<span class="mathjax-formula">$\mathcal {A}$</span>-Schemes and Zariski-Riemann Spaces

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A-Schemes and Zariski-Riemann Spaces

SATOSHITAKAGI

ABSTRACT - In this paper, we will investigate further properties ofA-schemes introduced in [Tak]. The category of A-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski-Riemann space in the category of A-schemes. We compare it with the classical Zariski- Riemann space, and characterize the latter by a left adjoint.

In this paper, we will investigate further properties ofA-schemes introduced in [Tak]. The first motivation of introducingA-schemes was to construct a scheme-like geometrical object from various kinds of algebraic systems, suchas commutative monoids, semirings, and etc. However, it happens to have advantages even in the case the algebraic system is that of rings. The category ofA-schemes is much more flexible than that of ordinary schemes: let us list up the properties ofA-schemes, and compare withordinary schemes:

(1) Let (Coh:Sch) be the category of coherent schemes and quasi-compact morphisms. Then, the category (A-Sch) of A-schemes contains (Coh:Sch) as a full subcategory. Also, (A-Sch) is a full subcategory of the category (LRCoh) of locally ringed coherent spaces and quasi- compact morphisms (Proposition 2.1.1).

(2) There is a spectrum functor, and is the left adjoint of the global section functorG:(A-Sch)^op!(Rng) ([Tak]).

(3) The inclusion functor (Coh:Sch)!(A-Sch) preserves fiber products (Corollary 2.3.4), and patchings via quasi-compact opens (Proposition 2.1.6).

(4) There is a valuative criterion of separatedness (Proposition 3.4.3) and that of properness (Proposition 3.4.4).

These imply that A-schemes behave much like ordinary schemes. By contrast, they have more virtues than ordinary schemes:

(5) The category (A-Sch) is small co-complete (Proposition 2.1.6) and small complete (Proposition 2.3.3).We don't need filteredness.

(6) There is a functorial epi-monic decomposition of morphisms of A- schemes (Theorem 2.2.12). In particular, we have the `image scheme' for eachmorphism.

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Therefore, we need not distinguish pro-schemes and ind-schemes from schemes anymore, if we work out on this category ofA-schemes. These properties give us various profits:

(7) We can consider quotientA-schemes whenever there is a group action on anA-scheme. We don't need any additional condition.

(8) Formal schemes can be treated on the same platform, asA-schemes (Example 3.1.7).

(9) We can think of universal `separation' ofA-schemes (Proposition 3.3.2).

(10) We can think of universal `compactification' ofA-schemes, namely, the Zariski-Riemann space (Theorem 4.1.2). The construction is the analog of the Stone-CÏechcompactification.

The key of this extension of the category of ordinary schemes is simple:to abandon the principal property of schemes, namely, `locally being a spectrum of a ring'. It is because this condition forces us only to use finite categorical operation in the category of ordinary schemes, and makes it very inconvenient. In particular, we have to give up Zariski-Riemann spaces in the category of ordinary schemes, although it is a fairly nice locally ringed space and there are various applications. On the other hand, Zariski-Riemann spaces can be treated equally withordinary schemes, when we extend our perspective to the category of A-schemes. Moreover, the construction of Zariski-Riemann spaces appears to be as natural as the spectrum of a ring.

Let us describe the contents of this paper. In § 1, we will prove some properties of coherent spaces, which we will need later. In particular, a quasi-compact morphism of coherent spaces is epic if and only if it is surjective, and its image is closed if and only if it is specialization closed.

In § 2, we will discuss the properties of the category of A-schemes, namely we will prove the co-completeness and completeness. The key lemma is the functorical decomposition of morphisms. This gives us the upper bound of the cardinality of the set of morphisms with fixed targets, and hence enables us to construct limits. This is the analog of the construction of co-limits in the category of algebras of various kinds.

In § 3, we define separated and proper morphisms ofA-schemes, and give valuative criteria for separatedness and properness. Unlike ordinary schemes, we don't have a canonical morphism SpecOX;x!X in the category ofA-schemes, wherexis a point of anA-schemeX. Hence, we had to modify the testing morphisms. The valuative criteria in the category of ordinary schemes check the right lifting properties of the com-

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mutative square

whereKis an arbitrary field andRis its valuation field. The left vertical arrow is the `testing morphism'. In the category ofA-schemes, we must replace the testing morphisms to formulate the valuative criteria: namely, just take the setfj;hgof the generic point and the closed point of Spec R witha natural induced A-scheme structure. Once we have the valuative criteria, we can construct the universal separation and the universal compactification; the latter is treated in § 4. We note here, that we will not include `of finite type' condition in the definition of proper morphisms, for we want to take limits. We emphasize the fact that separated morphisms and universally closed morphisms are closed under taking infinite limits, while morphisms of finite types are not.

