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Subspaces of Non-commutative Spaces

This paper examines various kinds of subspaces of the non-commutative spaces that are modelled on quasi-projective commutative schemes. It is shown how intersections and unions of weakly closed subspaces, closed subspaces, their weakly open and open complements, behave. Intersections with effective divisors are also examined, and so are closed points.


Introduction
This paper examines non-commutative analogues of some of the elementary ideas of topology and (algebraic) geometry. Our wish is to have a language for non-commutative algebraic geometry that is geometric and topological. This will, eventually, allow us to use our geometric intuition in situations that at present are viewed almost exclusively from an algebraic perspective. One can already see hints of this goal in Gabriel's thesis [5]. More recently, Rosenberg and Van den Bergh have given new life to this program. We further extend their ideas by examining the interactions between the following basic objects: weakly closed and weakly open subspaces, closed subspaces, open subspaces, closed points, and effective divisors.
We follow Rosenberg [17] [18] and Van den Bergh [23] in taking a Grothendieck category as our basic non-commutative geometric object. Thus a non-commutative space X is a Grothendieck category ModX. The idea for the notation X = ModX is Van den Bergh's; we also use his notion of X-X-bimodules, and denote the unit object in this monoidal category by o X .
The standard commutative example of a space is the category of quasi-coherent sheaves on a quasi-separated, quasi-compact scheme. If X is such a scheme we will speak of it as a space with the understanding that ModX is QcohX. The two non-commutative models are ModR, the category of right modules over a ring, and ProjA, the non-commutative projective spaces having a (not necessarily commutative) graded ring A as homogeneous coordinate ring (Definition 2.2).
The notion of a weakly closed subspace of a non-commutative space first appeared in Gabriel's thesis [5, page 395]-he calls a full subcategory of an abelian category closed if it is closed under subquotients and direct limits. We will modify his terminology, and say that a full subcategory of a Grothendieck category is weakly closed if it satisfies these conditions. Closure under subquotients ensures that the inclusion functor is exact, and closure under direct limits ensures that it has a right adjoint. If the inclusion functor also has a left adjoint we say that the subcategory is closed. This modification of Gabriel's terminology makes it compatible with the language of algebraic geometry: the closed subcategories (in our sense) of ModR are in bijection with the two-sided ideals of R; and, every closed subscheme of a scheme X determines a closed subspace of QcohX. If X is a quasiprojective scheme over a field then the closed subspaces are the same things as the closed subschemes (Theorem 4.1). Our terminology is also compatible with Van den Bergh's notion of weak ideals; weakly closed subspaces of a space X correspond to weak ideals in o X , the identity functor on ModX.
Proposition 4.7 shows that for noetherian non-commutative spaces that are close to being commutative there is a bijection between what we call the prime subspaces of X and the indecomposable injectives in ModX. This is extends Matlis' result for commutative notherian rings, and Gabriel and Cauchon's extension of Matlis' result to rings satisfying Gabriel's condition (H) (fully bounded right noetherian rings).
We define the complement to a weakly closed subspace in the obvious way (Definition 6.4), and define weakly open subspaces (Definition 2.5) in such a way that they are precisely the complements to the weakly closed subspaces when the ambient space is locally noetherian. The intersection of an arbitrary collection, and the union of a finite collection, of weakly closed subspaces (Definitions 3.2 and 3.4) have already been defined by Rosenberg; these are again weakly closed subspaces. Using the fact that a weakly open subspace is a complement to a weakly closed subspace allows us to define the intersection and union of weakly open subspaces. For the most part these ideas interact with each other in reasonable ways-for example, see Proposition 3.3, Lemmas 6.10, 6.16, 6.18, and 6.20, Corollary 7.4, and Propositions 7.1, 7.9, and 7.10. Furthermore, Proposition 7.5 shows that if U and V are weakly open subspaces of a locally noetherian space X, then U ∩ V is the fiber product U × X V in the category of non-commutative spaces with maps being the morphisms. Despite these positive results we do not know whether W ∩ (Y ∪ Z) is equal to (W ∩ Y ) ∪ (W ∩ Z) when W , Y , and Z, are weakly closed subspaces; the analogous formula for weakly open subspaces is valid. Another problem is that if W and Z are weakly closed subspaces of X such that W ∩ Z = φ, then Z need not be contained in X\W ; we determine exactly when Z is a weakly closed subspace of X\W . The fact that Z need not be contained in X\W is an essential feature of the non-commutativity of X. It reflects the fact that there can be non-split extensions between non-isomorphic simple modules.
In section 5 we define closed points of a space. A test case for our definition is the following. Let A be a connected graded k-algebra. Artin-Tate-Van den Bergh [2] have defined the notion of a point module for A with the idea behind their definition being that point modules for A should play the role of points on the non-commutative projective space having A as a homogeneous coordinate ring. However, to date there has been no definition of "point" in non-commutative geometry. In section 5 we show that point modules over A give closed points of the associated projective space ProjA.
To motivate the definition recall that a closed point p on a scheme X corresponds to a simple object O p = O X /m p in QcohX, where m p is the sheaf of ideals vanishing at p. This sets up a bijection between closed points and simple objects in QcohX. Since the geometric objects in non-commutative geometry are Grothendieck categories we define a closed point of a space X to be a certain kind of full subcategory of ModX. That subcategory is required to be closed, in the sense described above, and to consist of all direct sums of a simple X-module, say O p . The requirement that the subcategory be closed means that not all simple X-modules correspond to closed points. In particular, an infinite dimensional simple module over a finitely generated k-algebra does not correspond to a closed point; such simple modules behave like higher dimensional geometric objects (see the remarks after Proposition 5.8).
Sometimes even finite dimensional simple modules behave like higher dimensional geometric objects. The D-module approach to the representation theory of semisimple Lie algebras, and the dimension of the characteristic variety, yields a dimension function that gives finite dimensional representations dimension greater than zero, thus reflecting the fact that such modules do not always behave like points. The behavior of the higher Ext-groups of finite dimensional simples also indicates that they behave like higher dimensional geometric objects.
A further example of this phenomenon occurs when α :X → X is Van den Bergh's blowup for a non-commutative surface X. Sometimes X can have a module of Krull dimension one whose strict transform is a simpleX-module, say S. That simple module gives a closed point onX, but with regard to its intersection properties (defined in terms of Ext-groups) that simpleX-module behaves more like a curve than a point (see [11]). This is reinforced by the fact that α * S is the original module of Krull dimension one-in particular, α does not send this closed point ofX to a closed point of X. For many purposes it is better to think of S as a geometric object of dimension one.
In Section 8 we examine effective divisors (as defined by Van den Bergh) on a non-commutative space. These are analogues of effective Cartier divisors; an effective divisor is defined in terms of an invertible ideal, say o X (−Y ), in o X . That ideal cuts out a closed subspace Y on X. If W is a weakly closed subspace of X, we give precise conditions under which W ∩ Y is an effective divisor on W . We also show that if Y is ample, then W ∩ Y = φ for all W of positive dimension. We give conditions under which the set of closed points in X is the disjoint union of the sets of closed points in Y and X\Y . Effective divisors behave better than arbitrary closed (or weakly closed) subspaces with respect to intersection and containment of other subspaces.
Acknowledgements. We thank M. Artin, P. Jørgensen, S. Kovacs, M. Van den Bergh, and J. Zhang for helpful conversations. We thank C. Pappacena for asking questions that prompted Propositions 4.6 and 4.7. This work was begun while the author was visiting the University of Copenhagen during the summers of 1998 and 1999, and was continued during the workshop on Noncommutative Algebra at the Mathematical Sciences Research Institute at Berkeley in 2000. The author is grateful to both institutions for their hospitality and financial support.

Preliminaries
Throughout we work over a fixed commutative base ring k. All categories are k-linear and functors between them are k-linear.
Definition 2.1. A non-commutative space X is a Grothendieck category ModX. Objects in ModX are called X-modules. We say X is locally noetherian if ModX is locally noetherian (that is, if it has a set of noetherian generators).
The empty space φ is defined by declaring Modφ to be the zero category; that is, the abelian category having only one object and one morphism.
We write ModR for the category of right modules over a ring R.
We say X is affine if ModX has a progenerator, and in this case any ring R for which ModX is equivalent to ModR is called a coordinate ring of X. It is easy to see that an affine space is locally noetherian if and only if one, and hence all, of its coordinate rings is right noetherian.
If (X, O X ) is a scheme then the category ModO X of all sheaves of O X -modules is a Grothendieck category. If X is quasi-compact and quasi-separated (for example, if X is a noetherian scheme) the full subcategory of ModO X consisting of the quasicoherent O X -modules is a Grothendieck category [6, page 186]. We denote this category by QcohX. Whenever X is a quasi-compact and quasi-separated scheme we will speak of it as a space in our sense with the tacit understanding that ModX is QcohX. For example, Spec Z will denote the space whose module category is ModZ.
The non-commutative analogues of projective schemes were first studied in [3] and [25].
