aa r X i v : . [ m a t h . AG ] F e b Notes on schematic finite spaces
F. Sancho and P. SanchoFebruary 19, 2021
Abstract
The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give variouscharacterizations of these spaces and their natural morphisms. We show that schematicfinite spaces are strongly related to quasi-compact quasi-separated schemes.
Introduction
In Topology, Di ff erential Geometry and Algebraic Geometry, it is usual to study their geo-metric objects considering suitable finite open coverings and studying the associated finiteringed spaces. Let us remember how these finite ringed spaces are constructed: Let S be a topological space and let U = { U , . . . , U n } be a finite open covering of S .For each s ∈ S define U s : = ∩ s ∈ U i U i . Observe that the topology generated by U is equal to thetopology generated by { U s } s ∈ S . We shall say that U is a minimal open covering of S if U i , U j if i , j and U i = U s for some s ∈ S , for every i . Define the following equivalence relationon S : s ∼ s ′ if the covering U does not distinguish them, i.e., if U s = U s ′ . Consider on S thetopology generated by the covering U and let X : = S / ∼ be the quotient topological space. X is a finite T -topological space, then it is a finite poset as follows: [ s ] ≤ [ s ′ ] if U s ′ ⊆ U s .Let π : S → X , s [ s ] be the quotient morphism and let U [ s ] = { [ s ′ ] ∈ S : [ s ′ ] ≥ [ s ] } be theminimal open neighborhood of [ s ]. One has that π − ( U [ s ] ) = U s . Suppose now that ( S , O S ) isa ringed space (a scheme, a di ff erentiable manifold, an analytic space, etc.). We have then asheaf of rings on X , namely O : = π ∗ O S , so that π : ( S , O S ) → ( X , O ) is a morphism of ringedspaces. We shall say that ( X , O ) is the ringed finite space associated with the finite covering U . Observe that O [ s ] = O ( U [ s ] ) = O S ( U s ).To fix ideas, suppose that S is a quasi-compact quasi-separated scheme (see [5] 1.2.1).There exists a minimal a ffi ne open covering U = { U s , . . . , U s n } of S (see [9] 3.13). Considerthe associated ringed finite space X . It is easy to prove that the functors M π ∗ M , N π ∗ N O S -modulesand the category of quasi-coherent O X -modules. Besides, H i ( S , M ) = H i ( X , π ∗ M ) for anyquasi-coherent O S -module M . Observe that for every U s j , U s j ′ ⊆ U s i :- The restriction morphism O S ( U s i ) → O S ( U s j ) is a flat morphism, since the morphism O S ( U s i ) → ( O S ( U s i ) \ p ) − O S ( U s i ) = O S , p = ( O S ( U s j ) \ p ) − O S ( U s j ) is flat, for any p ∈ U s j = Spec O S ( U s j ).- The natural morphism O ( U s j ) ⊗ O S ( U si ) O S ( U s j ′ ) → O S ( U s j ∩ U s j ′ ) is an isomorphism,because U s j × U si U s j ′ = U s j ∩ U s j ′ .- The morphism O S ( U s j ∩ U s j ′ ) → Q U sk ⊆ U sj ∩ U sj ′ O S ( U s k ) is faithfully flat: it is flat be-cause U s j ∩ U s j ′ and U k are a ffi ne and it is faithfully flat because ` U sk ⊆ U sj ∩ U sj ′ U k → U s j ∩ U s j ′ is a surjective map.Therefore, for any points x j , x j ′ ≥ x i in X :a. The natural morphism O x i → O x j is flat.b. The natural morphism O x j ⊗ O xi O x j ′ → O ( U x j ∩ U x j ′ ) is an isomorphism.c. The morphism O ( U x j ∩ U x j ′ ) → Q x k ≥ x j , x j ′ O x k is faithfully flat. We shall say that a ringed finite space is schematic if it satisfies a., b. and c. In [8] 4.4,4.11, it is proved that a finite ringed space X is schematic i ff X satisfies a. and R n δ ∗ O X is aquasi-coherent module for any n , where δ : X → X × X is the diagonal morphism. In [10] 4.5,it is proved that X is schematic i ff X satisfies a. and R n i ∗ O U is quasi-coherent for any opensubset U i ⊂ X and for n ∈ N . In Algebraic Geometry, it is usual to approach the study ofschemes and their morphisms through the category of quasi-coherent modules, for example,the theory of intersection can be studied with the K -theory of quasi-coherent modules. Weshall denote by Qc - Mod X the category of quasi-coherent O X -modules. We prove that a finiteringed space X is schematic i ff Qc - Mod X satisfies minimal conditions:- A ringed finite space X is schematic i ff for any morphism f : M → N of quasi-coherent O X -modules Ker f is quasi-coherent, and δ ∗ ( M ) ∈ Qc - Mod X × X , for any M ∈ Qc - Mod X , where δ : X → X × X , δ ( x ) = ( x , x ) is the diagonal morphism.- A ringed finite space X is schematic i ff for any morphism f : M → N of quasi-coherent O X -modules Ker f is quasi-coherent, and i ∗ ( M ) ∈ Qc - Mod X for any M ∈ Qc - Mod U and any open subset U i ֒ → X .Likewise, we study and characterize a ffi ne finite spaces. Let us use the previous nota-tions. It can be proved that S is an a ffi ne scheme i ff the morphisms O ( S ) → Q i O s i and O ( U s i ) ⊗ O ( S ) O ( U s j ) → Q U sk ⊂ U si ∩ U sj O ( U s k ) are faithfully flat, for any i , j . We say that a finiteringed space X is a ffi ne if- The morphism O ( X ) → Q x ∈ X O x is faithfully flat.- The morphism O x ⊗ O ( X ) O x ′ → Q z ∈ U x ∩ U x ′ O z is faithfully flat, for any x , x ′ ∈ X .A ffi ne finite spaces are schematic and a finite ringed space X is schematic i ff U x is a ffi ne, forany x ∈ X . We prove that X is a ffi ne i ff O ( X ) → O x is flat, O X ( U x ∩ U y ) = O x ⊗ O ( X ) O y and X and U x ∩ U y are acyclic, for any x , y ∈ X . A schematic space X is a ffi ne i ff H ( X , M ) = M (see [8] 5.11), which is equivalent to saying that the functor M Γ ( X , M ) is exact (see Corollary 7.5). Next, we study the morphisms between schematic finite spaces. Let f : X → Y be amorphism of ringed spaces between schematic finite spaces. We say that f is a ffi ne if f ∗ O X is a quasi-coherent module and f − ( U ) is a ffi ne, for any a ffi ne open subset U ⊂ Y . We provethat f is a ffi ne if and only if f ∗ preserves quasi-coherence and it is an exact functor.We say that f is schematic if f ∗ O X is quasi-coherent and the morphism U x → U f ( x ) , x ′ f ( x ′ ) is a ffi ne, for any x ∈ X . We prove that the following statements are equivalent:- f is schematic.- f ∗ preserves quasi-coherence.- The natural flat morphism O x ⊗ O f ( x ) O y → Q z ∈ U x ∩ f − ( U y ) O z is faithfully flat, for any x ∈ X and y ≥ f ( x ).- R n Γ f ∗ O X ∈ Qc - Mod X × Y for any n , where Γ f : X → X × Y , Γ f ( x ) = ( x , f ( x )) is thegraph of f .- R n ( f ◦ i ) ∗ O U ∈ Qc - Mod X , for any open subset U i ֒ → X and any n ≥ Now, we ask ourselves whether schematic finite spaces are determined by the category oftheir quasi-coherent modules. A morphism of schemes F : S → T between quasi-compactquasi-separated schemes is an isomorphism if and only if the functors Qc - Mod
S F ∗ / / F ∗ o o Qc - Mod T are mutually inverse. Given a schematic morphism f : X → Y , we prove that the followingstatements are equivalent. Finite ringed spaces: basic notions 4- The functors Qc - Mod
X f ∗ / / f ∗ o o Qc - Mod Y are mutually inverse.- O Y = f ∗ O X and f is a ffi ne.- Essentially, f is the quotient morphism defined on X by a minimal a ffi ne open cover-ing.- The cylinder C ( f ) = X ` Y of f is a schematic space and f is a faithfully flat mor-phism.A morphism that satisfies any of these statements will be called quasi-isomorphism. Let us talk less accurately. Given a schematic finite space X , consider the ringed space˜ X : = lim → x ∈ X Spec O x . Spec O x is a subspace of ˜ X via the natural morphism i x : Spec O x → ˜ X and ˜ X = ∪ x ∈ X Spec O x .˜ X is quasi-compact, the set of its quasi-compact open subsets is a basis of its topology andthe intersection of two quasi-compact open subsets is quasi-compact. Given a quasi-coherent O X -module M , let ˜ M x be the O Spec O x -module of localizations of M x and consider the O ˜ X -module ˜ M : = lim ← x ∈ X i x , ∗ ˜ M x . We prove that H n ( X , M ) = H n ( ˜ X , ˜ M ), for any n ≥
0, and thatthe category of quasi-coherent modules of X is equivalent to the category of quasi-coherentmodules of ˜ X . Let X and Y be schematic finite spaces. We say that a morphism of ringedspaces f : ˜ X → ˜ Y is schematic if f ∗ preserves quasi-coherence. Let C W be the category ofschematic finite spaces localized by the quasi-isomorphisms. We prove thatHom C W ( X , Y ) = Hom sch ( ˜ X , ˜ Y ) . Let X be a finite set. It is well known ([1]) that giving a topology on X is equivalent to givinga preorder relation on X : x ≤ y ⇐⇒ ¯ x ⊆ ¯ y , where ¯ x , ¯ y are the closures of x and y . In addition, the topology is T if and only if the preorder is a partial order (i.e. it satisfiesantisymmetric property).Let X be a finite topological space. For each point p ∈ X , let us denote U x = smallest open subset containing x , . Finite ringed spaces: basic notions 5that is, U x = { y ∈ X : y ≥ x } . Then, x ≤ y ⇔ U y ⊆ U x . The family of open subsets { U x } x ∈ X constitutes a minimal basis of open subsets of X (any other base contains this one).A map f : X → Y between finite topological spaces is continuous if and only if it ismonotone (i.e. x ≤ y implies f ( x ) ≤ f ( y )).Let X be a finite topological space and let F be a sheaf of abelian groups (resp. rings,etc.) on X . The stalk of F at x ∈ X , F x , is an abelian group (resp. ring, etc.) and coincideswith the sections of F on U x . For each x ≤ y , the natural morphism r xy : F x → F y is just therestriction morphism F ( U x ) → F ( U y ), which satisfies: r xx = Id for any x , and r yz ◦ r xy = r xz for any x ≤ y ≤ z .Conversely, consider the following data:- An abelian group (resp. a ring, etc) F x for each x ∈ X .- A morphism of groups (resp. rings, etc) r xy : F x → F y for each x ≤ y , satisfying: r xx = Id for any x , and r yz ◦ r xy = r xz for any x ≤ y ≤ z .Let F be the following presheaf of groups (resp. rings, etc.): For each open subset U ⊂ X , F ( U ) : = lim ← x ∈ U F x . It is easy to prove that F x = F x and that F is a sheaf.
1. Definition :
A ringed space is a pair ( X , O ), where X is a topological space and O is asheaf of (commutative with unit) rings on X . A morphism of ringed spaces ( X , O ) → ( X ′ , O ′ )is a pair ( f , f ), where f : X → X ′ is a continuous map and f : O ′ → f ∗ O is a morphism ofsheaves of rings (equivalently, a morphism of sheaves of rings f − O ′ → O ). A finite ringedspace is a ringed space ( X , O ) whose underlying topological space X is finite.A morphism of ringed spaces ( X , O ) → ( X ′ , O ′ ) between two finite ringed spaces isequivalent to the following data:- a continuous (i.e. monotone) map f : X → X ′ ,- for each x ∈ X , a ring homomorphism f x : O ′ f ( x ) → O x , such that, for any x ≤ y , thediagram O ′ f ( x ) f x / / r f ( x ) f ( y ) (cid:15) (cid:15) O xr xy (cid:15) (cid:15) O ′ f ( y ) f y / / O y is commutative. We denote by Hom( X , Y ) the set of morphisms of ringed spaces betweentwo ringed spaces X and Y .
2. Example :
Let {∗} be the topological space with one element. We denote by ( ∗ , R ) thefinite ringed space whose underlying topological space is {∗} and the sheaf of rings is aring O ∗ = R . For any ringed space ( X , O ) there is a natural morphism of ringed spaces( X , O ) → ( ∗ , O ( X )).Let ( X , O ) be a finite ringed space. A sheaf M of O -modules (or O -module) is equivalent. Finite ringed spaces: basic notions 6to these data: an O x -module M x for each x ∈ X , and a morphism of O x -modules r xy : M x →M y for each x ≤ y , such that r xx = Id and r xz = r yz ◦ r xy for any x ≤ y ≤ z . Again, one hasthat M x = stalk of M at x = M ( U x )and r xy is the restriction morphism M ( U x ) → M ( U y ).For each x ≤ y the morphism r xy induces a morphism of O y -modules f r xy : M x ⊗ O x O y → M y . An O -module M is said to be quasi-coherent if for any x ∈ X there exist an open neighbour-hood U of x and an exact sequence of O | U -modules O I | U → O J | U → M | U → .
3. Theorem ([9] 3.6):
Let ( X , O ) be a finite ringed space. An O -module M is quasi-coherentif and only if for any x ≤ y the morphism f r xy : M x ⊗ O x O y → M y is an isomorphism.Proof. ⇒ ) Let U be an open neighbourhood of p such that there exists an exact sequence O I | U → O J | U → M | U → . We can suppose X = U . Obviously, ( O I ) x ⊗ O x O y = ( O x ) I ⊗ O x O y = ( O x ⊗ O x O y ) I = ( O y ) I = ( O I ) y and ( O J ) x ⊗ O x O y = ( O J ) y , then M x ⊗ O x O y = M y . ⇐ ) Given x ∈ X , consider an exact sequence of O x modules O Ix → O Jx → M x → ⊗ O x O y , for any x ≤ y , one has the exact sequence O Iy → O Jy → M y →
0. Then,one has a sequence of morphisms O I | U x → O J | U x → M | U x → y , for any y ∈ U x . Therefore, M is quasi-coherent. (cid:3) We shall denote by
Mod X the category of O -modules on a ringed space ( X , O ) and by Qc - Mod X the subcategory of quasi-coherent O -modules. Also, for any ring R , we shalldenote by Mod R the category of R -modules.. Finite ringed spaces: basic notions 7
4. Remarks : a) If f : X → Y is a morphism of ringed spaces and N is a quasi-coherent O Y -module, then f ∗ N : = f − N ⊗ f − O Y O X is a quasi-coherent O X -module. In particular, thisis true for morphisms between finite ringed spaces.b) If f : M → N is a morphism of O X -modules where M and N are quasi-coherent, thenCoker f is quasi-coherent. However, it is not always true that Ker f is quasi-coherent.
5. Example :
Let ( X , O ) be a finite ringed space and π : ( X , O ) → ( ∗ , O ( X )) the naturalmorphism of ring. If M is an O ( X )-module, then π ∗ M is a quasi-coherent O X -module, whichwe denote ˜ M . We say that ˜ M is the quasi-coherent module associated with M and we have afunctor π ∗ : Mod R → Qc - Mod X , M ˜ M . Note that ˜ M x = M ⊗ O ( X ) O x , for each x ∈ X .
6. Definition :
A finite ringed space ( X , O ) is a finite flat-restriction space (or finite fr-space )if the restriction morphisms r xy : O x → O y are flat, for any x ≤ y .
7. Proposition:
Let ( X , O ) be a finite ringed space. ( X , O ) is a finite fr-space ⇔ For any open subset U (resp. U x ) of X and any morphismf : M → N of quasi-coherent O U − modules (resp. O U x -modules), Ker f is quasi-coherent.Proof. ⇒ ) Let f : M → N be a morphism of quasi-coherent O U − modules. We have toprove that, for each x ≤ y ∈ U , the morphism˜ r xy : (Ker f ) x ⊗ O x O y → (Ker f ) y is an isomorphism. This follows from the next conmutative diagram of exact rows0 / / (Ker f ) x ⊗ O x O y ˜ r xy (cid:15) (cid:15) / / M x ⊗ O x O y f x / / ˜ r xy (cid:15) (cid:15) N x ⊗ O x O y ˜ r xy (cid:15) (cid:15) / / (Ker f ) y / / M y f y / / N y in which the first row is exact because O x → O y is a flat morphism and the second andthe third vertical morphisms are isomorphisms because M and N are quasi-coherent O U -modules. ⇐ ) Let x ≤ y ∈ X . Let f x : M x ֒ → N x be an injective morphism of O x -modules. We haveto prove that f x ⊗ M x ⊗ O x O y → N x ⊗ O x O y is still injective. Consider the open subset U x of X and the functor Mod O x → Qc - Mod U x , M x f M x . Then, the morphism f x gives us amorphism e f x : f M x → f N x of quasi-coherent O U x -modules. Note that ( e f x ) y = f x ⊗
1. Since byhypothesis Ker e f x is quasi-coherent, we have that:Ker( f x ⊗ = (Ker e f x ) y = Ker f x ⊗ O x O y = ⊗ O x O y = , so we conclude that f x ⊗ (cid:3) . A ffi ne finite spaces 8
8. Remark :
Let ( X , O ) be a finite ringed space. The proposition above says that Qc - Mod U is an abelian category for each open subset U of X if and only if ( X , O ) is a finite flat-restriction space. It is also true that in this case Qc - Mod U is a Grothendieck category (see[3]). ffi ne finite spaces
1. Notation :
Let ( X , O ) be a finite ringed space. For each x , y ∈ X , let us denote U xy = U x ∩ U y and O xy = O ( U xy ). If M is an O -module, we denote M xy = M ( U xy ).
2. Definition :
A finite ringed space ( X , O ) is called an a ffi ne (schematic) finite space if itsatisfies the following conditions:1. O ( X ) → Q x ∈ X O x is faithfully flat, for any x ∈ X .2. O x ⊗ O ( X ) O y = O xy , for any x , y ∈ X .3. O xy → Q z ∈ U xy O z is faithfully flat, for any x , y ∈ X .
3. Proposition: If ( X , O ) is an a ffi ne finite space, then it is a finite fr-space.Proof. By condition 3. of the definition above, O xx = O x → Q z ∈ U x O z is faithfully flat.Therefore, O x → O z is flat, for any z ≥ x . (cid:3)
4. Proposition ([8] 4.12):
Let ( X , O ) be a ringed finite space. X is a ffi ne i ff
1. The morphism O ( X ) → Q x ∈ X O x is faithfully flat.2. The morphism O y ⊗ O ( X ) O y ′ → Q z ∈ U yy ′ O z is faithfully flat, for any y , y ′ ∈ X.Proof. ⇒ ) It follows immediately from the definition. ⇐ ) In first place, note that for any x ≤ u , u ′ , the morphism O u ⊗ O ( X ) O u ′ → O u ⊗ O x O u ′ isan epimorphism and the composite morphism O u ⊗ O ( X ) O u ′ → O u ⊗ O x O u ′ → Y z ∈ U uu ′ O z is injective, because it is faithfully flat. Therefore, O u ⊗ O ( X ) O u ′ = O u ⊗ O x O u ′ .We only have to prove that O y ⊗ O ( X ) O y ′ = O yy ′ Let us prove it by reduction to absurdity. Let y , y ′ ∈ X be maximal such that the morphism O y ⊗ O ( X ) O y ′ → O yy ′ is not an isomorphism.. A ffi ne finite spaces 9First, if y ≤ y ′ , then U yy ′ = U y ′ and the epimorphism O y ⊗ O ( X ) O y ′ → O y ′ is faithfully flat,by 2. Therefore, O y ⊗ O ( X ) O y ′ = O y ′ = O yy ′ . So, neither y ≤ y ′ , nor y ′ ≤ y . The morphism B : = O y ⊗ O ( X ) O y ′ → Q z ∈ U yy ′ O z = : C is faithfully flat. Thus, the sequence of morphisms( ∗ ) B → C / / / / C ⊗ B C is exact. By the maximality of y and y ′ , given z , z ′ ∈ U yy ′ , O z ⊗ O ( X ) O z ′ = O z ⊗ O y O z ′ = O zz ′ .The natural morphism O z ⊗ O ( X ) O z ′ → O z ⊗ B O z ′ is surjective. The composite morphism O z ⊗ O ( X ) O z ′ → O z ⊗ B O z ′ → O zz ′ is an isomorphism, then O z ⊗ B O z ′ = O zz ′ . Therefore, C ⊗ B C = Q z , z ′ ∈ U yy ′ O zz ′ . Then, from the diagram ( ∗ ), B = O yy ′ and we have come to contradiction. (cid:3)
5. Corollary:
A finite ringed space ( U x , O ) is a ffi ne i ff the morphism O y ⊗ O x O y ′ → Q z ∈ U yy ′ O z is faithfully flat, for any y , y ′ ≥ x.Proof. It follows easily from the proposition above. (cid:3)
6. Proposition :
Let X be an a ffi ne finite space and U ⊂ X an open set. Then, U is a ffi ne i ff O ( U ) → Q q ∈ U O q is a faithfully flat morphism.Proof. ⇒ ) O ( U ) → Q q ∈ U O q is a faithfully flat morphism by definition of a ffi ne finite space. ⇐ ) We have to check that U satisfies the conditions 2. and 3. of Definition 2.2. Condition3. is clear, because X is a ffi ne. Now, let us check 2.: for each x , y ∈ U , the morphism O x ⊗ O ( X ) O y → O x ⊗ O ( U ) O y is surjective and the composite morphism O x ⊗ O ( X ) O y → O x ⊗ O ( U ) O y → O xy is an isomorphism, thus O x ⊗ O ( U ) O y ≃ O xy . (cid:3)
7. Corollary:
If X is an a ffi ne finite space, then U xy is a ffi ne for every x , y ∈ X.Proof.
It follows from condition 3. of Definition 2.2 and the proposition above. (cid:3)
8. Proposition :
Let X be an a ffi ne finite space and M a quasi-coherent O X -module. Thenatural morphism M ( V ) ⊗ O ( X ) O ( U ) → M ( U ∩ V ) is an isomorphism, for any open set V ⊆ X and any a ffi ne open set U ⊆ X.Proof.
1. The morphism O ( X ) → Q x ∈ X O x = : B is faithfully flat. The sequence of mor-phisms O ( X ) → B = Y x ∈ X O x / / / / B ⊗ O ( X ) B = Y x , y ∈ X O xy is a split sequence of morphisms under a faithfully flat base change ( O ( X ) → B ), then thissequence of morphisms is universally exact, i.e., if we tensor the sequence of morphisms. A ffi ne finite spaces 10by M ⊗ C − (where C is a commutative ring, M is a C -module and O ( X ) a C -algebra) thenwe obtain an exact sequence of morphisms. In particular, O xy ֒ → Q z ∈ U xy O z is universallyinjective and the sequence of morphisms( ∗ ) O ( X ) → Y x ∈ X O x / / / / Y x , y ∈ X , z ∈ U xy O z is universally exact.2. Let W ⊂ U x be an a ffi ne open set. Consider the universally exact sequence of mor-phisms O ( W ) → Y z ∈ W O z / / / / Y z , z ′ ∈ W , z ′′ ∈ U zz ′ O z ′′ . Tensoring by M x ⊗ O x − , we obtain the exact sequence of morphisms M x ⊗ O x O ( W ) → Y z ∈ W M z / / / / Y z , z ′ ∈ W , z ′′ ∈ U zz ′ M z ′′ , which shows that M x ⊗ O x O ( W ) = M ( W ). Therefore (using Corollary 2.7), M x ⊗ O ( X ) O y = M x ⊗ O x O x ⊗ O ( X ) O y = M x ⊗ O x O xy = M xy .