In § 4, we construct the Zariski-Riemann space as the universal compactification using the adjoint functor theorem. Later, we will also consider

`classical Zariski-Riemann space' on irreducible, reducedA-schemes, as it happens to be more easier to analyze than the previous Zariski-Riemann space. This construction is the analog of the conventional one, but with different flavor:we tried not to use valuation rings when defining it. Th is reveals the naturalness of the concept of valuation ringsÐ that it is as natural as the concept of `local rings' of spectra of rings. Also, note that what localization is to the spectrum is what separated, of finite type morphism is to the Zariski-Riemann space. These also imply that the way of constructing a topological object from rings is not at all uniqueÐwe can consider another `algebraic geometry'.

Here, we also compare our previous Zariski-Riemann space withthe classical one. Actually, the topology of the classical Zariski-Riemann space happens to be coarser: it is, in a sense, the coarsest possible topology. This property, which we will call `of profinite type', characterizes the classical Zariski-Riemann space. Though the classical Zariski-Riemann space has a weaker universality, it is valuable since it gives us concrete descriptions of its structure and morphisms. Any separated dominant morphism of ordinary integral schemes is of profinite type, so that they can be embedded into a universal proper, of profinite typeA-scheme.

In this paper, we only constructed Zariski-Riemann space for irreducible, reducedA-schemes, since this assumption makes the argument

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muchsimpler, and it will be sufficient for most of the applications. We believe that it is possible to extend it to arbitraryA-schemes with a little more effort. We have proved a variant of the Nagata embedding (Corollary 4.6.6). The original version of the Nagata embedding can also be proven, and will be shown in the forthcoming paper. We decided not to prove it here, since there are various proofs published already ([Nag], [Con], [Tem]), and it takes a little more detailed work which will make this paper more longer if we include it. However, the proof is rather natural and in- tuitive than the former ones.

We summarized the definition ofA-schemes at the end of this paper, as an appendix. This will be sufficient for the reader to go through this paper, though he hasn't looked over [Tak].

0.1 ±Notation and conventions

The reader is assumed to have standard knowledge of categorical theories; see for example, [CWM], [KS]. We fix a universe, and all sets are assumed to be small. The category of small sets (resp. sober spaces and continuous maps) is denoted by (Set) (resp. (Sob)).

When we talk of an algebraic system, all the operators are finitary, and all the axioms are identities. Any ring and any monoid is commutative, and unital. For a ring (or, other algebras with a structure of a multiplicative monoid)R, we denote byR_Sthe localization ofRalong the multiplicative systemSofR.

For any setS,P(S) is the power set ofS, andP^f(S) is the set of finite subsets ofS.

When given anA-scheme (or, any topological space with its structure sheaf)X, we denote the underlying space byjXj.

We frequently denote finite summations byP₅₁

, when the range of the index is not crucial. The same thing can be said for the notation[⁵¹ for finite unions.

1. Properties of coherent spaces

In this section, we will investigate some properties of sober and coherent spaces. Recall that a topological spaceXissober, if every irreducible closed subset ofXhas a unique generic point. For a sober spaceX,C(X) is the set of closed subsets ofX. This becomes a complete II-ring; see appendix § 5 for

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further details. A topological spaceXiscoherent, if it is sober, quasi-compact, quasi-separated, and has a quasi-compact open basis.

1.1 ±Monic and epic maps

LEMMA 1.1.1. Let s be any algebraic system, and f :A!B b e a homomorphism ofs-algebras. Then, f is monic if and only if f is injective.

PROOF. The `if' part is clear.

Suppose f is not injective. Then there are two distinct elements a₁;a₂2Asuchthat f(a₁)f(a₂). Let R₀ be the initial object in (s-alg).

Define two homomorphismsg_i:R₀[X]!Aby X7!a_i fori1;2. Then, fg1fg2butg16g2, a contradiction. p In the sequel, let f :X!Y be a morphism of sober spaces, and f^#:C(Y)!C(X) the corresponding homomorphism of complete idealic rings (for complete idealic rings, see Appendix).

PROPOSITION1.1.2. The followings are equivalent:

(i) f is injective.

(ii) f is monic.