Definition 2.2. Let k be a field. Let A be an N-graded k-algebra such that dim k A n < ∞ for all n. Define GrModA to be the category of Z-graded A-modules with morphisms the A-module homomorphisms of degree zero. We write FdimA for the full subcategory of direct limits of finite dimensional modules. We define the quotient category TailsA = GrModA/FdimA, and denote by π and ω the quotient functor and its right adjoint. The projective space X with homogeneous coordinate ring A is defined by ModX := TailsA.
We define its structure module, O X , to be πA, and we define This is an enriched space in the sense of [23, Section 3.6].
The cohomology groups of a module F over the enriched space X are defined in [25] and [3] to be H q (X, F ) := Ext q X (O X , F ).
The idea for the next definition is due to Rosenberg [ induces an affine map Spec k → Spec(k × k) of spaces that is not induced by a morphism between these schemes.
Definition 2.4. Let X be a non-commutative space. A weakly closed subspace, say W , of X is a full subcategory ModW of ModX that is closed under subquotients (and therefore under isomorphisms), and for which the inclusion functor α * : ModW → ModX has a right adjoint, denoted by α ! . We write α : W → X for the weak map corresponding to α * . A weakly closed subspace W is closed if α * also has a left adjoint, which is denoted by α * .
For example, X and φ are closed subspaces of X.
The inclusion of a closed subspace is an affine map. Let W be a weakly closed subspace of X and let α : W → X be the inclusion. Then ModW is a Grothendieck category and is locally noetherian if ModX is. Because ModW is closed under subquotients, α * is an exact functor. Because α ! has an exact left adjoint it sends injective X-modules to injective W -modules. The hypotheses imply that α ! α * ∼ = id W when W is a weakly closed, and α * α * ∼ = id W when W is closed.
When Z is a closed subscheme of a scheme X one often confuses the structure sheaf O Z which lies in QcohZ with its direct image sheaf which lies in QcohX. We will do something similar when W is a weakly closed subspace of a space X. If α : W → X is the inclusion, we will confuse the trivial W -W -bimodule o W with the weak X-X-bimodule corresponding to the left exact functor α * α ! . Thus, we will speak of the map o X → o W of weak X-X-bimodules, the kernel of which is the weak ideal defining W .
Definition 2.5. Let X be a non-commutative space. A weakly open subspace, say U , of X is a full subcategory ModU of ModX that is closed under isomorphism and is such that the inclusion functor j * : ModU → ModX has an exact left adjoint j * . If V is another weakly open subspace of X, we say that U is contained in V , and This section shows that there are reasonable notions of intersection and union for weakly closed subspaces.
The notion of a weakly closed subspace makes sense for schemes. For example, Spec Z has a weakly closed subspace W defined by declaring ModW to consist of the torsion abelian groups. There is a proper descending chain of weakly closed subspaces W = W 0 ⊃ W 1 ⊃ W 2 ⊃ . . . defined by declaring ModW i to be all torsion groups that have no torsion with respect to the first i positive primes.
If X is a scheme such that QcohX is a Grothendieck category, and Z is a closed subscheme of X, then the full subcategory of QcohX consisting of the modules whose support lies in Z is closed under subquotients and direct limits so is the module category of a weakly closed subspace of X; this subspace is rarely closed.
Let X be a space and M an X-module. Let W be the smallest weakly closed subspace such that M ∈ ModW . Then ModW consists of all subquotients of all direct sums of copies of M . This subcategory is closed under subquotients, so the inclusion functor is exact; it is closed under direct sums because direct sums are exact in ModX; it is therefore closed under direct limits, so the inclusion functor has a right adjoint.
The following is a triviality.
Lemma 3.1. If Z is a weakly closed subspace of X and W is a weakly closed subspace of Z, then W is a weakly closed subspace of X.
Definition 3.2. Let Z and W be weakly closed subspaces of X. We say that Z lies on W or is contained in W , if ModZ is contained in ModW . In this case we write Z ⊂ W . Their intersection W ∩ Z is defined by declaring that That is, ModW ∩ Z is the full subcategory of ModX consisting of those X modules that belong to both ModW and ModZ. Proof. Let α : W → X and β : Z → X be the inclusions. Let γ * : ModW ∩ Z → ModZ and δ * : ModW ∩ Z → ModW be the inclusion functors.
(1) To see that W is a space one checks that ModW is a Grothendieck category. By, for example, [23,Proposition 3.4.3], ModW is a cocomplete abelian category, and the inclusion α * is exact and commutes with direct limits. Since direct limits in ModW coincide with direct limits in ModX, ModW has exact direct limits.
Since ModX has a small set of generators, the collection of all submodules of any X-module is a small set, and hence so is the collection of all its subquotients. If {M i | i ∈ I} is a small set of generators for ModX, then the set of all subquotients of all M i that belong to ModW is small. This is a set of generators for ModW : if f : L → N is a non-zero morphism in ModW , then there is a morphism g : M i → L such that f g = 0; since L ∈ ModW g induces a morphism g ′ : M ′ → L from some subquotient M ′ of M i belonging to ModW such that f g ′ = 0.
(2) Since ModW and ModZ are both closed under subquotients and direct limits in ModX, so is ModW ∩ Z. It follows that ModW ∩ Z is an abelian category and that the inclusion of it in ModX is an exact functor commuting with direct limits. It therefore has a right adjoint. This proves (2).
(3) To see that W ∩ Z is closed in W it suffices to show that δ * commutes with products; that is, that ModW ∩ Z is closed under products in ModW . Let M λ be a family in ModZ ∩ W , and suppose they have a product in ModW . That product is α ! ( M λ ), where M λ is their product in ModX. But ModZ is closed under products in ModX, so M λ belongs to ModZ. Therefore its submodule α ! ( M λ ) is a Z-module, and hence is in ModW ∩ Z. This proves that δ * has a left adjoint δ * , whence W ∩ Z is closed in W .
We will now show that Since γ * δ * is a left adjoint to δ * γ ! and β * α * is a left adjoint to α ! β * , it suffices to show that δ * γ ! ∼ = α ! β * . If M is a Z-module, δ * γ ! M is the largest submodule of M that is in ModZ ∩ ModW ; since every submodule of M is in ModZ, δ * γ ! M is the largest submodule of M that is in ModW ; but that submodule is α ! β * M . It follows that and (3-2) follows from this. (4) If W and Z are closed subspaces of X, then ModW and ModZ are closed under products in ModX. It follows that ModW ∩ Z is closed under products in X, whence W ∩ Z is closed.
(5) and (6) follow from (2) and (3) Since Z is both weakly closed and weakly open in X, β * has an exact left adjoint β * . In particular, Z is closed in X. Hence by (3), W ∩ Z is closed in W , and γ * δ * ∼ = β * α * . The right-hand side of this is a composition of exact functors, so γ * δ * is exact. It follows that δ * is exact because γ is a weakly closed immersion.
Remarks. 1. The following two observations are easily checked. If W is a weakly closed subspace of Z, and Z is a weakly closed subspace of X, then W is a weakly closed subspace of X. If W is a closed subspace of Z, and Z is a closed subspace of X, then W is a closed subspace of X.
2. The binary operation ∩ on weakly closed subspaces is idempotent, commutative, and associative. One has W ∩ φ = φ and W ∩ X = W .

3.
One may define the intersection of an arbitrary collection of weakly closed subspaces W i by Mod(∩W i ) := ∩ModW i . It is clear that this is closed in ModX under subquotients and direct limits because each ModW i is. Hence ∩ i W i is a weakly closed subspace of X (cf. [17, Lemma 6.2.2, Chapter 3]).
Definition 3.4. Let W i , i ∈ I, be a finite collection of weakly closed subspaces of X. Their union, denoted ∪ i W i , is the smallest weakly closed closed subspace of X that contains all the W i . It is the intersection of all the weakly closed subspaces that contain all the W i .
Remarks. 1. We could first have defined the union of a pair of weakly closed subspaces, thus making ∪ a binary operation. One sees that ∪ is idempotent, commutative, and associative. Associativity could then be used to define arbitrary finite unions; this definition would agree with the one in Definition 3.4. It is clear that W ∪ φ = W and W ∪ X = X.
2. If W and Z are weakly closed, then ModW ∪ Z is the full subcategory consisting of all subquotients of modules of the form M ⊕ N where M ∈ ModW and N ∈ ModZ. Obviously this subcategory is closed under subquotients; it is closed under direct sums because ModW and ModZ are, and because direct sums are exact in ModX; it is therefore closed under direct limits, so is weakly closed; it contains ModW and ModZ, and any weakly closed subcategory containing ModW and ModZ must contain all modules of the form M ⊕ N ; hence this category equals ModW ∪ Z.
3. If X is the affine space with coordinate ring R, and W and Z are the closed subspaces cut out the two-sided ideals I and J, then W ∪ Z is the closed subspace cut out by I ∩ J. To see this first observe that ModR/I ∩ J is weakly closed and contains both ModR/I and ModR/J; secondly, any weakly closed subspace that contains both W and Z must contain R/I ⊕ R/J so must contain its submodule R/I ∩ J, and therefore must contain ModR/I ∩ J.
