , (iv) R has homogeneous components of every K -degree, i.e. deg K ( R + ) = K . Lemma 1.11.
In the above situation, let a ⊂ Q ( A ) be a fractional ideal and p aprime divisor of A . Then µ p ( R a ) = ν p ( a ) , in particular, µ p ( R p ′ ) = δ p , p ′ holds for any prime divisor p ′ of A .Proof. We calculate µ p ( R a ) ≥ max { µ p ( a ); a ∈ Q ( A ) ∗ , a ⊆ Aa } = ν p ( a ) = min { µ p ( a ); a ∈ a }≥ max { µ p ( r ); r ∈ Q + ( R ) + , R a ⊆ Rr } = µ p ( R a ) . (cid:3) Proof of Theorem 1.10.
Assertion (iv) follows from the Approximation Theoremfor Krull rings (see e.g. [15]). For (i) we observe that the canonical morphism Q + ( R ) → Q ( A )[ K ] is surjective. Indeed, for every w ∈ K there exists an element0 = aχ w ∈ R w by (iv) and thus aχ w /aχ is mapped to χ w .For (ii) we first observe that by Remark 1.5 α is a monomorphism of partiallyordered groups. Since the above Lemma and Remark 1.5 give α ( β ( a )) = a , α is alsosurjective and β is its inverse map. In particular, they induce bijections betweenthe sets of minimal positive elements of Div K ( R ) and Div ( A ). This means thatthe divisorial ideals β ( p ) are the K -prime divisors of R . For any prime divisor p of A and any K -prime divisor q ′ = β ( p ′ ) we have µ p ( q ′ ) = δ p , p ′ and therefore µ p = ν β ( p ) is the essential K -valuation corresponding to q = β ( p ) and we have q = h µ − p ( Z > ) ∩ R i . Since ν q restricts to ν p this implies q ∩ A = p .For (iii), let q be a K -prime divisor of R , and let p = q ∩ A be the correspondingprime divisor of A . First we show ( R q ) = A p . Let f /g ∈ ( R p ) . Then there are a, b ∈ A with f /g = a/b , and ν p ( a/b ) = ν q ( f /g ) ≥
0, i.e. a/b ∈ A p . Each generator t of the maximal ideal of A p satisfies ν q ( t ) = ν p ( t ) = 1 and is thus a generatorof the K -maximal ideal of R q . This implies the assertion. (cid:3) Remark 1.12.
From [14] we observe that if K is free, then for a K -prime divisor q of R and p = A ∩ q , each choice of a generator for the maximal ideal of A p givesan isomorphism R q ∼ = A p [ K ]. In particular, R q is a Krull ring and one concludesthat R is a Krull ring.The following well-behaved class of graded morphisms will be used in Section 3to describe the relation between the sections of O X (WDiv( X )) and Cox sheaves.Their properties, some of which are listed in Proposition 1.13 below, ensure thatCox sheaves inherit all graded properties from O X (WDiv( X )).A component-wise isomorphic epimorphism (CIE) is an epimorphism of gradedrings φ : R ′ → R accompanied by an epimorphism ψ : K ′ → K that restrictsto isomorphisms on homogeneous components. If ψ is fixed, then for each given R one obtains R ′ and φ constructively and uniquely by setting R ′ w ′ := R ψ ( w ′ ) for w ′ ∈ K ′ . The functor from K -graded rings to K ′ -graded rings thus defined is rightadjoint to the coarsening functor associated to ψ , see [24]. If R ′ and ψ are fixed,then each homomorphism χ : ker( ψ ) → R ′ + , ∗ with χ ( w ′ ) ∈ R ′ w ′ defines a CIE φ : R ′ → R ′ / h − χ ( w ′ ); w ′ ∈ ker( ψ ) i . Proposition 1.13.
Let φ : R ′ → R be a CIE. Then the following hold: ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 9 (i) R + ∼ = R ′ + /φ − (1) and there is a bijection of sets of graded ideals { a ′ E R ′ } −→ { a E R } a ′ φ ( a ′ ) h φ − ( a ) ∩ R ′ + i ←− [ a respecting inclusions, products, quotients, sums and intersections. (ii) Likewise, there is a bijection between the graded R ′ - resp. R -modules of Q + ( R ′ ) and Q + ( R ) . (iii) If M ⊆ R + is a submonoid and M ′ := φ − ( M ) , then M ′− R ′ → M − R isagain a CIE, in particular, for every f ′ ∈ R ′ + the map R ′ f ′ → R φ ( f ′ ) is aCIE. (iv) R ′ is K ′ -integral/-simple/-factorial resp. has units in every K ′ -degree ifand only if R is K -integral/-simple/-factorial resp. has units in every K -degree. (v) Let φ S : S ′ → S be a CIE with R ′ ⊆ S ′ and R ⊆ S extending the CIE φ : R ′ → R . Suppose that S ′ is K ′ -simple. Then the following hold: (a) S ′ = Q + ( R ′ ) if and only if S = Q + ( R ) . (b) Each K ′ -valuation ν on S ′ with ker( φ ) ⊆ ker( ν ) induces a K -valuation ν on S and vice versa. (c) R ′ is a K ′ -Krull ring defined by { ν j } j ∈ J if and only if R is a Krullring defined by { ν j } j ∈ J . Here, { ν j } j ∈ J are the essential K ′ -valuationsif and only if { ν j } j ∈ J are the essential K -valuations.Thus, R ′ and R share all of the graded properties defined in terms of graded idealsand all properties of R + /R + , ∗ . divisorial O X -algebras Let K be an abelian group. We begin with the prerequisites on graded sheavesneeded for the definition of K -Krull sheaves - the sheaf-theoretic analogon of K -Krull rings. Recall that a K -graded (pre-)sheaf of rings F on a topological space X is a (pre-)sheaf of K -graded rings with degree-preserving restriction maps. F isalso called a graded (pre-)sheaf with grading group K = gr ( F ). As a presheaf F then equals L w ∈ K F w where F w ⊆ F is the (pre-)sheaf of abelian groups assigning F ( U ) w to U . The monoid of K -homogeneous elements of F is the sheaf of monoids F + , := S w ∈ K F w . If all F ( U ) are K -integral, then F + denotes the sheaf ofmonoids given by F + , ( U ) \ { } . A morphism φ : G → F of graded (pre-)sheavescomes with a group homomorphism ψ : gr ( G ) → gr ( F ) such that each of themorphisms of graded rings φ U is accompanied by ψ . F is also called a graded G -algebra. Definition 2.1. (i) A discrete value sheaf is a sheaf of abelian groups Z withvalues in { , Z } such that Z ≥ ( U ) := Z ( U ) ≥ defines a subsheaf of Z .(ii) Let K be an abelian group. Let S be a sheaf of K -simple rings on X . A discrete K -valuation on S is a morphism ν : S + → Z to a discrete valuesheaf such that each ν U is surjective and either a discrete K -valuation orzero. The associated K -valuation sheaf is the graded subsheaf S ν ⊆ S ofrings generated by ν − ( Z ≥ ).(iii) A K -Krull sheaf in S is an intersection R = T j ∈ J S ν j of K -valuationsheaves such that for every U and every f ∈ R + ( U ) only finitely many ν j,U ( f ) are non-zero.If X is a scheme (or a graded scheme, see Section 4), then a definingfamily of K -valuations of a K -Krull sheaf R is called the family of essential K -valuations, if for each affine U ⊆ X , the family { ν j,U ; j ∈ J, ν j,U } isthe family of essential K -valuations of the K -Krull ring R ( U ). For K = 0 we usually omitt the prefix K . Recall that X is a Krull scheme if ithas a finite cover by spectra of Krull rings. A prime divisor is a point Y ∈ X withone-codimensional closure. Sums and intersections with subscript Y are taken overall prime divisors Y of X unless specified otherwise. Example 2.2.
Let X be a Krull scheme and K = 0. Each prime divisor defines avaluation ν Y : K ∗ → Z Y to the skyscraper sheaf ı Y ( Z ). The sections of its valuationsheaf K ν Y on U are O X,Y if Y ∈ U and K otherwise. This turns the structure sheaf O X = \ Y K ν Y into a Krull sheaf with essential valuations ν Y . Remark 2.3.