3. Consider the exact sequence of morphisms M ( V ) → Y y ∈ V M y / / / / Y y , y ′ ∈ V , z ∈ U yy ′ M z . Tensoring by ⊗ O ( X ) O x , we obtain the exact sequence of morphisms M ( V ) ⊗ O ( X ) O x → Y y ∈ V M xy / / / / Y y , y ′ ∈ V , z ∈ U yy ′ M xz . which shows that M ( V ) ⊗ O ( X ) O x = M ( V ∩ U x ).4. Consider the universally exact sequence ( ∗ ), where X = U . Tensoring by M ( V ) ⊗ O ( X ) ,we obtain the exact sequence of morphisms M ( V ) ⊗ O ( X ) O ( U ) → Y x ∈ U M ( V ∩ U x ) / / / / Y x , y ∈ U , z ∈ U xy M ( V ∩ U z ) , which shows that M ( V ) ⊗ O ( X ) O ( U ) = M ( V ∩ U ). (cid:3) . A ffi ne finite spaces 11
9. Theorem ([8] 2.5, 4.12):
Let ( X , O ) be an a ffi ne finite space. Consider the canonicalmorphism π : ( X , O ) → ( ∗ , O ( X )) , π ( x ) = ∗ , for any x ∈ X . The functors
Qc-Mod X π ∗ / / Mod O ( X ) , M ✤ / / π ∗ M = M ( X ) Mod O ( X ) π ∗ / / Qc-Mod X , M ✤ / / π ∗ M = ˜ Mestablish an equivalence between the category of quasi-coherent O X -modules and the cate-gory of O ( X ) -modules.Proof. The natural morphism π ∗ π ∗ M → M is an isomorphism because this morphism onstalks at x is the morphism M ( X ) ⊗ O ( X ) O x → M x , which is an isomorphism by Proposition2.8.The natural morphism M → π ∗ π ∗ M = ( π ∗ M )( X ) is an isomorphism: Tensoring the exactsequence of morphisms ( ∗ ), in the proof of Proposition 2.8, by M ⊗ O ( X ) − we obtain the exactsequence of morphisms M ⊗ O ( X ) O ( X ) → Y x ∈ X ( π ∗ M ) x / / / / Y x , y ∈ X , z ∈ U xy ( π ∗ M ) z which shows that M = M ⊗ O ( X ) O ( X ) = ( π ∗ M )( X ). (cid:3)
10. Lemma:
Let A → B and A ′ → B ′ be flat (resp. failthfully flat) morphisms of commuta-tive C-algebras. Then, A ⊗ C A ′ → B ⊗ C B ′ is a flat morphism (resp. faithfully flat).Proof. It follows from the equality M ⊗ A ⊗ C A ′ ( B ⊗ C B ′ ) = ( M ⊗ A B ) ⊗ A ′ B ′ . (cid:3)
11. Proposition:
The intersection of two a ffi ne open sets of an a ffi ne finite space is a ffi ne.Proof. Let U and U ′ be two a ffi ne open sets of the a ffi ne finite space X . Consider thefaithfully flat morphisms O ( U ) → Q x ∈ U O x , O ( U ′ ) → Q x ′ ∈ U ′ O x ′ . The composition of thefaithfully flat morphisms (recall Lemma 2.10) O ( U ′ ∩ U ) = O ( U ) ⊗ O ( X ) O ( U ′ ) → Y ( x , x ′ ) ∈ U × U ′ O xx ′ → Y ( x , x ′ ) ∈ U × U ′ , z ∈ U xx ′ O z , is faithfully flat, hence O ( U ′ ∩ U ) → Q z ∈ U ∩ U ′ O z is faithfully flat. By Proposition 2.6, U ∩ U ′ is a ffi ne. (cid:3) .1 Some conmutative algebra results. 12Let R be a commutative ring with a unit. A finite R-ringed space is a finite ringed space( X , O ) such that O is a sheaf of R -algebras; that is, for any x ∈ X , O x is an R -algebra and forany x ≤ x ′ , r xx ′ : O x → O x ′ is a morphism of R -algebras.Let X and Y be two finite R -ringed spaces. The direct product X × R Y is the finite R -ringedspace ( X × Y , O X × Y ), where ( O X × Y ) ( x , y ) : = O x ⊗ R O y , for each ( x , y ) ∈ X × Y and the morphismsof restriction are the obvious ones.
12. Proposition ([10] 5.27):
Let X and Y be a ffi ne finite R-ringed spaces. Then, X × R Y isan a ffi ne finite space and O ( X × R Y ) = O ( X ) ⊗ R O ( Y ) .Proof. Consider the universally exact sequence O ( X ) → Q x ∈ X O x / / / / Q x , x ′ ; z ∈ U xx ′ O z . Tensoringby ⊗ R O y we obtain the exact sequence O ( X ) ⊗ R O y → Y x ∈ X O ( U x × R U y ) / / / / Y x , x ′ ; z ∈ U xx ′ O ( U z × R U y ) . Hence, O ( X ) ⊗ R O y = O ( X × R U y ). Consider the universally exact sequence O ( Y ) → Q y ∈ Y O y / / / / Q y , y ′ ; z ∈ U yy ′ O z . Tensoring by O ( X ) ⊗ R we obtain the exact sequence O ( X ) ⊗ R O ( Y ) → Y y ∈ Y O ( X × R U y ) / / / / Y y , y ′ ; z ∈ U yy ′ O ( X × R U z ) . Hence, O ( X ) ⊗ R O ( Y ) = O ( X × R Y ). In particular, O xx ′ ⊗ R O yy ′ = O ( x , y )( x ′ , y ′ ) , for any x , x ′ ∈ X and y , y ′ ∈ Y . By Lemma 2.10, the morphism O ( X × R Y ) = O ( X ) ⊗ R O ( Y ) → Y x ∈ X O x ⊗ R Y y ∈ Y O y = Y ( x , y ) ∈ X × R Y O ( x , y ) is faithfully flat. By Lemma 2.10, the morphism O ( x , y )( x ′ , y ′ ) = O xx ′ ⊗ R O yy ′ → Y z ∈ U xx ′ O z ⊗ R Y z ′ ∈ U yy ′ O z ′ = Y ( z , z ′ ) ∈ U xx ′ × R U yy ′ O ( z , z ′ ) = Y ( z , z ′ ) ∈ U ( x , y )( x ′ , y ′ ) O ( z , z ′ ) is faithfully flat. Therefore, X × R Y is a ffi ne. (cid:3) If X is an a ffi ne finite space then, for each x ≤ y ∈ X , the morphism O x → O y is flat and O y ⊗ O x O y = O yy = O y . In this subsection we study this kind of morphisms. In this paper,we use well-known properties of flat morphisms and faithfully flat morphisms, that can befound in [7]..1 Some conmutative algebra results. 13
13. Notation :
Given p ∈ Spec R and an R -module M , we denote M p : = ( R \ p ) − · M .
14. Proposition :
Let f : A → B be a morphism of rings and f ∗ : Spec B → Spec
A theinduced morphism. The following conditions are equivalent:1. A → B is a flat morphism and B ⊗ A B = B.2. A f ∗ ( p ) = B f ∗ ( p ) , for all p ∈ Spec
B.3. The morphism f ∗ : Spec B → Spec
A is injective and A f ∗ ( p ) = B p , for any p ∈ Spec
B.Proof. . ⇒ . The morphism A f ∗ ( p ) → B f ∗ ( p ) is faithful flat, for any p . Besides, B f ∗ ( p ) = A f ∗ ( p ) ⊗ A B = A f ∗ ( p ) ⊗ A ( B ⊗ A B ) = B f ∗ ( p ) ⊗ A f ∗ ( p ) B f ∗ ( p ) , then A f ∗ ( p ) = B f ∗ ( p ) .2 . ⇒ . If A f ∗ ( p ) = B f ∗ ( p ) then B f ∗ ( p ) = B p and f ∗− ( f ∗ ( p )) = { p } , then f ∗ is injective.3 . ⇒ . The morphism A → B is flat: Given an injective morphism N ֒ → M of A -modules, N f ∗ ( p ) → M f ∗ ( p ) is injective, for any p . Then, N ⊗ A B p → M ⊗ A B p is injective, forany p and N ⊗ A B → M ⊗ A B is injective.Spec B p ⊆ Spec B f ∗ ( p ) ⊆ Spec A f ∗ ( p ) and Spec B p = Spec A f ∗ ( p ) . Hence, Spec B p = Spec B f ∗ ( p ) and B f ∗ ( p ) = B p . Then, ( B ⊗ A B ) p = ( B ⊗ A B ) ⊗ B B p = ( B ⊗ A B ) ⊗ B B f ∗ ( p ) = ( B ⊗ A B ) ⊗ A A f ∗ ( p ) = B f ∗ ( p ) ⊗ A f ∗ ( p ) B f ∗ ( p ) = B p , for any p ∈ Spec B . Therefore, B ⊗ A B = B . (cid:3)
15. Notation :
Given a morphism f : A → B and an ideal I ⊆ B denote A ∩ I : = f − ( I ).Denote ( I ) = { p ∈ Spec B : I ⊆ p } .
16. Proposition:
Let A → B be a flat morphism of rings such that B ⊗ A B = B. Then,1. ( I ∩ A ) · B = I, for any ideal I ⊆ B.2.
Spec
B is a topological subspace of
Spec
A, with their Zariski topologies.3. Let q ∈ Spec
A.(a) If q < Spec
B, then q · B = B.(b) If q ∈ Spec
B, then q · B ⊂ B is a prime ideal and ( q · B ) ∩ A = q .4. Spec B = ∩ Spec B ⊆ open set U ⊆ Spec A U.Proof.
1. Let p ∈ Spec B , q : = A ∩ p and M a B -module. By Proposition 2.14, M p = M ⊗ B B p = M ⊗ B B q = M q . Then,[( I ∩ A ) · B ] p = [( I ∩ A ) · B ] q = ( I q ∩ A q ) · B q = I q = I p . Hence, ( I ∩ A ) · B = I .. Schematic finite spaces 142. By Proposition 2.14, we can think Spec B as a subset of Spec A . Given an ideal I ⊆ B ,observe that ( I ) = (( I ∩ A ) · B ) = ( I ∩ A ) ∩ Spec B .3. (a) Suppose that there exists a prime ideal p ⊂ B that contains to q · B . Denote p ′ = p ∩ A . Then, q ∈ Spec A p ′ = Spec B p ⊆ Spec B . , which is contradictory. (b) Let p ∈ Spec B be a prime ideal such that p ∩ A = q . Then, p = ( p ∩ A ) · B = q · B .4. If q ∈ Spec A \ Spec B , then ( q ) ∩ Spec B = ( q · B ) = ( B ) = ∅ . Then, Spec B is equalto the intersection of the open sets U ⊆ Spec A such that Spec B ⊆ U . (cid:3)
1. Definition :
We say that a finite ringed space ( X , O ) is a schematic finite space if it islocally a ffi ne; i.e. if there exists an open covering { U i } i ∈ I on X , such that U i is an a ffi ne finitespace, for each i ∈ I .
2. Proposition :
Let X be a finite ringed space. X is a schematic finite space i ff the opensubsets U x are a ffi ne finite spaces for all x ∈ X.Proof. ⇒ ) Let { U i } i ∈ I be an a ffi ne open covering of X . For each x ∈ X , U x is an open subsetof one of the a ffi ne finite spaces U i . So, it follows from Corollary 2.7, that U x is also a ffi ne. ⇐ ) It is clear, since { U x } x ∈ X is an open covering of X . (cid:3)
3. Remarks :
1. All schematic finite spaces are finite fr-spaces.2. A ffi ne finite spaces are schematic.3. If X = U x , then X schematic if and only if it is a ffi ne.The finite ringed space associated with a minimal a ffi ne finite covering U of a quasi-compact and quasi-separated scheme is a schematic finite space, by Paragraph 0.1.
4. Examples :
Let us give some examples of schematic finite spaces (below we indicate theringed space constructed in Paragraph 0.5):.1 Definition, examples and first characterizations 15 k [ x ] ) ) ❙❙❙❙❙❙❙ k [ x , y ] / / & & ▲▲▲▲▲▲▲▲▲▲▲▲▲ k [ x , y , / y ] + + ❲❲❲❲❲❲❲❲ k [ x , / x ] k [1 / x , y / x ] / / ❤❤❤❤❤❤ k [1 / x , y / x , x / y ] / / k [ x , y , / x , / y ] k [1 / x ] ❦❦❦❦❦❦ k [1 / y , x / y ] / / ❤❤❤❤❤❤ k [ y , / y , x / y ] ❣❣❣❣❣❣❣
1. Projective line 2. Projective plane k [ x ] * * ❯❯❯❯❯❯❯❯❯❯ k [ x ] + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ k [ x , / x ] k ( x ) k [ x ] ✐✐✐✐✐✐✐✐✐✐ k [ x ] ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
3. A ffi ne line with a double point 4. Two lines glued at the generic pointIt can be proved that the first three examples are finite models of the schemes we indicate,but the fourth it is not the model of any scheme. Also note that none of these examples area ffi ne finite spaces.If X and Y are schematic finite R -ringed spaces, then X × R Y is an schematic finite space,by Proposition 2.12.
5. Proposition ([8] 4.11):
Let X be a finite ringed space. X is a schematic finite space ifand only if it satisfies the following two conditions:1. The natural morphism O y ⊗ O x O y ′ → O yy ′ is an isomorphism for any y , y ′ ≥ x.2. The natural morphism O yy ′ → Q z ∈ U yy ′ O z is faithfully flat, for any y , y ′ ∈ X for whichthere is an element x ∈ X such that y ≥ x and y ′ ≥ x.Proof. It follows easily from the definition of a ffi ne finite space that the open subsets { U x } x ∈ X are a ffi ne finite spaces i ff the conditions 1. and 2. above are satisfied. (cid:3)
6. Proposition ([8] 4.11):
Let ( X , O X ) be a ringed finite space. X is a schematic finite spacei ff the morphism O y ⊗ O x O y ′ → Y z ∈ U yy ′ O z is faithfully flat, for any x ≤ y , y ′ ∈ X.Proof.
It follows directly from Corollary 2.5. (cid:3) .2 More characterizations of schematic finite spaces 16
In this section, we see that schematic finite spaces can be characterized by the good behaviorof their quasi-coherent modules.
7. Proposition :
Let X be a schematic finite space, U i ⊆ X an open subset and N a quasi-coherent O U -module. Then, i ∗ N is a quasi-coherent O X -module.Proof. Let x ≤ y ∈ X . We have to see that the morphism ( i ∗ N ) x ⊗ O x O y → ( i ∗ N ) y is anisomorphism. This morphism is equal to the morphism N ( U ∩ U x ) ⊗ O ( U x ) O ( U y ) → N ( U ∩ U y ) , which is an isomorphism by Proposition 2.8. (cid:3)
8. Theorem :
Let X be a finite ringed space. X is a schematic finite space if and only if itsatisfies the next two conditions:1.
Ker f is quasi-coherent, for any morphism f : M → N of quasi-coherent O X -modules.2. For any open subset i : U x ֒ → X and any quasi-coherent O U x -module M , the O X -module i ∗ M is quasi-coherent.Proof. ⇒ ) We know that schematic finite spaces are finite fr-spaces. By Proposition 1.7,Ker f is quasi-coherent. The second condition follows from the proposition above. ⇐ ) First, let us prove that X is an fr-space. Let i : U x ֒ → X be an open subset and M → N a morphism of quasi-coherent O U x -modules. Ker[ M → N ] = Ker[ i ∗ M → i ∗ N ] | U x ,then it is quasi-coherent. By Proposition 1.7, X is an fr-space.If X is an fr-space and satisfies condition 2., then it is schematic:Consider x ≤ x ′ , let j : U x ′ ֒ → U x be the inclusion morphism and N a quasi-coherent O U x ′ -module. Since condition 2. is satisfied, the O U x -module j ∗ N = i ∗ (( i ◦ j ) ∗ N ) is quasi-coherent. It follows from this result that we can suppose X = U x (because being finitefr-space and schematic are local conditions).Now, by Corollary 2.5, we only have to prove that, for each y , y ′ ≥ x , the morphism O y ⊗ O x O y ′ → Y z ∈ U yy ′ O z is faithfully flat.Consider the open subset i : U y ֒ → X = U x . Since i ∗ O U y is quasi-coherent, O yy ′ = ( i ∗ O U y )( U y ′ ) = ( i ∗ O U y )( U x ) ⊗ O x O y ′ = O y ⊗ O x O y ′ . .2 More characterizations of schematic finite spaces 17In particular, O y = O yy = O y ⊗ O x O y . The morphism O y → O z is flat, for any z ≥ y , y ′ , thenthe morphism O y ⊗ O x O y ′ → O z ⊗ O x O y ′ = O zy ′ = O z is flat.If the morphism O y ⊗ O x O y ′ → Q z ∈ U yy ′ O z is not faithfully flat, there exists an ideal I ⊂ , O y ⊗ O x O y ′ such that I · Q z ∈ U yy ′ O z = Q z ∈ U yy ′ O z . Observe that the morphism O y → O y ⊗ O x O y ′ is flat since O x → O y ′ is flat. Besides,( O y ⊗ O x O y ′ ) ⊗ O y ( O y ⊗ O x O y ′ ) = O y ⊗ O x ( O y ′ ⊗ O x O y ′ ) = O y ⊗ O x O y ′ . By Proposition 2.16, there exists an ideal J ⊂ O y such that J · ( O y ⊗ O x O y ′ ) = I . Let M be the quasi-coherent O U y -module associated with the O y -module O y / J . Then, i ∗ M is thequasi-coherent O X -module associated with the O x -module O y / J and M ( U yy ′ ) = ( i ∗ M )( U y ′ ) = ( O y / J ) ⊗ O x O y ′ = ( O y ⊗ O x O y ′ ) / J · ( O y ⊗ O x O y ′ ) = ( O y ⊗ O x O y ′ ) / I , . However, M | U yy ′ =
0, since M z = ( O y / J ) ⊗ O y O z = O z / J · O z = O z / I · O z =
0, for any z ∈ U yy ′ .So we have a contradiction; therefore, the morphism O y ⊗ O x O y ′ → Q z ∈ U yy ′ O z is faithfullyflat. (cid:3)
9. Theorem :
Let ( X , O ) be a finite ringed space. Let δ : X → X × X, δ ( x ) = ( x , x ) be thediagonal morphism. Then, X is schematic i ff it satisfies these two conditions:1. Ker f is quasi-coherent, for any morphism f : M → N of quasi-coherent O X -modules.2. δ ∗ N is a quasi-coherent O X × X -module for any quasi-coherent O X -module N .Proof. ⇒ ) For any ( x , y ) ≤ ( x ′ , y ′ ), we have( δ ∗ N ) ( x , y ) ⊗ O ( x , y ) O ( x ′ , y ′ ) = N xy ⊗ O x ⊗O y ( O x ′ ⊗ O y ′ ) = N xy ⊗ O x O x ′ ⊗ O y O y ′ = N x ′ y ⊗ O y O y ′ = N x ′ y ′ = ( δ ∗ N ) ( x ′ , y ′ ) . ⇐ ) First, note that for any x ∈ X and any x ′ ≤ x ′′ , O xx ′ ⊗ O x ′ O x ′′ = ( δ ∗ O ) ( x , x ′ ) ⊗ O x O x ⊗ O x ′ O x ′′ = ( δ ∗ O ) ( x , x ′ ) ⊗ O ( x , x ′ ) O ( x , x ′′ ) = ( δ ∗ O ) ( x , x ′′ ) = O xx ′′ . In consequence, for any open subset i : U x ֒ → X and any quasi-coherent O U x -module M there exists a quasi-coherent O X -module N such that N | U x ≃ M : define N x ′ : = M x ⊗ O x O xx ′ ,for any x ′ ∈ X . N is quasi-coherent since for any x ′ ≤ x ′′ , N x ′ ⊗ O x ′ O x ′′ = M x ⊗ O x O xx ′ ⊗ O x ′ O x ′′ = M x ⊗ O x O xx ′′ = N x ′′ . Besides, N x = M x , so N | U x = M ..2 More characterizations of schematic finite spaces 18By Theorem 3.8, we have to prove that i ∗ M is a quasi-coherent O X -module. That is, wehave to prove that ( i ∗ M ) y ′ = ( i ∗ M ) y ⊗ O y O y ′ , for any y ≤ y ′ ∈ X :( i ∗ M ) y ′ = M ( U y ′ x ) = N ( U y ′ x ) = ( δ ∗ N )( U y ′ × U x ) = ( δ ∗ N )( U y × U x ) ⊗ O y ⊗O x O y ′ ⊗ O x = N ( U yx ) ⊗ O y O y ′ = M ( U yx ) ⊗ O y O y ′ = ( i ∗ M ) y ⊗ O y O y ′ . (cid:3)
10. Remark :
The theorem above can be restated by saying that a finite ringed space X isschematic i ff it is a finite fr-space and for any quasi-coherent module N , any x ∈ X and any x ′ ≤ x ′′ ∈ X , N xx ′ ⊗ O x ′ O x ′′ = N xx ′′ .
11. Proposition :
Let X be a finite fr-space and N a quasi-coherent O X -module. Then, N pq ⊗ O p O p ′ = N p ′ q , for any p ≤ p ′ ∈ X and for any q ∈ X i ff N p ′ ⊗ O p O p ′′ = N p ′ p ′′ , for anyp ≤ p ′ , p ′′ ∈ X.Proof. ⇒ ) N p ′ ⊗ O p O p ′′ = N pp ′ ⊗ O p O p ′′ = N p ′ p ′′ . ⇐ ) Let U ⊆ U p be an open set and p ≤ p ′ . Consider the exact sequence of morphisms N ( U ) → Y x ∈ U N x / / / / Y z ≥ x ∈ U N z . Tensoring by ⊗ O p O p ′ we obtain the exact sequence of morphisms N ( U ) ⊗ O p O p ′ → Y x ∈ U N p ′ x / / / / Y z ≥ x ∈ U N p ′ z , which shows that N ( U ) ⊗ O p O p ′ = N ( U ∩ U p ′ ). In particular, N pq ⊗ O p O p ′ = N p ′ q . (cid:3)
12. Corollary :
Let X be a finite ringed space. X is schematic i ff it is a finite fr-space andfor any quasi-coherent O X -module N and any x ≤ x ′ , x ′′ ∈ X, N x ′ ⊗ O x O x ′′ = N x ′ x ′′ Proof.
It follows directly from Theorem 3.9 and Proposition above. (cid:3) . A ffi ne morphisms 19 ffi ne morphisms
1. Definition :
Let X and Y be schematic finite spaces. A morphism f : X → Y of ringedspaces is said to be an a ffi ne morphism if f ∗ O X is a quasi-coherent O Y -module and the preim-age of any a ffi ne open subspace of Y is an a ffi ne open subspace of X .
2. Examples :
1. A schematic finite space X is a ffi ne i ff ( X , O ) → ( ∗ , O ( X )) is an a ffi nemorphism.2. If X is an a ffi ne finite space and U ⊆ X an a ffi ne open subset, the inclusion morphism i : U ֒ → X is an a ffi ne morphism: i ∗ O U is quasi-coherent by Proposition 2.8, and for anya ffi ne open subset V ⊆ Y , i − ( V ) = V ∩ U is a ffi ne by Proposition 2.11.3. Let X be a schematic finite space. Given x . x ′ ∈ X , we shall say that x ∼ x ′ if x ≤ x ′ and x ′ ≤ x . Let ¯ X : = X / ∼ be the Kolmogorov quotient of X and define O [ x ] : = O x , for any[ x ] ∈ ¯ X . Then, ¯ X is a schematic space, the quotient morphism π : ¯ X → X , π ( x ) : = [ x ] isa ffi ne and π ∗ O X = O ¯ X .
3. Proposition :
Let X and Y be a ffi ne finite spaces and f : X → Y an a ffi ne morphism. LetM be an O ( X ) -module (therefore, an O ( Y ) -module). Then,f ∗ ˜ M = ˜ M . Proof.
For any open set U y ,( f ∗ ˜ M )( U y ) = ˜ M ( f − ( U y )) = ˜ M ( X ) ⊗ O ( X ) O X ( f − ( U y )) = M ⊗ O ( X ) O X ( f − ( U y )) = M ⊗ O ( X ) ( f ∗ O X )( U y ) = M ⊗ O ( X ) O ( X ) ⊗ O ( Y ) O Y ( U y ) = M ⊗ O ( Y ) O Y ( U y ) = ˜ M ( Y ) ⊗ O ( Y ) O Y ( U y ) = ˜ M ( U y ) . (cid:3)
4. Proposition:
Let f : X → Y be an a ffi ne morphism and M a quasi-coherent O X -module.Then, f ∗ M is a quasi-coherent O Y -module.Proof. Being f ∗ M a quasi-coherent O Y -module is a local property. We can suppose that Y is a ffi ne. Then X is a ffi ne. The proof is completed by the previous proposition. (cid:3)
5. Proposition:
The composition of a ffi ne morphisms is a ffi neProof. It is obvious. (cid:3)
6. Proposition :
Let X and Y be schematic finite spaces. A morphism of ringed spacesf : X → Y is a ffi ne i ff f ∗ O X is quasi-coherent and f − ( U y ) is a ffi ne for any y ∈ Y. . A ffi ne morphisms 20 Proof. ⇒ ) It is obvious. ⇐ ) Let us proceed by induction on Y . If Y =
1, it is obvious. We can suppose that Y is a ffi ne and we only have to prove that X is a ffi ne. The morphism O ( Y ) → Q y ∈ Y O y isfaithfully flat, then the morphism O ( X ) → Y y ∈ Y O ( X ) ⊗ O ( Y ) O y = Y y ∈ Y ( f ∗ O X )( Y ) ⊗ O ( Y ) O y = Y y ∈ Y O ( f − ( U y ))is faithfully flat. Since f − ( U y ) is a ffi ne, the morphism O ( f − ( U y )) → Q x ∈ f − ( U y ) O x is faith-fully flat. The composition of faithfully flat morphisms is faithfully flat, then O ( X ) → Q y ∈ Y , x ∈ f − ( U y ) O x is faithfully flat. Therefore, the morphism O ( X ) → Y x ∈ X O x is faithfully flat.Let x , x ′ ∈ X . Given an open set V ⊆ Y denote ¯ V : = f − ( V ). ¯ U f ( x ) f ( x ′ ) is an a ffi ne opensubset of ¯ U f ( x ) (by induction hypothesis), then ¯ U f ( x ) f ( x ′ ) ∩ U x is a ffi ne, and it is included in¯ U f ( x ′ ) , hence ¯ U f ( x ) f ( x ′ ) ∩ U x ∩ U x ′ is a ffi ne. Then, U xx ′ = ¯ U f ( x ) f ( x ′ ) ∩ U x ∩ U x ′ is a ffi ne and the morphism O xx ′ → Q z ∈ U xx ′ O z is faithfully flat. Since f ∗ O X is quasi-coherent(and Proposition 2.8), O ( ¯ U f ( x ) ) = f ∗ O X ( U f ( x ) ) = O ( X ) ⊗ O ( Y ) O f ( x ) , O ( ¯ U f ( x ′ ) ) = O ( X ) ⊗ O ( Y ) O f ( x ′ ) . O ( ¯ U f ( x ) f ( x ′ ) ) = f ∗ O X ( U f ( x ) f ( x ′ ) ) = O ( X ) ⊗ O ( Y ) O f ( x ) f ( x ′ ) = O ( X ) ⊗ O ( Y ) O f ( x ) ⊗ O ( Y ) O f ( x ′ ) = ( O ( X ) ⊗ O ( Y ) O f ( x ) ) ⊗ O ( X ) ( O ( X ) ⊗ O ( Y ) O f ( x ′ ) ) = O ( ¯ U f ( x ) ) ⊗ O ( X ) O ( ¯ U f ( x ′ ) . ) ( ∗ )Now it is easy to prove that O xx ′ = O ( ¯ U f ( x ) f ( x ′ ) ∩ U x ∩ U x ′ ) = ( O ( ¯ U f ( x ) f ( x ′ ) ) ⊗ O ( ¯ U f ( x ) ) O x ) ⊗ O ( ¯ U f ( x ′ ) ) O x ′ ( ∗ ) = O x ⊗ O ( X ) O x ′ . Therefore, X is a ffi ne. (cid:3)
7. Corollary :
Let X and Y be schematic finite spaces and let f : X → Y be a morphism ofringed spaces. Then, being f a ffi ne is a local property on Y.