4. Let W and Z be weakly closed subspaces of X. Their Gabriel product W • Z is the weakly closed subspace defined by declaring ModW •Z to be the full subcategory of ModX consisting of the modules M for which there is an exact sequence 0 → L → M → N → 0 with N ∈ ModW and L ∈ ModZ [5, p. 395]. Van den Bergh has shown that W • Z is weakly closed, and is closed if both Z and W are [23,Prop. 3.4.5]. Since W • Z contains both W and Z, it follows that In the situation of the previous remark, Mod(W • Z ∩ Z • W ) = ModR/IJ + JI, so in general the inclusion in (3-4) is strict. Example 6.11 exhibits closed points p and q in a an affine space such that p ∪ q = p • q = q • p. 5. If W, Y and Z are weakly closed subspaces of X, we do not know whether W ∩(Y ∪Z) = (W ∩Y )∪(W ∩Z). The analogous equality for weakly open subspaces does hold (Lemma 6.16).

Closed subspaces
The first evidence that the notion of a closed subspace in Definition 2.4 is appropriate for non-commutative geometry is the following result of Rosenberg [17,Proposition 6.4.1,p.127]. If X is an affine space, say X = ModR, then each two-sided ideal I cuts out a closed subspace, namely ModR/I, and this sets up a bijection between the closed subspaces of X and the two-sided ideals of R (see also [19,Proposition 2.3]). We call the closed subspace ModR/I the zero locus of I. Thus, if X is an affine scheme, its closed subschemes are in natural bijection with the closed subspaces. We prefer to say that the closed subspaces are the same things as the closed subschemes. Further evidence is provided by [22,Theorem 3.2] which shows that a two-sided ideal in an N-graded k-algebra A cuts out a closed subspace of the projective space with homogeneous coordinate ring A. There is further evidence. If X is a scheme such that QcohX is a Grothendieck category (for example, if X is noetherian) and Z is a closed subscheme of X, then QcohZ is a closed subspace of QcohX in the sense of Definition 2.4. Indeed, this is the motivation for the definition of a closed subspace.
The main result in this section (Theorem 4.1 shows that if X is a quasi-projective scheme over a field k, then the closed subschemes give all the closed subspaces. We also prove a non-commutative geometric version of Matlis' result on the bijection between prime ideals in a commutative noetherian ring and indecomposable injectives over that ring (Proposition 4.7).
Remarks. 1. An interesting and ubiquitous non-commutative space is the graded line L 1 , defined by ModL 1 = GrModk[x] where k is a field and deg x = 1. We associate to every subset D ⊂ Z a closed subspace Z D of L 1 defined by declaring ModZ D to be the full subcategory consisting of those modules M such that M n = 0 for all n / ∈ D. Since there is a proper descending chain of subsets D, L 1 does not have the descending chain condition on closed subspaces. If D consists of n consecutive integers, then Z D is isomorphic to the affine space with coordinate ring the n × n upper triangular matrices over k.
If X is an affine space with coordinate ring the enveloping algebra of a nonabelian solvable Lie algebra over a field k of characteristic zero, then X contains many weakly closed subspaces isomorphic to L 1 [19]. IfX is the blowup of a noncommutative surface X at a closed point p lying on an effective divisor Y such that {O p (nY ) | n ∈ Z} is infinite, then the exceptional curve onX is a weakly closed subspace ofX isomorphic to L 1 [23].
2. The category of right comodules over a coalgebra is a Grothendieck category, so is a space. Let C and D be coalgebras and let X and Y be the spaces defined by ModX = ComodC and ModY = ComodD. If ϕ : C → D is a homomorphism of coalgebras, then the rule (M, ρ) → (M, (id M ⊗ϕ)ρ) gives an exact functor f * : ComodC → ComodD. This is the direct image functor for an affine map f : X → Y .
The right adjoint to f * is f ! = − D C, the co-tensor product. Joyal and Street [12,Proposition 2,p. 454] show that the closed subspaces of Y are in bijection with the subcoalgebras of D. Actually they don't quite prove that, but using the fact that in a locally finite space weakly closed and closed subspaces coincide (Proposition 4.8), their result implies this. The closed points of Y , in the sense of Section 5, are in bijection with the simple subcoalgebras of D.
Theorem 4.1. Let X be a noetherian scheme having an ample line bundle (for example, let X be quasi-projective over Spec k with k a field). Then closed subschemes and closed subspaces of X are the same things.
We will prove the theorem after the next lemma.
Let O X (1) denote an ample line bundle on X. If i : Z → X is the inclusion of a closed subscheme, then the restriction i * (O X (1)) to Z is ample. We will denote it by O Z (1).
Let M be an O X -module. Its annihilator is the kernel of the natural map O X → Hom OX (M, M). This is also the largest ideal I in O X such that MI = 0, where MI is defined as the image of the natural map M ⊗ OX I → M. We denote it by Ann M. If M is coherent, its scheme-theoretic support is the closed subscheme of X cut out by its annihilator, i.e., it is ( Lemma 4.2. Let M be a coherent O X -module, and Z its scheme-theoretic support. If W is a weakly closed subspace of X such that M ∈ ModW , then Z ⊂ W , i.e., QcohZ ⊂ ModW .
Proof. We show first that some finite direct sum of copies of M has a submodule that is an invertible By the preceding paragraph, O Z (−n) belongs to ModW for some n, and hence for all n ≫ 0. Since every coherent O Z -module is a quotient of a suitable finite direct sum of copies of O Z (−n) for some sufficiently large n, cohZ ⊂ ModW . Since ModW is closed under direct limits it follows that QcohZ ⊂ ModW .
Suppose that F ∈ modW , and let Y denote its scheme-theoretic support. By Lemma 4.2, QcohY ⊂ ModW . Since Y is also the scheme theoretic support of F (n), F (n) ∈ ModW for all n ∈ Z. For some n ≫ 0 and some suitably large finite direct sum, there is an epimorphism O X → F (n), hence an epimorphism It follows that F (n) is in QcohZ. Hence F ∈ QcohZ also, and we conclude that ModW = QcohZ as desired.
This result suggests the following definitions.
Definition 4.3. Let X be a locally noetherian space. The support of a noetherian Xmodule M is the smallest closed subspace Z such that M ∈ ModZ. Such a smallest closed subspace exists because an intersection of closed subspaces is closed. A nonzero X-module is prime if all its non-zero submodules have the same support. A non-empty closed subspace Z of X is prime if it is the support of a prime noetherian X-module.
The definition of a prime module emerged in discussions with C. Pappacena and agrees with his definition in [15].
If X has the descending chain condition on closed subspaces, then every non-zero X-module has a prime submodule: simply take a submodule whose support is as small as possible. As remarked at the beginning of this section, the graded line, , does not have the descending chain condition on closed subspaces; it is easy to see that k[x] does not contain a prime submodule.
If X is affine with coordinate ring R, then the support of an R-module is the zero locus of its annihilator because R/ Ann M embeds in a product of copies of M , so is a module over any closed subspace that contains M . A closed subspace of X is prime if and only if it is the zero locus of a prime ideal; thus prime subspaces of X are in bijection with the prime ideals in R.
A non-zero submodule of a prime module is prime. A direct sum of copies of a prime module is prime. An auto-equivalence of ModX sends prime modules to prime modules.
By Lemma 4.2, if X is a noetherian scheme having an ample line bundle, the support as we have just defined it is the same as the scheme-theoretic support.
Proposition 4.4. Let X be a noetherian scheme having an ample line bundle.
1. The (scheme-theoretic) support of a prime module is integral.
Proof. We will use the fact that a closed subscheme Z of X is reduced and irreducible if and only if the product of non-zero ideals of O Z is non-zero.
(1) Let Z denote the support (= scheme-theoretic support) of a prime module M. Without loss of generality we may assume that M is coherent. By the arguments in Lemma 4.2, O Z (−n) embeds in a finite direct sum of copies of M. It follows that O Z (−n), and hence O Z , is prime. Hence, the scheme-theoretic support of every non-zero coherent ideal of O Z is Z. In other words, if I is a non-zero coherent ideal of O Z , then IJ = 0 for all non-zero ideals J of O Z . Hence Z is reduced and irreducible.
(2) If Z is integral, every non-zero ideal of O Z has annihilator equal to zero, so has scheme-theoretic support Z, thus showing that O Z is prime. Conversely, if O Z is prime, then every non-zero ideal of it has support equal to Z so has zero annihilator, thus showing that Z is integral.
The conclusion of Lemma 4.2 captures one of the essential aspects of commutativity: it says that X has a lot of closed subspaces. This is similar to saying that a ring has lots of two-sided ideals. The next definition singles out this property and we think of it as saying that the space is "almost commutative". Definition 4.5. We say that a locally noetherian space X has enough closed subspaces if every noetherian module M has the following property: the smallest weakly closed subspace Z ⊂ X such that M ∈ ModZ is equal to the support of M .
In other words, X has enough closed subspaces if every weakly closed subspace W ⊂ X has the following property: if M is a noetherian W -module, then there is a closed subspace Z of X such that Z ⊂ W and M ∈ ModZ. Lemma 4.2 implies that a quasi-projective scheme has enough closed subspaces.