(iii) Any prime element of C(X)is in the image of f^#. PROOF. (i),(ii) follows from Lemma 1.1.1.

(i))(iii): Letxbe a point ofX. It suffices to show that f ¹(f(x)) fxg.

Let w2f ¹(f(x)) be the generic point of an irreducible component of f ¹(f(x)). Then we havef(w)f(x). Sincef is injective,wx. This shows that f ¹(f(x))x.

(iii))(i): Note that SpecC(X)Imf^# shows thatfxg f ¹(f(x)) for any pointxofX. Iff(x)f(x⁰) for two points ofX, then

fxg f ¹(f(x))f ¹(f(x⁰)) fx⁰g:

SinceXis sober,xcoincidesx⁰. p

(i) f^#is surjective.

(ii) Imf is homeomorphic to X.

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PROOF. (i))(ii): Since f^# is surjective, it is epic, hence f is monic.

This shows that f is injective. The surjectivity of f^# shows that f ¹(f(z)\Imf)z for any closed subset z of X. Hence f(z)\Imf f(z), which implies f(z) is closed in Imf.

(ii))(i): Since f is injective,f ¹(f(z))zfor any closed subsetzofX.

On the other hand, f(z) is closed in Imf sincef is homeomorphic onto the image. Therefore, f ¹(f(z))z, which implies f^#is surjective. p Next, we will investigate the condition when a morphism f :X!Y of sober spaces becomes epic. Let (IIRng^y) be the catogory of complete idealic semirings (see Appendix). First, note that the functorC:(Sob)^op! (IIRng^y) is fully faithful, since it is the right adjoint and left inverse of Spec ([Tak]). An objectRof (IIRng^y) isspatial, ifRis isomorphic toC(X) for some sober spaceX.

REMARK 1.1.4. There exist some non-spatial complete II-rings. Let C(R) be the complete II-rings of closed subsets of the real line with the standard topology. Let RC(R)= be the quotient complete II-ring, whereis the congruence generated byZZ^ofor anyZ, whereZ^ois the open kernel ofZ. Then, Ris a non-trivial II-ring, but has no points; see [Ste].

LEMMA1.1.5. Let R be an object of(IIRng^y), and R[t]be a polynomial complete idealic semiring with idempotent multiplication.

(1) An element of R[t]can be expressed by abt, where a;bare elements of R and ab.

(2) A prime element p of R[t]is either of the following:

(a) pabt, where a;bare prime elements of R and ab.

(b) pat, where a is a prime element.

(3) If an object R of(IIRng^y)is spatial, then so is R[t].

(4) In particular, a morphism f :X!Y of sober spaces is epic if and only if f^#:C(Y)!C(X)is monic.

PROOF. (1) Easy.

(2) Letpabtbe a prime element. It is easy to see thata2Rmust be prime. Also,bmust be prime or 1, sincextytxyt.

(3) Easy.

(4) The `if' part is obvious, since (Sob)^op is can be regarded as a full subcategory of (IIRng^y) via C. If f^# is not monic, we have two distinct

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morphismsg;h:F1[t]!C(Y) suchthat f^#gf^#h, whereF1is the initial object of (IIRng^y). But sinceF1[t] is spatial, we conclude thatf is not epic.

(1) f is epic.

(2) f^#is injective.

(3) f^#is monic (in(IIRng^y)).

(4) Imf \z is dense in z, for any closed subset z of Y.

PROOF. (i),(ii),(iii) is a consequence of Lemma 1.1.1 and 1.1.5.

(ii))(iv): For any closed subsetzofX, letz⁰be the closure of Imf\z.

Then we have f^#(z⁰)f^#(z). Since f^#is injective, we havez⁰z, hence the result follows.

(iv))(ii): Suppose f^#(z)f^#(z⁰) for some closed subsetsz;z⁰ofY. Th e equation ff ¹(z)Imf\zinduces

zImf \zImf\z⁰z⁰:

p

1.2 ±Coherent spaces

PROPOSITION1.2.1. Let f :X!Y be an epimorphism of sober spaces, and X be noetherian. Then f is surjective.

PROOF. Letybe any point ofY, andZ fygbe the irreducible subset corresponding toy. SinceXis noetherian,f ¹(Z) can be covered by a finite number of irreducible closed subsets: f ¹(Z) [⁵¹_i Wi. Let j_i be the generic point ofW_ifor eachi. Since the image of f is dense inZandZis irreducible, at least one of thej_i's must be mapped toy. p The proof of the next theorem requires some preliminaries on ultrafilters (see [CN], for example). The reader who knows well may skip and go on to the next theorem.