8. Example :
Let ( X , O ) be a schematic finite space and O → O ′ a morphism of sheavesof rings, such that O ′ is a quasi-coherent O -module. ( X , O ′ ) is a schematic finite space:Given x ≤ y , y ′ , the morphism O y ⊗ O x O y ′ → Q z ∈ U yy ′ O z is faithfully flat, by Proposition 3.6.Tensoring by ⊗ O x O ′ x we obtain the faithfully flat morphism O ′ y ⊗ O ′ x O ′ y ′ → Q z ∈ U yy ′ O ′ z . Hence,( X , O ′ ) is a schematic finite space by Proposition 3.6. The obvious morphism Id : ( X , O ′ ) → ( X , O ) is a ffi ne.. Schematic morphisms 21
1. Definition :
Let X , Y be schematic finite spaces. A morphism of ringed spaces f : X → Y is said to be a schematic morphism if for any x ∈ X the morphism f x : U x → U f ( x ) , f x ( x ′ ) : = f ( x ′ ) is a ffi ne.
2. Example : If X is a schematic finite space, X → ( ∗ , O ( X )) is a schematic morphism.
3. Example : If U is an open subspace of a schematic finite space X , then the inclusionmorphism U ֒ → X is schematic.
4. Remark :
Let X and Y be schematic finite spaces and f : X → Y be a morphism of ringedspaces. Then, being f schematic is a local property on Y and on X .
5. Proposition:
The composition of schematic morphisms is schematic.Proof.
It is a consequence of Proposition 4.5. (cid:3)
6. Proposition : A ffi ne morphisms between schematic finite spaces are schematic mor-phisms.Proof. Let f : X → Y be an a ffi ne morphism. Then, f − ( U f ( x ) ) is a ffi ne, U x ֒ → f − ( U f ( x ) )is an a ffi ne morphism and f − ( U f ( x ) ) → U f ( x ) is a ffi ne, by Corollary 4.7. The composition U x ֒ → f − ( U f ( x ) ) → U f ( x ) is a ffi ne, by Proposition 4.5. Hence, f is a schematic morphism. (cid:3)
7. Proposition :
Let f : X → Y be a schematic morphism and M a quasi-coherent O X -module. Then, f ∗ M is a quasi-coherent O Y -module.Proof. We can suppose that Y is a ffi ne. Consider an open set U x i ⊆ X and denote M U x = i ∗ M | U x . Observe that f ∗ M U x = ( f ◦ i ) ∗ M | U x is a quasi-coherent O Y -module, because thecomposite morphism f ◦ i : U x → U f ( x ) ֒ → Y is a ffi ne and by Proposition 4.4.Let { U x , . . . , U x n } be an open covering of X and { U x ijk } k an open covering of U x i ∩ U x j ,for each i , j . Consider the exact sequence of morphisms M → Y i M U xi / / / / Y i , j , k M U xijk Taking f ∗ , we obtain an exact sequence of morphisms, then f ∗ M is a quasi-coherent O Y -module. (cid:3)
8. Corollary:
Let X be a schematic finite space and U i ֒ → X an open subset. Given a quasi-coherent O U -module N , there exists a quasi-coherent O X -module M , such that M | U ≃ N . . Schematic morphisms 22 Proof.
Define M : = i ∗ N . (cid:3)
9. Lemma :
Let X be an a ffi ne finite space and U ⊂ X an open set. Then, U is a ffi ne i ff U ∩ U x is a ffi ne, for any x ∈ X.Proof. If U is a ffi ne, then U ∩ U x is a ffi ne, for any x ∈ X , by Proposition 2.11. Let us provethe converse implication. The inclusion morphism i : U ֒ → X is an a ffi ne morphism, byProposition 4.6. Hence, U = i − ( X ) is a ffi ne. (cid:3)
10. Proposition :
A morphism of ringed spaces f : X → Y between a ffi ne finite spaces isa ffi ne i ff it is a schematic morphism.Proof. ⇒ ) It is known (see Proposition 5.6). ⇐ ) By Proposition 5.7, we only have to prove that f − ( U ) is a ffi ne, for any a ffi ne opensubset U ⊆ Y . By the previous lemma, we only have to prove that f − ( U ) ∩ U x is a ffi ne.The composition of a ffi ne morphisms is a ffi ne, then U x → U f ( x ) ֒ → Y is a ffi ne. Hence, f − ( U ) ∩ U x is a ffi ne. (cid:3)
11. Corollary:
Let f : X → Y be a schematic morphism. Then, f is a ffi ne i ff there exists ana ffi ne open covering of Y, { U i } , such that f − ( U i ) is a ffi ne, for any i.Proof. Recall that being f a ffi ne is a local property on Y . (cid:3) In [10] 5.6, it is proved that a morphism of ringed spaces f : X → Y is schematic i ff R i f ∗ M is quasi-coherent for any quasi-coherent module M and any i .
12. Theorem :
Let X and Y be schematic finite spaces and f : X → Y a morphism ofringed spaces. Then, f is a schematic morphism i ff f ∗ M is quasi-coherent, for any quasi-coherent O X -module M .Proof. ⇒ ) Recall Proposition 5.7. ⇐ ) We can suppose that X and Y are a ffi ne. We only have to prove that f − ( U y ) isa ffi ne, for any y ∈ Y , by Proposition 4.6. By Proposition 2.6, we only have to prove that themorphism O X ( f − ( U y )) → Q x ∈ f − ( U y ) O x is faithfully flat. Observe that O X ( f − ( U y )) = ( f ∗ O X )( U y ) = ( f ∗ O X )( Y ) ⊗ O Y ( Y ) O y = O X ( X ) ⊗ O Y ( Y ) O y and for any x ∈ f − ( U y ) O x ⊗ O Y ( Y ) O y = O x ⊗ O y O y ⊗ O Y ( Y ) O y = O x ⊗ O y O y = O x . . Schematic morphisms 23The morphism O X ( X ) → O x is flat, then tensoring by ⊗ O Y ( Y ) O y the morphism O X ( f − ( U y )) →O x is flat, for any x ∈ f − ( U y ).If the morphism O X ( f − ( U y )) → Q x ∈ f − ( U y ) O x is not faithfully flat, there exists an ideal I ⊂ , O X ( f − ( U y )) such that I · Q x ∈ f − ( U y ) O x = Q x ∈ f − ( U y ) O x . Observe that the morphism O X ( X ) → O X ( X ) ⊗ O Y ( Y ) O y = O X ( f − ( U y )) is flat since O Y ( Y ) → O y is flat. Besides, O X ( f − ( U y )) ⊗ O X ( X ) O X ( f − ( U y )) = O X ( X ) ⊗ O Y ( Y ) O y ⊗ O X ( X ) O X ( X ) ⊗ O Y ( Y ) O y = O X ( X ) ⊗ O Y ( Y ) O y ⊗ O Y ( Y ) O y = O X ( X ) ⊗ O Y ( Y ) O y = O X ( f − ( U y )) . By Proposition 2.16, there exists an ideal J ⊂ O X ( X ) such that J · O X ( f − ( U y )) = I . Let M be the quasi-coherent O X -module associated with the O X ( X )-module O X ( X ) / J . Then, f ∗ M is the quasi-coherent O Y -module associated with the O Y ( Y )-module O X ( X ) / J and M ( f − ( U y )) = f ∗ M ( U y ) = f ∗ M ( Y ) ⊗ O Y ( Y ) O y = ( O X ( X ) / J ) ⊗ O Y ( Y ) O y = ( O X ( X ) ⊗ O Y ( Y ) O y ) / J · ( O X ( X ) ⊗ O Y ( Y ) O y ) = O X ( f − ( U y )) / I , . However, M | f − ( U y ) = M x = O x / J · O x = O x / I · O x =
0, for any x ∈ f − ( U y ). This iscontradictory, then the morphism O X ( f − ( U y )) → Q x ∈ f − ( U y ) O x is faithfully flat. (cid:3)
13. Notation :
Let f : X → Y be a morphism of ringed spaces, between ringed finite spaces, x ∈ X and y ∈ Y . We shall denote U xy : = U x ∩ f − ( U y ) and O xy : = O ( U x ∩ f − ( U y )).
14. Proposition:
A morphism of ringed spaces f : X → Y between schematic finite spacesis schematic i ff for any x ∈ X and y ≥ f ( x )
1. U xy is a ffi ne.2. O xy = O x ⊗ O f ( x ) O y .Proof. Consider the morphism f x : U x → U f ( x ) . Then, f x ∗ O U x is a quasi-coherent modulei ff condition 2. is satisfied. By Proposition 4.6, f x is a ffi ne i ff the conditions 1. and 2. aresatisfied. Then, f is schematic i ff the conditions 1. and 2. are satisfied. (cid:3)
15. Theorem :
Let f : X → Y be a morphism of ringed spaces between schematic finitespaces. Then, f is schematic i ff the induced morphism on spectra by the morphism of rings O x ⊗ O f ( x ) O y → Q z ∈ U xy O z is surjective, for any x and y ≥ f ( x ) .Proof. ⇒ ) By Proposition 5.14, U xy is a ffi ne and O x ⊗ O f ( x ) O y = O xy . Therefore, the morphism O x ⊗ O f ( x ) O y = O xy → Y z ∈ U xy O z . Removable points. Minimal schematic space 24is faithfully flat, and the induced morphism on spectra is surjective. ⇐ ) Let z ∈ U xy . Since O x → O z is a flat morphism, the morphism O x ⊗ O f ( x ) O y → O z ⊗ O f ( x ) O y = O z ⊗ O f ( z ) O f ( z ) ⊗ O f ( x ) O y = O z ⊗ O f ( z ) O f ( z ) = O z is flat. Denote B = O x ⊗ O f ( x ) O y and C = Q z ∈ U xy O z . The morphism B → C is faithfully flat.Let z , z ′ ∈ U xy . The morphism O z ⊗ O x O z ′ → O z ⊗ B O z ′ is surjective and the compositemorphism O z ⊗ O x O z ′ → O z ⊗ B O z ′ → O zz ′ is an isomorphism. Therefore, O z ⊗ B O z ′ = O zz ′ . The exact sequence of morphisms B → C = Y z ∈ U xy O z / / / / C ⊗ B C = Y z , z ′ ∈ U xy O zz ′ shows that B = O xy . Therefore the morphism O xy → Q z ∈ U xy O z is faithfully flat. By Proposi-tion 2.6, U xy is a ffi ne. By Proposition 5.14, f is schematic. (cid:3)
1. Proposition :
Let X be a schematic finite space. Let p ∈ X be a point such that themorphism O p → Q q > p O q is faithfully flat . Consider the ringed subspace Y = X − { p } of X ( O Y , y : = O X , y , for any y ∈ Y). Then, Y is a schematic finite space, the inclusion mapi : Y ֒ → X is an a ffi ne morphism and i ∗ O Y = O X .Proof. Y is a schematic finite space by Proposition 3.6, because X is a schematic finitespace. Let us prove that i : Y ֒ → X is a ffi ne and i ∗ O Y = O X : Consider U x ⊂ X . If x , p ,then i − ( U x ) = U x and ( i ∗ O Y )( U x ) = O Y , x = O X , x = O X ( U x ). If x = p denote U : = U p − { p } .Observe that O X | U = O Y | U . The morphism A = O p → Q x ∈ U O x = B is faithfully flat. Theexact sequence of morphisms A → B / / / / B ⊗ A B and the equality B ⊗ A B = Q x , y ∈ U O xy show that A = O X ( U ) = O Y ( U ). By Proposition 2.6, U is a ffi ne. Then, i − ( U p ) = U is a ffi ne and ( i ∗ O Y )( U p ) = O Y ( U ) = A = O X ( U p ). (cid:3)
2. Remark : If U is a ffi ne then the morphism O U → Q q ∈ U O q is faithfully flat. We haveproved that O p → Q q > p O q is faithfully flat i ff U : = U p − { p } is a ffi ne and O p = O ( U ).
3. Lemma :
Let X be an a ffi ne finite space and U ⊆ X an a ffi ne open subset. Then, therestriction morphism O ( X ) → O ( U ) is flat. If I : = { q ∈ X : q > p } = ∅ , define Q q ∈ I O q : = { } . . Removable points. Minimal schematic space 25 Proof.
The morphism O ( U ) → Q x ∈ U O x is faithfully flat and the composite morphism O ( X ) → O ( U ) → Q x ∈ U O x is flat. Then, the morphism O ( X ) → O ( U ) is flat. (cid:3)
4. Proposition :
Let X be a schematic finite space. Let p ∈ X be a point such that themorphism O p → Q q > p O q is faithfully flat and let Y : = X − { p } . An open set V ⊆ X is a ffi nei ff V ∩ Y is a ffi neProof. We only have to prove the converse implication. We can suppose that V = Y and wehave to prove that X is a ffi ne.By Lemma 6.3, the morphism O X ( X ) = O Y ( Y ) → O Y ( U x ∩ Y ) = O X ( U x ) = O x is flat,for any x ∈ X . Then, the morphism O ( X ) = O ( Y ) → Q y ∈ Y O Y , y = Q y ∈ Y O X , y is faithfully flat.Therefore the morphism O ( X ) → Q x ∈ X O X , x is faithfully flat. Likewise, given U x , U x ′ ⊆ X ,the morphism O X ( U x ∩ U x ′ ) → Q x ′′ ∈ U x ∩ U x ′ O X , x ′′ is faithfully flat. Besides, O x ⊗ O ( X ) O x ′ = O Y ( U x ∩ Y ) ⊗ O ( Y ) O Y ( U x ′ ∩ Y ) = O Y ( U x ∩ Y ∩ U x ′ ∩ Y ) = O X ( U x ∩ U x ′ )Therefore, X is a ffi ne. (cid:3)
5. Definition :
Let X be a schematic finite space. We shall say that x ∈ X is removable if O x → Q x ′ > x O x ′ is faithfully flat.If O x = x is a removable point.
6. Proposition :
Let X be a schematic finite space and let p , p ′ ∈ X be two points. Then,p , p ′ are removable points of X i ff p is a removable point of X and p ′ is a removable point ofX − p.Proof. It is immediate. (cid:3)
7. Proposition:
Let U ′ ⊆ U be a ffi ne open subsets of a schematic finite space X and supposethat the morphism O ( U ) → O ( U ′ ) is faithfully flat. Then, U − U ′ is a set of removable pointsof X.Proof. O ( U ) = O ( U ′ ) because the morphism O ( U ) → O ( U ′ ) is faithfully flat and O ( U ) ⊗ O ( U ) O ( U ′ ) = O ( U ′ ) = O ( U ′ ) ⊗ O ( U ) O ( U ′ ) . Let x ∈ U − U ′ , then O x = O ( U ) ⊗ O ( U ) O x = O ( U ′ ) ⊗ O ( U ) O x = O ( U ′ ∩ U x ). By Proposition2.11, U ′ ∩ U x is a ffi ne, then the morphism O x = O ( U ′ ∩ U x ) → Q x ′ ∈ U ′ ∩ U x O x ′ is faithfullyflat. Hence, O x → Q y > x O y is faithfully flat, and x is a removable point. (cid:3)
8. Remark :
In addition, we have proved that O ( U ) = O ( U ′ ).. Removable points. Minimal schematic space 26
9. Definition :
A schematic finite space X is said to be minimal if there are no removablepoints in X and it is T . Let ˜ X the Kolmogorov space of X and P be the set of all theremovable points of ˜ X , we shall denote X M : = ˜ X \ P .By Proposition 6.1, X M is a schematic finite space, the natural morphism X M i ֒ → ˜ X isa ffi ne and O ˜ X = i ∗ O X M .
10. Proposition :
Let f : X → Y be a schematic morphism. If x ∈ X is not a removablepoint, then f ( x ) ∈ Y is not a removable point. Then, we have the commutative diagramX f (cid:15) (cid:15) / / ˜ X ˜ f (cid:15) (cid:15) X M ? _ o o f M (cid:15) (cid:15) Y / / ˜ Y Y M ? _ o o where ˜ X and ˜ Y are the Kolmogorov spaces of X and Y respectively, and ˜ f and f M are theinduced morphisms.Proof. Consider the a ffi ne morphism f x : U x → U f ( x ) , f x ( x ′ ) : = f ( x ′ ). Since f x ∗ O U x is aquasi-coherent O U f ( x ) -module, then O x ⊗ O f ( x ) O y = O ( f − x ( U y )), for any y ∈ U f ( x ) . If f ( x )is a removable point, then O f ( x ) → Q y > f ( x ) O y is faithfully flat. Tensoring by O x ⊗ O f ( x ) , onehas the faithfully flat morphism O x → Q y > f ( x ) O ( f − x ( U y )). The open sets f − x ( U y ) are a ffi ne,because f x : U x → U f ( x ) is a ffi ne. Then, the morphisms O ( f − x ( U y )) → Q x ′ ∈ f − x ( U y ) O x ′ arefaithfully flat. Hence, the morphism O x → Q x ′ ∈ f − x ( U y ) , y > f ( x ) O x ′ is faithfully flat. Therefore, O x → Q x ′ > x O x ′ is faithfully flat and x is a removable point. (cid:3)
11. Proposition:
Let p ∈ Y be a removable point, i : Y − { p } → Y be the inclusion morphismand f : X → Y a schematic morphism. If f ( X ) ⊆ Y − p and g : X → Y − { p } is the morphismof ringed spaces such that f = i ◦ g, then g is a schematic morphism.Therefore, f M : X M → Y M is a schematic morphism.Proof. It is an immediate consequence of Theorem 5.15. (cid:3)
12. Proposition:
Let X be a schematic finite space and U ⊂ X an a ffi ne open subspace. LetX ′ : = X ` { u } be the ringed finite space defined by1. The preorder on X ⊂ X ′ is the pre-established preorder. Given x ∈ X, then u < x i ff x ∈ U, and x < u i ff x ≤ x ′ , for any x ′ ∈ U.2. O X ′ , x : = O X , x for any x ∈ X, and O X ′ , u : = O X ( U ) . The restriction morphisms are theobvious morphisms. . Serre Theorem 27 Then, X ′ is a schematic finite space and u is a removable point of X ′ .Proof. Let us denote U u : = U ⊂ X and ˜ U x ′ : = { y ∈ X ′ : y ≥ x ′ } , for any x ′ ∈ X ′ . ByProposition 2.11, the morphism O X ′ , y ⊗ O X ′ , x ′ O X ′ , y ′ = O X ( U y ∩ U y ′ ) → Y x ∈ U y ∩ U y ′ O X , x = Y x ∈ U y ∩ U y ′ O X ′ , x is faithfully flat, for any y , y ′ ≥ x ′ . If U ⊆ U y ∩ U y ′ , the morphism O X ( U y ∩ U y ′ ) → O X ( U ) isflat, by Lemma 6.3. Hence, the morphism O X ′ , y ⊗ O X ′ , x ′ O X ′ , y ′ → Q x ∈ ˜ U y ∩ ˜ U y ′ O X ′ , x is faithfullyflat. By Proposition 3.6, X ′ is a schematic finite space.The morphism O X ′ , u = O X ( U ) → Y x ∈ U O X , x = Y x ′ > u O X ′ , x ′ is faithfully flat, because U is a ffi ne. Hence, u is removable. (cid:3) Let X be a finite topological space and F a sheaf of abelian groups on X .
1. Proposition :
If X is a finite topological space with a minimum, then H i ( X , F ) = forany sheaf F and any i > . In particular, for any finite topological space one hasH i ( U p , F ) = for any p ∈ X, any sheaf F and any i > .Proof. Let p be the minimum of X . Then U p = X and, for any sheaf F , one has Γ ( X , F ) = F p ;thus, taking global sections is the same as taking the stalk at p , which is an exact functor. (cid:3) Let f : X → Y a continuous map between finite topological spaces and F a sheaf on X .The i-th higher direct image R i f ∗ F is the sheaf on Y given by:[ R i f ∗ F ] y = H i ( f − ( U y ) , F ) . Let F be a sheaf on a finite topological space X . We define C n F as the sheaf on X whosesections on an open subset U are ( C n F )( U ) = Y U ∋ x < ··· < x n F x n . Serre Theorem 28and whose restriction morphisms ( C n F )( U ) → ( C n F )( V ) for any V ⊆ U are the naturalprojections.One has morphisms d : C n F → C n + F , a = ( a x < ··· < x n ) d ( a ) = ( d ( a ) x < ··· < x n + ) definedin each open subset U by the formula(d a ) x < ··· < x n + = X ≤ i ≤ n ( − i a x < ··· b x i ··· < x n + + ( − n + ¯ a x < ··· < x n where ¯ a x < ··· < x n denotes the image of a x < ··· < x n under the morphism F x n → F x n + . There is alsoa natural morphism d : F → C F . One easily checks that d =
2. Theorem ([10] 2.15): C · F is a finite and flasque resolution of F (in fact, it is the Gode-ment resolution of F).Proof.
By definition, C n F = n > dim X . It is also clear that C n F are flasque. Let ussee that 0 → F → C F → · · · → C dim X F → C · F )( U p ) is a resolution of F ( U p ). One has adecomposition( C n F )( U p ) = Y p = x < ··· < x n F x n × Y p < x < ··· < x n F x n = ( C n − F )( U ∗ p ) × ( C n F )( U ∗ p )with U ∗ p : = U p − { p } ; via this decomposition, the di ff erential d : ( C n F )( U p ) → ( C n + F )( U p )becomes: d( a , b ) = ( b − d ∗ a , d ∗ b )with d ∗ the di ff erential of ( C · F )( U ∗ p ). If d ( a , b ) =
0, then b = d ∗ a and d (0 , a ) = ( a , b ).It is immediate now that every cycle is a boundary. (cid:3) This theorem, together with De Rham’s theorem ([4], Thm. 4.7.1), yields that the co-homology groups of a sheaf can be computed with the standard resolution, i.e., H i ( U , F ) = H i Γ ( U , C · F ), for any open subset U of X and any sheaf F of abelian groups on X .
3. Theorem ([8] 4.3, 4.12):
Every quasi-coherent module of an a ffi ne finite space is acyclic.Proof. Let X be an a ffi ne finite space.Proceed by induction over the order of X . The open sets U xy are a ffi ne by Corollary 2.7. If X = U xy , then X = U x and every sheaf is acyclic. We can suppose that every quasi-coherentmodule on U xy is acyclic, by induction hypothesis. Let us prove that any quasi-coherent O X -module M is acyclic. We have to prove that the sequence of morphisms M ( X ) → Y x ∈ X M x → Y x < x M x → · · · . Serre Theorem 29is exact. It is su ffi cient to check that tensoring the previous sequence by ⊗ O ( X ) O z , for any z ∈ X , the sequence of morphisms M z / / Q z ≤ x M x × Q z (cid:10) x M x z / / Q z ≤ x < x M x × Q x < x z (cid:10) x , z ≤ x M x × Q x < x z (cid:10) x M x z / / · · · is exact. That is, we have to prove that the sequence of morphisms ( S ) M ( U z ) / / C ( U z , M ) × Q z (cid:10) x M ( U x z ) / / C ( U z , M ) × Q z (cid:10) x C ( U x z , M ) × Q x < x z (cid:10) x M ( U x z ) / / · · · is exact. Let D · r : = ⊕ x < ··· < x r , z (cid:10) x r ( M ( U x r z ) ⊕ C · ( U x r z , M ))[ − r ] and let d r be the di ff erentialsuch that over each direct summand ( M ( U x r z ) ⊕ C · ( U x r z , M ))[ − r ] is the known di ff erentialof M ( U x r z ) ⊕ C · ( U x r z , M ) multiplied by ( − r . H i ( D · r ) = i ≥
0. The sequenceof morphisms ( S ) is equal to the di ff erential complex D · : = D · ⊕ D · ⊕ · · · ⊕ D · n with thedi ff rential d = d · · · − d · · · ... ... . . . ... ...... ... ... . . . ... ... − − − · · · d n − − − − · · · − d n Let D · > = ⊕ i > D · i . Consider the exact sequence of morphisms of complexes0 → D · > → D · → D · → . Then, H i ( D · ) = H i ( D · > ), for any i ≥
0. Let D · > = ⊕ i > D · i . Consider the exact sequence ofmorphisms 0 → D · > → D · > → D · →
0. Then, H i ( D · ) = H i ( D · > ) = H i ( D · > ). Recursively H i ( D · ) = H i ( D · > n ) = i ≥
0, and the sequence of morphisms ( S ) is exact. (cid:3) Let R and R ′ be commutative rings and R → R ′ a flat morphism of rings. Let ( X , O ) bean R -ringed finite space. Let O ⊗ R R ′ be the sheaf of rings on X defined by ( O ⊗ R R ′ )( U ) : = O ( U ) ⊗ R R ′ . Consider the obvious morphism π : ( X , O ⊗ R R ′ ) → ( X , O ), π ( x ) = x . Let M bea sheaf of O -modules. Then, H i ( X , π ∗ M ) = H i ( X , M ) ⊗ R R ′ . . Serre Theorem 30If N is a quasi-coherent O ⊗ R R ′ -module, then π ∗ N = N is a quasi-coherent O -module.Let S ⊂ R be a multiplicative system, R ′ = S − · R and N a quasi-coherent O ⊗ R R ′ -module. Then, π ∗ N is a quasi-coherent O -module, N = π ∗ π ∗ N and H i ( X , N ) = H i ( X , π ∗ N ) .