A quasi-compact scheme X is a Krull scheme if and only if O X is aKrull sheaf.For the remainder of this section, X is a Krull scheme. Recall that the presheafof Weil divisors is WDiv := L Y Z Y and there is a morphismdiv := X Y ν Y : K ∗ → WDiv . Its image PDiv is the presheaf of principal divisors of X and its cokernel is thepresheaf Cl of divisor class groups. For each prime divisor Y there is a naturalprojection pr Y : WDiv → ı Y ( Z ). The support | D | of a Weil divisor D ∈ WDiv( U )is the union over the closures of { Y } in U , where Y runs through the prime divisorsoccuring with non-zero coefficient in D . Example 2.4.
For a subgroup L ≤ WDiv( X ) the constant sheaf of L -graded groupalgebras S := K [ L ] = L D ∈ L K · χ D is a sheaf of L -simple rings. Each prime divisor Y ∈ X defines an L -valuation µ Y : S + −→ ı Y ( Z ) S + ( U ) ∋ f χ D µ Y,U ( f χ D ) := ν Y,U ( f ) + pr Y,U ( D | U )Then O X ( L ) := T Y S µ Y is a L -Krull sheaf on X , called the divisorial O X -algebra associated to L . Its homogeneous parts have sections O X ( L ) D ( U ) = O X ( D )( U ) · χ D ,where O X ( D ) is the O X -submodule of K associated to D with sections O X ( D )( U ) = { f ∈ K ( U ); f = 0 or div U ( f ) + D | U ≥ } , in particular O X ( L ) = O X . The sum over all µ Y defines a morphismdiv L := X Y µ Y : S + −→ WDiv S + ( U ) ∋ f χ D div L,U ( f χ D ) = div U ( f ) + D | U with kernel O X ( L ) + , ∗ . In particular, O X ( L )( X ) + , ∗ is the set of all elements f χ D with div X ( f ) = − D . Proposition 2.5.
In the above notation, the divisorial algebra R := O X ( L ) hasthe following properties: (i) there is a canonical isomorphism S ( X ) ∼ = R ξ , and for affine U we have S ( U ) ∼ = Q + ( R ( U )) and deg L ( R ( U ) + ) = L , (ii) { µ Y } Y are the essential L -valuations of R , (iii) The sections of S µ Y on U equal R Y if Y ∈ U and S otherwise, (iv) The stalk at x ∈ X is the L -local L -Krull ring R x = \ x ∈{ Y } S ( X ) µ Y,X ⊆ S ( X ) ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 11 whose L -maximal ideal a x has homogeneous elements a x ∩ R + x = { g ∈ R + x ; there is U ∋ x with g ∈ R ( U ) , x ∈ | div L,U ( g ) |} . Its homogeneous units are R + , ∗ x = { g ∈ S ( X ) + ; there is U ∋ x with div L,U ( g ) = 0 } = \ x ∈{ Y } ker( µ Y,X ) and deg L ( R + , ∗ x ) is the subgroup of Weil divisors in L that are principalnear x . The stalk at a prime divisor Y has units in every degree and a Y has a generator in ( R Y ) = O X,Y . (v) The image of div
L,U (resp. div
L,U |R ( U ) + ) consists of (the non-negative ele-ments in) the union over all Cl( U ) -classes of divisors in L | U , in particular,if L | U maps onto Cl( U ) , then R ( U ) is L -factorial, Remark 2.6. [3, Remark I.3.1.6] For an open set U ⊂ X each g ∈ O X ( L )( U ) + defines an open subset U g := U \ | div L,U ( g ) | and a canonical isomorphism O X ( L )( U ) g ∼ = O X ( L )( U g ) . In particular, O X ( L ) is quasi-coherent. Remark 2.7.
As a subsheaf of a constant sheaf, O X ( L ) has injective restrictionmaps, and thus all canonical maps O X ( L )( U ) → O X ( L ) x for x ∈ U and O X ( L ) x →O X ( L ) x ′ for x ∈ x ′ are injectve as well. Proof of Proposition 2.5.
For (i) note that the inclusions ı V : R ( V ) ⊆ S ( X ) inducean injection ı ξ : R ξ → S ( X ). For affine open U ⊆ X we consider the canonicalmonomorphism α U : Q + ( R ( U )) → S ( X ) , g/h ı U ( g ) ı U ( h ) − . For surjectivity of ı ξ and α U let f χ D ∈ S ( X ) D and set W := X \ | div X ( f ) + D | .Then f | W χ D ∈ R ( W ) D and ı ξ (( f | W χ D ) ξ ) = f χ D . Furthermore, there exists h ∈O X ( U ) = R ( U ) with U h ⊆ W ∩ U , and by Remark 2.6 there are m > gχ D ∈R ( U ) D such that gχ D ( hχ ) − m = f | U h χ D and hence α U ( gχ D / ( hχ ) m ) = f χ D .For the supplement and assertion (ii), we invoke Theorem 1.10 with A := O ( U ), φ : L → L | U and R = R ( U ).For (iii) note that if Y / ∈ U then µ Y,U = 0 and therefore S ( U ) µ Y,U = S ( U ). If Y ∈ U , then (iv) gives R Y = S ( X ) µ Y,X = S ( U ) µ Y,U .For (iv) first note, that in S ( X ) we have R + x = { f χ D ∈ S ( X ) + ; there is U ∋ x with f ∈ O X ( D )( U ) } If f ∈ O X ( D )( U ) and x ∈ U , then for every prime divisor Y containing x in itsclosure we have Y ∈ U and thus µ Y,X ( f χ D ) = µ Y,U ( f χ D ) ≥
0. Conversely, if f χ D ∈ S ( X ) + satisfies µ Y,X ( f χ D ) ≥ Y with x ∈ { Y } ,then for the complement W of all prime divisors Y ′ with µ Y ′ ,X ( f χ D ) < f ∈ O X ( D )( W ) and x ∈ W . This establishes that R x is the L -Krull ringin S ( X ) defined by all µ Y,X with x ∈ { Y } . Its homogeneous units are thereforeobtained as the intersection of the kernels of the defining L -valuations. For thesecond representation, let g ∈ R + , ∗ x ⊆ S ( X ) + and let W be the complement of allprime divisors Y ′ with µ Y ′ ,X ( g ) = 0. None of these Y ′ contain x in their closure,therefore x ∈ W . The equation div L,W ( g ) = 0 holds by definition of W . Conversely,if g ∈ S ( X ) + satisfies div L,U ( g ) = 0 for some U containing x , then g is invertiblein R ( U ) and hence in R x . In particular, deg L ( R + , ∗ x ) is contained in the subgroupof Weil divisors in L that are principal near x . For the converse inclusion, let D | U = div U ( f ). Then f − χ D ∈ R ( U ) D is a unit and thus ( f − χ D ) x is a unit. Let a x be the ideal generated by the set { g ∈ R + x ; there is U ∋ x with g ∈ R ( U ) , x ∈ | div L,U ( g ) |} Due to Lemma 5.1, this set is closed under addition of elements of the same degreeand therefore coincides with a x ∩ R + x . Its complement in R + x is R + , ∗ x and thus a x is the only L -maximal ideal of R x .For the supplement on the stalks of prime divisors first note that ( R Y ) = O X,Y because taking stalks commutes with direct sums. Now, let U ⊆ X be affine with Y ∈ U and D ∈ L . By the Approximation Theorem for Krull rings there exists f ∈ O X ( D )( U ) with pr Y (div U ( f )) = 0 and pr Y ′ (div U ( f )) = pr Y ′ ( D | U ) for all Y ′ ∈ | D | U | and pr Y ′′ (div U ( f )) ≥ f χ D ) Y is aunit of degree D in R Y .In (v) the statements on the image of div L,U follow from the definition of div L .If div L,U is surjective, then R ( U ) + / R ( U ) + , ∗ ∼ = WDiv ≥ ( U ) is a factorial monoid,and thus also R ( U ) + is a factorial monoid, meaning that R ( U ) is L -factorial. (cid:3) Remark 2.8.