The next result shows that this notion of "almost commutative" is compatible with an older idea of what it means for a ring to be "almost commutative". Gabriel's condition (H) [5, page 422] is expressed as part (2) of the next result; Gabriel (loc. cit., page 423) showed that (2) implies (3) and Cauchon [4] proved the converse. The equivalent condition (1) is geometric and therefore has meaning for non-affine spaces. For example, if A is a connected graded noetherian algebra that is a finite module over its center, then property (1) is satisfied by the projective space X having A as a homogeneous coordinate ring. Proposition 4.6. Let X be a locally noetherian affine space with coordinate ring R. The following are equivalent: 1. X has enough closed subspaces; Ann(m i ) for some finite set of elements m i ∈ M ; intersection 3. there is a bijection between the prime ideals in R and the indecomposable injective R-modules.
Proof. Gabriel and Cauchon have proved the equivalence of (2) and (3).
(2) ⇒ (1) Let M be a noetherian X-module, and W a weakly closed subspace containing M . Then In the next proof we use the notion of Krull dimension as defined by Gabriel. A noetherian X-module is critical if Kdim M/N < Kdim M for all non-zero submodules N ⊂ M . The definition of Krull dimension is such that every non-zero module has a non-zero critical submodule. A submodule of a critical module is critical. Hence a prime subspace is the support of a critical module, and that module may also be chosen to be prime and noetherian. Proposition 4.7. Let X be a locally noetherian space having the descending chain condition on closed subspaces. If X has enough closed subspaces there is a bijection Proof. Let Z be a prime subspace of X and let M be a critical noetherian prime module having support equal to Z. Since M is critical it is uniform, so its injective envelope, say E, is indecomposable. We will show that the rule Z → E is a welldefined map setting up the claimed bijection.
To show this is well-defined, we must show that E ∼ = E ′ whenever E ′ is an injective envelope of another critical noetherian prime module, say M ′ , whose support is equal to Z.
The hypothesis on X implies that ModZ consists of all subquotients of all direct sums of copies of M . Of course, the same is true with M ′ in place of M . Since each of M and M ′ is a subquotient of a finite direct sum of copies of the other they have the same Krull dimension.
Choose the smallest integer n such that M ′ is isomorphic to L/K where K ⊂ L ⊂ M 1 ⊕ . . . ⊕ M n and each M i ∼ = M . The minimality of n ensures that L ∩ M i is non-zero for all i. If K ∩ M i were non-zero for all i, then the Krull dimension of L/K would be strictly smaller that that of M ; this is not the case so K ∩ M i is zero for some i. Hence L ∩ M i embeds in M ′ . In particular, M and M ′ have a common non-zero submodule, from which we conclude that To see that this map is injective, suppose that Z and Z ′ are prime subspaces for which the corresponding indecomposable injectives, say E and E ′ , are isomorphic. Let M and M ′ be critical noetherian prime modules whose supports are Z and Z ′ respectively. Thus E and E ′ are injective envelopes of M and M ′ respectively. Since E and E ′ are isomorphic and indecomposable, M and M ′ have a common non-zero submodule, say N . However, since M and M ′ are prime they have the same support as N . Thus Z = Z ′ .
To show that the map is surjective, suppose that E is an indecomposable injective. Because X has the descending chain condition on closed subspaces, E has a prime submodule and hence a prime critical noetherian submodule, say M . If Z is the support of M , then the map defined above sends Z to E.
The next result gives another class of spaces which have enough closed subspaces. It applies for example to the category of comodules over a coalgebra that is defined over a field. Proof. Let X be a locally finite space; that is, every X-module is a direct limit of finite length X-modules. Let W be a weakly closed subspace of X. It suffices to show that ModW is closed under products because then the inclusion i * : ModW → ModX will have a left adjoint.
Let {M i | i ∈ I} be a collection of W -modules. By hypothesis, i∈I M i , their product in ModX, is a sum of finite length X-modules. Let N be a finite length submodule of i∈I M i . For each j ∈ I let ρ j : N → M j be the composition of the inclusion N → i∈I M i with the projection i∈I M i → M j . Write K j = ker ρ j . Since ∩ j∈I K j is the kernel of the inclusion N → i∈I M i it is zero; since N has finite length there is a finite subset J ⊂ I such that ∩ j∈J K j = 0. It follows that the composition is monic. Since J is finite, j∈J M j is isomorphic to ⊕ j∈J M j , and therefore belongs to ModW . Hence N belongs to ModW . We have shown that every finite length submodule of i∈I M i belongs to ModW . Since ModW is closed under direct limits we conclude that i∈I M i belongs to ModW .

Closed points
Notation. If D is a division ring, we write Spec D for the space ModD of right D-modules.
Definition 5.1. A closed point of a space X is a closed subspace that is isomorphic to Spec D for some division ring D. We call it a D-rational point of X.
We denote a closed point by a single letter, say p, and often write p ∈ X to indicate that p is a closed point of X. Since Modp is equivalent to the category of modules over a division ring it contains a unique simple module up to isomorphism, and because Modp is closed under subquotients that simple is simple as an Xmodule. We denote any module in this isomorphism class by O p . Thus D is isomorphic to End X O p , and every p-module is a direct sum of copies of O p .
Although a closed point determines a simple module, a simple module need not determine a closed point.
We will say that an X-module M is compact if Hom X (M, −) commutes with direct sums. If X is locally noetherian, this is equivalent to the condition that Hom(M, −) commute with direct limits, and to the condition that M be noetherian [16,Section 5.8].
is a finitely generated module over End X S for all compact X-modules M . Lemma 5.3. Let S be a tiny simple X-module. Then every compact submodule of a direct product of copies of S is isomorphic to a finite direct sum of copies of S.
Therefore ker ρ n+1 contains ker ρ 1 ∩ · · · ∩ ker ρ n . It follows that 0 = M n = ker ρ 1 ∩ · · · ∩ ker ρ n . Hence the map Notation. If S is a simple X-module, we write SumS for the full subcategory consisting of all modules that are isomorphic to a direct sum of copies of S. If D denotes the endomorphism ring of S, then SumS is equivalent to ModD.
In a Grothendieck category a sum of simple modules is isomorphic to a direct sum of simple modules. This is not true for (ModZ) op .
Lemma 5.4. Let S be a simple X-module. The inclusion i * : SumS → ModX has a right adjoint i ! given by i ! N := the sum of all submodules of N that are isomorphic to S.
Proof. If f : M → N is a map, then f (i ! M ) ⊂ i ! N , so i ! can be defined on morphisms by sending a morphism to its restriction. Thus i ! really is a functor, and it takes values in SumS because a sum of simple modules is a direct sum. It is clear that Alternatively, since SumS is closed under subquotients, i * is exact, and since SumS is closed under direct sums i * commutes with direct sums, whence i * has a right adjoint.
Theorem 5.5. Suppose that X is locally noetherian. Let S be a simple X-module. The following are equivalent: 1. there is a closed point p ∈ X such that S ∼ = O p ; 2. i * : SumS → ModX has a left adjoint; 3. every direct product of copies of S is isomorphic to a direct sum of copies of S; 4. S is tiny.
Proof. (1) ⇒ (2) Since Modp consists of all direct sums of copies of O p , Modp = SumS. Hence the fact that p is closed ensures that i * has a left adjoint.
(2) ⇒ (1) By Lemma 5.4, i * has a right adjoint, so the hypothesis ensures that SumS satisfies the requirements to be the module category of a closed point.
(2) ⇔ (3) Since i * is exact, it has a left adjoint if and only if it commutes with products. That is, if and only if SumS is closed under products. But this is equivalent to condition (3). ( But it is also in SumS, so is a finite direct sum of copies of S. Thus Hom X (M, S) is a finite direct sum of copies of End X S.
(4) ⇒ (3) Let P be a product of copies of S. Since X is locally noetherian, P is a direct limit of noetherian submodules. Each of those submodules is of compact, and hence semisimple by Lemma 5.3. Therefore P is semisimple.
A locally noetherian space X over a field k is Ext-finite if dim k Ext i X (M, N ) < ∞ for all noetherian X-modules M and N and all integers i. If A is a connected graded k-algebra such that dim k A n < ∞ for all n, then [3, Section 3] gives conditions which ensure that the projective space with homogeneous coordinate ring A is Ext-finite.
Corollary 5.6. Let X be a locally noetherian space over Spec k. If X is Ext-finite, then every simple X-module is isomorphic to O p for some closed point p. If k is algebraically closed, then p is k-rational.
is a finite dimensional division algebra over k, and therefore equal to k if k is algebraically closed.
The following is routine but we record it for later reference.
Proposition 5.7. Let X be an affine space with coordinate ring R. There is a bijection between 1. closed points in X; 2. simple R-modules S such that S has finite length over End R S; 3. maximal two-sided ideals m such that R/m is artinian.
If R is a countably generated algebra over an uncountable algebraically closed field k, then all closed points are k-rational, and they are in bijection with the finite dimensional simple R-modules.