4. Serre Theorem ([8] 5.11) :
Let X be a schematic finite space. X is a ffi ne i ff every quasi-coherent O X -module M is acyclic (or H ( X , M ) = ).Proof. ⇐ ) Let R : = O ( X ). Recall Notation 2.13. Given p ∈ Spec R , consider the sheaf ofrings on X , O ⊗ R R p . Obviously, ( X , O ⊗ R R p ) is a schematic finite space and ( X , O ) is a ffi nei ff ( X , O ⊗ R R p ) is a ffi ne for any p . Hence, we can suppose R is a local ring. We can supposethat X is minimal. Let X ′ be the set of the closed points of X . Let x ′ ∈ X ′ . The morphism O x ′ → Q x > x ′ O x is flat but it is not faithfully flat, then there exists a prime ideal I x ′ ⊂ O x ′ such that I x ′ · Q x > x ′ O x = Q x > x ′ O x . Let p be the quasicoherent ideal defined by p x ′ : = I x ′ if x ′ ∈ X ′ and p x : = O x if x < X ′ . Observe that ( O X / p ) x =
0, for any x ∈ X \ X ′ , then( O X / p )( X ) = Q x ′ ∈ X ′ O X / I x ′ . Consider the exact sequence of morphisms0 → p → O → O X / p → R = O ( X ) → ( O X / p )( X ) = Q x ′ ∈ X ′ O X / I x ′ is surjective, because H ( X , p ) = R is a local ring, then Q x ′ ∈ X ′ O X / I x ′ is a local ring, hence X ′ = { x ′ } . Therefore, X = U x ′ ,which is a ffi ne. (cid:3) This theorem yields the usual Serre’s criterion on algebraic varieties (see [10] 4.13 and[11]).
5. Corollary:
A schematic finite space X is a ffi ne i ff the functor Γ : Qc - Mod X → Mod O ( X ) , M 7→ Γ ( X , M ) is exact.Proof. ⇒ ) By the Serre Theorem H ( X , M ) =
0, for any quasi-coherent module M . Hence Γ is exact. ⇐ ) It has been proved in the proof of Serre Theorem. (cid:3)
6. Corollary :
A schematic finite space X is a ffi ne i ff H ( X , I ) = for any quasi-coherentideal I ⊆ O .Proof. ⇐ ) Let R = O ( X ). We have just proved this implication when R is a local ring, inthe proof of the Serre Theorem. Let p ∈ Spec R and let I p ⊂ O ⊗ R R p be a quasi-coherentideal. Consider the obvious morphism O → O ⊗ R R p . J : = O × O⊗ R R p I p is a quasi-coherentideal of O and π ∗ J = I p (where π : ( X , O ⊗ R R p ) → ( X , O ) is defined by π ( x ) : = x ). Then,. Cohom. characterization of schematic finite spaces 31 H ( X , I p ) = H ( X , J ) ⊗ R R p =
0. Then, ( X , O ⊗ R R p ) is a ffi ne, for any p . Therefore, X isa ffi ne. (cid:3)
7. Corollary :
Let X be a minimal a ffi ne finite space and suppose that O ( X ) is a local ring.Then, there exists a point p ∈ X such that X = U p .
8. Theorem :
Let X and Y be schematic finite spaces. A ringed space morphism f : X → Yis a ffi ne i ff f ∗ M is quasi-coherent and R f ∗ M = , for any quasi-coherent O X -module M .Proof. ⇒ ) It is obvious. ⇐ ) Let U ⊆ Y be an a ffi ne open subspace. By Corollary 5.8, any quasi-coherent mod-ule M on f − ( U ) is the restriction of a quasi-coherent module on X . H ( f − ( U ) , M ) = H ( U , f ∗ M ) =
0, for any quasi-coherent O X -module M . By Serre Theorem 7.4, f − ( U ) isa ffi ne. Hence, f is a ffi ne. (cid:3)
9. Theorem:
Let f : X → Y be a schematic morphism. The functorf ∗ : Qc - Mod X → Qc - Mod Y , M f ∗ M is exact i ff f is a ffi ne.Proof. ⇒ ) 1. Let U ⊆ Y be an open subset, V = f − ( U ) and f | V : V → U , f | V ( x ) : = f ( x ). Then f | V ∗ is exact: Any short exact sequence of quasi-coherent modules N • on V isa restriction of a short exact sequence of quasi-coherent modules M • on X ; and f V ∗ N • = ( f ∗ M • ) | U .2. We can suppose that Y is a ffi ne. We can suppose that Y = ( ∗ , A ). We can suppose that A = O X ( X ).3. We have to prove that X is a ffi ne. The functor, Qc - Mod X → Mod O ( X ) , M f ∗ M = Γ ( X , M )is exact. By Corollary 7.5, X is a ffi ne. ⇐ ) By Theorem 7.8, R f ∗ M =
0, for any quasi-coherent module M . Then, f ∗ is exact. (cid:3) We say that an R -ringed space ( X , O X ) is a flat R -ringed space if the morphism R → O x isflat, for any x ∈ X .. Cohom. characterization of schematic finite spaces 32
1. Theorem:
Let ( X , O ) be a flat R-ringed finite space and M an R-module. If H i ( X , O ) is aflat R-module, for any i > , then O ( X ) is a flat R-module andH i ( X , ˜ M ) = H i ( X , O ) ⊗ R M , ∀ i . Proof.
Let C i : = Ker[ C i ( X , O ) → C i + ( X , O )] and B i : = Im[ C i − ( X , O ) → C i ( X , O )]. Let n = dim X . C n = C n ( X , O ) is R -flat. The sequence 0 → B n → C n → H n ( X , O ) → B n is flat. The sequence 0 → C n − → C n − ( X , O ) → B n → C n − is R -flat.Recursively, C i and B i are R -flat for any i . Hence, O ( X ) = C is R -flat and H i ( X , ˜ M ) = H i ( X , C ·O ⊗ R M ) = H i ( X , C ·O ) ⊗ R M = H i ( X , O X ) ⊗ R M . (cid:3)
2. Corollary :
Let X be a flat O ( X ) -ringed finite space. Assume H i ( X , O ) is a flat O ( X ) -module, for any i > . Then, the morphism O ( X ) → Q x ∈ X O x is faithfully flat.Proof. Let M be an O ( X )-module. By Theorem 8.1, ˜ M ( X ) = M . Then, the morphism M = ˜ M ( X ) ֒ → Y x ∈ X ˜ M x = M ⊗ O ( X ) Y x ∈ X O x is injective. Therefore, the flat morphism O ( X ) → Q x ∈ X O x is faithfully flat. (cid:3)
3. Theorem:
Let X be a flat O ( X ) -ringed finite space. Then, X is a ffi ne i ff O x ⊗ O ( X ) O y = O xy , for any x , y,3’. U xy is acyclic, for any x , y.Proof. ⇒ ) U xy is a ffi ne by Corollary 2.7. By Theorem 7.3, X and U xy are acyclic. ⇐ ) Let z ∈ U xy . The morphism O xy → O z ⊗ O ( X ) O xy = O z ⊗ O ( X ) O x ⊗ O ( X ) O y = O zx ⊗ O ( X ) O y = O z ⊗ O ( X ) O y = O zy = O z is flat, since the morphism O ( X ) → O z is flat.By Corollary 8.2, the morphisms O ( X ) → Q x ∈ X O x and O ( U xy ) → Q z ∈ U xy O z are faith-fully flat. Hence, X is a ffi ne. (cid:3)
4. Theorem:
A finite fr-space X is schematic i ff for any x ≤ y , y ′ , . Cohom. characterization of schematic finite spaces 33 O y ⊗ O x O y ′ = O yy ′ .2. U yy ′ is acyclic.Proof. ⇒ ) U x is an a ffi ne finite space. By Theorem 8.3, we are done. ⇐ ) U x is an a ffi ne finite space by Theorem 8.3, then X is schematic. (cid:3)
5. Proposition :
Let X be an a ffi ne finite space. An open subset U ⊆ X is a ffi ne i ff it isacyclic.Proof. ⇐ ) U satisfies 1’. and 3’ of Theorem 8.3. The composite morphism of the epimor-phism O x ⊗ O ( X ) O y → O x ⊗ O ( U ) O y and the morphism O x ⊗ O ( U ) O y → O xy is an isomorphism,then O x ⊗ O ( U ) O y → O xy is an isomorphism.Besides, the morphism O ( U ) = O ( X ) ⊗ O ( X ) O ( U ) → O x ⊗ O ( X ) O ( U ) = O x is flat. (cid:3)
6. Proposition :
Let f : X → Y be a schematic morphism and M a quasi-coherent O X -module. Then, R i f ∗ M is a quasi-coherent O Y -module, for any i ≥ . If Y is a ffi ne, R i f ∗ M = g H i ( X , M ) .Proof. We can suppose that Y is a ffi ne. Given y ∈ Y , the inclusion morphism f − ( U y ) j ֒ → X is a ffi ne: Let U ⊂ X be an a ffi ne subset and i be the composite morphism U ֒ → X f → Y ,which is an a ffi ne morphism. Then, j − ( U ) = f − ( U y ) ∩ U = i − ( U y ) is a ffi ne.Observe that R n f ∗ ( j ∗ N ) = R n ( f ◦ j ) ∗ N = n > N , since j and f ◦ j are a ffi ne morphisms. Likewise, H n ( X , j ∗ N ) = H n ( f − ( U y ) , N ) = H n ( Y , ( f ◦ j ) ∗ N ) =
0, for any n > M f − ( U y ) = j ∗ M | f − ( U y ) and consider the obvious exact sequence of morphisms0 → M → ⊕ y ∈ Y M f − ( U y ) π → M ′ → . Then, H ( X , M ) = Coker π X and H n − ( X , M ′ ) = H n ( X , M ) for any n >
1. Besides, R f ∗ M = Coker π , which is quasi-coherent, and R n f ∗ M = R n − f ∗ M ′ for any n >
1. There-fore, R f ∗ M = g H ( X , M ) since Y is a ffi ne. Hence, R f ∗ M ′ = g H ( X , M ′ ), since M ′ isquasi-coherent. By induction on n , R n f ∗ M = R n − f ∗ M ′ = g H n − ( X , M ′ ) = g H n ( X , M )for any n > (cid:3) This proposition, Theorem 5.12 and [10] Theorem 5.6 show that the definitions of schematicmorphism given in this paper and in [10] are equivalent.. Cohom. characterization of schematic finite spaces 34
7. Lemma :
Let X be an f r-space and δ : X → X × X, δ ( x ) : = ( x , x ) be the diagonalmorphism. Let M be an O X -module. R i δ ∗ M is quasi-coherent i ff H i ( U pq , M ) ⊗ O p O p ′ = H i ( U p ′ q , M ) , for any p ≤ p ′ and for any q.Proof. ( R i δ ∗ M ) ( q , q ′ ) = H i ( U qq ′ , M ) and( R i δ ∗ M ) ( p , p ′ ) ⊗ O ( p , p ′ ) O ( q , q ′ ) = H i ( U pp ′ , M ) ⊗ O p ⊗O p ′ ( O q ⊗ O q ′ ) = ( H i ( U pp ′ , M ) ⊗ O p O q ) ⊗ O p ′ O q ′ , for any ( p , p ′ ) ≤ ( q , q ′ ). Now, the proof is easily completed. (cid:3)
8. Proposition :
Let X be an f r-space and δ : X → X × X the diagonal morphism. Let M be an O X -module. Then, δ ∗ M is a quasi-coherent O X × X -module i ff M p ′ ⊗ O p O p ′′ = M p ′ p ′′ ,for any p ≤ p ′ , p ′′ .Proof. It is a consequence of Lemma 8.7 and Proposition 3.11. (cid:3)
9. Cohomological characterization of schematic finite spaces ([8] 4.7, 4.4 ):
Let X be an f r -space and δ : X → X × X the diagonal morphism. X is a schematic finite space i ff R i δ ∗ O X is a quasi-coherent module, for any i ≥ Proof. ⇐ ) We have to prove that U p is a ffi ne. U p is acyclic, and satisfies the property 2’ ofTheorem 8.3, by the previous proposition. We only need to prove that U qq ′ is acyclic, forany q , q ′ ∈ U p : 0 = H i ( U q , O ) ⊗ O p O q ′ = H i ( U pq , O ) ⊗ O p O q ′ = H i ( U q ′ q , O ) . ⇒ ) The diagonal morphism δ is schematic by Theorem 3.9 and Theorem 5.12. By Propo-sition 8.6, we are done. (cid:3)
10. Corollary ([10] 4.5):
An f r-space X is schematic i ff for any open set j : U q ֒ → X,R i j ∗ O U q is a quasi-coherent O X -module, for any i.Proof. Let δ : X → X × X be the diagonal morphism. Then, X is a schematic finite space i ff R i δ ∗ O X is a quasi-coherent module, for any i , which is equivalent to say that H i ( U pq , O ) ⊗ O p O p ′ = H i ( U p ′ q , O ), for any p ≤ p ′ , and any q , that is to say, R i j ∗ O U q is a quasi-coherent O X -module, for any i and any open set j : U q ֒ → X . (cid:3) . Cohom. characterization of schematic finite spaces 35A scheme is said to be a semiseparated scheme if the intersection of two a ffi ne open setsis a ffi ne. For example, the line with a double point is a semiseparated scheme (but it is notseparated). The plane with a double point is not semiseparated, but it is quasi-separated.
11. Definition :
A ringed finite space X is said to be semiseparated if the open sets U pq areacyclic, for any p , q ∈ X .
12. Proposition :
Let X be a ringed finite space and let δ : X → X × X be the diagonalmorphism. X is seimiseparated i ff R i δ ∗ O X = , for any i > .Proof. ( R i δ ∗ O X ) ( p , q ) = H i ( U pq , O ), and R i δ ∗ O X = ff ( R i δ ∗ O X ) ( p , q ) = p , q ∈ X .Hence, X is semiseparated i ff R i δ ∗ O X =
0, for any i > (cid:3)
13. Theorem :
Let X be an f r-space and δ : X → X × X be the diagonal morphism. X isa semiseparated schematic finite space i ff R i δ ∗ O X = , for any i > and δ ∗ O X is a quasi-coherent module.Proof. ⇒ ) By 8.9, δ ∗ O X is quasi-coherent, and R i δ ∗ O X = ⇐ ) X is a schematic finite space, by 8.9, and it is semiseparated by the previous proposi-tion. (cid:3)
14. Proposition:
A schematic finite space is semiseparated i ff it satisfies any of the followingequivalent conditions:1. The intersection of any two a ffi ne open subspaces is a ffi ne.2. There exists an a ffi ne open covering of X, U = { U , . . . , U n } such that U i ∩ U j is a ffi nefor any i , j.Proof. Assume X is a semiseparated schematic finite space. Let δ : X → X × X , δ ( x ) = ( x , x )be the diagonal morphism and U , U ′ two a ffi ne open subspaces. Since R i δ ∗ O X =
0, for any i > H i ( U ∩ U ′ , O ) = H i ( U × U ′ , δ ∗ O ) =
0. By 8.5 U ∩ U ′ is a ffi ne.Assume that there exists an a ffi ne open covering of X , U = { U , . . . , U n } such that U i ∩ U j is a ffi ne for any i , j . R i δ ∗ O X is quasi-coherent, by 8.9. R i δ ∗ O X ( U i × U j ) = H i ( U i ∩ U j , O X ) = R i δ ∗ O X = X is semiseparated. (cid:3) All the examples in Examples 3.4 are semiseparated finite spaces.Finally, let us give some cohomological characterizations of schematic morphisms.
15. Proposition:
Let X be an a ffi ne finite space and Y a schematic finite space. A morphismof ringed spaces f : X → Y is a ffi ne i ff f ∗ O X is quasi-coherent and R i f ∗ O X = , for any i > . . Cohom. characterization of schematic finite spaces 36 Proof. ⇒ ) A ffi ne finite spaces are acyclic, then ( R i f ∗ O X ) y = H i ( f − ( U y ) , O X ) =
0, for any i > y ∈ Y . Hence, R i f ∗ O X =
0, for any i > ⇐ ) Let U ⊆ Y be an a ffi ne open subspace. H i ( f − ( U ) , O X ) = H i ( U , f ∗ O X ) =
0, for any i >
0, then f − ( U ) is acyclic, therefore it is a ffi ne by Proposition 8.5. (cid:3)
16. Proposition :
A morphism of ringed spaces f : X → Y between schematic finite spacesis a ffi ne i ff f ∗ O X is quasi-coherent, R i f ∗ O X = for any i > and there exists an opencovering { U i } of Y such that f − ( U i ) is a ffi ne, for any i.Proof. ⇒ ) ( R i f ∗ O X ) y = H i ( f − ( U y ) , O X ) =
0, for any i > y ∈ Y . Hence, R i f ∗ O X =
0, for any i > ⇐ ) The morphisms f − ( U i ) → U i are a ffi ne, by the previous proposition. Then, f isa ffi ne. (cid:3)
17. Proposition :
Let f : X → Y be a morphism of ringed spaces between schematic finitespaces. Then, f is schematic i ff Γ f : X → X × Z Y, Γ f ( x ) = ( x , f ( x )) is schematic.Proof. ⇐ ) It is easy to check that π : X × Z Y → Y , π ( x , y ) = y is schematic. Then, f isschematic because f = π ◦ Γ f and π and Γ f are schematic. ⇒ ) Let x ∈ X and ( x ′ , y ) ∈ X × Z Y (where ( x , f ( x )) ≤ ( x ′ , y )). U x ( x ′ , y ) = U x ′ ∩ U xy is a ffi nebecause it is the intersection of two a ffi ne open subsets of the a ffi ne finite space U x . Observethat O x ⊗ O ( x , f ( x )) O ( x ′ , y ) = O x ⊗ O x ⊗ Z O f ( x ) O x ′ ⊗ Z O y = O x ′ ⊗ O f ( x ) O y = O x ′ y = O x ( x ′ , y ) Then, Γ f is schematic, by Proposition 5.14. (cid:3)
18. Theorem :
A morphism of ringed spaces f : X → Y between schematic finite spaces isschematic i ff R i Γ f ∗ O X is a quasi-coherent O X × Y -module, for any i ≥ .Proof. ⇒ ) By Proposition 8.6, R i Γ f ∗ O X is a quasi-coherent O X × Y -module, for any i ≥ ⇐ ) R i Γ f ∗ O X is a quasi-coherent O X × Y -module, for any i ≥
0. Then, H ( U x , O X ) ⊗ f ( x ) O y = H ( U x f ( x ) , O X ) ⊗ O f ( x ) O y = H ( U x f ( x ) , O X ) ⊗ O ( x , f ( x )) O ( x , y ) = H ( U xy , O X ) , for any x and y ≥ f ( x ). Therefore, O x ⊗ O f ( x ) O y = O xy . Besides,0 = H i ( U x , O X ) ⊗ O f ( x ) O y = H i ( U x f ( x ) , O X ) ⊗ O f ( x ) O y = H i ( U x f ( x ) , O X ) ⊗ O ( x , f ( x )) O ( x , y ) = H i ( U xy , O X ) , for any i >
0. Then, the open subsets U xy are acyclic, hence f is schematic by Proposition5.14. (cid:3) . Quasi-isomorphisms 37
19. Theorem :
Let f : X → Y be a morphism of ringed spaces between schematic finitespaces. Let x ∈ X and let f U x be the composite morphism U x ֒ → X → Y. Then, f isschematic i ff R i f U x ∗ O U x is a quasi-coherent O Y -module, for any i ≥ and any x ∈ X.Proof. ⇒ ) If f is schematic, f U x is schematic and R i f U x ∗ O U x is a quasi-coherent O Y -module,for any i ≥
0, by Proposition 8.6. ⇐ ) R i f U x ∗ O U x is a quasi-coherent O Y -module. Then, H ( U x f ( x ) , O X ) ⊗ O f ( x ) O y = H ( U xy , O X ) , for any x and y ≥ f ( x ). Therefore, O x ⊗ O f ( x ) O y = O xy . Besides,0 = H i ( U x , O X ) ⊗ O f ( x ) O y = H i ( U x f ( x ) , O X ) ⊗ O f ( x ) O y = H i ( U xy , O X ) , for any i > . Then, the open sets U xy are acyclic. By Proposition 5.14, f is schematic. (cid:3)
1. Definition :
A schematic morphism f : X → Y is said to be a quasi-isomorphism if1. f is a ffi ne.2. f ∗ O X = O Y .If f : X → Y is a quasi-isomorphism we shall say that X is quasi-isomorphic to Y .
2. Examples :
1. If X is an a ffi ne finite space, the morphism X → ( ∗ , O ( X )) is a quasi-isomorphism.2. Let X be a schematic finite space and let ˜ X be the Kolmogorov quotient of X . Thequotient morphism π : X → ˜ X is a quasi-isomorphism (see Example 4.2.3.).3. If X is a schematic finite T -topological space, X M ֒ → X is a quasi-isomorphism.4. Let f : X → Y be a schematic morphism. ( Y , f ∗ O X ) is a schematic finite space byExample 4.8. Let us prove that the obvious ringed morphism f ′ : X → ( Y , f ∗ O X ), f ′ ( x ) = f ( x ) is schematic. By Theorem 5.15, the morphism O x ⊗ ( f ∗ O X ) f ( x ) ( f ∗ O X ) y = O x ⊗ ( f ∗ O X ) f ( x ) ( f ∗ O X ) f ( x ) ⊗ O f ( x ) O y = O x ⊗ O f ( x ) O y → Y z ∈ U xy O z . Quasi-isomorphisms 38is surjective on spectra. Again by Theorem 5.15, f ′ is schematic. We have the obviouscommutative diagram X f / / f ′ $ $ ❍❍❍❍❍❍❍❍❍ Y ( Y , f ∗ O X ) Id : : ✈✈✈✈✈✈✈✈✈ Id is a ffi ne. If f is a ffi ne, then f ′ is a quasi-isomorphism.
3. Definition :
Let f : X → Y be a schematic morphism. We shall say that f is flat if themorphism O Y , f ( x ) → O X , x is flat, for any x ∈ X . We shall say that f is faithfully flat if themorphism O Y , y → Q x ∈ f − ( U y ) O X , x is faithfully flat, for any y ∈ Y .If { U i } is an open covering of X , the natural morphism ` i U i → X is faithfully flat.
4. Remark :
Quasi-isomorphisms are faithfully flat morphisms: Given y ∈ Y , f − ( U y ) isa ffi ne, then the morphism O y = ( f ∗ O X ) y = O X ( f − ( U y )) → Y x ∈ f − ( U y ) O x is faithfully flat.
5. Proposition :
Let X be a schematic finite space, U = { U , . . . , U n } a minimal a ffi ne opencovering of X and Y the ringed finite space associated with U . Then, Y is a schematic finitespace and the quotient morphism π : X → Y is a quasi-isomorphism.Proof. Y = { y , . . . , y n } , where π − ( U y i ) = U i . Recall that π ∗ O X = O Y . Let y ≤ y , then O y = O X ( U ) → O X ( U ) = O y is a flat morphism by Lemma 6.3. Let y i , y j ≥ y k , then O y i ⊗ O yk O y j = O X ( U i ) ⊗ O X ( U k ) O X ( U j ) = O X ( U i ∩ U j ) = O X ( π − ( U y i ∩ U y j )) = O Y ( U y i ∩ U y j ) = O y i y j . U i ∩ U j is an a ffi ne finite space by Proposition 2.11. U i ∩ U j = ∪ U k ⊂ U i ∩ U j U z . The morphisms O ( U k ) → Q x ∈ U k O x , O ( U i ∩ U j ) → Q U k ⊂ U i ∩ U j , x ∈ U k O x are faithfully flat. Then, the morphism O ( U i ∩ U j ) → Q U k ⊂ U i ∩ U j O ( U k ) is faithfully flat. Therefore, the morphism O y i y j → Q y k ∈ U yiyj O y k isfaithfully flat. Then, Y is a schematic finite space.Finally, π is a ffi ne by Proposition 4.6. (cid:3) . Quasi-isomorphisms 39
6. Proposition:
The composition of quasi-isomorphisms is a quasi-isomorphism.