By arguments from [14], each R ( U ) is a Krull ring. Thus, if L | U maps onto Cl( U ), then R ( U ) is a factorial ring by [1]. However, the sections of Coxsheaves will in general not be factorial. Integrality and normality for the sectionsof Cox sheaves are proven in the case of normal (pre-)varieties, see [3, Sect. I.5.1]or [6], but neither proof seems to be applicable in the more general setting of Krullschemes. 3. characterization of Cox sheaves Intuitively, a Cox sheaf should be a Cl( X )-graded O X -algebra R whose [ D ]-homogeneous parts are of type O X ( D ), i.e. there should exist isomorphisms of O X -modules π D : O X ( D ) → R [ D ] for all Weil divisors D ∈ WDiv( X ). This requirementfixes the O X -module structure of Cox sheaves. There is however no canonical way toequip such an O X -module with an O X -algebra structure. But we can and do requirethat the multiplication in R is natural in the sense that up to the isomorphisms π D it is given by the multiplication in K ; meaning that to multiply homogeneoussections of degree [ D ] and [ D ′ ] in R is the same as to apply π − D resp. π − D ′ , multiplythe resulting sections of O X ( D ) and O X ( D ′ ) in K and then apply π D + D ′ . Thistranslates into the condition that the morphism of O X -modules O X (WDiv( X )) = M D ∈ WDiv( X ) O X ( D ) · χ D π −−−−→ R = M [ D ] ∈ Cl( X ) R [ D ] . defined by the sum of the π D is a morphism of graded O X -algebras. Summing up, aCox sheaf is defined as a Cl( X )-graded O X -algebra R posessing a graded morphismfrom O X (WDiv( X )) to R which restricts to isomorphisms of the homogeneousparts. Such a kind of morphism between two graded sheaves has useful propertiesand thus justifies the following definition. Definition 3.1.
A morphism π : F → G of graded presheaves of rings with accom-panying map ψ : L → K of abelian groups is called a component-wise isomorphicepimorphism (CIE) if ψ is an epimorphism and the restriction π |F w is an isomor-phism for every w ∈ L . Equivalently, every pair ( π U , ψ ) is a CIE of rings. Remark 3.2.
Let ψ : L → K be an epimorphism of abelian groups.(i) Let π : F → G be a CIE accompanied by ψ . Then there is an exactsequence 0 / / ker( ψ ) χ / / F + π / / G + / / ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 13 where ker( ψ ) is considered as a constant presheaf and χ U maps w ∈ L tothe unique preimage of 1 G ( U ) in F ( U ) w . Furthermore, F is a sheaf if andonly if G and ker( π ) are sheaves.(ii) A morphism π : F → G of graded sheaves of rings is a CIE of sheaves ifand only if every π x : F x → G x is a CIE of graded rings.(iii) Let F be a L -graded presheaf of rings. For a morphism χ : ker( ψ ) → F + , ∗ with χ U ( w ) ∈ F ( U ) w the cokernel of the K -graded presheaf of ideals I χ : U
7→ h F ( U ) − χ U ( w ); w ∈ ker( ψ ) i is a CIE. Conversely, every CIE with prescribed F and ψ is of this formwith χ as in (i).In this terminology, the precise definition of Cox sheaves is the following. Definition 3.3.
Let X be a Krull scheme. A Cox sheaf on X is a Cl( X )-gradedsheaf of rings R such that there exists a CIE π : O X (WDiv( X )) → R that isaccompanied by the canonical map WDiv( X ) → Cl( X ). R is then automatically a O X -algebra with R = O X . Remark 3.4.
For a Cl( X )-graded sheaf R on a Krull scheme X the following areequivalent:(i) R is a Cox sheaf,(ii) for every subgroup L ≤ WDiv( X ) mapping onto Cl( X ) there exists a CIE π : O X ( L ) → R ,(iii) for some subgroup L ≤ WDiv( X ) mapping onto Cl( X ) there exists a CIE π : O X ( L ) → R . Proof.
Assume that (iii) holds. Let π : O X ( L ) → R be a CIE. Let D j , j ∈ J be abasis of WDiv( X ). Then there exist D ′ j ∈ L, j ∈ J and f j ∈ O X ( D ′ j − D j )( X ) withdiv( f j ) + D ′ j = D j , and the isomorphisms O X ( D j ) · f j −−→ O X ( D ′ j )fit together to a homomorphism Φ : O X (WDiv( X )) → O X ( L ) with accompanyinghomomorphism φ : WDiv( X ) → L, D j D ′ j . The composition π ◦ Φ : O X ( L ) → R is the epimorphism requested in assertion (i). (cid:3) Existence of Cox sheaves follows from Remark 3.2 because for any L mappingonto Cl( X ) a suitable map χ is defined by assigning arbitrary f j ∈ O X ( D j )( X )with div X ( f j ) = − D j to the elements of a basis { D j } j ∈ J of L ∩ PDiv( X ), and thusthe presheaf R = O X ( L ) / I χ is a Cox sheaf, compare [3]. As seen by this discussion,our definition is the axiomatic version of the constructive approach of Hausen andArzhantsev.Uniqueness is a matter of caution. In the case that Cl( X ) is free, it is well-knownthat all Cox sheaves are isomorphic. A further condition enforcing uniqueness inthe case of prevarieties over an algebraically closed field K is O ( X ) ∗ = K ∗ whichholds e.g. if X is projective, see [3, Sect. I.4.3]. In general Cox sheaves are only weakly unique in the following sense: Proposition 3.5.
Let X be a Krull scheme and let R and R ′ be Cox sheaves on X and let U ⊆ X be open. Then the following hold: (i) R + / R + , ∗ ∼ = O X (WDiv( X )) + / O X (WDiv( X )) + , ∗ ∼ = R ′ + / R ′ + , ∗ . (ii) There are bijections respecting sums, intersections, inclusions, productsand ideal quotients between the sets of graded ideals of R ( U ) and R ′ ( U ) . (iii) R ( U ) is finitely generated as a O X ( U ) -algebra if and only if R ′ ( U ) is so.If X is a scheme over S = Spec( B ) , then R ( U ) is finitely generated over B if and only if R ′ ( U ) is so. Proof.
Everything but the last assertion follows directly from Proposition 1.13.Suppose that R ( U ) is finitely generated by homogeneous sections g , . . . , g m . ThenCl( X ) is finitely generated by Theorem 1. Let L ≤ WDiv( X ) be a finitely generatedsubgroup mapping onto Cl( X ) and let π : O X ( L ) → R and π ′ : O X ( L ) → R ′ beCIE. Let χ be the kernel character of π and let D , . . . , D n be a basis of L ∩ PDiv( X ). Then O X ( L )( U ) is generated by χ U ( ± D ) , . . . , χ U ( ± D n ) and any choiceof homogeneous preimages under π U f , . . . , f m for g , . . . , g m . Thus, R ′ ( U ) isgenerated by their images under π ′ U . (cid:3) The above shows that the question of uniqueness is of little practical consequencesince all Cox sheaves on a given X behave in the same way. We now proceed withthe proof of Theorem 1. The general ideal for showing that Cox sheaves have theasserted properties is to show that they are inherited from O X (WDiv( X )) becauseCIEs preserve most graded properties (even in both directions). The second partof the proof adepts the arguments of [3, Thm. I.6.4.3 and Prop. I.6.4.5]. Proof of Theorem 1.