The hypotheses on k and R in the last part of the proposition are needed because it is an open problem whether there exists a division algebra D over an algebraically closed field k such that D = k but D is finitely generated as a k-algebra. If there were such a D, then ModD would have a single closed point but that point would not be k-rational and would not correspond to a simple module of finite dimension over k.
The next result says that point modules over a reasonable graded algebra yield closed points in the associated projective space. Proof. The χ 1 hypothesis implies that Hom X (O X , πM (n)) has finite dimension for all n ∈ Z; The hypothesis on M/N ensures that πM is a simple X-module. If N is any right noetherian X-module, then there is an epimorphism from a finite direct sum of copies of O X (n) for various integers n to N . Hence Hom X (N , πM ) is a subspace of a finite direct sum of various Hom X (O X (n), πM ), so finite dimensional. Thus πM is a tiny simple.
We have the following consequences of Proposition 3.3. If W is a weakly closed subspace of X, and p is a closed point of X such that O p is a W -module, then p is a closed point of W . If Z is a closed subspace of X and p is a closed point of Z, then p is a closed point of X. However, a closed point of a weakly closed subspace of X need not be a closed point of X; for example, if M is an infinite dimensional simple module over a ring R such that R/ Ann M is not artinian, then M does not correspond to a closed point of ModR even though SumM is a closed point in SumM .
Remark. Closed points do not always behave like zero-dimensional geometric objects. The following example has been emphasized by Artin and Zhang.
Fix an algebraically closed field k. Let E be an elliptic curve over Spec k, σ an automorphism of E having infinite order, and L a line bundle of degree ≥ 3 on E. Let B = B(E, σ, L) be the twisted homogenous coordinate ring associated to this data [1]. Then B is a connected graded k-algebra of Gelfand-Kirillov dimension two, so A := B⊗ k B is the homogeneous coordinate ring of a non-commutative projective space X that is like a non-commutative 3-fold. If F and G are noetherian Xmodules then Ext i X (F , G) is finite dimensional for all i. The ring A is Artin-Schelter-Gorenstein so X satisfies Serre duality. We use the B-B-bimodule structure on B to view it as a graded right A-module. Applying the quotient functor π : GrModA → TailsA = ModX to this module produces an X-module M. Since σ has infinite order every non-zero two-sided graded ideal in B has finite codimension [1]; hence M is a simple X-module. By Corollary 5.6, there is a closed k-rational point p ∈ X such that M ∼ = O p . Because the Gelfand-Kirillov dimension of B is two, and A is Cohen-Macaulay with respect to GK-dimension, Ext 2 A (B, A) = 0. By [3, Theorem 8.1(5)], Ext 2 X (O p , O X ) = 0. Serre duality now gives H 1 (X, O p (i)) = 0 for some integer i. The non-vanishing of this cohomology group indicates that p is behaving like a geometric object of dimension greater than zero.
Another example of this sort of phenomenon is given in [10]. Let k be the complex field. There is a smooth non-commutative projective surface X satisfying Serre duality, and a closed k-rational point p ∈ X such that Ext 1 (O p , O p ) = 0 and Ext 2 X (O p , O p ) = k. Thus p behaves like a −2-curve on X. This analogy is strengthened by the fact that X is constructed from the sheaf of differential operators on P 1 (and the projectivized cotangent bundle to P 1 is a surface with a −2-curve). An alternative construction of X is to take a projective compactification (as described before Proposition 8.11) of the affine space with coordinate ring a nonsimple primitive factor ring of the enveloping algebra U (sl 2 ); the associated graded ring of this factor is the coordinate ring of a singular quadric surface in A 3 , and the exceptional fiber in its minimal resolution is a −2-curve. All this suggests that p behaves both like a point and like a −2-curve. Van den Bergh has suggested that we should perhaps think of X as its own blowup at p. The plausibility of this idea is reinforced by the fact that the ideal annihilating the simple sl 2 -module corresponding to p is idempotent (because sl 2 is semisimple), so the Rees ring construction of the blowup would simply reproduce the original surface. In that case there is a weak map g : W → p such that ig = f α, where α : W → Z and i : p → X are the inclusions; its direct image functor is g * = i * f * α * .
The map f : Spec k → X induced by the inclusion (1) Using the exactness of some of the functors, we have Combining these two computations shows that Taking left derived functors of both sides, and using the fact that (−) * and −⊗ D O p are exact, gives (2).
If p and q are distinct closed points, then o p ⊗ X o q = 0.
6. The complement to a weakly closed subspace Throughout this section X is a locally noetherian space. Weakly closed and weakly open subspaces were defined in section two. We begin this section by defining the complement to a weakly closed subspace. Results in Gabriel's thesis show that the complement to a weakly closed subspace is weakly open, and that every weakly open subspace arises as such a complement.
Definition 6.1. Let X be a locally noetherian space, and let W be a weakly closed subspace. The category of X-modules supported on W is the full subcategory consisting of those X-modules which are the directed union of a family of submodules M each of which has a finite fitration by submodules 0 We call the modules M i /M i−1 the slices of the filtration. Definition 6.3. The saturation of a weakly closed subspace W of X is the weakly closed subspace W sat defined by ModW sat = Mod W X. We say that W is saturated if W = W sat .
Remarks. 1. It is obvious that (W sat ) sat = W sat . 2. The proof of Lemma 6.10 shows that if W and Z are weakly closed subspaces of a locally noetherian space X, then Z sat ∩ W sat = (Z ∩ W ) sat . This implies that Definition 6.4. If W is a weakly closed subspace of a locally noetherian space X, its complement, denoted X\W , is the non-commutative space defined by We write j : X\W → X for the map consisting of the adjoint pair of functors (j * , j * ) where j * : ModX → ModX\W is the quotient functor, and j * is its right adjoint (which exists because Mod W X is a localizing subcategory).
One sees at once that X\X = φ and X\φ = X.
Because j * is fully faithful we have the following result.
Proposition 6.5. Let W be a weakly closed subspace of a locally noetherian space X. Then j : X\W → X is a weakly open immersion.
We will identify ModX\W with the full subcategory of ModX consisting of those X-modules M for which the canonical map M → j * j * M is an isomorphism. In this way we can speak of X\W as a weakly open subspace of X.
Since Mod W X is a localizing subcategory of ModX, its inclusion has a right adjoint τ : ModX → Mod W X. This sends an X-module to its largest submodule that is supported on W . We sometimes call it the W -torsion functor. By [5], for every X-module M there is an exact sequence and τ j * j * = 0.
Proposition 6.6. Every weakly open subspace of X is of the form X\W for some weakly closed subspace W .
If Z is a closed subspace of X, we call X\Z an open subspace of X.
The following two results are used in section 7. Proof. This result is a restatement of condition (b) in [5, Lemme 1, p. 370]; condition (c) of that result holds because j * j * ∼ = id U and j * is a left adjoint to j * . Proof. This is a restatement of [5, Coroll. 1, p. 379]. Definition 6.9. Let U i , i ∈ I, be a small set of weakly open subspaces of X. Let j i : U i → X be the inclusion maps. We define their union, ∪ i∈I U i , to be the weakly open subspace of X whose module category is ModX/T where T is the localizing subcategory consisting of the X-modules M such that j * i M = 0 for all i. We say that the U i provide a weakly open cover of X if their union is X; i.e., if T = 0. This is equivalent to the requirement that the diagonal ModX → i ModU i is faithful. If each U i is open, we say that they provide an open cover of X.
We could first define the union of a pair of weakly open subspaces thus making ∪ a binary operation on weakly open subspaces. This operation is idempotent, commutative, and associative. We can use the associativity to define the union of a finite collection of weakly open subspaces in the obvious way, and this agrees with the definition above. One has U ∪ φ = U and U ∪ X = X. Proof. By definition, ModU ∪ V is the quotient of ModX by the localizing subcategory Mod W X ∩ Mod Z X, so it suffices to show that this is equal to Mod W ∩Z X. It suffices to prove that the full subcategories of noetherian modules are the same. It is obvious that mod W ∩Z X ⊂ mod W X ∩ mod Z X.
To prove the reverse inclusion, suppose that M is in mod W X∩mod Z X. There are finite filtrations 0 = L 0 ⊂ L 1 ⊂ . . . ⊂ L r = M and 0 = N 0 ⊂ N 1 ⊂ . . . ⊂ N s = M such that L i /L i−1 ∈ modW and N j /N j−1 ∈ modZ for all i and j. Taking a common refinement of these filtrations produces a finite filtration of M all of whose slices belong to modW ∩ Z, thus showing that M ∈ mod W ∩Z X.
Remarks. 1. It is straightforward to show that if S ⊂ T ⊂ A are Serre subcategories of a Grothendieck category A, then T/S is a Serre subcategory of A/S, and A/T is equivalent to (A/S)/(T/S) [16,Ex. 6,p. 174].
This has the following geometric interpretation. If W ⊂ Z ⊂ X are weakly closed subspaces of X, then X\Z is a weakly open subspace of X\W (Corollary 7.4). Example 6.11 shows that X\Z is not the complement to Z\W in X\W ; indeed, the last phrase does not always make sense because Z\W need not be a weakly closed subspace of X\W .