7. Theorem:
Let f : X → Y be a schematic morphism. The functorsf ∗ : Qc - Mod X → Qc - Mod Y and f ∗ : Qc - Mod Y → Qc - Mod X are mutually inverse (i.e., the natural morphisms M → f ∗ f ∗ M , f ∗ f ∗ N → N are isomor-phisms) i ff f is a quasi-isomorphism.Proof. ⇐ ) The morphism f ∗ f ∗ M → M is an isomorphism: It is a local property on Y . Wecan suppose that Y is an a ffi ne finite space (then X is a ffi ne). Consider a free presentation of M , ⊕ I O X → ⊕ J O X → M →
0. Taking f ∗ , which is an exact functor because R f ∗ =
0, onehas the exact sequence of morphisms ⊕ I O Y → ⊕ J O Y → f ∗ M →
0. Taking f ∗ , one has theexact sequence of morphisms ⊕ I O X → ⊕ J O X → f ∗ f ∗ M →
0, then f ∗ f ∗ M = M .Likewise, f ∗ f ∗ N = N . ⇒ ) O Y = f ∗ f ∗ O Y = f ∗ O X . Obviously, f ∗ is an exact functor. By Theorem 7.9, f is a ffi ne. (cid:3)
8. Corollary:
Let f : X → Y be a quasi-isomorphism. Y is a ffi ne i ff X is a ffi ne.Proof. ⇐ ) For any quasi-coherent O Y -module N , N = f ∗ f ∗ N , then H ( Y , N ) = H ( X , f ∗ N ) = . By the Serre Theorem Y is a ffi ne. (cid:3)
9. Corollary :
Let X f → Y g → Z be schematic morphisms and assume g ◦ f is a quasi-isomorphism. Then,1. If g is a quasi-isomorphism, then f is a quasi-isomorphism.2. If f is a quasi-isomorphism, then g is a quasi-isomorphism.Proof.
1. Considering the diagram Qc - Mod
X f ∗ / / ( g ◦ f ) ∗ + + Qc - Mod
Y g ∗ / / f ∗ o o Qc - Mod Zg ∗ o o ( g ◦ f ) ∗ g g it is easy to prove that f ∗ and f ∗ are mutually inverse functors.2. Proceed likewise. (cid:3) . Quasi-isomorphisms 40
10. Corollary:
Let X and Y be a ffi ne finite spaces. Then, a schematic morphism f : X → Yis a quasi-isomorphism i ff O Y ( Y ) = O X ( X ) .Proof. ⇐ ) Observe that the diagram X f / / (cid:15) (cid:15) Y (cid:15) (cid:15) ( ∗ , O X ( X )) / / ( ∗ , O Y ( Y ))is commutative, ( X , O X ) is quasi-isomorphic to ( ∗ , O X ( X )) and ( Y , O Y ) is quasi-isomorphic to( ∗ , O Y ( Y )). (cid:3)
11. Corollary :
Let f : X → Y be a schematic morphism and let f M : X M → Y M be theinduced morphism. Then, f is a quasi-isomorphism i ff f M is a quasi-isomorphism.Proof. It is an immediate consequence of Corollary 9.9. (cid:3)
12. Corollary :
Let f : X → X ′ be a quasi-isomorphism , X ′′ a schematic finite space andg : X ′ → X ′′ a morphism of ringed spaces. Then, g is schematic (resp. a ffi ne) i ff g ◦ f isschematic (resp. a ffi ne)Proof. Recall Theorem 5.12 (resp. Theorem 7.9). (cid:3)
13. Proposition :
Let f : X → X ′ be an a ffi ne morphism of schematic spaces. Assume thatX ′ is T . Let U : = { f − ( U x ′ ) } x ′ ∈ X ′ , let X / ∼ be the schematic space associated with theopen covering U , and π : X → X / ∼ the quotient morphism. The morphism f ′ : X / ∼ → X ′ ,f ′ ([ x ]) = f ( x ) , induced by f , is a ffi ne and Y is homeomorphic to Im f .Proof. By Corollary 9.12, we only have to prove that Y is homeomorphic to Im f . Themorphism f ′ : X / ∼ → Im f is clearly bijective and continuous. Given [ x ] , [ x ′ ] ∈ X / ∼ , if f ′ ([ x ]) ≤ f ′ ([ x ′ ]), then f ( x ) ≤ f ( x ′ ) and U f ( x ′ ) ⊆ U f ( x ) . Hence, f − ( U f ( x ′ ) ) ⊆ f − ( U f ( x ) ) and U [ x ′ ] ⊆ U [ x ] . Therefore, [ x ] ≤ [ x ′ ]. That is, f ′ is a homeomorphism. (cid:3)
14. Lemma :
Let h : X → Y be a quasi-isomorphism. Then, Y \ h ( X ) is a set of removablepoints of Y.
0. Change of base and flat schematic morphisms 41
Proof.
Let y ∈ Y \ h ( X ). Since h − ( U y ) is a ffi ne, the morphism O Y ( U y ) = O X ( h − ( U y )) → Y x ′ ∈ h − ( U y ) O X , x ′ is faithfully flat. This morphism factors through the morphism O Y ( U y ) → Y h ( x ′ ) ∈ U y O Y , h ( x ′ ) , then this last morphism is faithfully flat. Hence, y is a removable point of Y . (cid:3)
15. Theorem :
Let f : X → Y be a quasi-isomorphism. Assume that Y is T . Considerthe a ffi ne open covering of X, { f − ( U y ) } y ∈ Y and let X / ∼ be the associated schematic fi-nite space. Then, f is the composition of the quotient morphism π : X → X / ∼ and anisomorphism f ′ : X / ∼ → Y \ P, f ′ ([ x ]) = f ( x ) , where P is a set of removable points of Y.Therefore, if f : X → Y is a quasi-isomorphism and Y is minimal, f is the composition ofthe quotient morphism X → X / ∼ and an isomorphism X / ∼ ≃ Y.Proof.
By Lemma 9.14, we can suppose that f is surjective. By Proposition 9.13, f ′ : Y ′ → Y is a homeomorphism and it is a ffi ne. Finally, O Y , f ′ ([ x ]) = ( f ∗ O X ) f ( x ) = O Y ′ , [ x ] , for any [ x ]. (cid:3)
10 Change of base and flat schematic morphisms
1. Proposition ([10] 5.27):
Let X, X ′ and Y schematic finite spaces and f : X → Y andf ′ : X ′ → Y schematic morphisms. Then,1. X × Y X ′ is a schematic finite space.2. If X, X ′ and Y are a ffi ne, then O ( X × Y X ′ ) = O ( X ) ⊗ O ( Y ) O ( X ′ ) and X × Y X ′ is a ffi ne.3. Given a commutative diagram of schematic morphismsU g & & ◆◆◆◆◆◆ h / / X f & & ▼▼▼▼▼▼ V / / YU ′ g ′ qqqqqq h ′ / / X ′ f ′ qqqqqq the morphism h × h ′ : U × V U ′ → X × Y X ′ , h × h ′ ( u , u ′ ) : = ( h ( u ) , h ′ ( u ′ )) is schematic.4. π : X × Y X ′ → X, π ( x , x ′ ) = x, is schematic.
0. Change of base and flat schematic morphisms 42
5. If h : X → X ′ is a schematic Y-morphism, then Γ h : X → X × Y X ′ , Γ h ( x ) : = ( x , h ( x )) isschematic.6. The diagonal morphism δ : X → X × Y X, δ ( x ) = ( x , x ) is schematic.Proof.
1. We only need to prove 2.2. X × O ( Y ) X ′ is an a ffi ne schematic space and O ( X × O ( Y ) X ′ ) = O ( X ) ⊗ O ( Y ) O ( X ′ ), byProposition 2.12. Let ( x , x ′ ) ∈ X × O ( Y ) X ′ . If f ( x ) = f ′ ( x ′ ), then O x ⊗ O ( Y ) O x ′ = O x ⊗ O f ( x ) ( O f ( x ) ⊗ O ( Y ) O f ( x ) ) ⊗ O f ( x ) O x ′ = O x ⊗ O f ( x ) O f ( x ) ⊗ O f ( x ) O x ′ = O x ⊗ O f ( x ) O x ′ If f ( x ) , f ′ ( x ′ ), then ( x , x ′ ) ∈ X × O ( Y ) X ′ is a removable point: Consider the morphism f x : U x → Y , f x ( z ) : = f ( z ). Then, O xy = ( f x ∗ O U x ) y = ( f x ∗ O U x )( Y ) ⊗ O ( Y ) O y = O x ⊗ O ( Y ) O y .Observe that U x ′ f ( x ) ⊂ , X ′ and it is a ffi ne since f ′ x ′ : U x ′ → Y , f ′ x ′ ( z ) : = f ′ ( z ) is a ffi ne. Themorphism O x ⊗ O ( Y ) O x ′ = O x f ( x ) ⊗ O ( Y ) O x ′ = O x ⊗ O ( Y ) O f ( x ) ⊗ O ( Y ) O x ′ = O x ⊗ O ( Y ) O x ′ f ( x ) → Y z ∈ U x ′ f ( x ) O x ⊗ O ( Y ) O z is faithfully flat. Therefore, ( x , x ′ ) is removable. In conclusion, X × Y X ′ = ( X × O ( Y ) X ′ ) \{ A setof removable points } , then X × Y X ′ is a ffi ne and O ( X × Y X ′ ) = O ( X × O ( Y ) X ′ ) = O ( X ) ⊗ O ( Y ) O ( X ′ ).3. By Proposition 5.14, given ( u , u ′ ) ∈ U × V U ′ , ( x , x ′ ) ∈ X × Y X ′ (where ( x , x ′ ) ≥ ( h ( u ) , h ′ ( u ′ ))), we have to prove that U ( u , u ′ )( x , x ′ ) is acyclic and O ( u , u ′ )( h ( u ) , h ( u ′ )) ⊗ O ( h ( u ) , h ( u ′ )) O ( x , x ′ ) = O ( u , u ′ )( x , x ′ ) : U ( u , u ′ )( x , x ′ ) = U ux × U g ( u ) f ( x ) U u ′ x ′ , which is a ffi ne (then acyclic) and O ( u , u ′ ) ⊗ O ( h ( u ) , h ( u ′ )) O ( x , x ′ ) = ( O u ⊗ O g ( u ) O u ′ ) ⊗ O h ( u ) ⊗ O f ( h ( u )) O h ′ ( u ′ ) ( O x ⊗ O f ( x ) O x ′ ) = ( O u ⊗ O h ( u ) O x ) ⊗ O g ( u ) ⊗ O f ( h ( u )) O f ( x ) ( O u ′ ⊗ O h ′ ( u ′ ) O x ′ ) = O ux ⊗ O g ( u ) f ( x ) O u ′ x ′ = O ( u , u ′ )( x , x ′ ) . (cid:3) Obviously, Hom Y ( Z , X × Y X ′ ) = Hom Y ( Z , X ) × Hom Y ( Z , X ′ )for any schematic finite Y -spaces Z , X , X ′ .
2. Proposition: A ffi ne morphisms and quasi-isomorphisms are stable by base change.Proof. Let f : X → Y be an a ffi ne morphism and Y ′ → Y a schematic morphism. In orderto prove that the schematic morphism X × Y Y ′ → Y ′ is a ffi ne, it is su ffi cient to prove that X × Y U y ′ is a ffi ne, for any y ′ ∈ Y ′ . Observe that X × Y U y ′ = f − ( U f ( y ′ ) ) × U f ( y ′ ) U y ′ , which isa ffi ne because f − ( U f ( y ′ ) ) is a ffi ne.0. Change of base and flat schematic morphisms 43Let f be a quasi-isomorphism. We only have to prove that O ( X × Y U y ′ ) = O y ′ : O ( X × Y U y ′ ) = O ( f − ( U f ( y ′ ) )) × U f ( y ′ ) U y ′ ) = O ( f − ( U f ( y ′ ) )) ⊗ O f ( y ′ ) O y ′ = O f ( y ′ ) ⊗ O f ( y ′ ) O y ′ = O y ′ . (cid:3)
3. Lemma:
Let f : X → Y and g : Y ′ → Y be schematic morphisms and let g ′ : X × Y Y ′ → Xbe defined by g ′ ( x , y ′ ) : = x. Let M be a quasi-coherent O X -module. If X , Y and Y ′ are a ffi ne,then Γ ( X × Y Y ′ , g ′∗ M ) = Γ ( X , M ) ⊗ O ( Y ) O ( Y ′ ) . Proof.
Consider an exact sequence of O X -modules ⊕ I O X → ⊕ J O X → M → g ′∗ , ⊕ I O X × Y Y ′ → ⊕ J O X × Y Y ′ → g ′∗ M → ⊕ I O ( X × Y Y ′ ) → ⊕ J O ( X × Y Y ′ ) → g ′∗ M ( X × Y Y ′ ) → O ( X × Y Y ′ ) = O ( X ) ⊗ O ( Y ) O ( Y ′ ). Hence, the sequence of morphisms ⊕ I O ( X ) ⊗ O ( Y ) O ( Y ′ ) → ⊕ J O ( X ) ⊗ O ( Y ) O ( Y ′ ) → g ′∗ M ( X × Y Y ′ ) → O ( X )-modules ⊕ I O ( X ) → ⊕ J O ( X ) → M ( X ) → ⊕ I O ( X ) ⊗ O ( Y ) O ( Y ′ ) → ⊕ J O ( X ) ⊗ O ( Y ) O ( Y ′ ) → M ( X ) ⊗ O ( Y ) O ( Y ′ ) → Γ ( X , M ) ⊗ O ( Y ) O ( Y ′ ) → Γ ( X × Y Y ′ , g ′∗ M ) is anisomorphism. (cid:3)
4. Theorem :
Let f : X → Y be a schematic morphism and g : Y ′ → Y a flat schematicmorphism. Denote f ′ : X × Y Y ′ → Y ′ , f ′ ( x , y ′ ) = y ′ , g ′ : X × Y Y ′ → X, g ′ ( x , y ′ ) = x theinduced morphisms. Then, the natural morphismg ∗ R i f ∗ M → R i f ′∗ ( g ′∗ M ) is an isomorphism.Proof. We have to prove that the morphism is an isomorphism on stalks at z , for any z ∈ Y ′ .That is to say, we have to prove that the morphism H i ( f − ( U g ( z ) ) , M ) ⊗ O g ( z ) O z → H i ( f − ( U g ( z ) ) × U g ( z ) U z , g ′∗ M )0. Change of base and flat schematic morphisms 44is an isomorphism. We can suppose that Y ′ = U z , Y = U g ( z ) and X = f − ( U g ( z ) ). Then, g ′ isan a ffi ne morphism and H i ( X × Y Y ′ , g ′∗ M ) = H i ( X , g ′∗ g ′∗ M ). By Lemma 10.3,( g ′∗ g ′∗ M ) x = Γ ( U x × Y Y ′ , g ′∗ M ) = M x ⊗ O ( Y ) O ( Y ′ ) . Hence, C · ( g ′∗ g ′∗ M ) = ( C ·M ) ⊗ O ( Y ) O ( Y ′ ) and H i ( X , g ′∗ g ′∗ M ) = H i ( X , M ) ⊗ O ( Y ) O ( Y ′ ). Thatis, H i ( X × Y Y ′ , g ′∗ M ) = H i ( X , g ′∗ g ′∗ M ) = H i ( X , M ) ⊗ O ( Y ) O ( Y ′ ) . (cid:3)
5. Proposition :
Let X and Y be schematic finite spaces and f : X → Y a schematic mor-phism. Then, f is flat (resp. faithfully flat) i ff the functorf ∗ : Qc - Mod Y → Qc - Mod X , M f ∗ M is exact (resp. faithfully exact).Proof. Obviously, if f is flat then f ∗ is exact. If f is faithfully flat then f ∗ is faithfullyexact: We only have to check that M = f ∗ M =
0. Given y ∈ Y , the morphism O y ֒ → Q x ∈ f − ( U y ) O x is faithfully flat. Tensoring by M y ⊗ O y one has the injective morphism M y ֒ → Y x ∈ f − ( U y ) M y ⊗ O y O x = Y x ∈ f − ( U y ) M y ⊗ O y O f ( x ) ⊗ O f ( x ) O x = Y x ∈ f − ( U y ) M f ( x ) ⊗ O f ( x ) O x = Y x ∈ f − ( U y ) ( f ∗ M ) x = . Hence, M = f ∗ is exact f is flat: Let f ( x ) ∈ Y . Consider an ideal I ⊆ O f ( x ) and let ˜ I ⊆ O U f ( x ) be the quasi-coherent O U f ( x ) -module associated with I . Consider the inclusion morphism i : U f ( x ) ֒ → Y . The morphism i ∗ ˜ I ֒ → i ∗ O U f ( x ) is injective, then f ∗ i ∗ ˜ I ֒ → f ∗ i ∗ O U f ( x ) is injective.Hence, I ⊗ O f ( x ) O x = ( f ∗ i ∗ ˜ I ) x ֒ → ( f ∗ i ∗ O U f ( x ) ) x = O x is injective. Therefore, the morphism O f ( x ) → O x is flat. Finally, if f ∗ is faithfully exact f is faithfully flat: Let y ∈ Y be a maximalpoint of Y , if it exists, such that the flat morphism O y → Q x ∈ f − ( U y ) O x is not faifhfully flat.Then, there exists an ideal I ⊂ , O y such that I · Q x ∈ f − ( U y ) O x = Q x ∈ f − ( U y ) O x . Let M be thequasi-coherent O U y -module associated with O y / I and let f ′ : f − ( U y ) → U y be the morphismdefined by f ′ ( x ) : = f ( x ). Obviously, f ′∗ M =
0. Let i : U y ֒ → Y and i ′ : f − ( U y ) ֒ → X bethe inclusion morphisms. By Theorem 10.4, 0 = i ′∗ f ′∗ M = f ∗ i ∗ M , then i ∗ M =
0. Hence,0 = i ∗ i ∗ M = M which is contradictory. (cid:3)
0. Change of base and flat schematic morphisms 45
6. Proposition :
Let f : X → Y be a schematic morphism. Then, f is a quasi-isomorphismi ff it is faithfully flat and the natural morphism f ∗ f ∗ M → M is an isomorphism, for anyquasi-coherent O X -module M .Proof. ⇒ ) It is Remark 9.4 and Theorem 9.7. ⇐ ) f ∗ is an exact functor since f ∗ is faithfully exact and Id = f ∗ f ∗ . By Theorem 7.9, f isa ffi ne. Finally, the morphism O Y → f ∗ O X is an isomorphism, since f ∗ O Y = O X → f ∗ f ∗ O X isan isomorphism. (cid:3)
7. Proposition :
Let f : X → Y be a schematic morphism. Then, f is faithfully flat i ff f M : X M → Y M is surjective and flat.Proof. Let g : X ′ → X be a quasi-isomorphism. Then, f is faithfully flat i ff f ◦ g is faithfullyflat, since f ∗ is faithfully exact i ff g ∗ ◦ f ∗ is faithfully exact. Let g : Y → Y ′ be a quasi-isomorphism. Likewise, f is faithfully flat i ff g ◦ f is faithfully flat.Therefore, we have to prove that f M is a faithfully flat i ff is surjective and flat. Theconverse implication is obvious. Let us prove the direct implication. If f is not surjective,let y ∈ Y M be maximal satisfying f − ( y ) = ∅ . Consider the commutative diagram of obviousmorphisms O y i / / i (cid:15) (cid:15) Q f M ( x ) ≥ y O x Q f M ( x ) > y O xi v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ Q y ′ ≥ y O y ′ i / / Q y ′ ≥ y Q f M ( x ) ≥ y ′ O x The morphisms i and i are faithfully flat since f M is faithfully flat, i is obviously faithfullyflat, hence i is faithfully flat and y is a removable point, which is contradictory. (cid:3)
8. Proposition :
Let f : X → Y be a schematic morphism and g : Y ′ → Y a faithfully flatschematic morphism. Let f ′ : X × Y Y ′ → Y ′ be the morphism defined by f ′ ( x , y ′ ) = y ′ . Then,1. f is a ffi ne i ff f ′ is a ffi ne.2. f is a quasi-isomorphism i ff f ′ is a quasi-isomorphism.Proof. We can suppose that X , Y and Y ′ are minimal schematic spaces. The morphism g ′ : X × Y Y ′ → X , g ′ ( x , y ′ ) : = x is faithfully flat since it is flat and surjective.1. ⇐ ) The functor g ∗ f ∗ = f ′∗ g ′∗ is exact since f ′∗ and g ′∗ are exact. Hence, f ∗ is exactsince g ∗ is faithfully exact and f is a ffi ne.1. Quasi-open immersions 462. ⇐ ) We only have to prove that the morphism O Y → f ∗ O X is an isomorphism. Taking g ∗ , we obtain the isomorphism O X → g ∗ f ∗ O X = f ′∗ g ′∗ O X = f ′∗ O X × Y Y ′ = O X . Hence, O Y → f ∗ O X is an isomorphism. (cid:3)
11 Quasi-open immersions
1. Definition :
We shall say that a schematic morphism f : X → Y is a quasi-open immer-sion if it is flat and the diagonal morphism X → X × Y X is a quasi-isomorphism.
2. Example : If X is a schematic finite space and U ⊆ X an open subset, then the inclusionmorphism U ֒ → X is a quasi-open immersion.
3. Proposition:
If f : X → Y is a quasi-isomorphism, then it is a quasi-open immersionProof.
Quasi-isomorphisms are faithfully flat morphisms, by Observation 9.4. The mor-phism X × Y X → X is a quasi-isomorphism by Proposition 10.2. The composite morphism X → X × Y X → X is the identity morphism, then X → X × Y X is a quasi-isomorphism, byCorollary 9.9. (cid:3)
4. Proposition:
If f : X → Y is a quasi-open immersion and Y ′ → Y a schematic morphism,then X × Y Y ′ → Y ′ is a quasi-open immersion.Proof. The morphism X → Y is flat. Taking × Y Y ′ , the morphism X × Y Y ′ → Y ′ is flat. Themorphism X → X × Y X is a quasi-isomorphism. Taking × Y Y ′ , the morphism X × Y Y ′ → ( X × Y Y ′ ) × Y ′ ( X × Y Y ′ ) is a quasi-isomorphism, by Proposition 10.2. Hence, X × Y Y ′ → Y ′ is a quasi-open immersion. (cid:3)
5. Proposition :
Let f : X → Y be a schematic morphism. Let Y ′ → Y be a faithfully flatschematic morphism and f ′ : X × Y Y ′ → Y ′ , f ′ ( x , y ′ ) : = f ( x ) the induced morphism. Then fis a quasi-open immersion i ff f ′ is a quasi-open immersion.
6. Proposition:
The composition of two quasi-open immersions is a quasi-open immersion.Proof.
The composition of two flat morphisms is flat. Let f : X → Y , g : Y → Z be quasi-open immersions. Consider the commutative diagram X δ X / / X × Y X Id × Id / / (cid:15) (cid:15) X × Z X f × f (cid:15) (cid:15) Y δ Y / / Y × Z Y
1. Quasi-open immersions 47Observe that X × Y X = Y × Y × Z Y ( X × Z X ) and δ Y is a quasi-isomorphism. Then, Id × Id is aquasi-isomorphism by Proposition 11.4. The morphism (Id × Id) ◦ δ X is a quasi-isomorphismby Proposition 9.6, hence g ◦ f is a quasi-open immersion. (cid:3)
7. Proposition :
Let f : X → Y, g : Y → Z be schematic morphisms and suppose g ◦ f is aquasi-open immersion.1. If g is a quasi-open immersion, then f is a quasi-open immersion.2. If f is a quasi-isomorphism, then g is a quasi-open immersion.Proof.
1. Consider the commutative diagram X δ X / / X × Y X Id × Id / / (cid:15) (cid:15) X × Z X f × f (cid:15) (cid:15) Y δ Y / / Y × Z Y Id × Id is a quasi-isomorphism since δ Y is a quasi-isomorphism. (Id × Id) ◦ δ X is a quasi-isomorphism, since g ◦ f is a quasi-open immersion. Hence, δ X is a quasi-isomorphism, byCorollary 9.9, that is, f is a quasi-open immersion.2. The obvious morphism X × Z X → Y × Z Y is a quasi-isomorphism, since is the com-position of the quasi-isomorphisms X × Z X → X × Z Y , X × Z Y → Y × Z Y . Consider thecommutative diagram X δ X / / f (cid:15) (cid:15) X × Z X f × f (cid:15) (cid:15) Y δ Y / / Y × Z Y Then, δ Y is a quasi-isomorphism since f , δ X and f × f are quasi-isomorphisms. That is, g isa quasi-open immersion. (cid:3)
8. Definition :
Let X and Y be ringed finite spaces and f : X → Y a morphism of ringedspaces. C ( f ) : = X ` Y is a finite ringed space as follows: the order relation on X and on Y is the pre-stablished order relation, and given x ∈ X and y ∈ Y we shall say that x > y if f ( x ) ≥ y ; O C ( f ) , x : = O X , x for any x ∈ X , O C ( f ) , y : = O Y , y for any y ∈ Y ; the morphisms betweenthe stalks of O C ( f ) are defined in the obvious way.1. Quasi-open immersions 48Observe that X is an open subset of C ( f ) and F : C ( f ) → Y , F ( x ) : = f ( x ), for any x ∈ X and F ( y ) : = y , for any y ∈ Y is a morphism of ringed spaces. F ∗ O C ( f ) = O Y because( F ∗ O C ( f ) ) y = O C ( f ) , y = O Y , y Besides, R i F ∗ O C ( f ) = i >
0, because( R i F ∗ O C ( f ) ) y = H i ( U y , O C ( f ) ) = .