Let R be a Cox sheaf on X and let π : O X ( L ) → R be a CIE.Then we have commutative diagrams with CIE downward arrows (1) O X ( L )( U ) / / π U (cid:15) (cid:15) O X ( L ) xπ x (cid:15) (cid:15) O X ( L ) x / / π U (cid:15) (cid:15) O X ( L ) x ′ π x ′ (cid:15) (cid:15) (2) R ( U ) / / R x R x / / R x ′ and the lower arrows inherit injectivity from the upper ones. Considering diagram(1) for the generic point ξ we see that R ξ is K -simple with degree zero part O X,ξ and deg K ( R + ξ ) = K because O X ( L ) ξ is L -simple with degree zero part O X,ξ anddeg L ( O X ( L ) + ξ ) = L . Denote by S L the constant sheaf assigning O X ( L ) ξ . Then π ξ defines a CIE of sheaves π S : S L → S and the constant sheaf S := R ξ has thedesired properties. R is a subsheaf of S and hence K -integral.For (ii), consider the diagram with exact rows and injective upward arrows0 / / ker( ψ ) χ S / / S + L π S / / S + / / / / ker( ψ ) χ / / O X ( L ) + ?(cid:31) O O π / / R + / / ?(cid:31) O O π S ,X ) = im( χ S ,X ) is contained in O X ( L )( X ) + , ∗ , its elements are triv-ially valuated by all µ Y,X . Consequently, ker( π S ,U ) = im( χ S ,U ) is valuated triviallyby all µ Y,U . Thus, each µ Y induces a K -valuation µ Y : S + → Z Y which alsorestrict to ν Y on K ∗ . Since O X ( L ) is defined in S L by the family { µ Y } Y , the equal-ity R = T Y S µ Y now follows by applying Proposition 1.13(v) to the sections overarbitrary open U and assertion (ii) is proven.The first part of assertion (iii) is due to the fact that µ Y and µ Y have thesame image. For the second part, consider an element π X ( f χ D ) of the kernel ofdiv K,X = P Y µ Y , i.e. a global homogeneous unit. Then div L,X ( f χ D ) = 0, i.e. D = − div X ( f ) is a principal divisor, hence π X ( f χ D ) has degree [ D ] = [0].Concerning the supplement we calculate[ D ] = deg K ( π ξ ( f χ D )) = [deg L ( f χ D )] = [div L,X ( f χ D )] = [div K,X ( π ξ ( f χ D ))] . It remains to show that a K -graded sheaf R satisfying (i)-(iii) is a Cox sheaf.Recall the notation div K := P Y µ Y : S + → WDiv and note that by (ii) we havediv K |K ∗ = div. In order to show that K is canonically isomorphic to Cl( X ) we firstnote that by (ii) the degree map deg K induces an isomorphism S ( X ) + / S ( X ) ∗ ∼ = K . ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 15
The homomorphism δ : S ( X ) + → Cl( X ) , f [div K,X ( f )] thus induces the map δ : K −→ Cl( X ) , deg K ( f ) [div K,X ( f )]which has cokernel Cl( X ) / im(div K,X ) and kernel deg K ( R ( X ) + , ∗ ). Thus, condition(iii) precisely says that δ is an isomorphism.Next, we show that the ismorphism K = S λ f −−→ S deg K ( f ) given by multiplicationwith f ∈ S ( X ) + restricts to an isomorphism λ f : O X (div X,K ( f )) −→ R deg K ( f ) .Indeed, for a non-zero g ∈ O X (div K,X ( f ))( U ) we calculatediv U,K ( f | U g ) = div U ( g ) + (div X,K ( f )) | U ≥ , i.e. f | U g ∈ R ( U ) + . Conversely, each non-zero h ∈ R ( U ) deg K ( f ) satisfiesdiv U (( f | U ) − h ) + div X,K ( f ) | U = div K,U ( h ) ≥ . Now, let L ≤ WDiv( X ) be any subgroup mapping onto Cl( X ) and let { D j } j ∈ J bea basis of L . Then there exist f j ∈ S + , j ∈ J with div K,X ( f j ) = D j . We set f D := Q j ∈ J f m j j for D = P j ∈ J m j D j . By our first claim we know that δ (deg K ( f D )) =[ D ]. By our second claim, every f D defines an isomorphism λ f D : O X ( D ) → R [ D ] . By construction, the isomorphisms respect the multiplication of homogeneous com-ponents in O X ( L ) and thus define a graded epimorphism O X ( L ) → R . Hence, R is a Cox sheaf. (cid:3) Remark 3.6.
One may argue that the characterizing condition (ii) of Theorem 1is not purely intrinsic because the condition that R is a Cox sheaf in S contains anexistence statement for certain discrete K -valuations which are not derived from R .A more obviously intrinsic characterization is obtained by replacing (ii) with(ii’) R is a sheaf of K -Krull rings and every affine open U satisfies:(a) Q + ( R ( U )) = S ( U ),(b) the essential K -valuations { µ Y,U } Y ∈ U of R ( U ) restrict on K ( U ) ∗ = Q ( O X ( U )) ∗ to the essential valuations { ν Y,U } Y ∈ U of O X ( U ),(c) for every Y ∈ U and every affine open V ∋ Y we have µ Y,U = µ Y,V .Since the family of essential K -valuations of a K -Krull ring R may be derived from R , condition (ii’) is indeed intrinsic. Proof of Theorem 2.
Quasi-coherence of R follows from the more general observa-tion that for every non-zero g ∈ R ( U ) + there is a canonical isomorphism R ( U ) g ∼ = R ( U g ) , where U g := U \ | div U,K ( g ) | . This in turn follows directly from the correspondingobservation on O X ( L ) using that π U and π U g are CIEs and Proposition 1.13(iii).For (i) note that K -factoriality of R ( U ) follows via Proposition 1.13(iv) from L -factoriality of O X ( U ) which was proven in Proposition 2.5(v). Now, let D = D + − D − be any Weil divisor on X , written as a difference of effective divisors. Choose asubgroup L which contains D + and D − mapping onto Cl( X ) and a componentwiseisomorphic epimorphism π : O X ( L ) → R . Then for all open U we have[ D ] = deg K ( π U ( χ D + )) − deg K ( π U ( χ D − )) ∈ h deg K ( R ( U ) + ) i . For the case that U is affine, observe that in the diagram of CIEs O X ( L )( U ) / / (cid:15) (cid:15) Q + ( O X ( L )( U )) / / (cid:15) (cid:15) O X ( L ) ξ (cid:15) (cid:15) R ( U ) / / Q + ( R ( U )) / / R ξ the lower right arrow is a graded isomorphism because the upper right arrow is one.Furthermore, deg K ( R ( U ) + ) = deg L ( O X ( L )( U ) + ) = L = K. The first part of assertion (ii) follows directly from Proposition 2.5(ii) and Propo-sition 1.13(v). The second statement is obvious for
Y / ∈ U and follows from (iii)otherwise.For assertion (iii), consider the diagram (2) of inclusions and CIEs from the be-ginning of the proof of Theorem 1 with x ′ = ξ . Since O X ( L ) x is the intersectionover all K [ L ]( X ) µ Y,X with x ∈ { Y } , Proposition 1.13(v) implies that R x is the inter-section over all S ( X ) µ Y,X with x ∈ { Y } . Moreover, R + , ∗ x is the image of O X ( L ) + , ∗ x and thus has the requested description. Furthermore, the unique L -maximal idealof O X ( L ) x is mapped onto a unique K -maximal ideal a x by Proposition 1.13(i). Itshomogeneous elements, which were calculated in Proposition 2.5(iv), are mappedonto the homogeneous elements of a x which establishes the desired description.For (iv), observe that the stalk R Y at a prime divisor Y has units in every degreebecause O X ( L ) Y does. (cid:3) Remark 3.7.
One of the starting points for the present considerations on thevaluative structure of Cox sheaves was [3, Sect. I.5]. The [ D ]-divisor of a non-zero f ∈ R ( X ) [ D ] defined there is div Cl( X ) ,X ( f ) in our notation, and it is shown thatthe assignment f div [ D ] ( f ) is homomorphic and encodes the divisibility relationin R ( X ), which in our setting is due to the definition of div Cl( X ) ,X as the sum ofall µ Y,X . 4. graded schemes
Graded schemes are implicitely already well-known from the proj construction,see Example 4.10. Related examples of graded schemes in the context of toric goodquotients have been studied in [23]. More generally, the categories of K -gradedresp. noetherian Z -graded schemes with degree-preserving morphisms have beendiscussed in [9, 27]. The category of graded schemes introduced below includes theaforementioned and has more morphisms, in particular good quotients, which areaffine morphisms from K -graded to 0-graded schemes satisfying a natural conditionon their structure sheaves, see Definition 4.8. Good quotients behave very naturallyin that they respect intersections of closed sets and are surjective with distinguishedpoints in each fibre, see Proposition 4.9. We also introduce the other conceptsneeded for Theorem 4, namely K -Krull schemes which are the most general objectswith well-behaved notions of Weil and principal divisors. Definition 4.1.