2. It follows from Lemma 6.10 and the previous remark that U and V are weakly open subspaces of U ∪ V . It is easy to see that U ∪ V is the smallest weakly open subspace of X that contains both U and V . Example 6.11. There is a space X with closed subspaces W ⊂ Z ⊂ X for which there is no weak map from Z\W to X\W making the following diagram commute: This example is typical of what can happen in non-commutative geometry. Any space having a non-split extension of two non-isomorphic tiny simple modules has a closed subspace isomorphic to X (for example, the affine space with coordinate ring the enveloping algebra of any non-abelian solvable Lie algebra). The behavior is due to the fact that Ext 1 X (O q , O p ) = 0 even though p and q are distinct closed points. Although Z consists of two closed points and W consists of only one of them, ModZ ∩ ModX\W = φ. (2) Let β * : ModV → ModU be the inclusion. Clearly i * = j * β * . Since j * and i * have left adjoints, ModU and ModV are closed under products in ModX. It follows that ModV is closed under products in ModU . Since j * j * ∼ = id U , β * ∼ = j * i * , and this is left exact. Therefore β * has a left adjoint, say β * . Notice that i * ∼ = β * j * . To see that β * is exact, suppose that 0 → L → M → N → 0 is exact in ModU . By [5, Coroll. 1, p. 368], there is an exact sequence 0 → L ′ → M ′ → N ′ → 0 in ModX such that the first sequence is obtained by applying j * to the second. It follows that β * applied to the first sequence is isomorphic to i * applied to the second; since i * is exact, we deduce that β * is exact. Thus β is the inclusion of an open subspace of U . Lemma 6.13. Let W and Z be weakly closed subspaces of a locally noetherian space X. Then  where W and Z are weakly closed subspaces such that U = X\W and V = X\Z.
Remarks. 1. The definition of U ∩ V does not depend on the choice of W and Z. Although U may be the complement of many different weakly closed subspaces W , Mod W X depends only on U because, if j : U → X is the inclusion, Mod W X consists of the modules on which j * vanishes; by Lemma 6.13, Mod W ∪Z X is the smallest localizing category containing Mod W X and Mod Z X.
2. It follows from the remark before Example 6.11 that U ∩ V is a weakly open subspace of both U and V . Proposition 6.15 shows that it is the largest weakly open subspace of X that is contained in both U and V .
(1) Because ModU is a full subcategory of ModX it is also a full subcategory of ModZ. Because ν * := j * δ * is an exact left adjoint to δ ! j * ∼ = ν * , U is a weakly open subspace of Z. Definition 6.19. The weak closure of a weakly open subspace U of X is the smallest weakly closed subspace of X that contains U ; we denote it byŪ . This makes sense because an intersection of weakly closed subspaces is weakly closed-see the remarks after Proposition 3.3.
If U is a weakly open subspace of a locally noetherian space X, then ModŪ consists of all X-modules N for which there exists a U -module P and X-submodules L ⊂ M ⊂ j * P such that N ∼ = M/L. It is clear that this subcategory of ModX contains ModU , is closed under subquotients, and is closed under direct sums because direct sums are exact in ModX and j * commutes with direct sums. (2) The hypothesis implies that X\Ū = φ, whence (Ū ) sat = X.
If X is the affine space in Example 6.11, then U = X\q is an open subspace such thatŪ = X, but U ∩ (X\p) = U ∩ q = φ. It would be worthwhile to find conditions which ensure that U ∩ V = φ for all non-empty weakly open subspaces V whenŪ = X.
There is not a useful analogue of quasi-compactness for weakly open covers: if W i is the weakly closed subspace of Spec Z defined at the beginning of section three and U i is its weakly open complement, then Spec Z is the union of the U i but is not the union of any finite subset of these.
However, the same proof as for the commutative case shows that affine spaces satisfy an analogue of quasi-compactness for open covers. Proof. Suppose that R is a coordinate ring for X, and that Z i , i ∈ I, are closed subspaces of X such that U i = X\Z i . Let K i be the two-sided ideal of R corresponding to Z i ; that is, ModZ i = ModR/K i . Let K denote the sum of all the K i . If j i : U i → X is the inclusion, then j * i (R/K i ) = 0, so j * i (R/K) = 0. Hence R/K = 0, and it follows that 1, and hence R, is contained in i∈F K i for some finite set F ⊂ I. Now we show that the U i , i ∈ F , provide an open cover of X. If M is a non-zero module such that j * i M = 0 for all i ∈ F , then j * i N = 0 for all i ∈ F for every simple subquotient N of M . It follows that N ∈ ModZ i for all i ∈ F , so N K i = 0 for all i ∈ F ; this contradicts the fact that R = i∈F K i .

Containment and intersection of subspaces
We have defined containment, intersection, and union, for a pair of weakly closed subspaces, and for a pair of weakly open subspaces. We would also like to define containment and intersection for pairs of subspaces where one of the subspaces is weakly open and the other is weakly closed.
We begin with the following question: if Z and W are weakly closed subspaces of X such that W ∩ Z = φ, is Z a weakly closed subspace of X\W ? Unfortunately it need not be.
To determine exactly when Z is a weakly closed subspace of X\W we need to address the following general question. If β : Z → X is a weak map, and j : X\W → X is the inclusion of a weakly open subspace, when is there a weak map γ : Z → X\W such that the diagram commutes? The case when Z is a weakly closed subspace such that Z ∩ W = φ is relevant to the question posed in the previous paragraph.
Suppose that these conditions hold. Then 1. if β is a map, so is γ; 2. if β is an affine map, so is γ; 3. if β is a weakly closed (resp. weakly open, closed) immersion, so is γ.
Define γ * : ModZ → ModX\W by γ * = j * β * . This is an exact functor so defines a weak map γ : Z → X\W . (It is not necessarily true that jγ = β.) Let τ : ModX → Mod W X denote the torsion functor that takes the W -support of a module. If N ∈ ModZ, then there is an exact sequence The vanishing of Hom implies that β * N cannot have a non-zero Wsubmodule, so τ (β * N ) = 0. The vanishing of Ext 1 implies that R 1 τ (β * N ) = 0. Hence the natural transformation β * → j * j * β * is an isomorphism. ( If there is a weak map γ : Z → X\W such that jγ = β, then β * ∼ = j * γ * , so j * β * ∼ = j * j * γ * ∼ = γ * , so we may assume that γ * = j * β * . It follows that j * j * β * ∼ = j * γ * ∼ = β * , whence (2) holds. But j * j * β * N is the largest essential extension of β * N by a module in Mod W X. It follows that one cannot extend β * N in an essential way by a W -module. In other words, if 0 → β * N → E 0 → E 1 → . . . is a minimal injective resolution in ModX, then Hom X (α * M, E 1 ) = 0 for all Wmodules M ; (1) follows. Now suppose that the three equivalent conditions hold.
4. If V is contained in X\W , so are the weakly closed and weakly open subspaces of V . Proof. Let β : p → X denote the inclusion. By Proposition 7.1, there is a closed immersion γ : p → X\W such that β = jγ. Thus we can view p as a closed subspace of X\W . The fact that p is a closed point of X\W follows from the fact that γ * O p = j * β * O p is simple in ModX\W (a localization functor sends a simple module to a simple module or zero). Part (3) follows from the fact that j * is fully faithful.
Corollary 7.4. Let W ⊂ Z ⊂ X be weakly closed subspaces of X. Then X\Z is a weakly open subspace of X\W . Precisely, there is a weakly open immersion γ : X\Z → X\W such that the following diagram commutes: Proof. By Lemma 6.7, Ext 1 X (−, β * N ) vanishes on all Z-modules. Thus the first condition in Proposition 7.1 is satisfied, and it follows that such a γ exists.  , we cannot conclude that U ∩ V is affine even if U and V are open affine subspaces of an affine space X. This is not a surprise. Suppose that X is affine with coordinate ring R. If i and j are affine maps, then U and V are affine, with coordinate rings R U and R V say; we can choose R such that i and j are induced by ring homomorphisms R → R U and R → R V , both of which are necessarily epimorphisms in the category of rings. However, when R is not commutative, there may not be a ring structure on R U ⊗ R R V such that the natural map R → R U ⊗ R R V is a ring homomorphism. Proposition 7.5. Let U and V be weakly open subspaces of a locally noetherian space X. Then U ∩ V is the fiber product U × X V of U and V in the category of spaces (with maps as the morphisms).
Proof. Consider the diagram (7-2) above. Suppose that f : Y → U and g : Y → V are maps such that if = jg. We must show that there is a unique map h : Y → U ∩V such that f = αh and g = βh.
Because the image of i * f * is the same as that of j * g * , it is contained in ModU ∩ ModV = ModU ∩ V , whence i * f * ∼ = γ * γ * i * f * . Thus The proof of that result shows that h * = α * f * . The "same" argument shows there is a unique map h ′ : Y → U ∩ V such that g = βh ′ , and h ′ * = β * g * . Therefore and we conclude that h = h ′ , completing the proof.