9. Theorem :
Let f : X → Y be a schematic morphism. Then, f is a quasi-open immersioni ff C ( f ) = X ` Y is a schematic finite space. If f is a quasi-open immersion, then it is thecomposition of the open inclusion X ֒ → C ( f ) and the quasi-isomorphism F : C ( f ) → Y.Proof. ⇒ ) Given, x ≥ x ′ ∈ X ⊂ C ( f ), the morphism O C ( f ) , x ′ = O X , x ′ → O X , x = O C ( f ) , x is flat. Given, y ≤ y ′ ∈ Y ⊂ C ( f ), the morphism O C ( f ) , y = O Y , y → O Y , y ′ = O C ( f ) , y ′ is flat. Given x ∈ X ⊂ C ( f ) and f ( x ) ≥ y ∈ Y ⊂ C ( f ), the morphism O C ( f ) , y → O C ( f ) , f ( x ) = O Y , f ( x ) → O X , x = O C ( f ) , x is flat.Given c ∈ C ( f ), we shall denote ˜ U c : = { z ∈ C ( f ) : z ≥ c } . We have to prove that ˜ U c isa ffi ne. Recall Theorem 8.3.a. If c = x ∈ X , then ˜ U c = U x ⊆ X is a ffi ne. If c = y ∈ Y , ˜ U y is acyclic.b. Given x , x ′ ∈ ˜ U y ∩ X , ˜ U x ∩ ˜ U x ′ = U x ∩ U x ′ which is quasi-isomorphic to U x × U y U x ′ ,then ˜ U x ∩ ˜ U x ′ is acyclic and O C ( f ) , xx ′ = O xx ′ = O x ⊗ O y O x ′ = O C ( f ) , x ⊗ O C ( f ) , y O C ( f ) , x ′ c. Given, y ′ , y ′′ ∈ ˜ U y ∩ Y , ˜ U y ′ ∩ ˜ U y ′′ = F − ( U y ′ ∩ U y ′′ ), which is acyclic because U y ′ ∩ U y ′′ is acyclic, and O C ( f ) , y ′ y ′′ = O y ′ y ′′ = O y ′ ⊗ O y O y ′′ = O C ( f ) , y ′ ⊗ O C ( f ) , y O C ( f ) , y ′′ d. Given x , y ′ ∈ ˜ U y , where x ∈ X and y ′ ∈ Y . Observe that ˜ U x ∩ ˜ U y ′ = U xy ′ and U xy ′ = f − x ( U f ( x ) ∩ U y ′ ), which is a ffi ne since f x : U x → U f ( x ) is a ffi ne. and U f ( x ) ∩ U y ′ ⊂ U y is a ffi ne. Finally, O C ( f ) , xy ′ = O xy ′ ∗ = O x ⊗ O y O y ′′ = O C ( f ) , x ⊗ O C ( f ) , y O C ( f ) , y ′
1. Quasi-open immersions 49( ∗ observe that U x × U y U y ′ = U xy ′ ).Therefore C ( f ) is schematic.If M is a O C ( f ) -quasi-coherent module, F ∗ M is a quasi-coherent O Y -module since( F ∗ M ) y ⊗ O Y , y O Y , y ′ = M y ⊗ O C ( f ) , y O C ( f ) , y ′ = M y ′ = ( F ∗ M ) y ′ . By Theorem 5.12, F is schematic. F is a quasi-isomorphism since F ∗ O C ( f ) = O Y and F − ( U y ) = ˜ U y , for any y ∈ Y . ⇐ ) The morphism f is the composition of the open immersion X ֒ → C ( f ) and the quasi-isomorphism F : C ( f ) → Y , hence f is a quasi-open immersion. (cid:3)
10. Proposition:
Let f : X → Y be a schematic morphism. Then, f is a quasi-isomorphismi ff it is a faithfully flat quasi-open immersion.Proof. ⇒ ) It is Remark 9.4 and Proposition 11.3. ⇐ ) If y ∈ Y , then y is a removable point of C ( f ), since the morphism O C ( f ) , y = O Y , y → Y x ∈ f − ( U y ) O X , x = Y x > y , x ∈ X O C ( f ) , x is faithfully flat. The morphism X → C ( f ) is a quasi-isomorphism, since X = C ( f ) − Y and C ( f ) − Y is quasi-isomorphic to C ( f ). Finally, X is quasi-isomorphic to Y , since C ( f ) isquasi-isomorphic to Y . (cid:3)
11. Theorem:
Let f : X → Y be a schematic morphism. Then, f is a quasi-open immersioni ff f is flat and the morphism f ∗ f ∗ M → M is an isomorphism for any quasi-coherent O X -module.Proof. ⇒ ) The diagonal morphism δ : X → X × Y X is a quasi-isomorphism. Then, δ isa ffi ne. Consider the projections, π , π : X × Y X → X . The morphism f is flat, then f ∗ f ∗ M = π ∗ π ∗ M , by Theorem 10.4. Observe that( π ∗ π ∗ M ) x = Γ ( X × Y U x , π ∗ M ) = Γ ( X × Y U x , δ ∗ δ ∗ π ∗ M ) = Γ ( X × Y U x , δ ∗ M ) = Γ ( U x , M ) = M x , for any x ∈ X . Therefore, the morphism f ∗ f ∗ M → M is an isomorphism. ⇐ ) Let i : U x → X be the obvious inclusion and denote i ∗ M | U x = M U x . The naturalmorphism f ∗ f ∗ M U x → M U x is an isomorphism. Then,( π ∗ M ) ( x , x ′ ) = M x ⊗ O f ( x ) O x ′ = ( f ∗ M U x ) f ( x ) ⊗ O f ( x ) O x ′ = ( f ∗ f ∗ M U x ) x ′ = ( M U x ) x ′ = M ( U xx ′ ) = ( δ ∗ M ) ( x , x ′ ) .
1. Quasi-open immersions 50Hence, δ ∗ is an exact functor, since π is flat. By Theorem 7.9, δ is an a ffi ne morphism.Besides, O X × Y X = π ∗ O X = δ ∗ O X . Hence, δ is a quasi-isomorphism and f is a quasi-openimmersion. (cid:3)
12. Lemma :
Let f : X → Y be a schematic morphism and suppose that X is a ffi ne andf ∗ O X = O Y . Then, f is a quasi-open immersion.Proof. We have to prove that C ( f ) is a schematic finite space. By Proposition 3.6, we haveto prove that O z ⊗ O z O z → Q w ∈ U z z O w is faithfully flat, for any z ≤ z , z ∈ C ( f ).Suppose that z , z ∈ X . The epimorphism O z ⊗ O ( X ) O z → O z ⊗ O z O z is an isomorphism,since the composite morphism O z ⊗ O ( X ) O z → O z ⊗ O z O z → O z z is an isomorphism.Besides, the morphism O z z → Q w ∈ U z z O w is faithfully flat, since X is a ffi ne.Suppose that z ∈ X and that z ∈ Y (then z ∈ Y ). Observe that U z ∩ f − ( U z ) = U z z = { c ∈ C ( f ) : c ≥ z , z } . Then, O z ⊗ O z O z = O z ⊗ O z O z ⊗ O z O z = O z ⊗ O z ( O z ⊗ O f ( z ) O z ) ⊗ O z O z = O z ⊗ O f ( z ) O z = O z z → Y w ∈ U z z O w is faithfully flat, since U z z ⊂ X is a ffi ne, by Proposition 5.14.Suppose that z , z ∈ Y . Observe that U C ( f ) , z z = U Y , z z ` f − ( U Y , z z ) and O Y ( U Y , z z ) = O X ( f − ( U Y , z z )). The morphism O z ⊗ O z O z = O Y , z z → Q w ∈ U Y , z z O w is faithfully flat,since U Y , z z is a ffi ne. For any open subset V ⊂ X and any x ∈ V , the morphism O X ( V ) = O ( X ) ⊗ O ( X ) O ( V ) → O x ⊗ O ( X ) O ( V ) = O x is flat. Hence, the morphism O z ⊗ O z O z = O Y , z z = O X ( f − ( U Y , z z )) → Q x ∈ f − ( U Y , z z ) O x is flat. Therefore, the morphism O z ⊗ O z O z → Y w ∈ U C ( f ) , z z O w = Y w ∈ U Y , z z O w × Y x ∈ f − ( U Y , z z ) O x is faithfully flat. (cid:3)
13. Proposition :
Let f : X → Y be a schematic morphism and suppose that X is a ffi ne.Then, there exist an open inmersion i : X → Z such that i ∗ X = O Z and an a ffi ne morphismg : Z → Y such that f = g ◦ i.Proof. The obvious morphism f ′ : X → ( Y , f ∗ O X ), f ′ ( x ) = f ( x ) is a quasi-open inmersionby the lemma above and Example 9.2. Let i : X → C ( f ′ ), π : C ( f ′ ) → ( Y , f ∗ O X ) and letId : ( Y , f ∗ O X ) → Y be the obvious morphism. Observe that i is an open immersion, g : = Id ◦ π is a ffi ne, since π is a quasi-isomorphism and Id is a ffi ne, and f = Id ◦ f ′ = Id ◦ π ◦ i = g ◦ i . (cid:3)
2. Quasi-closed immersions 51
12 Quasi-closed immersions
Let
I ⊂ O X be a quasi-coherent ideal and ( I ) : = { x ∈ X : ( O X / I ) x , } , which is aclosed subspace of X . Consider the schematic space ( X , O X / I ) and observe that x ∈ X \ Y i ff ( O X / I ) x =
0. Hence, X \ Y is a set of removable points of ( X , O X / I ) and the obvious mor-phism (( I ) , O X / I ) → ( X , O X / I ) is a quasi-isomorphism. We shall say that the compositionof the a ffi ne morphisms (( I ) , O X / I ) → ( X , O X / I ) → ( X , O X )is a closed immersion.Let f : X ′ → X be a schematic morphism. Let I = Ker[ O X → f ∗ O X ′ ]. The obviousmorphism f ′ : X ′ → ( X , O X / I ), f ′ ( x ) = f ( x ) is schematic since f ′∗ M = f ∗ M is a quasi-coherent O X / I -module, for any quasi-coherent O X ′ -module M , because it is a quasi-coherent O X -module. Obviously, f is the composition of the morphisms X ′ → ( X , O X / I ) → ( X , O X ).Assume that O X ′ , x ′ , x ′ ∈ X ′ (recall that if O X ′ . x ′ = x ′ is a removable point).The closure of f ( X ′ ) in X is ( I ) : x ∈ X \ ( I ) i ff ( O X / I ) x =
0, which is equivalent to sayingthat ( f ∗ O X ′ ) x = O X / I → f ∗ O X ′ is inyective.( f ∗ O X ′ ) x = O X ′ ( f − ( U x )) = ff f − ( U x ) = ∅ . The morphism X ′ → (( I ) , O X / I ), x f ( x )is schematic and f ′ is the composition of the morphisms X ′ → (( I ) , O X / I ) → ( X , O X / I ) (seeProposition 6.11).
1. Definition :
Let f : X ′ → X be a schematic morphism. We shall say that f is a quasi-closed immersion if it is a ffi ne and the morphism O X → f ∗ O X ′ is an epimorphism.Suppose that O X ′ , x ′ ,
0, for any x ′ ∈ X ′ , let f : X ′ → X be a quasi-closed immersi´on and I = Ker[ O X → f ∗ O X ′ ]. Then f is the composition of a quasi-isomorphism X ′ → (( I ) , O X / I )and a closed immersion (( I ) , O X / I ) → ( X , O X ). Spec O X Let { X i , f i j } i , j ∈ I (where I < ∞ ) be a direct system of morphisms of ringed spaces. Letlim → i X i be the direct limit of the topological spaces X i : lim → i X i = ` i X i / ∼ , where ∼ is theequivalence relation generated by the relation x i ∼ f i j ( x i ), and U ⊆ lim → i X i is an open subseti ff f − j ( U ) is an open subset for any j ∈ I , where f j : X j → lim → i X i is the natural map.Define O lim → i X i ( V ) : = lim ← i ∈ I O X i ( f − i ( V )), for any open set V ⊆ lim → i X i . It is well known that3. Spec O X
52( lim → i X i , O lim → i X i ) is the direct limit of the direct system of morphisms { X i , f i j } in the categoryof ringed spaces.
1. Definition :
Given a schematic finite space X we shall denoteSpec O X : = lim → x ∈ X Spec O x . (the sheaf of rings considered on Spec O x is the sheaf of localizations of O x , ˜ O x .)Observe that O Spec O X (Spec O X ) = lim ← x ∈ X O x = O X ( X ).
2. Example :
Obviously, if X = U x , then Spec O X = Spec O x and O Spec O X = ˜ O x .Consider the following relation on ` Spec O x i : Given p ∈ Spec O x and q ∈ Spec O y weshall say that p ≡ q if there exist u ≥ x , y and r ∈ Spec O u such that the given morphismsSpec O u ֒ → Spec O x and Spec O u ֒ → Spec O y map r to p and r to q , respectively (recallProposition 2.14). Let us prove that ≡ is an equivalence relation: Let p ≡ q , ( r p , q ) and q ≡ q ′ ( q ′ ∈ Spec O z , there exist u ′ ≥ y , z and r ′ ∈ Spec O u ′ such that r ′ q , q ′ ). Recall that O uu ′ = O u ⊗ O y O u ′ and that O uu ′ → Q w ∈ U uu ′ O w is faithfully flat. Then,Spec O u ∩ Spec O u ′ = Spec O uu ′ = ∪ w ∈ U uu ′ Spec O w Since, q = r = r ′ ∈ Spec O u ∩ Spec O u ′ there exists v ∈ U uu ′ and r ′′ ∈ Spec O v such that r ′′ r , r ′ . Then, r ′′ p , q ′ and p ≡ q ′ .Observe that Spec O X = ` x ∈ X Spec O x / ≡ as topological spaces. Besides, the morphismsSpec O x → Spec O X are injective, Spec O X = ∪ x ∈ X Spec O x as topological spaces ( U ⊆ Spec O X is an open set i ff U ∩ Spec O x is an open set, for any x ∈ X ).
3. Lemma :
Let A → B be a flat morphism and assume B ⊗ A B = B. If I ⊆ A is a radicalideal, then I · B is a radical ideal of B.Proof.
Let p ∈ Spec B ⊂ Spec A and recall Notation 2.13. Then,(rad( I · B )) p = rad( I · B p ) = rad( I · A p ) = rad( I ) · A p = I · A p = I · B p = ( I · B ) p Therefore, rad( I · B ) = I · B . (cid:3)
4. Proposition :
Let X be a schematic finite space. Let
I ⊆ O X be a quasi-coherent ideal.The ideal rad I ⊂ O X , defined by (rad I ) x : = rad I x , for any x ∈ X, is a quasi-coherent idealof O X .Proof. We only have to prove that given a flat morphism A → B such that B ⊗ A B = B andan ideal I ⊆ A , then (rad I ) · B = rad( I · B ). This is a consequence of Lemma 13.3. (cid:3)
3. Spec O X
5. Remark : (rad I )( U ) = rad( I ( U )), for any open subset U ⊂ X :(rad I )( U ) = lim ← x ∈ U (rad I ) x = lim ← x ∈ U (rad I x ) = rad lim ← x ∈ U I x = rad I ( U ) .
6. Definition :
Let X be a schematic finite space. We shall say that a quasi-coherent ideal I ⊆ O X is radical if I = rad( I ).
7. Notation :
Let X be a schematic finite space. Given a quasi-coherent ideal I ⊂ O X , weshall denote ( I ) : = ∪ x ∈ X { p ∈ Spec O x : I x ⊆ p } ⊆ Spec O X Given a closet subset C ⊂ Spec O X , let I C ⊂ O X be the radical quasi-coherent ideal definedby I C , x : = ∩ p ′ ∈ C ∩ Spec O x p ′ ⊂ O x , for any x ∈ X .
8. Proposition:
The maps { Closed subspaces of
Spec O X } ←→ { Radical quasi-coherent ideals of O X } C ✤ / / I C ( I ) I ✤ o o are mutually inverse.
9. Notation :
Given a ring B and b ∈ B we denote B b = { , b − , b − , · · · } · B .
10. Proposition:
If X is an a ffi ne finite space, then Spec O X = Spec O ( X ) .Proof. The morphism O ( X ) → O x is flat and O x ⊗ O ( X ) O x = O x , then Spec O x ֒ → Spec O ( X )is a subspace. The morphism O ( X ) → Q x ∈ X O x is faithfully flat, then the induced morphism ` x ∈ X Spec O x → Spec O ( X ) is surjective. The sequence of morphisms O ( X ) → Y x ∈ X O x / / / / Y x ≤ x ′ ∈ X O x ′ is exact. Then, the natural morphism f : Spec O X → Spec O ( X ) is continuous and bijective.Given a closed set C ⊂ Spec O X , let I C be the radical quasi-coherent ideal of O X associatedwith C . I C = ˜ I C ( X ) since X is a ffi ne. Recall Notation 2.15. Then, C ∩ Spec O x = ( I C , x ) = ( I C ( X ) · O x ) . Hence f ( C ) = ( I C ( X )) and f is a homeomorphism.Also observe that ( lim ← x O x ) ( a x ) = lim ← x ∈ X O x , a x , for any ( a x ) ∈ lim ← x ∈ X O x ⊆ Q x ∈ X O x , hence O lim → x ∈ X Spec O x = g O ( X ). (cid:3) If C ∩ Spec O x = ∅ , then ∩ p ′ ∈ C ∩ Spec O x p ′ : = O X .
3. Spec O X
11. Definition :
Let f : X → Y be a schematic morphism. Consider the morphisms O f ( x ) →O x , which induce the scheme morphisms Spec O x → Spec O f ( x ) , which induce a morphismof ringed spaces ˜ f : Spec O X → Spec O Y . We shall say that ˜ f is the morphism induced by f .
12. Proposition :
Let f : X → Y be a quasi-isomorphism. Then, the morphism induced byf , ˜ f : Spec O X → Spec O Y , is an isomorphism.Proof. Observe thatSpec O X = lim → x ∈ X Spec O x = lim → y ∈ Y lim → x ∈ f − ( U y ) Spec O x = lim → y ∈ Y Spec O X ( f − ( U y )) = lim → y ∈ Y Spec O Y ( U y ) = Spec O Y . (cid:3)
13. Proposition :
Let X be a schematic finite space, U i ⊂ X an open subset and
I ⊆ O U aquasi-coherent ideal. Then, there exists a quasi-coherent ideal J ⊆ O X such that J | U = I .Proof. J : = Ker[ O X → i ∗ ( O U / I )] holds J | U = I . (cid:3)
14. Notation :
Given a schematic finite space X we shall denote ˜ X = Spec O X .
15. Proposition:
Let X be a schematic finite space and U ⊂ X an open subset. Then,1. ˜ U is a topological subspace of ˜ X . ˜ U = ∩ ˜ U ⊆ open subset ¯ V ⊆ ˜ X ¯ V . Proof.
1. Given a closed set C ⊂ ˜ U , let I C ⊆ O U be the radical quasi-coherent idealassociated. Let J ⊆ O X be the quasi-coherent ideal such that J | U = I . Then, the closedsubset D = ( J ) = (rad J ) of ˜ X holds that D ∩ ˜ U = C .2. Let p ∈ ˜ X − ˜ U . Let P ⊂ O X be the sheaf of ideals defined by P x = p ⊂ O x if p ∈ Spec O x and P x = O x if p < Spec O x . By Proposition 2.16, P is quasi-coherent. ( P ) ⊂ Spec O X isthe closure of p and ( P ) ∩ ˜ U x = ∅ , for any ˜ U x ⊂ ˜ U , hence ( P ) ∩ ˜ U = ∅ . Then, ˜ U is equalto the intersection of the open subsets ¯ V ⊆ ˜ X , such that ˜ U ⊆ ¯ V . (cid:3)
16. Definition :
Let X be a schematic finite space. We shall say that a quasi-coherent O X -module M is finitely generated if M x is a finitely generated O x -module, for any x ∈ X .3. Spec O X
17. Proposition :
Let X be an a ffi ne finite space and M a a quasi-coherent O X -module.Then, M is finitely generated i ff M ( X ) is a finitely generated O ( X ) -module.Proof. ⇒ ) Given x ∈ X , M x = M ( X ) ⊗ O ( X ) O x . Let N x ⊂ M ( X ) be a finitely generated O ( X )-submodule such that N x ⊗ O ( X ) O x = M x and N : = P x ∈ X N x . Then N = M ( X ), since N ⊗ O ( X ) O x = M x for any x ∈ X . ⇐ ) M x = M ( X ) ⊗ O ( X ) O x is a finitely generated O x -module, for any x ∈ X . (cid:3)
18. Proposition :
Let X be a schematic finite space. Any quasi-coherent O X -module is thedirect limit of its finitely generated submodules.Proof. Let M be a quasi-coherent O X -module. Let us fix x ∈ X and a finitely generatedsubmodule N ⊂ M x . Consider the inclusion morphism i : U x ֒ → X and let M : = Ker[
M → i ∗ ( M | U x / ˜ N )]. Observe that M ⊂ M and M | U = ˜ N . Given x ∈ X , let N ⊂ M , x be a finitely generated submodule such that N ⊗ O x O y = M , y , for any y ∈ U x ∩ U x . Let U = U x ∪ U x and let N ⊂ M | U be the finitely generated O U -modulesuch that N | U x = ˜ N and N | U x = ˜ N . Consider the inclusion morphism i : U ֒ → X and let M : = Ker[ M → i ∗ ( M | U / N )]. Observe that M | U = N . Given x ∈ X ,let N ⊂ M , x be a finitely generated submodule such that N ⊗ O x O y = M , y for any y ∈ U x ∩ U . Let U : = U ∪ U x and let N ⊂ M | U be the finitely generated O U -modulesuch that N | U = N and N | U x = ˜ N . Consider the inclusion morphism i : U ֒ → X and let M : = Ker[ M → i ∗ ( M | U / N )]. Observe that M | U = N . So on we shall get a finitelyquasi-coherent O X -submodule M n ⊂ M such that M n , x = N . Now it is easy to prove thisproposition. (cid:3)
19. Corollary :
Let X be a schematic finite space. Any quasi-coherent ideal
I ⊂ O X is thedirect limit of its finitely generated ideals I i ⊂ I .
20. Lemma:
Let X be a schematic finite space, ¯ U ⊂ ˜ X an open subset and C = ˜ X − ¯ U. Then, ¯ U is quasi-compact i ff there exists a finitely generated ideal I ⊂ O X such that ( I ) = C.Proof. ⇒ ) Consider the quasi-coherent ideal I C ⊂ O X . Let J = {I j } j ∈ J the set of finitelygenerated ideals of O X contained in I C . By Corollary 13.19, I C = lim → j ∈ J I j . Then, C = ( I C ) = ( lim → j ∈ J I j ) = ∩ j ∈ J ( I j ) and ¯ U = ∪ j ∈ J ( ˜ X − ( I j ) ). There exists j ∈ J such that¯ U = ˜ X − ( I j ) , since ¯ U is quasi-compact. Hence, C = ( I j ) . ⇐ ) Let x ∈ X , then I x = ( a , . . . , a n ) ⊂ O x is finitely generated. C ∩ ˜ U x = ( I x ) = ∩ i ( a i ) ,then ¯ U ∩ ˜ U x = ∪ i Spec O x , a i is quasi-compact. Therefore, ¯ U = ∪ x ( ¯ U ∩ ˜ U x ) is quasi-compact. (cid:3)
21. Proposition:
Let X be a schematic finite space. Then,
3. Spec O X
1. The intersection of two quasi-compact open subsets of ˜ X is quasi-compact.2. The family of quasi-compact open subsets of ˜ X is a basis for the topology of ˜ X.3. If ¯ V ⊆ ˜ X is a quasi-compact open subset then ¯ V ∩ ˜ U is quasi-compact, for any opensubset U ⊂ X.Proof.