DEFINITION1.2.2. LetSbe a non-empty set.

(1) AfilterF onSis a non-empty subset ofP(S) satisfying:

(i) ;2=F .

(ii) IfA2F andAB, thenB2F .

(iii) IfAandBare elements ofF , thenA\B2F .

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The set of filters becomes a poset by inclusions.

(2) A maximal filter with respect to inclusions is called anultrafilter.

A filterUis an ultrafilter if and only ifa2Uora^c 2Ufor any subset aofS. Also, note that exactly one ofaora^cis inU.

For anys2 S,

U_s fa S js2ag

becomes an ultrafilter, which is calledprincipal. An ultrafilter is principal, if and only if it contains a finite subset ofS.

LetS be a subset ofP(S) satisfying (i) ;2= S .

(ii) IfAandBare elements ofS , thenA\B2S .

Then there exists a ultrafilter containingS , using the axiom of choice.

A filter F on a non-empty set is prime, if a[b2F implies either a2F orb2F . One can easily prove that the notion of prime filters is equivalent to that of ultrafilters.

LEMMA 1.2.3. Let X fxlg_l be a set, and Xl fxlg be one-pointed spaces, regarded as coherent spaces. Let X1 `

lXlbe the coproduct of X_l's in the category of coherent spaces. Then, any point of X1corresponds to a ultrafilter on X.

PROOF. First, note thatXis isomorphic to Spec (Q

lF1), whereF1is the initial object in the category of (IIRng). Hence, a closed subset ofX1

corresponds to a filter onX, and any point ofX1corresponds to a prime

filter, in other words, a ultrafilter onX. p

THEOREM1.2.4. Let f :X!Y be an epimorphism of coherent spaces.

Then f is surjective.

PROOF. Let y₀2Y be any point ofY, andY₀be the closure offy₀g in Y. Since f is epic, fy_lg_lImf \Y₀ is dense in Y₀. Assume that y₀2=Imf\Y₀. Th en, Imf\Y₀must be an infinite set, since if it is finite, then its closure is equal to[⁵¹_l fylg, which is a proper closed subset of Y0. Choose xl2X suchthat f(xl)yl for each l, and set S fxlg_l. Also, let X~ `

lfx_lg be the coproduct of fx_lg's in the category of coherent spaces. By Lemma 1.2.3, the points of X~ correspond to the ultrafilters onS. We have the natural commutative diagram

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Next, we define a map W:C(Y0)n fY0g !PSn f;g by sending Z to fx2Sjf(x)2=Zg. This is well defined, since f(S) is dense in Y₀. Also, note that ImW is stable under taking finite intersections: W(Z₁)\ W(Z2)W(Z1[Z2). It follows that there is an ultrafilter U on S, con- taining the image ofW. Letx0be the point ofX, corresponding to~ U. We claim that~f(x0)y0, from wh ich f(i(x0))y0follows. Indeed, the image of x₀ is the generic point of the intersections of Z2C(X)_cpt suchthat x₀2~f ¹(Z). Hence, it suffices to show thatY₀ is the only closed subset satisfying the condition. If Z6Y₀, th en Z is a proper closed subset of Y0, andx02~f ¹(Z) impliesW(Z)^c2U. But on the other hand,W(Z)2U, which is a contradiction to U being a ultrafilter. p COROLLARY1.2.5. Let f :X!Y be a morphism of coherent spaces.

Then, the image of f with its induced topology is also coherent.

PROOF. Th e morph ism f corresponds to the homomorphism f^#:C(Y)_cpt ! C(X)_cpt of II-rings. Let R be the image of f^#. Th en, C(Y)_cpt!Ris surjective, andR!C(X)_cpt is injective. SetWSpecR.

Then, f factors throughW, withX!Wepic (hence surjective by Theo- rem 1.2.4) andWYan immersion by Proposition 1.1.3. This tells thatW

coincides withthe image of f. p

EXAMPLE1.2.6. (1) LetY jA¹_Cjbe the underlying space of the affine line overC, andXY(C) be the set of closed points ofY, endowed witha discrete topology. Then, the natural mapX!Y is a morphism of sober spaces. This is an epimorphism by Proposition 1.1.6, but not surjective, since the image does not contain the generic point ofY.