We now generalize the question posed prior to Proposition 7.1. If Z and W are weakly closed subspaces of a space X, when is Z\(Z ∩ W ) a weakly closed subspace of X\W ? Example 6.11 shows that the answer is not "always". Proposition 7.6. Let W and Z be weakly closed subspaces of a locally noetherian space X. Denote the inclusion maps by There is a weakly closed immersion γ : Z\(Z ∩W ) → X\W such that jγ = δε if and only if Ext 1 X (α * M, δ * ε * N ) = 0 for all M ∈ ModW and all N ∈ ModZ\(Z ∩ W ). In that case, if δ is a closed immersion, so is γ.
To show that γ is a weakly closed immersion we must show that that γ * (ModZ\(Z∩ W )) is closed under subquotients in ModX\W and that γ * is exact, fully faithful, and has a right adjoint.
Since it is a composition of left exact functors, γ * is left exact. To show that γ * is exact, it suffices to show that j * δ * (R 1 ε * ) = 0. Now R 1 ε * takes values in Mod Z∩W Z and j * δ * vanishes on this because j * vanishes on Mod W X. Hence j * δ * (R 1 ε * ) = 0 and γ * is exact.
Because j * and δ * have right adjoints, they commute with direct limits. By Lemma 6.7, ε * commutes with direct limits. Therefore γ * commutes with direct limits. Since γ * is exact it therefore has a right adjoint, which we denote by γ ! .
Hence γ * := ε * δ * j * is a left adjoint to γ * , showing that γ is a closed immersion. Proof. Since δ * and j * are fully faithful, the category on the right-hand side is a full subcategory of ModX. So is the category on the left-hand side, because γ * is fully faithful. Hence to verify this equality it is enough to check it on objects. Because j * γ * ∼ = δ * ε * , the left-hand side is contained in the right-hand side. Conversely, a module on the right-hand side is of the form δ * L for some L ∈ ModZ and it satisfies δ * L ∼ = j * j * δ * L. The proof of Proposition 7.6 showed that j * δ * ∼ = γ * ε * , so δ * L ∼ = j * j * δ * L ∼ = j * γ * ε * L, which is in the left-hand side. Remarks. In these remarks, W and Z are weakly closed subspaces of X, U denotes X\W , and we use the notation of Proposition 7.6.
1. Example 6.11 shows that Z ∩ U is not always defined. In that example, Z ∩ (X\q) is not defined because Z\(Z ∩ q) = p and Ext 1 X (O q , O p ) = 0. This shows that Z ∩ U need not be defined even if U → X is affine.
The language we have been using is close to that of topology. For this paragraph only we will say that the notions of weakly closed and weakly open subspaces provide a space with an almost-topological structure. If Z and U are respectively a weakly closed and weakly open subspace of X, it (almost) makes sense to ask if the almost-topological structures on Z and U (which they have by virtue of the fact that they are spaces) are the same as the induced almost-topological structures on them. The next result shows this is true for U , but the example that follows it shows this is false for Z. Proposition 7.10. Let U be a weakly open subspace of a locally noetherian space X. Then every weakly closed subspace of U is of the form Z ∩ U for some weakly closed subspace Z of X.
Proof. Let V be a weakly closed subspace of U , and let γ : V → U and j : U → X be the inclusions. We seek a weakly closed subspace Z of X such that V = Z ∩ U . That is, if δ : Z → X denotes the inclusion, we seek a commutative diagram By hypothesis, j * and γ * are fully faithful, so we can think of them as inclusions of full subcategories. Let A be the full subcategory of ModX consisting of all X-modules N that can be written as N ∼ = M/L where L ⊂ M ⊂ j * γ * P are X-submodules of j * γ * P for some P ∈ ModV .
Clearly A is closed under subquotients in ModX. Hence the inclusion δ * : A → ModX is exact.
where the final direct sum is taken in ModV and the final isomorphism is due to the fact that j * and γ * commute with direct limits. It follows that ⊕ i N i is in A. Thus A is closed under direct limits in ModX. Hence δ * commutes with direct limits, and therefore has a right adjoint δ ! . So we can define a weakly closed subspace Z of X by declaring ModZ to be A. We now show that j * γ * (ModV ) = δ * (ModZ) ∩ j * (ModU ). One inclusion is easy: if P ∈ ModV , then j * γ * P is in ModZ by definition, thus showing that the righthand side contains the left-hand side. Now suppose that N is in the right-hand side. Thus N = M/L where L ⊂ M ⊂ j * γ * P for some P ∈ ModV . There are inclusions To show that Z ∩ U is defined we must show that Ext 1 X (α * −, δ * ε * −) = 0. Since δ * ε * = j * δ * this follows from Lemma 6.7. Hence Z ∩ U is defined, and it follows from the previous paragraph and Lemma 7.7 that V = Z ∩ U .
We should probably call the Z constructed in the previous proof the weak closure of V in X, and extend Definition 6.19 accordingly. If we do that, then Proposition 7.10 says that V =V ∩ U whenever V is a weakly closed subspace of a weakly open subspace U . Thus V is the analogue of a locally closed subscheme in the sense that it is a weakly open subspace of its weak closureV .
If products are exact in ModX (for example, if X is affine), there is a version of Proposition 7.10 with "weakly closed" replaced by "closed". In that case, the equalityV ∩ U = V extends the result that if R S is an Ore localization of a ring R, and J is a right ideal of R S , then J = (J ∩ R)R S .

Effective Divisors
Van den Bergh [23] has defined the notion of a divisor on a non-commutative space. The reader is referred there for the definition and basic results. Further properties of effective divisors can be found in [9].
Definition 8.1. [9] An effective divisor Y on a non-commutative space X is an invertible proper subobject o X (−Y ) of o X in the category of X-X-bimodules.
If X is a scheme such that QcohX is a Grothendieck category, then there is a bijection between effective Cartier divisors and effective divisors in the sense of Definition 8.1.
If Y is an effective divisor, then o X (−Y ) is a proper ideal in o X , so the results in [23,Section 3.5] show that Y determines, and is determined by, a non-empty closed subspace of X which we also denote by Y . The category ModY is the full subcategory of ModX consisting of those M such that the natural map M (−Y ) → M is zero.
We write i : Y → X for the inclusion of the effective divisor. The left and right adjoints to the inclusion i * : ModY → ModX are related via their derived functors as follows: and Because i * is right exact, R 2 i ! = 0. Therefore the spectral sequence (2-1) degenerates, giving a long exact sequence The associated exact sequence of bimodules is Applying M ⊗ X − to this produces an exact sequence We note that Tor X r (−, o Y ) = 0 for r ≥ 2.
The last condition is equivalent to i * α * α ! vanishing on all injective X-modules.
Proof. By Proposition 3.3, there is a commutative diagram Before proceeding we establish the equivalence claimed in the last sentence of the Proposition. By definition of Tor X , if E is an injective X-module, then , so δ * vanishes on all injective W -modules. Since α ! sends injectives to injectives, γ * δ * α ! vanishes on all injective X-modules. Hence i * α * α ! vanishes on all injective X-modules.
Conversely, suppose that W ∩ Y = φ and that Tor However, the exact sequence gives rise to the exact sequence E) is characterized by the fact that it is the kernel in the exact sequence which is zero by hypothesis. It follows that and hence that on injective W -modules, and hence on all W -modules. However, if F, G ∈ Lex(X, X) are mutual quasi-inverses and each sends ModW to itself, then α ! F α * and α ! Gα * are in Lex(W, W ) and are mutual quasi-inverses. Therefore Hom W (I, −) is an auto-equivalence of ModW . The next two lemmas will be used later.
If Y is an effective divisor on X, then Mod Y X is closed under injective envelopes in ModX.
is an invertible bimodule. Therefore 0 = R 2 τ ∼ = (R 1 j * )j * . Since every X\Y -module is of the form j * M , it follows that R 1 j * = 0 and we conclude that j * is exact. But j * commutes with direct sums (Proposition 6.8), so it has a right adjoint. Since j * j * ∼ = id X\Y , j * is fully faithful. Hence j is an affine map.
(2) The definition of j * ensures that i ! j * is zero. Let N ∈ ModX\Y , and suppose that i * j * N is non-zero. By , there is an exact sequence It follows from the construction of j * , that j * N is not an essential extension of (j * N )(−Y ). Hence, if V denotes the image of (j * N )(−Y ) in j * N , V ∩ L = 0 for some non-zero submodule L ⊂ j * N . But then L embeds in i * j * N , so is a Y -module, contradicting the fact that τ (j * N ) = 0. From this contradiction we deduce that i * j * N = 0.
Proposition 8.6. Let Y be an effective divisor on X. If Z is a weakly closed subspace of X such that Z ∩ Y = φ, then 1. Z is contained in X\Y , and 2. Y is contained in X\Z.
Proof. Let i : Y → X and β : Z → X denote the inclusions. Because Z ∩ Y = 0, both i ! and i * vanish on all Z-modules. Let M and N be respectively a Y -module and Z-module.
(1) Since i ! and R 1 i ! vanish on all Z-modules, (8-2) gives Ext 1 X (i * M, β * N ) = 0. This is precisely the criterion for Z to be contained in X\Y .