1. Let ¯ U , ¯ U ⊂ ˜ X be two quasi-compact open subsets, C : = ˜ X − ¯ U , C : = ˜ X − ¯ U ,and I , I ⊂ O X two finitely generated ideals such that C = ( I ) and C = ( I ) . Then, C ∪ C = ( I ) ∪ ( I ) = ( I · I ) and ¯ U ∩ ¯ U = ˜ X − ( I · I ) . By Lemma 13.20, ¯ U ∩ ¯ U is quasi-compact.2. Let ¯ U ⊂ ˜ X be an open subset and C = ˜ X − ¯ U . I C = lim → j ∈ J I j , where {I j } j ∈ J is the set ofthe finitely generated of O X contained in I C . Then, C = ( I C ) = ( lim → j ∈ J I j ) = ∩ j ∈ J ( I j ) and¯ U = ∪ j ∈ J ( ˜ X − ( I j ) ), where the open subsets ˜ X − ( I j ) are quasi-compact by Lemma 13.20.3. Let C = ˜ X − ¯ V · and let I ⊂ O X be a finitely generated ideal such that C = ( I ) . Then, C ∩ ˜ U = ( I | U ) and ¯ V ∩ ˜ U = ˜ U − ( I | U ) . By Lemma 13.20, ¯ V ∩ ˜ U is quasi-compact. (cid:3)
22. Corollary :
Let X be a schematic finite space, U ⊆ X an open subset and ¯ U ⊂ ˜ U aquasi-compact open subset. Then,1. There exists a quasi-compact open subset ¯ W ⊂ ˜ X, such that, ¯ W ∩ ˜ U = ¯ U.2. ¯ U is equal to the intersection of the quasi-compact open subsets of ˜ X which contain it.Proof.
1. By Proposition 13.15, there exists an open subset ¯ W ′ ⊆ ˜ X such that ¯ W ′ ∩ ˜ U = ¯ U .Given p ∈ ¯ U , there exists a quasi-compact open subset ¯ W p ⊂ ¯ W ′ such that p ∈ ¯ W p . Thereexist p , . . . , p n ∈ ¯ U such that ¯ U ⊂ ∪ ni = ¯ W p i ⊂ ¯ W ′ . Hence, ¯ W : = ∪ ni = ¯ W p i holds ¯ W ∩ ˜ U = ¯ U .2. Given an open subset ¯ V ⊂ ˜ X such that ¯ U ⊂ ¯ V , there exists a quasi-compact opensubset ¯ V ′ ⊂ ˜ X such that ¯ U ⊂ ¯ V ′ ⊂ ¯ V . By Proposition 13.15, we are done. (cid:3)
23. Lemma :
Let X be a schematic finite space, U , U ⊂ X open subsets, ¯ V ⊂ ˜ U and ¯ V ⊂ ˜ U quasi-compact open subsets and ¯ W ⊂ ˜ X an open subset such that ¯ V ∩ ¯ V ⊂ ¯ W.Then, there exist open subsets ¯ W , ¯ W ⊂ ˜ X such that ¯ V ⊂ ¯ W , ¯ V ⊂ ¯ W and ¯ W ∩ ¯ W ⊂ ¯ W.Proof.
By the quasi-compactness of ¯ V and ¯ V , to prove this theorem we can easily reduceourselves to the case in which ¯ V = Spec O x , a ⊂ ˜ U x ( a ∈ O x ) and ¯ V = Spec O x , a ⊂ ˜ U x ( a ∈ O x ).3. Spec O X V ∩ ¯ V = ∅ . Let O U x , a be the quasi-coherent O U x -module defined by O U x , a ( U z ) = O z , a , for any z ∈ U x . Let i x : U x ⊂ X be the inclusion morphism. Let I bethe kernel of the natural morphism O X → i x ∗ O U x , a . Likewise, define O U x , a , i x and I .Observe that i x ∗ O U x , a ( U z ) = O x z , a for any z ∈ X , and the natural morphism i x ∗ O U x , a ( U z ) → Y y ∈ U x z O y , a is injective. Then, the sequence of morphisms( ∗∗ ) 0 → I , z → O z → Y y ∈ U x z O y , a is exact. Observe that ¯ V ∩ ˜ U z = ∪ y ∈ U x z Spec O y , a . Then, ( I ) is equal to the closure Cl ( ¯ V )of ¯ V in ˜ X .Let z = x , tensoring ( ∗∗ ) by ⊗ O x O x , a , we obtain the exact sequence0 → I , x ⊗ O x O x , a → O x , a → Y y ∈ U x x O y , a ⊗ O x O x , a . O y , a ⊗ O x O x , a = O y , a ∩ Spec O x , a ⊂ ¯ V ∩ ¯ V = ∅ . Then, I , x ⊗ O x O x , a = O x , a ,hence I , x · O x , a = O x , a . Therefore, ( I ) ∩ ¯ V = ∅ , that is, Cl ( ¯ V ) ∩ ¯ V ∗ = ∅ . Let J ⊂ I be a finitely generated ideal such that J , x · O x , a = O x , a . Again, ( J ) ∩ ¯ V = ∅ and¯ V ⊂ ( J ) . Likewise, define a finitely generated ideal J such that ( J ) ∩ ¯ V = ∅ and¯ V ⊂ ( J ) .Given a subset Y ⊂ ˜ X denote Y c : = ˜ X − Y . Let ¯ W : = ( Cl (( J · J ) c ) ∪ ( J ) ) c and¯ W : = ( Cl (( J · J ) c ) ∪ ( J ) ) c . Obviously, ¯ W ⊂ (( J · J ) c ∪ ( J ) ) c = ( J ) − ( J ) and¯ W ⊂ ( J ) − ( J ) . Then, ¯ W ∩ ¯ W = ∅ . We only have to prove that ¯ V ⊂ ¯ W . We knowthat ¯ V ∩ ( J ) = ∅ , it remains to prove that ¯ V ∩ Cl (( J · J ) c ) = ∅ . ¯ V ∩ ( J · J ) c = ∅ and ( J · J ) c is the union of a finite set of subsets Spec O y , b ⊂ ˜ U y ⊂ ˜ X (with b ∈ O y and y ∈ X ). As we have proved above ( ∗ = ), ¯ V ∩ Cl (Spec O y , b ) = ∅ since ¯ V ∩ Spec O y , b = ∅ . Then,¯ V ∩ Cl (( J · J ) c ) = ∅ .2. Suppose that V ∩ V , ∅ . Let I : = I ˜ X − ¯ W ⊂ O X . ˜ X − ¯ W ⊂ ˜ X is equal to ˜ Y : = Spec O X / I . ¯ V ∩ ˜ Y = Spec ( O X / I ) x , [ a ] and ¯ V ∩ ˜ Y = Spec( O X / I ) x , [ a ] . By 1., there existopen subsets ¯ W ′ , ¯ W ′ ⊂ ˜ Y such that ¯ W ′ ∩ ¯ W ′ = ∅ , ¯ V ∩ ˜ Y ⊂ ¯ W ′ and ¯ V ∩ ˜ Y ⊂ ¯ W ′ . Then,¯ W = ¯ W ∪ ¯ W ′ and ¯ W = ¯ W ∪ ¯ W ′ are the searched open subsets. (cid:3)
24. Lemma :
Let X be a schematic finite space and B the family of quasi-compact opensubsets of ˜ X. Let F ′ be a presheaf on ˜ X and F the sheafification of F ′ . If for any ¯ V ∈ B and
3. Spec O X any finite open covering { ¯ V i ∈ B } of ¯ V the sequence of morphisms F ′ ( ¯ V ) → Y i F ′ ( ¯ V i ) / / / / Y i j F ′ ( ¯ V i ∩ ¯ V j ) is exact, then F ′ ( ¯ V ) = F ( ¯ V ) .Proof. It is well known. (cid:3)
25. Corollary :
Let X be a schematic finite space and let { F i } i be a direct system of sheavesof abelian groups on ˜ X. Then, H n ( ˜ X , lim → i ∈ I F i ) = lim → i ∈ I H n ( ˜ X , F i ) , for any n ≥ .
26. Corollary :
Let X be a schematic finite space, U ⊂ X an open subset, ¯ V ⊂ ˜ U a quasi-compact open subset and F a sheaf of abelian groups on ˜ X. Then, F | ˜ U ( ¯ V ) = lim → ¯ V ⊂ ¯ W F ( ¯ W ) . Therefore, H n ( ¯ V , F | ˜ U ) = lim → ¯ V ⊂ ¯ W H n ( ¯ W , F ) , for any n ≥ .Proof. Let G be the presheaf on ˜ U defined by G ( ¯ V ) : = lim → ¯ V ⊂ ¯ W F ( ¯ W ). F | ˜ U is the sheafificationof G . Let { ¯ V i } a finite quasi-compact open covering of ¯ V . Let I i be the family of open subsets¯ W i ⊂ ˜ X such that ¯ V i ⊂ ¯ W i and I = Q i I i . The sequence of morphisms F ( ∪ i ¯ W i ) → Y i F ( ¯ W i ) / / / / Y i , j F ( ¯ W i ∩ ¯ W j )is exact. Taking direct limits, we obtain the exact sequencelim → ( ¯ W i ) ∈ I F ( ∪ i ¯ W i ) → lim → ( ¯ W i ) ∈ I Y i F ( ¯ W i ) / / / / lim → ( ¯ W i ) ∈ I Y i j F ( ¯ W i ∩ ¯ W j )Observe that lim → ( ¯ W i ) ∈ I F ( ∪ i ¯ W i ) = G ( ∪ i ¯ V i ), lim → ( ¯ W i ) ∈ I Q i F ( ¯ W i ) = Q i G ( ¯ V i ), and by Lemma 13.23,lim → ( ¯ W i ) ∈ I Q i j F ( ¯ W i ∩ ¯ W j ) = Q i j G ( ¯ V i ∩ ¯ V j ). Hence, G ( ∪ i ¯ V i ) → Y i G ( ¯ V i ) / / / / Y i , j G ( ¯ V i ∩ ¯ V j )4. H n ( X , M ) = H n ( ˜ X , ˜ M ) 59is exact. By Lemma 13.24, G ( ¯ V ) = F | ˜ U ( ¯ V ).Finally, let F → C ·F be the Godement resolution, then H n ( ¯ V , F | ˜ U ) = H n Γ ( ¯ V , ( C ·F ) | ˜ U ) = lim → ¯ V ⊂ ¯ W H n Γ ( ¯ W , C ·F ) = lim → ¯ V ⊂ ¯ W H n ( ¯ W , F ) . (cid:3)
27. Theorem:
Let X be a schematic finite space and U ⊂ X an open subset. Then, O ˜ X | ˜ U = O ˜ U . Let x ∈ X and p ∈ ˜ U x ⊆ ˜ X. Then, O ˜ X , p = O x , p . Proof.
Let i : U ֒ → X be the inclusion morphism and ˜ i : ˜ U ֒ → ˜ X the induced morphism.The natural morphism O ˜ X → ˜ i ∗ O ˜ U defines by adjunction the morphism O ˜ X | ˜ U → O ˜ U and wehave to prove that the morphism O ˜ X , p = O ˜ X | ˜ U , p → O ˜ U , p is an isomorphism, for any p ∈ ˜ U .Let I : = { ( ¯ W , ¯ V ), where ¯ V is any quasicompact open subset of ˜ U such that p ∈ ¯ V and ¯ W isany open subset of ˜ X such that ¯ V ⊂ ¯ W } . Then, O ˜ X , p = lim → ( ¯ W , ¯ V ) ∈ I O ˜ X ( ¯ W ) = lim → p ∈ ¯ V lim → ¯ V ⊂ ¯ W O ˜ X ( ¯ W ) = lim → p ∈ ¯ V O ˜ U ( ¯ V ) = O ˜ U , p . Finally, O ˜ X , p = O ˜ U x , p = O x , p . (cid:3) H n ( X , M ) = H n ( ˜ X , ˜ M )
1. Notation :
Given an a ffi ne scheme Spec R and an R -module M we shall denote ˜ M thesheaf of localizations of the R -module M .
2. Definition :
Let X be a schematic finite space and ˜ X = Spec O X . We shall say that an O ˜ X -module ¯ M is quasi-coherent if ¯ M | ˜ U x is a quasi-coherent O ˜ U x -module for any x ∈ X .I warn the reader that this definition is not the usual definition of quasi-coherent module.Let X be a schematic finite space and ¯ M a quasi-coherent O ˜ X -module. Let M be the O X -module defined by M x = ¯ M | ˜ U x ( ˜ U x ), then it easy to check that M is a quasi-coherent O X -module (see [6] II 5.1 (d) and 5.2 c.)Let M be a quasi-coherent O X -module. Define ˜ M : = lim ← x ∈ X ˜ i x ∗ g M x , where ˜ i x : ˜ U x → ˜ X is the morphism induced by the inclusion morphism i x : U x ֒ → X . Observe that ˜ M ( ˜ X ) = lim ← x ∈ X M x = M ( X ).4. H n ( X , M ) = H n ( ˜ X , ˜ M ) 60
3. Proposition :
Let X be an a ffi ne finite space and M a quasi-coherent O X -module. then ˜ M = g M ( X ) .Proof. Observe that ˜ i x ∗ g M x = g M x , then˜ M = lim ← x ∈ X ˜ i x ∗ g M x = lim ← x ∈ X g M x = g lim ← x ∈ X M x = g M ( X ) . (cid:3)
4. Proposition :
Let X be a schematic finite space, U ⊂ X an open subset and M a quasi-coherent O X -module. Then, ˜ M | ˜ U = g M | U . In particular, ˜ M is a quasi-coherent O ˜ X -module.Proof. Proceed like in the proof of Theorem 13.27. (cid:3)
Let M and M ′ be quasi-coherent O X -modules. Any morphism of O X -modules M → M ′ induces a natural morphism ˜ M = lim ← x ∈ X ˜ i x ∗ g M x → lim ← x ∈ X ˜ i x ∗ g M ′ x = ˜ M ′ .Let ¯ M and ¯ N be quasi-coherent O ˜ X -modules. Any morphism of O ˜ X -modules ¯ M → ¯ N induces natural morphisms M x : = ¯ M | ˜ U x ( ˜ U x ) → ¯ N | ˜ U x ( ˜ U x ) = : N x and then a morphism M → N .
5. Theorem:
Let X be a schematic finite space. The category of quasi-coherent O X -modulesis equivalent to the category of quasi-coherent O ˜ X -modules.Proof. The functors ¯ M { ¯ M | ˜ U x ( ˜ U x ) } x ∈ X and M ˜ M are mutually inverse. (cid:3)
6. Proposition :
Let f : X → Y be a schematic morphism and ˜ f : ˜ X → ˜ Y the inducedmorphism. Let M be a quasi-coherent O X -module and N a quasi-coherent O Y -module.Then,1. ˜ f ∗ ˜ M = g f ∗ M .2. ˜ f ∗ ˜ N = g f ∗ N .Proof. Consider the obvious commutative diagram˜ X ˜ f / / ˜ Y ˜ U x ?(cid:31) ˜ i x O O ˜ f xy / / ˜ U y ?(cid:31) ˜ i y O O ( f ( x ) ≥ y )4. H n ( X , M ) = H n ( ˜ X , ˜ M ) 611. Observe that ( f ∗ M )( U y ) = M ( f − ( U y )) = lim ← x ∈ f − ( U y ) M x , then g f ∗ M = lim ← y ∈ Y ˜ i y ∗ ( lim ← x ∈ f − ( U y ) g M x ) = lim ← y ∈ Y ˜ i y ∗ ( lim ← x ∈ f − ( U y ) ˜ f xy ∗ g M x ) = lim ← y ∈ Y lim ← x ∈ f − ( U y ) ˜ i y ∗ ˜ f xy ∗ g M x = lim ← y ∈ Y lim ← x ∈ f − ( U y ) ˜ f ∗ ˜ i x ∗ g M x = ˜ f ∗ lim ← y ∈ Y lim ← x ∈ f − ( U y ) ˜ i x ∗ g M x = ˜ f ∗ lim ← x ∈ X ˜ i x ∗ g M x = ˜ f ∗ ˜ M .
2. Observe that ( f ∗ N ) x = N f ( x ) ⊗ O f ( x ) O x , then g ( f ∗ N ) | ˜ U x = g ( f ∗ N ) | U x = g N f ( x ) ⊗ O f ( x ) O x .On the other hand, ( ˜ f ∗ ˜ N ) | ˜ U x = ˜ f ∗ x f ( x ) ( ˜ N | ˜ U f ( x ) ) = ˜ f ∗ x f ( x ) ( g N f ( x ) ) = g N f ( x ) ⊗ O f ( x ) O x . (cid:3)
7. Lemma :
Let X be a schematic finite space and F a sheaf of abelian groups on X. LetU , V ⊂ X be two open subsets. Consider the obvious commutative diagram ˜ U (cid:31) (cid:127) i / / ˜ X g U ∩ V = ˜ U ∩ ˜ V (cid:31) (cid:127) ¯ i / / ?(cid:31) ¯ j O O ˜ V ?(cid:31) j O O Then, j ∗ ( R n i ∗ F ) = R n ¯ i ∗ ( ¯ j ∗ F ) , for any n ≥ .Proof. Let p ∈ ˜ V . Then,( j ∗ ( R n i ∗ F )) p = ( R n i ∗ F ) p = lim → p ∈ ¯ W ⊂ ˜ X H n ( ¯ W ∩ ˜ U , F | ˜ U ) = lim → p ∈ ¯ W ′ ⊂ ˜ V H n ( ¯ W ′ ∩ ˜ U ∩ ˜ V , F | ˜ U ∩ ˜ W ) = ( R n ¯ i ∗ ( ¯ j ∗ F )) p . (cid:3)
8. Theorem :
Let X be a semiseparated schematic finite space and M a quasi-coherent O X -module. Then, H n ( X , M ) = H n ( ˜ X , ˜ M ) , for any n ≥ .Proof. Let ˜ i : ˜ U x ֒ → ˜ X be the inclusion morphism. The morphism ¯ i : ˜ U y ∩ ˜ U x ֒ → ˜ U y , p i ( p ), is an a ffi ne morphism of schemes since ˜ U x ∩ ˜ U y is an a ffi ne scheme because X is semiseparated. Let ˜ N be a quasi-coherent O ˜ U x -module and p ∈ ˜ U y . By Lemma 14.7,( R n ˜ i ∗ ˜ N ) p = ( R n ¯ i ∗ ˜ N | ˜ U y ∩ ˜ U x ) p =
0, for any n >
0. Hence, R n ˜ i ∗ N = H n ( ˜ X , ˜ i ∗ N ) = H n ( ˜ U x , N ) =
0, for any n >
0. That is, ˜ i ∗ N is acyclic.5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 62Given a quasi-coherent O X -module M denote ˜ M ˜ U x = ˜ i ∗ ˜ i ∗ ˜ M . Observe that ˜ M ˜ U x isacyclic and ˜ M ˜ U x ( ˜ X ) = ˜ M | ˜ U x ( ˜ U x ) = g M | U x ( ˜ U x ) = M ( U x ) = M x . The obvious sequenceof morphisms ˜ M → Y x ∈ X ˜ M ˜ U x → Y x < x ˜ M ˜ U x → Y x < x < x ˜ M ˜ U x → · · · is exact. Denote this resolution ˜ M → ˜ C · ˜ M and let M → C ·M be the standard resolution of M . Then, H n ( ˜ X , ˜ M ) = H n ( Γ ( ˜ X , ˜ C · ˜ M )) = H n ( Γ ( X , C ·M )) = H n ( X , M ) . (cid:3) Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y )
1. Proposition :
Let f : X → Y be a schematic morphism. Then, the induced morphism ˜ f : ˜ X → ˜ Y is quasi-compact, that is, ˜ f − ( ¯ V ) is quasi-compact for any quasi-compact opensubset ¯ V ⊂ ˜ YProof.
Any a ffi ne scheme morphism is quasi-compact. Given x ∈ X , denote ˜ f x : ˜ U x → ˜ U f ( x ) the morphism induced by f x : U x → U f ( x ) , f x ( x ′ ) : = f ( x ′ ). By Proposition 13.21.2., ¯ V ∩ ˜ U f ( x ) is quasi-compact. Then,˜ f − ( ¯ V ) = ∪ x ∈ X ˜ f − ( ¯ V ) ∩ ˜ U x = ∪ x ∈ X ˜ f − x ( ¯ V ∩ ˜ U f ( x ) )is quasi-compact. (cid:3)
2. Proposition :
Let f : X → Y be a schematic morphism and ˜ f : ˜ X → ˜ Y the inducedmorphism. Then, ˜ f − ( ˜ U ) = g f − ( U ) , for any open subset U ⊂ X.Proof.
Given two open subsets V , V ′ of a schematic finite space, observe that ˜ V ∩ ˜ V ′ = g V ∩ V ′ and ˜ V ∪ ˜ V ′ = g V ∪ V ′ .Given y ≥ f ( x ), Spec Q z ∈ U xy O z → Spec( O x ⊗ O f ( x ) O y ) = Spec O x × Spec O f ( x ) Spec O y issurjective, by Theorem 5.15. Hence, ˜ U x ∩ ˜ f − ( ˜ U y ) = ∪ z ∈ U xy ˜ U z = ˜ U xy = g U x ∩ f − ( U y ).Obviously, g f − ( U ) ⊆ ˜ f − ( ˜ U ). Let p ∈ ˜ f − ( ˜ U ) and x ∈ X such that p ∈ ˜ U x . Then,˜ f ( p ) ∈ ˜ U f ( x ) ∩ ˜ U . Let y ∈ U f ( x ) ∩ U such that ˜ f ( p ) ∈ ˜ U y . Then, p ∈ ˜ U x ∩ ˜ f − ( ˜ U y ) = g U x ∩ f − ( U y ) ⊂ g f − ( U ). Therefore, ˜ f − ( ˜ U ) ⊆ g f − ( U ). (cid:3)
3. Definition :
Let X and Y be schematic finite spaces. We shall say that a morphism ofringed spaces f ′ : ˜ X → ˜ Y is a schematic morphism if f ′∗ ˜ M is a quasi-coherent O ˜ Y -modulefor any quasi-coherent O ˜ X -module ˜ M .5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 63
4. Example : If f : X → Y is a schematic morphism, then ˜ f : ˜ X → ˜ Y is a schematicmorphism, by Proposition 14.6.
5. Proposition :
Let f : Spec B → Spec
A, f ′ : ˜ A → f ∗ ˜ B be a morphism of ringed spaces.If f ∗ ˜ M is a quasi-coherent ˜ A-module for any B-module M, then ( f , f ′ ) is a morphism ofschemes.Proof. Let f ′′ : Spec B → Spec A be the morphism defined on spectra by f ′ Spec A : A → B .We only have to prove that f = f ′′ .Let p ∈ Spec B , a ∈ A and U a = Spec A \ ( a ) . By the hypothesis, f ∗ g B / p is a quasi-coherent ˜ A -module. Then,( B / p ) a = (( f ∗ g B / p )(Spec A )) a = ( f ∗ g B / p )( U a ) = g B / p ( f − ( U a )) . Then, f ′ Spec A ( a ) ∈ p ⇐⇒ ( B / p ) a = ⇐⇒ g B / p ( f − ( U a )) = ⇐⇒ f − ( U a ) ∩ ( p ) = ∅⇐⇒ p < f − ( U a ) ⇐⇒ f ( p ) < U a ⇐⇒ a ∈ f ( p ) . Therefore, f ′ Spec A − ( p ) = f ( p ), that is to say, f ′′ = f . (cid:3)
6. Proposition :
Any schematic morphism f ′ : ˜ X → ˜ Y is a morphism of locally ringedspaces.Proof.
Let p ∈ ˜ U x ⊂ ˜ X . Let g be the composition of the schematic morphismsSpec O x , p ֒ → Spec O x ֒ → ˜ X → ˜ Y . Let y ∈ Y be a point such that f ( p ) ∈ ˜ U y , then g − ( ˜ U y ) = Spec O x , p . Consider the continuousmorphism h : Spec O x , p → ˜ U y , q g ( q ) and let ˜ i y : ˜ U y ֒ → ˜ Y be the inclusion morphism.Consider the morphism O ˜ Y → g ∗ ˜ O x , p . Taking i ∗ , we obtain a morphism φ : O ˜ U y → h ∗ ˜ O x , p .The morphism of ringed spaces ( h , φ ) is schematic, since h ∗ M = ˜ i ∗ y ˜ i y ∗ h ∗ M = ˜ i ∗ y g ∗ M is quasi-coherent, for any quasi-coherent O ˜ U y -module M . By Proposition 15.5, h is a morphism oflocally ringed spaces. We are done. (cid:3)
7. Lemma :
Let M be a finitely generated O X -module. For any p ∈ ˜ X, there exist an openneighbourhood ¯ U of p and an epimorphism of sheaves O n ¯ U → ˜ M | ¯ U .
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 64 Proof. ¯ V : = { q ∈ ˜ X : ˜ M q = } is an open subset of ˜ X , since ¯ V ∩ ˜ U x = { q ∈ ˜ U x : M x q = } isan open subset of ˜ U x . Hence, given another quasi-coherent module ˜ M ′ , and a morphism of O ˜ X -modules ˜ M ′ → ˜ M , if the morphism on stalks at p , ˜ M ′ p → ˜ M p is an epimorphism thenthere exists an open neighbourhood ¯ V of p such that the morphism of sheaves ˜ M ′| ¯ V → ˜ M | ¯ V is an epimorphism.Let m , . . . , m r be a generator system of the O p -module ˜ M p . Let ¯ W ⊂ ˜ X be an openneighbourhood of p , such that there exist n , . . . , n r ∈ ˜ M ( ¯ W ) satisfying n i , p = m i . Themorphism of O ¯ W -modules O r ¯ W → ˜ M | ¯ W , ( a i ) P i a i · n i is an epimorphism on stalks at p .Hence, it is an epimorphism in an open neighbourhood ¯ U of p . (cid:3)
8. Proposition:
Any schematic morphism f ′ : ˜ X → ˜ Y is quasi-compact.Proof.