The K -spectrum of a K -graded ring R is the set X := Spec K ( R )of K -prime ideals of R , endowed with the topology whose closed sets are of the form V ( a ) = { p ∈ X ; a ⊆ p } with K -graded ideals a E R . Its K -graded structure sheaf O X (with K -local stalks) is defined on the basis X f := X \ V ( h f i ) of principal opensets for f ∈ R + , by O X ( X f ) := R f and on arbitrary open U ⊆ X by O X ( U ) := lim ←− X f ⊆ U O X ( X f ) . The pair (Spec K ( R ) , O Spec K ( R ) ) is the affine K -graded scheme corresponding to R .A graded scheme is a pair ( X, O X ) consisting of a topological space X anda graded sheaf of rings O X that has a cover by affine gr ( O X )-graded schemes( U, O X | U ). ( X, O X ) is also called a gr ( O X )-graded scheme. A morphism of thegraded schemes ( X, O X ) and ( X ′ , O X ′ ) is a continuous map φ : X → X ′ to-gether with a morphism of graded sheaves φ ∗ : O X ′ → φ ∗ O X such that foreach x ∈ X the induced graded homomorphism φ ∗ x : O X ′ ,φ ( x ) → O X,x satisfies
ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 17 h ( φ ∗ x ) − ( m x ) ∩ O + X ′ ,φ ( x ) i = m φ ( x ) , where m x and m φ ( x ) are the respective unique gr ( O X )-/ gr ( O X ′ )-maximal ideals.Schemes are the same as 0-graded schemes, they form a full subcategory of thecategory of graded schemes. We will often only write X for the graded scheme( X, O X ). When talking about a morphism of graded schemes we will also write thecontinuous map φ : X → X ′ of the underlying topological spaces in place of thepair ( φ, φ ∗ ). Remark 4.2.
For an affine K -graded scheme X = Spec K ( R ), the stalk at p ∈ X is the graded localization R p . Morphisms between affine graded schemes are givenby maps of graded rings: If φ : X → X ′ is a morphism of affine graded schemes,then φ ∗ : R ′ = O ( X ′ ) → R = O ( X ) is a graded morphism, and φ maps the point p ∈ X to the point h ( φ ∗ ) − ( p ) ∩ R ′ + i . Example 4.3.
Let M be an abelian monoid contained in an abelian group K andlet k be a field. Let R := k [ M ] be the canonically K -graded monoid algebra of M over k . Recall that a face of an abelian monoid is a submonoid τ ⊆ M such that w + w ′ ∈ τ implies w, w ′ ∈ τ for all w, w ′ ∈ M . Then there is an order reversingbijection faces( M ) ←→ Spec K ( R ) =: Xτ p τ := h χ w ; w ∈ M \ τ i deg K ( R + \ p ) =: τ p ←− [ p where Spec K ( R ) is ordered by inclusion. Furthermore, O X ( X χ w ) = k [ M − Z ≥ w ]and O X, p = k [ M − τ p ]. If ψ : K ′ → K is a group homomorphism mapping thesubmonoid M ′ into M , then it induces a graded map e ψ : R ′ := k [ M ′ ] → R and amap faces( M ) → faces( M ′ ) , τ ψ − ( τ ) ∩ M ′ . The corresponding map of graded schemes is φ : X −→ X ′ := Spec K ′ ( R ′ ) , p τ p ′ ψ − ( τ ) ∩ M ′ . This consideration links graded schemes to combinatorics when applied to finitelygenerated monoids. On the other hand, we observe that τ ⊆ M is a face if and onlyif τ c ⊔ {∞} ⊆ M ⊔ {∞} is a prime ideal in the sense of [13]. Moreover, a subgroup p of a K -graded ring R generated by homogeneous elements is a K -prime ideal ifand only if R + , \ p is a face of the multiplicative monoid R + , . Thus, there is alsoa canonical homeomorphism between Spec K ( R ) and the space of prime ideals of M ⊔ {∞} which is a scheme over the field F with one element in the sense of [13].This line of thought is continued in Remark 6.6. Remark 4.4.
The category of graded rings is a subcategory of the category of monoidal pairs or sesquiads from [12] via the assignment R = L w ∈ K R w ( R + , , R ).Graded ideals of R correspond to ideals of ( R + , , R ) and graded localizations of R correspond to localizations of ( R + , , R ). Therefore, the affine graded scheme(Spec K ( R ) , O Spec K ( R ) ) is naturally identified with the set of prime ideals of ( R + , , R )equipped with the structure sheaf induced by ( R + , , R ), and the category of gradedschemes becomes a subcategory of the category of sesquiad Zariski schemes which isobtained by gluing prime spectra of sesquiads. Within this category (non-trivially)graded schemes take an intermediate position between schemes, whose affine chartsare given by monoid pairs of the form ( R, R ), and F -schemes, whose affine chartsare given by pairs of the form ( M, Z [ M ]).Having introduced the core definitions we will from now on assume that basic con-cepts like generic points , open and closed graded subschemes and embeddings have been introduced as well, naturally extending the well-known notions for schemes.Furthermore, K -reduced resp. K -integral K -graded schemes are defined by the ab-sence of homogeneous nilpotent elements resp. homogeneous zero divisors in allsections. For a K -integral K -graded scheme X , the constant sheaf K K assigningthe stalk O X,ξ at the generic point is a sheaf of K -simple rings. A quasi-compact K -graded scheme is K -noetherian resp. K -Krull if the sections O X ( U ) are K -noetherian resp. K -Krull for all (or equivalently, some cover of X by) affine U . Fora K -Krull scheme X , a K -prime divisor is a point with one-codimensional closure. Remark 4.5. If X is a K -Krull scheme, then the stalks at K -prime divisorsare K -valuation rings. Thus, each K -prime divisor Y defines a K -valuation ν Y : K + K → ı Y ( Z ) and these are the essential K -valuations of the K -Krull sheaf O X = T Y K K ν Y . Therefore, a K -graded scheme X is a K -Krull scheme if and only if itis quasi-compact and O X is a K -Krull sheaf.If X is a K -Krull scheme, then the direct sum over the skyscraper sheaves ı Y ( Z )is the presheaf WDiv K of K -Weil divisors and the direct sum over the K -valuations ν Y defines a morphism div K : K + K → WDiv K . Its image PDiv K is the presheaf of K -principal divisors and its cokernel is the presheaf of K -divisor class groups Cl K .The support | D | of a Weil divisor D ∈ WDiv( U ) is the union over all K -primedivisors Y ∈ U occuring with non-zero coefficient in D . Proposition 4.6.
Let X be a K -Krull scheme. Then the stalk of O X at x is O X,x = \ x ∈{ Y } ( O X,ξ ) ν Y,X ⊆ O
X,ξ where Y runs through all K -prime divisors Y containing x in their closure. Since the category of graded rings over a fixed graded ring has the graded tensorproduct as a coproduct, the category of graded schemes has fiber products. A gradedscheme X is called separated resp. of affine intersection if the diagonal morphism∆ X : X → X × X is a closed embedding resp. affine. The latter property isequivalent to affineness of the intersection of any two affine opens subsets. Proposition 4.7.
In a K -Krull scheme X of affine intersection the following hold: (i) Every open affine U ⊆ X is the complement of a K -divisor on X . (ii) If X is affine, then V ( f ) = | div K ( f ) | for every f ∈ O ( X ) + . Next, we introduce good quotients of graded schemes.
Definition 4.8.
A morphism from a K -graded scheme to a scheme is called K -invariant . A good quotient by K is an affine K -invariant morphism q : X → Y suchthat the pullback O Y → ( q ∗ O X ) is an isomorphism. Proposition 4.9. If q : X → Y is a good quotient, then the following hold: (i) q is surjective and closed, (ii) if A i ⊆ X, i ∈ I are closed then q ( T i A i ) = T i q ( A i ) , (iii) every preimage q − ( y ) contains a unique element which is contained in allclosures of elements of q − ( y ) ; and y is a closed point if and only if thiselement is a closed point.Proof. For closedness we use the fact that h a i R ∩ R = a holds for a E R . For(ii) we use ( P i a i ) ∩ R = P i ( a i ∩ R ). Surjectivity and (iii) follow from the factthat for a prime ideal q of R ,there exists a unique K -prime p that is maximal with p ∩ R = q , and q is maximal if and only if p is K -maximal. Indeed, if q is maximalthen the sum m over all K -graded ideals a of R with a ∩ R = q has the desiredproperties. The general case reduces to this case via localization. (cid:3) ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 19
For the Z -graded case, properties (i) and (iii) were observed in [9, Lemma 1.1.2].A good quotient that is bijective, i.e. a homeomorphism, is called geometric . Awell-known example for a graded geometric quotient is the proj construction of a Z -graded ring. Example 4.10.