(2) Since i ! vanishes on all Z-modules, the only Z-module belonging to Mod Y X is the zero module. Hence Hom X (β * N, −) is zero on Mod Y X. By Lemma 8.3, all the terms in the minimal injective resolution of i * M belong to Mod Y X; hence Ext n X (β * N, i * M ) = 0 for all n. Hence Y is contained in X\Z.
Corollary 8.7. Let Y be an effective divisor on X. A closed point in X is contained in either Y or X\Y but not both, and is a closed point in whichever of these spaces contains it.
Proof. Let p be a closed point of X and suppose that p does not lie on Y . By Proposition 8.6, p is contained in X\Y . As remarked in the proof of Corollary 7.3, p is closed in X\Y and hence is a closed point of X\Y .
It is not true, even in the commutative case, that the set of closed points in X is the disjoint union of the set of closed points in Y and the set of closed points in X\Y ; consider the spectrum of a discrete valuation ring. The next result gives conditions under which that is true. Proposition 8.8. Let X be a locally noetherian space. Suppose that j : X\Y → X is the inclusion of the complement to an effective divisor. Let p ∈ X\Y be a closed k-rational point of this complement. If j * O p is a noetherian X-module, then it is simple, and j(p) (see Definition 5.9) is a closed k-rational point of X.
Proof. We will show first that if j * O p is not a simple X-module, then it is not noetherian. To that end, suppose that M is a non-zero X-submodule of j * O p that is not equal to j * O p .
Let i : Y → X be the inclusion and let τ : ModX → Mod Y X be the Y -torsion functor. Since i ! j * = 0, i ! M = 0 and τ M = 0. Hence there is an exact sequence . Then applying  to the other two terms in  gives the other columns in the commutative diagram below: Repeating this argument produces a chain of submodules such that M n ∼ = M (nY ) and M n /M n−1 ∼ = (i * M )(nY ). By the Snake Lemma, Hence the chain of submodules is strictly increasing, thus showing that j * O p is not noetherian.
When j * O p is noetherian there can be no such M , so j * O p is simple. Now we show that it is a tiny simple. First, Hom Remark. Even in the commutative case the hypothesis in Proposition 8.8 does not always hold: for example, let X = Spec R where R is a discrete valuation ring. However, if X is a scheme of finite type over a field k, then every closed k-rational point in the complement of an effective divisor is a closed k-rational point of X. In the non-commutative case this sort of finiteness hypothesis does not ensure that a k-rational closed point of X\Y is a closed point of X.
For example, consider the graded line defined by ModL 1 = GrModk[x] where deg x = 1. Then x cuts out an effective divisor Y defined by ModY = GrModk[x]/(x), and L 1 \Y is isomorphic to Spec k (ModL 1 \Y = Tailsk[x] = P 0 k ). However, if j : L 1 \Y → L 1 denotes the inclusion, then j * O L 1 \Y ∼ = k[x, x −1 ]. Since L 1 \Y is the generic point of L 1 (see [20]), it is no surprise that it is not a closed point of L 1 .
Corollary 8.9. Let X be a locally noetherian space. Suppose that j : X\Y → X is the inclusion of the complement to an effective divisor. If j * O p is a noetherian X-module for every closed point p ∈ X\Y , then the set of closed points on X is the disjoint union of the sets of closed points on Y and on X\Y .
Proof. It is clear that a closed point on Y is a closed point on X, so the result follows from Proposition 8.8 and Corollary 8.7.
Another way of stating the conclusion of this is that j * and j * set up a bijection between the closed points of X that do not lie on Y , and the closed points of X\Y .
Effective divisors on a space typically arise from regular elements in a (homogeneous) coordinate ring. For example, if z is a regular element in a ring R such that zR = Rz, then ModR/(z) is an effective divisor on ModR. Proof. We use the notation and terminology in [23].
Define Let π : GrModA → ModX be the quotient functor. There is an injective Q in GrModA such that I = πQ. It is easy to see that there is an exact sequence where γ(q) = qz. Since πK ∼ = i * i ! I, applying π to this gives (8-7). Therefore Hom . When A is connected and generated by A 1 , A[z −1 ] is strongly graded, meaning that 1 belongs to the product of the degree n and degree −n components for all n; this implies that GrModA[z −1 ] is equivalent to ModA[z −1 ] via the functor that takes the degree zero component of a module.
An affine scheme can be embedded in a projective scheme in such a way that it is the open complement of an ample divisor. It is well-known that a non-commutative affine space can be embedded in a non-commutative projective space in a similar way. We show next that closed points behave well with respect to this embedding.
First we recall the embedding just alluded to. Let R be an algebra over a field k. Fix a set of generators for R and write R n for the k-linear span of all products of ≤ n of those generators. Let t denote an indeterminate commuting with R, and setR := R 0 ⊕ R 1 t ⊕ R 2 t 2 ⊕ · · · ⊂ R[t]. Define U = ModR and X = TailsR. We write π : GrModR → ModX and ω : ModX → GrModR for the quotient functor and its right adjoint.
The closed subspace of X where t ∈R 1 vanishes is denoted by H and is called the hyperplane at infinity. Proposition 8.10 shows that H is an effective divisor because ModH ∼ = ModR/(t). We also have ModH ∼ = Tails gr R, where gr R is the associated graded ring with respect to the filtration R n . Because R[t −1 ] 0 ∼ = R, there is a natural map j : U → X sending U isomorphically to X\H. Proposition 8.11. Suppose thatR is a finitely generated right noetherian k-algebra. The functors j * and j * set up a bijection between 1. the finite dimensional simple R-modules V such that End R V ∼ = k and 2. the closed k-rational points of X that do not lie on H.
Proof. This will follow from Corollary 8.9 once we show that j * V is a noetherian X-module for every finite dimensional simple right R-module V . By definition, j * V ∼ = πṼ , whereṼ is the gradedR-module with degree n componentṼ n = V t n and action (vt n ).(rt i ) := (vr)t n+i for v ∈ V and r ∈ R i . Since every component ofṼ has the same finite dimension, πṼ is noetherian. The last term of this is zero by Proposition 8.5, and the first is zero because i ! j * = 0. Hence there is a map β : i * K → i * M such that β • i * (α) = id M . However, i * (α) = γ • α where γ : M → i * M is the canonical epimorphism, so βγα = id M , which shows that the original exact sequence splits. Hence the result.
The next result is an analogue of the result that an ample divisor on a scheme meets every subscheme of positive dimension.
Let X be a locally noetherian space and modX its category of noetherian modules. We denote the Krull dimension, in the sense of Gabriel [5], of M ∈ modX by Kdim M . We say that a noetherian X-module M is d-critical if Kdim M = d but Kdim M/N < d for all 0 = N ⊂ M . A noetherian module of Krull dimension d has a d-critical quotient. We define the dimension of a weakly closed subspace W of X to be dim W := max{Kdim M | M ∈ modW }.
Definition 8.13. Let X be a locally noetherian space endowed with a structure map π : X → Spec k. Set O X = π * k and H q (X, −) = Ext q X (O X , −). An effective divisor Y on X is ample if for every M ∈ modX • there are integers n i ≤ 0 and an epimorphism O X (n i Y ) → M , and • H q (X, M (nY )) = 0 if q = 0 and n ≫ 0.
Proposition 8.14. Let X be a locally noetherian space over Spec k, and let Y be an ample effective divisor on X. Suppose that dim k Ext n X (M, N ) < ∞ for all M, N ∈ modX. If W is a weakly closed subspace of X and dim W > 0, then W ∩ Y = φ.
Proof. We will assume that W ∩ Y = φ and get a contradiction. Let d = dim W and let M be a d-critical noetherian W -module. By , there is an exact sequence 0 → i ! M (−Y ) → M (−Y ) → M → i * M → 0. Since W ∩ Y = φ, i ! M = i * M = 0, whence M ∼ = M (Y ). Hence dim k H 0 (X, M (nY )) is a finite number independent of n. By [3], ModX is equivalent to TailsB where B is the graded algebra ⊕ ∞ n=0 H 0 (X, O X (nY )). Under this equivalence, M ∼ = πV , where V is the graded B-module ⊕ ∞ n=0 H 0 (X, M (nY )). But dim V n is finite and independent of n so πV has finite length. This contradicts our hypothesis that Kdim M = d ≥ 1.
The next result generalizes the surprising result in [13,Lemma 6.4]. In that result, Y is an effective divisor and X\Y is an affine space having a coordinate ring that is an infinite dimensional C-algebra; such an algebra cannot have a finite dimensional simple module, so X\Y has no closed points. Proposition 8.15. Let k be an algebraically closed field and X a locally noetherian Ext-finite space over k. Let i : Y → X be the inclusion of an effective divisor. Suppose that X\Y has no closed k-rational points. If F is a noetherian X-module such that i * F = 0, then F = 0.
Proof. Suppose that F = 0. Since F is noetherian it has a simple quotient. By Corollary 5.6, that simple is of the form O p for a closed k-rational point p ∈ X. By the hypothesis and Corollary 8.7, p ∈ Y ; this implies that i * F = 0.