Let ¯ V ⊂ ˜ Y be a quasi-compact open subset, we have to prove that f ′− ( ¯ V ) is quasi-compact. We only have to prove that ˜ U x ∩ f ′− ( ¯ V ) is quasi-compact, for any x ∈ X . Hence,we can suppose that ˜ X = ˜ U x .Let C : = ˜ Y − ¯ V . By Lemma 13.20, there exists a finitely generated ideal I ⊂ O Y suchthat ( I ) = C . Consider the exact sequence of morphisms 0 → I → O ˜ Y → O ˜ Y / I →
0. ByLemma 15.7, there exist an open covering { ¯ U i } of ˜ Y and epimorphisms O n i ¯ U i → I | ¯ U i . Taking f ∗ , one has an exact sequence of morphisms f ′∗ I → ˜ O x → f ′∗ ( O ˜ Y / I ) → J : = Im[ f ′∗ I → ˜ O x ] is a finitely generated quasi-coherent ideal of ˜ O x , since it is solocally (over f ′− ( ¯ U i )). ( J ) = f ′− ( C ), therefore f ′− ( ¯ V ) = ˜ X − ( J ) is a quasi-compactopen subset. (cid:3) Let C sch be the category of schematic finite spaces and W the family of quasi-isomorphisms.Let us construct the localization of C sch by W , C sch [ W − ].
9. Definition :
A schematic pair of morphisms from X to Y , X f −− > Y is a pair of schematicmorphisms ( φ ′ , f ′ ) X ′ φ ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ f ′ ❅❅❅❅❅❅❅❅ X Y where φ ′ is a quasi-isomorphism.
10. Example :
A schematic morphism f : X → Y can be considered as a schematic pair ofmorphisms: Consider the pair of morphisms ( Id X , f ) X Id X (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ f (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ X Y
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 65
11. Definition :
Let f = ( φ ′ , f ′ ) : X − − > Y y g = ( ϕ ′ , g ′ ) : Y − − > Z be two schematic pairsof morphisms, where φ ′ : X ′ → X is a quasi-isomorphism and f ′ : X ′ → Y is a schematicmorphism and ϕ ′ : Y ′ → Y is a quasi-isomorphism and g ′ : Y ′ → Z is a schematic morphism.Let π , π : X ′ × Y Y ′ → X ′ , Y ′ be the two obvious projection maps (observe that π is a quasi-isomorphism). We define g ◦ f : = ( φ ′ ◦ π , g ′ ◦ π ) : X − − > ZX ′ × Y Y ′ π z z ✉✉✉✉✉✉✉✉✉ π $ $ ■■■■■■■■■ X ′ φ ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ f ′ $ $ ■■■■■■■■■■ Y ′ ϕ ′ z z ✉✉✉✉✉✉✉✉✉✉ g ′ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ X f / / ❴❴❴❴❴❴❴❴❴ Y g / / ❴❴❴❴❴❴❴❴❴ Z Let f : X → Y and g : Y → Z be schematic morphisms. Then,( Id X , f ) ◦ ( Id Y , g ) = ( Id X , f ◦ g ) .
12. Definition :
Two schematic pairs of morphisms ( φ ′ , f ′ ) , ( φ ′′ , f ′′ ) : X − − > YX ′ φ ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ f ′ ❅❅❅❅❅❅❅❅ X ′′ φ ′′ ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ f ′′ ❆❆❆❆❆❆❆❆ X Y X Y are said to be equivalent, ( φ ′ , f ′ ) ≡ ( φ ′′ , f ′′ ), if there exist a schematic space T and twoquasi-isomorphisms π ′ : T → X ′ , π ′′ : T → X ′′ such that the diagram T π ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦ π ′′ ❆❆❆❆❆❆❆❆ X ′ φ ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ f ′ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ X ′′ φ ′′ t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ f ′′ ❆❆❆❆❆❆❆❆ X Y is commutative.In order to prove the associative property the reader should consider the following com-5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 66mutative diagram (where the double arrows are quasi-isomorphisms) T × X ′′ T ′ $ $ ■■■■■■■■■ $ $ ■■■■■■■■■ z z ✉✉✉✉✉✉✉✉✉✉ z z ✉✉✉✉✉✉✉✉✉✉ T $ $ ■■■■■■■■■■ $ $ ■■■■■■■■■■ ~ ~ ⑦⑦⑦⑦⑦⑦⑦ ~ ~ ⑦⑦⑦⑦⑦⑦⑦ T ′ ! ! ❈❈❈❈❈❈❈❈ ! ! ❈❈❈❈❈❈❈❈ z z tttttttttt z z tttttttttt X ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ , , ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ X ′′ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ X ′′′ ! ! ❇❇❇❇❇❇❇❇ r r ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ r r ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ X Y
13. Definitions and notations:
Let f = ( φ, f ′ ) : X − − > Y be a schematic pair of mor-phisms. The equivalence class of f (resp. ( φ, f ′ ) ) will be denoted [ f ] (resp. [ φ, f ′ ] ). We shallsay that [ f ] (or [ f ] : X → Y) is a [schematic] morphism from X to Y.Let f : X → Y be a schematic morphism. The equivalence class of ( Id , f ) will be denoted [ f ] .
14. Proposition:
Let ( φ ′ , f ′ ) : X − − > Y be a schematic pair of morphisms, where φ ′ : X ′ → X is a quasi-isomorphism and f ′ : X ′ → Y is a schematic morphism. Let ϕ : X ′′ → X ′ be aquasi-isomorphism. Then, [ φ ′ , f ′ ] = [ φ ′ ◦ ϕ, f ′ ◦ ϕ ] .Proof. Consider the commutative diagram X ′′ ϕ } } ⑤⑤⑤⑤⑤⑤⑤⑤ Id ! ! ❈❈❈❈❈❈❈❈ X ′ φ ′ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ f ′ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ X ′′ φ ′ ◦ ϕ t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ f ′ ◦ ϕ ❆❆❆❆❆❆❆❆ X Y (cid:3)
15. Theorem :
Let f , F : X − − > Y and g , G : Y − − > Z be schematic pairs of morphisms.If [ f ] = [ F ] and [ g ] = [ G ] , then [ g ◦ f ] = [ G ◦ F ] .Proof.
1. Let us prove that [ g ◦ f ] = [ g ◦ F ]. Write f = ( φ, f ′ ), φ : X ′ → X , f ′ : X ′ → Y .Let ϕ : X ′′ → X ′ be a quasi-isomorphism and let F ′ : = ( φ ◦ ϕ, f ′ ◦ ϕ ). By Proposition 15.14,5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 67[ f ] = [ F ′ ]. Consider the commutative diagram X ′′ (cid:15) (cid:15) ϕ (cid:15) (cid:15) X ′′ × Y Y ′ (cid:15) (cid:15) (cid:15) (cid:15) o o o o X ′ f ′ * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ (cid:15) (cid:15) φ (cid:15) (cid:15) X ′ × Y Y ′ o o o o / / Y ′ (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ X f , F ′ / / ❴❴❴❴❴❴❴❴❴❴❴ Y g / / ❴❴❴ Z By Proposition 15.14, [ g ◦ f ] = [ g ◦ F ′ ]. Finally, since [ f ] = [ F ], there exists F ′ such that[ g ◦ f ] = [ g ◦ F ′ ] = [ g ◦ F ].2. Likewise, [ g ◦ f ] = [ G ◦ f ].3. The theorem is a consequence of 1. and 2. (cid:3) Let f : X − − > Y and g : Y − − > Z be two schematic pairs of morphisms. We define[ g ] ◦ [ f ] : = [ g ◦ f ] .
16. Proposition:
Let ( φ, f ′ ) : X − − > Y be a schematic pair of morphisms. Then,1. If f ′ is a quasi-isomorphism, then [ φ, f ′ ] − = [ f ′ , φ ] .2. [ φ, f ′ ] = [ f ′ ] ◦ [ φ ] − .Proof. We have the quasi-isomorphism φ : X ′ → X and the morphism f ′ : X ′ → Y . Let π , π : X ′ × X X ′ → X ′ , π ( x ′ , x ′ ) : = x ′ , π ( x ′ , x ′ ) : = x ′ , which are quasi-isomorphisms.1. Let δ : X ′ → X ′ × X X ′ be the diagonal morphism, which is a quasi-isomorphismbecause π and π ◦ δ = Id are quasi-isomorphisms. Then,[ φ, f ′ ] ◦ [ f ′ , φ ] = [ f ′ ◦ π , f ′ ◦ π ] = [ f ′ ◦ π ◦ δ, f ′ ◦ π ◦ δ ] = [ f ′ , f ′ ] = [ Id Y , Id Y ] = [ Id Y ]Likewise, [ f ′ , φ ] ◦ [ φ, f ′ ] = [ Id X ].2. It is easy to check that [ f ′ ] ◦ [ φ ] − = [ Id X ′ , f ′ ] ◦ [ φ, Id X ′ ] = [ φ, f ′ ] . (cid:3)
17. Proposition :
Let X be a minimal schematic finite space. Let f , g : X → Y be twoschematic morphisms. Then, [ f ] = [ g ] i ff f = g.Proof. ⇒ ) There exists a (surjective) quasi-isomorphism π : T → X such that f ◦ π = g ◦ π .Then f and g are equal as continuous maps. Finally, the morphism O Y → f ∗ π ∗ O T = f ∗ O X coincides with the morphism O Y → g ∗ π ∗ O T = g ∗ O X . (cid:3)
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 68
18. Proposition :
Let X be a minimal schematic finite space. Let f , g : X → Y be twoschematic morphisms and π : Y → Y ′ a quasi-isomorphism. Then, π ◦ f = π ◦ g i ff f = g.Proof. ⇒ ) Observe that [ f ] = [ g ] since [ π ] ◦ [ f ] = [ π ◦ f ] = [ π ◦ g ] = [ π ] ◦ [ g ]. By Proposition15.17, f = g . (cid:3)
19. Proposition :
A [schematic] morphism [ φ, f ] : X → Y is an isomorphism i ff f is aquasi-isomorphism.Proof. ⇐ ) [ φ, f ] − = [ f , φ ], by Proposition 15.16. ⇒ ) [ φ, f ] = [ f ] ◦ [ φ ] − is invertible, then [ f ] is invertible. Put f : Z → Y and let[ ϕ, g ] : Y → Z be the inverse morphism of [ f ], where ϕ : T → Y is a quasi-isomorphism(and we can assume that T is minimal) and g : T → Z is a schematic morphism. Then,[Id Y ] = [ f ] ◦ [ ϕ, g ] = [ f ] ◦ [ g ] ◦ [ ϕ ] − and [ ϕ ] = [ f ◦ g ]. Hence, ϕ = f ◦ g , by Proposition15.17. Besides, [Id Z ] = [ ϕ, g ] ◦ [ f ]. If we consider the commutative diagram Z × Y T π / / π (cid:15) (cid:15) T ϕ (cid:15) (cid:15) g (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ Z f / / Y Z then [Id Z ] = [ ϕ, g ] ◦ [ f ] = [ π , g ◦ π ]. Let i : ( Z × Y T ) M ⊆ Z × Y T be the natural inclusion, π ′ ; = π ◦ i and π ′ = π ◦ i . Then, [ π ′ , g ◦ π ′ ] = [Id Z ] and [ π ′ ] = [ g ◦ π ′ ]. By Proposition15.17, π ′ = g ◦ π ′ . Let ˜ g : T → ( Z × Y T ) M , ˜ g ( t ) = ( g ( t ) , t ). Observe that π ′ ◦ ˜ g ◦ π ′ = g ◦ π ′ = π ′ . By Proposition 15.18, ˜ g ◦ π ′ = Id ( Z × Y T ) M . Obviously, π ′ ◦ ˜ g = Id T . Hence, π ′ is an isomor-phism and π a quasi-isomorphism. Finally, f is a quasi-isomorphism since π , φ and π arequasi-isomorphisms and f ◦ π = φ ◦ π . (cid:3) Given a [schematic] morphism g = [ φ, f ] : X → Y , consider the functors g ∗ : Qc - Mod X → Qc - Mod Y , g ∗ M : = f ∗ φ ∗ M g ∗ : Qc - Mod Y → Qc - Mod X , g ∗ M : = φ ∗ f ∗ M Recall [ φ, f ] = [ φ ◦ ϕ, f ◦ ϕ ], where ϕ is a quasi-isomorphism. Then we have canonicalisomorphisms f ∗ φ ∗ M = f ∗ ϕ ∗ ϕ ∗ φ ∗ M = ( f ◦ ϕ ) ∗ ( φ ◦ ϕ ) ∗ M and φ ∗ f ∗ M = φ ∗ ϕ ∗ ϕ ∗ f ∗ M = ( φ ◦ ϕ ) ∗ ( f ◦ ϕ ) ∗ M .
20. Proposition :
Let g = [ φ, f ] : X → Y be a [schematic] morphism. The functors g ∗ andg ∗ are mutually inverse i ff g is a [schematic] isomorphism.
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 69 Proof. ⇒ ) If Id = g ∗ g ∗ and Id = g ∗ g ∗ , then Id = f ∗ φ ∗ φ ∗ f ∗ = f ∗ f ∗ and Id = φ ∗ f ∗ f ∗ φ ∗ , hence f ∗ f ∗ = φ ∗ φ ∗ = Id. By Theorem 9.7, f is a quasi-isomorphism and g is invertible. ⇐ ) By Proposition 15.19, f is a quasi-isomorphism, then g ∗ g ∗ = f ∗ φ ∗ φ ∗ f ∗ = f ∗ f ∗ = Idand g ∗ g ∗ = φ ∗ f ∗ f ∗ φ ∗ = φ ∗ φ ∗ = Id. (cid:3)
21. Notation :
Let X and Y be two schematic finite spaces. Hom [ sch ] ( X , Y ) will denote thefamily of [schematic] morphisms from X to Y . Hom sch ( ˜ X , ˜ Y ) will denote the set of schematicmorphisms from ˜ X to ˜ Y .
22. Lemma :
Let g : Y ′ → Y be a [schematic] isomorphism and X a schematic finite space.Then, the maps
Hom [ sch ] ( X , Y ′ ) → Hom [ sch ] ( X , Y ) , [ f ] [ g ] ◦ [ f ]Hom [ sch ] ( Y , X ) → Hom [ sch ] ( Y ′ , X ) , [ f ] [ f ] ◦ [ g ] are biyective.
23. Proposition:
Let X be a schematic finite space and Y an a ffi ne finite space. Then, Hom [ sch ] ( X , Y ) = Hom rings ( O ( Y ) , O ( X )) . Proof.
For any schematic finite space T , Hom sch ( T , ( ∗ , A )) = Hom rings ( A , O ( T )) . Consider the natural morphism π : Y → ( ∗ , O ( Y )), which is a quasi-isomorphism. Then,Hom [ sch ] ( X , Y ) = Hom [ sch ] ( X , ( ∗ , O ( Y )) = Hom rings ( O ( Y ) , O ( X )) . (cid:3) Let [ φ, f ] : X → Y be a [schematic] morphism, where φ : X ′ → X is a quasi-isomorphismand f : X ′ → Y a schematic morphism. Consider the morphisms˜ X ′ ˜ φ ˜ f (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ ˜ X ˜ Y where ˜ φ is an isomorphism, by Proposition 13.12. The morphismHom [ sch ] ( X , Y ) → Hom sch ( ˜ X , ˜ Y ) , [ φ, f ] ˜ f ◦ ˜ φ − is well defined.
24. Lemma :
Let X be a minimal schematic space, Y a schematic T -space, f : X → Y aschematic morphism and ˜ f : ˜ X → ˜ Y the induced morphism. Given x ∈ X, y = f ( x ) i ff y isthe greatest element of Y such that ˜ f ( ˜ U x ) ⊆ ˜ U y .
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 70 Proof.
Obviiously f ( U x ) ⊂ U f ( x ) and ˜ f ( ˜ U x ) ⊂ ˜ U f ( x ) . If ˜ f ( ˜ U x ) ⊆ ˜ U y ′ , then ˜ U x ⊂ ˜ f − ( ˜ U y ′ ) = g f − ( U y ′ ) Hence, ˜ U x = ˜ U x ∩ g f − ( U y ′ ) = g U x ∩ f − ( U y ′ ). Therefore, x ∈ U x ∩ f − ( U y ′ ), since x is not a removable point. That is, f ( x ) ∈ U y ′ and f ( x ) ≥ y ′ . (cid:3)
25. Proposition:
Let X and Y be schematic finite spaces. The natural morphism
Hom [ sch ] ( X , Y ) → Hom sch ( ˜ X , ˜ Y ) , [ φ, f ] ˜ f ◦ ˜ φ − is injective.Proof. Let [ φ, f ] , [ φ ′ , f ′ ] : X → Y be [schematic] morphisms such that ˜ f ◦ ˜ φ − = ˜ f ′ ◦ ˜ φ ′− .We can suppose that φ = φ ′ , then ˜ f = ˜ f ′ . Let us say that f and f ′ are morphisms from X ′ to Y . By Proposition 13.12 and Lemma 15.22 we can suppose that X ′ and Y are minimalschematic spaces.By Lemma 15.24, the map f is determined by ˜ f . The morphism of rings O f ( x ′ ) → O x ′ isdetermined by the morphism of schemes Spec O x ′ → Spec O f ( x ′ ) . Therefore, f = f ′ . (cid:3)
26. Definition :
Let U i → U and U i → U be quasi-open immersions. We denote U ∪ U U : = C ( i ) ` U C ( i ). Observe that C ( i ) and C ( i ) are open subsets of U ∪ U U , C ( i ) ∪ C ( i ) = U ∪ U U , C ( i ) ∩ C ( i ) = U and the natural morphisms C ( i j ) → U j are quasi-isomorphisms, for j = , U , U ⊂ X be open subsets. Then, the natural morphism U ∪ U ∩ U U → U ∪ U isa quasi-isomorphism.Let V / / U V qqqqqq & & ▼▼▼▼▼▼ ˜ / / U qqqqqq & & ▼▼▼▼▼▼ V / / U be a commutative diagram of quasi-open immersions, where the arrows ˜ −→ are quasi-isomorphisms. Then, the natural morphism V ∪ V V → U ∪ U U is a quasi-isomorphism.
28. Theorem:
Let U i → U and U i → U be quasi-open immersions. Then, Hom [ sch ] ( U ∪ U U , Y ) = Hom [ sch ] ( U , Y ) × Hom [ sch ] ( U , Y ) Hom [ sch ] ( U , Y ) . In other words (by 15.27), let U , U ⊂ X be open subsets. Then,
Hom [ sch ] ( U ∪ U , Y ) = Hom [ sch ] ( U , Y ) × Hom [ sch ] ( U ∩ U , Y ) Hom [ sch ] ( U , Y ) .
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 71 Proof.
Let U j ֒ → U ∪ U and U j ֒ → U ∪ U be the obvious inclusion morphisms . Wehave to prove thatHom [ sch ] ( U ∪ U , Y ) → Hom [ sch ] ( U , Y ) × Hom [ sch ] ( U ∩ U , Y ) Hom [ sch ] ( U , Y ) (cid:2) f (cid:3) ([ f ] ◦ [ j ] , [ f ] ◦ [ j ])is bijective.Let [ f ] , [ g ] such that ([ f ] ◦ [ j ] , [ f ] ◦ [ j ]) = ([ g ] ◦ [ j ] , [ g ] ◦ [ j ]). There exist a minimalschematic finite space W , a quasi-isomorphism φ : W → U ∪ U and morphisms f ′ , g ′ : W → Y such that [ f ] = [ φ, f ′ ] and [ g ] = [ φ, g ′ ]. Then,[ f ] ◦ [ j ] = [ φ | φ − ( U ) , f ′| φ − ( U ) ] and [ g ] ◦ [ j ] = [ φ | φ − ( U ) , g ′| φ − ( U ) ] . By Proposition 15.17, f ′| φ − ( U ) = g ′| φ − ( U ) . Likewise, f ′| φ − ( U ) = g ′| φ − ( U ) . That is, f ′ = g ′ and[ f ] = [ g ].Let ([ f ] , [ f ]) ∈ Hom [ sch ] ( U , Y ) × Hom [ sch ] ( U , Y ) Hom [ sch ] ( U , Y ). Write [ f ] = [ φ , g ] and[ f ] = [ φ , g ]. Since [ f ] ◦ [ i ] = [ f ] ◦ [ i ], there exist a schematic finite space V and acommutative diagram V × U U ∩ U (cid:15) (cid:15) / / V g (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂❂❂ φ (cid:15) (cid:15) U V % % U ∩ U i ✐✐✐✐✐✐✐✐✐✐✐ i * * ❯❯❯❯❯❯❯❯❯❯❯ YU V × U U ∩ U O O / / V φ O O g @ @ ✁✁✁✁✁✁✁✁✁✁✁ (where the arrows · · · > are quasi-isomorphisms). Then, we have a schematic morphism g : V ∪ V V → Y and the composition φ of the quasi-isomorphisms V ∪ V V → U ∪ U ∩ U U → U ∪ U . The reader can check that [ φ, g ] ([ f ] , [ f ]). (cid:3)
29. Theorem:
Let X and Y be schematic finite spaces and suppose that Y is semiseparated.Then, the morphism
Hom [ sch ] ( X , Y ) → Hom sch ( ˜ X , ˜ Y ) , [ φ, f ] ˜ f ◦ ˜ φ − is bijective.
5. Hom sch ( ˜ X , ˜ Y ) = Hom [ sch ] ( X , Y ) 72 Proof.
By Proposition 15.25, it is injective. Let f ′ ∈ Hom sch ( ˜ X , ˜ Y ).1. Assume that ˜ X = Spec A is a ffi ne. We can suppose that Y is a T -space. Observe that f ′∗ ˜ A is a quasi-coherent O ˜ Y -module, then f ′∗ ˜ A = ˜ A , where A is a quasi-coherent O Y -moduleand an O Y -algebra. Let us prove that ( Y , A ) is a ffi ne. We only have to prove that ( Y , A p )is a ffi ne for any p ∈ Spec A . Given an open subset U ⊂ Y , let I U : = { quasi-compact opensubsets ¯ V ⊂ ˜ Y : ˜ U ⊂ ¯ V } . Recall A ( U ) = A | U ( U ) = g A | U ( ˜ U ) = ˜ A | ˜ U ( ˜ U ) = lim → ¯ V ∈ I U ˜ A ( ¯ V ) = lim → ¯ V ∈ I U ˜ A ( f ′− ( ¯ V )) . Denote by f ′ p the composition of the morphisms Spec A p ֒ → Spec A f ′ → ˜ Y . Observe that A p ( U ) : = A ( U ) p = lim → ¯ V ∈ I U ˜ A ( f ′− ( ¯ V )) p = lim → ¯ V ∈ I U ˜ A p ( f ′ p − ( ¯ V )) . Then, we can suppose that A = A p . The obvious morphism Id : ( Y , A ) → ( Y , O Y ) is a ffi neand ( U yy ′ , O Y | U yy ′ ) is a ffi ne, for any y , y ′ ∈ Y , since Y is semiseparated. Hence, ( U yy ′ , A | U yy ′ )is a ffi ne. Then, the morphism A yy ′ → Q k ∈ U yy ′ A k is faithfully flat. Let y be the greatest pointof Y such that f ′ ( p ) ∈ ˜ U y and let y ′ ∈ Y be another point. Observe that A y ′ = lim → ¯ V ∈ I Uy ′ ˜ A ( ¯ V ) ∗ = lim → ¯ V ∈ I Uy ′ , ¯ W ∈ I Uy ˜ A ( ¯ V ∩ ¯ W ) = lim → ¯ W ∈ I Uyy ′ ˜ A ( ¯ W ) = ˜ A | ˜ U yy ′ ( ˜ U yy ′ ) = A yy ′ ( ∗ = since f ′− ( ¯ V ) = f ′− ( ¯ V ∩ ¯ W )). Then, A y ′ = A yy ′ → Q k ∈ U yy ′ A k is faithfully flat. Hence, y ′ is a removable point of ( Y , A ) if y (cid:10) y ′ . Therefore, ( Y , A ) is quasi-isomorphic to ( U y , A | U y ),then it is a ffi ne.Finally, Spec A = Spec A ( Y ) = Spec A = ˜ X and the morphism ˜ X = Spec
A →
Spec O Y = ˜ Y induced by the obvious morphism Id : ( Y , A ) → ( Y , O Y ) is f ′ . Therefore,Hom [ sch ] ( X , Y ) = Hom sch ( ˜ X , ˜ Y ).2. Now, in general.Hom [ sch ] ( X , Y ) = Hom [ sch ] ( lim → x ∈ X U x , Y ) = lim ← x ∈ X Hom [ sch ] ( U x , Y ) = lim ← x ∈ X Hom sch ( ˜ U x , ˜ Y ) + = Hom sch ( lim → x ∈ X ˜ U x , ˜ Y ) = Hom sch ( ˜ X , ˜ Y )( + = given ( f x ) ∈ lim ← x ∈ X Hom sch ( ˜ U x , ˜ Y ) the induced morphism of ringed spaces f : lim → x ∈ X ˜ U x → ˜ Y is schematic, since for any quasi-coherent O ˜ X -module ˜ M = lim ← x ∈ X ˜ i x ∗ f M x , the O ˜ Y -module f ∗ ˜ M = lim ← x ∈ X f ∗ ˜ i x ∗ f M x = lim ← x ∈ X f x ∗ f M x is quasi-coherent). (cid:3) EFERENCES 73
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