Let R be a Z -graded ring with deg Z ( R + ) ⊆ Z ≥ and consider theproper Z -graded ideal a = L n> R n . Let X := Spec Z ( R ) and set b X := X \ V ( a ).Then the quotients X f → Spec(( R f ) ) for f ∈ a ∩ R + glue to a good quotient q : b X → Proj( R ) by Z which is even bijective, i.e. geometric. However, unless R = R , the structure sheaves of b X and Proj( R ) will be different. K -graded schemes also occur naturally as relative K -spectra of a quasi-coherent K -graded sheaves on schemes: Construction 4.11.
Let X be a scheme and let R be a quasi-coherent K -graded O X -algebra. Then Spec K ( R ( U )) is open in Spec K ( R ( V )) for any two affine open U ⊆ V ⊆ X and hence the K -prime spectra of R ( U ) for all affine U glue to a K -graded scheme, called the relative K -spectrum Spec
K,X ( R ) of R , and there is acommutative diagram Spec X ( R ) / / ' ' ◆◆◆◆◆◆◆◆◆◆◆ X Spec
K,X ( R ) q : : tttttttttt and q ∗ O Spec
K,X ( R ) = R holds. If R = O X , then q is a good quotient. Remark 4.12. L ≤ WDiv( X ) be any subgroup mapping onto Cl( X ) andlet e X := Spec L,X ( O X ( L )). For any Cox sheaf R , every CIE π : O X ( L ) → R inducesa graded homeomorphism b X := Spec Cl( X ) ,X ( R ) → e X which is an isomorphism ifand only if L maps isomorphically onto Cl( X ) (which in turn can only occur ifCl( X ) is free). 5. Proofs of Theorems 4 and 5
Proof of Theorem 4.
First suppose that conditions (i)-(iii) hold. Consider the coverof X by all affine open U . Then b X is covered by their preimages b U = q − ( U ) andthis cover has a finite subcover. Consequently, X has a finite affine cover and each O ( U ) = O ( b U ) is a Krull ring because O ( b U ) is a K -Krull ring. Thus, X is a Krullscheme. We show that R := q ∗ O b X is a Cox sheaf by applying Theorem 1. Firstly,since q is affine and hence R is quasi-coherent, we obtain R ξ = Q + ( q ∗ O b X ( U )) = Q + ( O b X ( q − ( U ))) = O b X, b ξ for any affine open U . Thus, R ξ is K -simple and deg K ( R + ξ ) = K . Because q isa good quotient, we also have ( R ξ ) = (( q ∗ O b X ) ) ξ = O X,ξ and thus the constantsheaf S := R ξ has all properties asserted in Theorem 1(i). Furthermore, there is acanonical isomorphism q ∗ : S ∼ = q ∗ K K .For the verification of Theorem 1(ii), consider a K -prime divisor b Y and its image Y := q ( b Y ). Due to (ii), we have ı Y ( Z ) = q ∗ ı b Y ( Z ) and WDiv = q ∗ WDiv K , and µ Y := q ∗ ν b Y : S + → ı Y ( Z ) restricts to ν Y on K . By definition, the family { µ Y } Y defines R as a K -Krull sheaf in S .For Theorem 1(iii), we first observe that div K,X := P Y µ Y,X = q ∗ div KX is sur-jective because div K b X is so, due to Cl K ( b X ) = 0. Furthermore, we have R ( X ) + , ∗ = R ( X ) ∗ and thus Theorem 1 implies that R is a Cox sheaf on X .Now suppose that R is a Cox sheaf and q : b X = Spec K,X ( R ) → X its characteris-tic space. Then the canonical isomorphism q ∗ : R ∼ = q ∗ O b X induces an isomorphism q ∗ : S ∼ = q ∗ K K as in the first part of the proof. By construction the K -spectra b U of R ( U ) for affine U ⊆ X form a cover of b X by affine K -Krull schemes. Since { µ Y } Y are the essential K -valuations of R and they restrict to { ν Y } Y , there are naturalbijections α : h µ − Y,U ( Z > ) ∩ O ( b U ) + i 7→ h ν − Y,U ( Z > ) ∩ O ( U ) i = Y between the K -prime divisors of b U = Spec K ( R ( U )) and the prime divisors of U .Because g is affine, these glue to an isomorphism α : WDiv K ( b X ) → WDiv( X ) , b Y q ( b Y )which in turn induces an isomorphisms of presheaves β : q ∗ WDiv K ∼ = WDiv and β b Y : q ∗ ı b Y ( Z ) ∼ = ı q ( b Y ) ( Z ). By construction, we have β b Y ◦ q ∗ ν b Y ◦ q ∗ = µ q ( b Y ) which gives β ◦ q ∗ div K ◦ q ∗ = div K , the first supplement. In particular, β ◦ q ∗ div K ◦ q ∗|K ∗ = divholds and assertions (i)-(iii) are verified.For the second supplement, consider a K -prime divisor b Y and set Y := q ( b Y ).We claim that q ( { b Y } ) = { Y } . Indeed, q ( { b Y } ) is closed, irreducible and contains { Y } . The other inclusion follows because { Y } has codimension one and q ( { b Y } ) is aproper subset of X . For any point x ∈ X let b x ∈ b X be the unique point containedin all closures of points mapped to x . Firstly, we claim that b x ∈ { b Y } . If b x ∈ { b Y } then x ∈ Y by the previous considerations. Conversely, if x ∈ { Y } = q ( { b Y } ) then { b Y } contains a point z with q ( z ) = x . By definition, b x ∈ { z } ⊆ { b Y } . Thus, weobtain R x = \ x ∈{ Y } S ( X ) µ Y,X = \ b x ∈{ b Y } K K ( b X ) ν b Y , c X = O b X, b x in R ξ = O b X, b ξ . In particular, for Y = x the point b x lies in the closure of { b Y } and O b X, b x = R Y = O b X, b Y which implies b x = b Y . (cid:3) The following Lemma was needed to show graded locality of the stalks O X ( L ) x and R x in the respective proofs. Lemma 5.1.
Let R be a Cox sheaf on X and let f, f ′ ∈ R ( U ) [ D ] with f + f ′ = 0 .Then | div K,U ( f ) | ∩ | div K,U ( f ′ ) | ⊆ | div K,U ( f + f ′ ) | . Accordingly, for g, g ′ ∈ O X ( D )( U ) with g + g ′ = 0 we have | div U ( g ) + D | ∩ | div U ( g ′ ) + D | ⊆ | div U ( g + g ′ ) + D | . Proof.
It suffices to consider the case that U is affine. Using Proposition 4.7 wecalculate | div K,U ( f ) | ∩ | div K,U ( f ′ ) | = q ( | div Kq − ( U ) ( q ∗ ( f )) | ) ∩ q ( | div Kq − ( U ) ( q ∗ ( f ′ )) | )= q ( | div Kq − ( U ) ( q ∗ ( f )) | ∩ | div Kq − ( U ) ( q ∗ ( f ′ )) | )= q ( V q − ( U ) ( q ∗ ( f )) ∩ V q − ( U ) ( q ∗ ( f ′ ))) ⊆ q ( V q − ( U ) ( q ∗ ( f + f ′ ))) = | div K,U ( f + f ′ ) | . (cid:3) Proof of Theorem 5.
First, let R = R ( X ) be the Cox ring of X . Then (i), (ii) and(iii) follow from Theorem 1. For assertion (iv) we additionally suppose that X hasan affine cover by complements of Weil divisors. Let p be a K -prime divisor. By K -factoriality we have p = h f i with some K -prime f ∈ R . Then Y := div K,X ( f ) iscontained in some affine open set U = X \ | D | with some effective Weil divisor D .Let O X ( L ) map onto R . Then there exists g ∈ O X ( L )( X ) with D = div L,X ( g ) =div K,X ( π X ( g )). Let h := π X ( g ). Since div K,X ( f ) (cid:2) div K,X ( h ), f does not divide ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 21 h which means h / ∈ p . Hence, R p = ( R h ) p and the second ring has units in every K -degree by Theorem 1.10 (iii) and Proposition 1.13 (iv).Now, let K be finitely generated, and R an algebraic Cox ring, i.e. a K -gradedring satisfying conditions (i), (ii) and (iv). We claim that there is a set of K -primes f , . . . , f r such that h deg K ( f j ); j = k i = K for every k = 1 , . . . , r . Since K is finitely generated, there are pairwise non-associated K -primes f , . . . , f m with h deg K ( f ) , . . . , deg K ( f m ) i = K . The localization R h f i has units in every degree by(iv), so there are fractions g /h , . . . , g n /h n , where none of the g j , h j are divisible by f , whose degrees together generate K . Decomposing gives f m +1 , . . . , f t such that f , . . . , f t are pairwise non-associated K -primes and h deg K ( f ) , . . . , deg K ( f t ) i = K .Proceeding in this way for k = 2 , . . . , m , we arrive at a set f , . . . , f r with therequested properties.For j = 1 , . . . , r let R j be the localization by the product of all f k with k = j . Let b X be the K -graded scheme with affine charts b X j := Spec K ( R j ). Then by choice of f , . . . , f r all X j = Spec(( R j ) ) contain X ′ = Spec(( R f ··· f r ) ) as a principal opensubset and thus glue to a scheme X . The maps b X j → X j glue to a good quotient q : b X → X by K .We verify that R is the Cox ring of X by showing that q is a graded characteristicspace. b X is a K -Krull scheme because every R j is a K -Krull ring (they are even K -factorial). Each b X j contains all K -prime divisors of X := Spec K ( R ) exceptthe K -principal divisors of those f k with k = j . Thus, b X contains every K -primedivisor of X , which implies O ( b X ) = R and O ( b X ) + , ∗ = O ( b X ) ∗ by (ii). Moreover, K K ( b X ) = K K ( b X j ) = Q + ( R ) and hence Cl K ( b X ) = Cl( R ) = 0 by (i).By (iv), each R j has units of every K -degree. Firstly, this yields Q + ( R j ) = Q (( R j ) ) and deg K ( R j ) = K . There q ∗ : K → ( K K ) is an isomorphism anddeg K ( K + K ) = K . Secondly, there are bijections respecting sums, products, intersec-tions and inclusions between the K -graded ideals of R j and the ideals of ( R j ) . Inparticular, there is an isomorphism WDiv K ( b X j ) → WDiv( X j ) , b Y g ( b Y ) and forevery K -prime divisor b Y ∈ b X j the generator of the maximal ideal of O X j ,q ( b Y ) alsogenerates the K -maximal ideal of O b X j , b Y . Consequently, the induced isomorphism α : WDiv K ( b X ) → WDiv( X ) satisfies α ◦ div K ◦ q ∗ = div. Thus, Theorem 4 givesthe assertion. (cid:3) Remark 5.2.
Conditions (i), (ii) and (iv) irredundantly characterize Cox ringsof Krull schemes with affine cover by divisor complements and finitely generatedclass group. Indeed, [5] offers examples of rings satisfying (i) and (ii) but not (iv).Also, one may take any K -graded Cox ring R and extend the grading trivially to a K ⊕ Z -grading. The monoid R + stays the same and its units remain in degree 0,but deg K ( R + ) = K ⊕ = K ⊕ Z , so the units of R p cannot attain degrees outsideof K ⊕ A ) is not free and φ := id Div ( A ) is chosen. Examples with (ii) and (iv) but not(i) are also obtained from Example 1.9 whenever Cl( A ) is free and φ is the inclusionof a subgroup K ⊂ Div ( A ) which maps injectively but not surjectively to Cl( A ).6. graded schemes and diagonalizable actions The defining algebraic data of a graded scheme ( X, O X ) also define a scheme:By equipping each O X ( U ) with the trivial 0-grading, one obtains a 0-graded quasi-coherent O X -algebra O (0) X whose 0-graded relative spectrum X (0) := Spec X ( O (0) X )is a scheme. The canonical affine morphism X (0) → X which on affine charts isgiven as Spec( R ) → Spec K ( R ) , p
7→ h p ∩ R + i is surjective. In this section, let K be an algebraically closed field. We will show that thefunctor f : X X (0) induces an equivalence between the category A of gradedreduced schemes X of finite type over K with finitely generated grading groups gr ( X ) and the category B of prevarieties over K with quasi-torus actions admittingaffine invariant covers. Definition 6.1.
Let Z be a prevariety over K with the action of a diagonaliz-able group H := Spec max ( K [ K ]). Then Z has an induced H -invariant topology Ω Z,H consisting of the H -invariant Zariski open sets. The K -graded sheaf of ringsobtained by restricting O Z to Ω Z,H is denoted O Z,H and called the H -invariantstructure sheaf .If Z is affine, then the Ω Z,H -closed (and Ω
Z,H -irreducible) subsets are preciselythe Zariski closed sets with K -homogeneous ( K -prime) vanishing ideal. Equiva-lently, a subset of Z is Ω Z,H -closed and Ω
Z,H -irreducible if it is Ω Z -closed and itsΩ Z -irreducible components are permuted by H . Proposition 6.2.
Let t denote the functor sending a ringed space to its space ofclosed irreducible subsets with induced structure sheaf. Let g be the equivalence fromschemes of finite type over K to prevarieties over K . Then we have an equivalenceof categories A −→ B r : X g ( f ( X )) = Spec max ,X ( O (0) X ) t ( Z, Ω Z,H , O Z,H ) ←− [ [ H × Z → Z ] : s Here r ( X ) comes with the action by Spec max ( K [ gr ( X )]) which is induced on affinecharts by the map R → K [ gr ( X )] ⊗ K R, f w χ w ⊗ f w . Remark 6.3.
Let Z be an affine H -variety. Then H -orbits correspond naturallyto K -prime ideals of R := O ( Z ) of the form h m ∩ R + i where m is a maximal ideal of R . Let X := Spec K ( R ) and for a point z ∈ Z set S z := deg K ( R + \ I ( Hz )). The set V X ( I ( Hz )) of K -prime ideals containing I ( Hz ) fits into the known correspondence(e.g. [17, Prop. 3.8]) between the orbits contained in Hz ∼ = Spec max ( K [ S z ]) andthe faces of S z in the following way:orbits( Hz ) ←→ V X ( I ( Hz )) ←→ faces( S z ) Hz I ( Hz ) , p deg K ( R + \ p ) O p ←− [ p , p τ ←− [ τ where O p := V Z ( p ) \ S p = q ∈ V X ( p ) V Z ( q ) and p τ := I ( Hz ) + P w ∈ S z \ τ R w . The abovebijections are order-reversing, where the order on orbits is defined as Hz < Hz : ⇔ Hz ⊆ Hz and the other sets are ordered by inclusion.For a Ω Z,H -closed-irreducible subset Y ⊆ Z the stalk of O Z,H at Y is defined as( O Z,H ) Y := lim −→ U ∈ Ω Z,H U ∩ Y = ∅ O Z ( U ) . It coincides with the stalk of O s ( Z ) at the point Y ∈ s ( Z ). If Z is Ω Z,H -irreduciblethen we denote by K H the constant sheaf assigning the stalk at Z . Remark 6.4. Ω Z,H has a basis of affine H -invariant open sets if and only if Z ∈ B (e.g. Z allows a good quotient by H ). In this case, the following statements hold:(i) The stalk ( O Z,H ) Y at Y coincides with the graded localization O ( U ) I ( Y ) for every affine invariant open set U meeting Y . In particular, if Z isΩ Z,H -irreducible, then K H is a K -simple sheaf. ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 23 (ii) The generic isotropy group of an Ω
Z,H -closed Ω
Z,H -irreducible set Y isSpec max ( K [ K/ deg K (( O Z,H ) + , ∗ Y )]). In particular, H acts freely on a bigopen subset of Z if and only if H acts freely on non-empty open subsets ofall H -prime divisors, i.e. if and only if the stalks ( O Z,H ) Y at all H -primedivisors have units in every degree.The graded schemes in A are of finite type, in particular K -noetherian, hencethey have a cover by K -spectra of K -Krull rings if and only if they are K -normal , i.e.they have a cover by K -spectra of K -normal rings. Correspondingly, a H -prevariety Z ∈ B is called H -normal if and only if the sections of O Z,H over affine invariantsubsets are X ( H )-normal. A H -prime divisor is a Ω Z,H -closed Ω
Z,H -irreduciblesubset Y which is maximal among the proper subsets of Z with these properties.An equivalent condition is that Y is closed and the Ω Z -irreducible components of Y are 1-codimensional and are permuted by H , compare [3, Sect. I.6.4]. Each H -primedivisor Y on a H -normal prevariety Z defines a discrete value sheaf ı Y ( Z ) on Ω Z,H ,which takes values Z if U intersects Y and 0 otherwise, and a discrete K -valuation ν Y : K + H → ı Y ( Z ). These define the K -Krull sheaf O Z,H = T Y ( K H ) ν Y ⊆ K H .Their sum defines a morphismdiv H := X Y ν Y : K + H → WDiv H := M Y ı Y ( Z )to the presheaf of H -Weil divisors. Image and cokernel presheaf are the H -principaldivisors PDiv H resp. the H -class group Cl H . In this terminology, Theorem 4translates into the following characterization of characteristic spaces Spec Z ( R ) → Z of Cox sheaves of finite type. Theorem 6.5.
Let q : b Z → Z be a H -invariant morphism of prevarieties. Then Z is normal and q is a characteristic space if and only if the following hold: (i) b Z is H -normal with deg K ( K H ( b Z ) + ) = K , (ii) q is a good quotient and induces a commutative diagram of presheaves K ∗ div / / ∼ = q ∗ (cid:15) (cid:15) WDiv( q ∗ K H ) ∗ q ∗ div H / / q ∗ WDiv H b Y q ( b Y ) ∼ = O O (iii) Cl H ( b Z ) = 0 , and O ( b Z ) + , ∗ = O ( b Z ) ∗ .If b Z = Spec Z ( R ) with a Cox sheaf R then with div K := P Y µ Y the followingcommutative diagram extends the diagram of (ii): S + div K / / / / ∼ = q ∗ (cid:15) (cid:15) WDiv q ∗ K + H q ∗ div H / / / / q ∗ WDiv H b Y q ( b Y ) ∼ = O O Each prime divisor Y is the image q ( b Y ) of a uniqe H -prime divisor b Y . If H b z ⊆ b Z isthe unique closed orbit in q − ( z ) , then H b z ⊆ b Y if and only if z ∈ Y . In particular, ( O b X,H ) H b z = R z . This result only required those properties of normal prevarieties which they sharewith Krull schemes. Using the fact that X reg is big in X and WDiv( X reg ) =CaDiv( X reg ) combined with noetherianity yields additional properties of character-istic spaces, e.g. irreducibility and normality, and the property that q − ( X reg ) is abig subset on which H acts freely and q is geometric, see [3, Sect. I.5, I.6] for proofsand further properties of characteristic spaces. Remark 6.6.
Recall that a lattice is a finitely generated free abelian group. By aseparated toric graded scheme over K we mean a quasi-compact separated gradedscheme X of finite type over K such that(i) the grading group M = gr ( X ) is a lattice,(ii) X is M -normal,(iii) X contains Spec M ( K [ M ]) as an open subset,(iv) X is effectively graded , i.e. h deg M ( O ( U )) i = M for all affine open U ⊆ X . Z ∈ B is a (separated) toric variety if and only if X = s ( Z ) ∈ A is a separatedtoric graded scheme. If Σ is the fan in N = M ∗ describing Z where M = gr ( X ),then Ω Z,T is finite and its basis consists of the affine invariant charts { Z σ } σ ∈ Σ .The Ω Z,T -irreducible Ω
Z,T -closed subsets are the orbit closures { V ( σ ) } σ ∈ Σ . For σ ∈ Σ let I σ E K [ M ∩ σ ∨ ] be the vanishing ideal of the closed orbit of Z σ . Then O X,I σ = K [ M ∩ σ ∨ ] I σ = K [ M ∩ σ ∨ ] and there is a natural bijectionΣ −→ Xσ I σ deg M ( O X,p ) ∨ ←− [ p Furthermore, applying Example 4.3 to each monoid σ ∨ ∩ M and its monoid alge-bra K [ σ ∨ ∩ M ], we see that X is canonically homeomorphic to the F -scheme A Σ obtained by gluing the spectra of ( σ ∨ ∩ M ) ⊔ {∞} . For a cone σ ∈ Σ we denote by p σ the closed point ( σ ∨ ∩ M ) ⊔ {∞} \ ( σ ⊥ ∩ M ) of Spec(( σ ∨ ∩ M ) ⊔ {∞} ). Thecanonical bijection between A Σ and Σ is Σ −→ A Σ σ p σ ( O A Σ ,p \ {∞} ) ∨ ←− [ p A different kind of connection between the categories of F -schemes of finite typeand toric varieties is established in [13]. References [1] D.F. Anderson: Graded Krull domains. Comm. Algebra 7 (1979), no. 1, 79–106.[2] I.V. Arzhantsev: On the factoriality of Cox rings. Mat. Zametki 85 (2009), no. 5, 643–651(Russian); English transl.: Math. Notes 85 (2009), no. 5, 623–629.[3] I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface: Cox rings, arXiv:1003.4229, see also theauthors’ webpages.[4] H. B¨aker: Good quotients of Mori dream spaces, Proc. Amer. Math. Soc. 139 (2011), 3135–3139.[5] B. Bechtold: Factorially graded rings and Cox rings. J. Algebra 369 (2012), 351–359.[6] F. Berchtold, J. Hausen: Homogeneous coordinates for algebraic varieties. J. Algebra 266(2003), no. 2, 636–670.[7] F. Berchtold, J. Hausen: Cox rings and combinatorics. Trans. Amer. Math. Soc. 359 (2007),no. 3, 1205–1252.[8] M. Brion: The total coordinate ring of a wonderful variety. J. Algebra 313 (2007), no. 1, 61–99.[9] A. Canonaco: The Beilinson Complex and Canonical Rings of Irregular Surfaces. Memoirs ofthe Amer. Math. Soc. 183 (2006), no. 862.[10] A.-M. Castravet, J. Tevelev: M ,n is not a Mori Dream Space, arXiv:1311.7673.[11] D.A. Cox: The homogeneous coordinate ring of a toric variety. J. Alg. Geom. 4 (1995), no.1, 17–50.[12] A. Deitmar: Congruence schemes. International Journal of Math. 24 (2013), no. 2, 46 p.[13] A. Deitmar: F1-schemes and toric varieties. Beitr. Alg. Geom. 49 (2008), no. 2, 517–525.[14] E.J. Elizondo, K. Kurano, K. Watanabe: The total coordinate ring of a normal projectivevariety. J. Algebra 276 (2004), no. 2, 625–637.[15] R.M. Fossum: The Divisor Class of a Krull Domain. Springer (1973).[16] R. Hartshorne: Algebraic Geometry. New York: Springer-Verlag, 1977[17] J. Hausen: Cox rings and combinatorics II. Mosc. Math. J. 8 (2008), no. 4, 711–757.[18] J. Hausen, H. S¨uß: The Cox ring of an algebraic variety with torus action. Advances inMathematics 225 (2010), 977–1012. ALUATIVE AND GEOMETRIC CHARACTERIZATIONS OF COX SHEAVES 25 [19] Y. Hu, S. Keel: Mori dream spaces and GIT. Michigan Math. J. 48 (2000), 331–348.[20] Y. Kamoi: Noetherian Rings Graded by an Abelian Group. Tokyo J. Math. 18 (1995), no. 1,31–48.[21] M.D. Larsen, P.J. McCarthy: Multiplicative theory of ideals. Academic Press, New York,1971.[22] H. Lee, M. Orzech: Brauer groups, class groups and maximal orders for a Krull scheme.Canad. J. Math. 34 (1982), 996–1010.[23] M. Perling: Toric Varieties as Spectra of Homogeneous Prime Ideals. Geom. Dedicata 127(2007), 121–129.[24] F. Rohrer: Coarsenings, injectives and Hom functors. Rev. Roumaine Math. Pures Appl. 57(2012), 275–287.[25] F. Rohrer: Graded integral closures. Beitr. Algebra Geom. (2013), 1–18.[26] B. Sturmfels, M. Velasco: Blow-ups of P n − at n points and spinor varieties. J. Commut.Algebra Volume 2 (2010), no. 2, 223–244.[27] M. Temkin: On local properties of non-Archimedean spaces II, Isr. J. of Math. 140 (2004),1–27. Mathematisches Institut, Universit¨at T¨ubingen, Auf der Morgenstelle 10, 72076T¨ubingen, Germany
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