On the other hand, we have a morphism alg(X)!Yof coherent spaces:

here, alg is the left adjoint of the underlying functor (Coh)!(Sob) ([Tak]). In this case, alg(X) is just the coproduct`

x2Xfxgin the category of coherent spaces. This is also epic, hence surjective by 1.2.4. The non- principal points of alg(X) maps onto the generic point ofY.

(2) LetXbe as above, and setV jA²_Cj, the affine plane overC. Since there is a non-canonical bijection W:C!C², there exists a map

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W:X!V, the image of which is the set of closed points ofV. When we algebraizeX, we again obtain a surjective map alg(X)!V. Some of the non-principal points ofXmap to a generic point of a curve onV, others map to the generic point ofV. These two examples tell us that the non-principal points of alg(X) behave like `universal generic points' ofX, although the Krull dimension of alg(X) is zero.

The next theorem is important when we consider valuative criteria.

THEOREM1.2.7. Let f :X!Y be a dominant morphism of coherent spaces. Then, any minimal point of Y (that is, the generic point of an irreducible component of Y) is contained in the image of f .

PROOF. Let y₀ be any minimal point ofY, andfU^lg_l be the filtered system of quasi-compact open neighborhood of y₀. For each l, f ¹(U^l)!U^lis dominant sincefis so. SetU¹ limlU^l. This is a pointed space fy0g, since the underlying functor (Coh)!(Set) preserves limits ([Tak]). We claim that f ¹(U¹)!U¹ is dominant (in particular, f ¹(U¹)6 ;). SinceC(U¹)_cptandC(f ¹(U¹))_cptare naturally isomorphic to lim_!C(U^l)_cpt and lim_!C(f ¹(U^l))_cptrespectively, it suffices to show that the homomorphism

f^#:lim_!C(U^l)_cpt!lim_! C(f ¹(U^l))_cpt

satisfies f^#(Z)0)Z0. Let Z be an element satisfying f^#(Z)0.

Since fC(U^l)_cptg_l is a filtered inductive system, Z and f^#(Z) can be represented by an element of C(U^l)_cpt and C(f ¹(U^l))_cpt for some l, respectively. Since C(U^l)_cpt !C(f ¹(U^l))_cpt is dominant, Z must be

zero. p

COROLLARY1.2.8. LetXbe a coherent subspace of a coherent spaceY. Then, the closure of Xconsists of all points which are specializations of points onX.

PROOF. LetZbe the set of points which are specializations of points on X. It is clear thatZX, so we will show the converse. Letybe any point in the closure of XinY. SinceXis also a coherent subspace, there exists a minimal point y₀ of X which is a generalization of y. Since X!X is dominant,y0 is contained inXby Theorem 1.2.7. SinceZis stable under

specializations,ymust be inZ. p

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2. The category ofA-Schemes

In [Tak], we introduced the definition ofA-schemes. The advantage of the notion ofA-schemes is not only generalizing the concept of schemes to other algebraic systems, but also giving the way to infinite categorical operations: in fact, the category ofA-schemes is small complete and co- complete. A finite patching over quasi-compact open sets, and fiber products commute withthose ofQ-schemes, namely, ordinary schemes.

Notations: from now on, the homomorphism C(Y)_cpt!C(X)_cpt associated to a morphism f :X!YofA-schemes is denoted byf ¹, orjfj ¹. We use the notation f^# for the morphism of structure sheaves O_Y!fO_X.

2.1 ±Co-completeness

First, we begin withdescribing what A-schemes is like when the algebraic system is that of rings.

PROPOSITION 2.1.1. Let s be an algebraic system of rings, and (LRCoh) be the category of locally ringed coherent spaces and quasi- compact morphisms. Then, there is a natural underlying functor (A-Sch)!(LRCoh) defined by (X;O_X;b_X)7!(X;O_X). Further, this functor is fully faithful.

PROOF. LetX(X;O_X;b_X) be anA-scheme. What we have to show first is thatXis a locally ringed space, i.e.O_X;xis a local ring for anyx2X.

LetM_xbe a subset ofO_X;x, consisting of germsasuchthatb_Xa₂(a)3x. We will show that this is the unique maximal ideal ofOX;x.

Leta;bbe two elements ofMx. We havea2(ab)a2(a)a2(b); recall thata2(a) is the principal ideal generated bya. Hence,

b_Xa₂(ab)b_Xa₂(a)b_Xa₂(b)xxx;

which shows thatab2M_X. It is easier to show thatca2M_x for any c2OX;xanda2Mx. Hence,Mxis an ideal ofOX;x.

SupposehU;ai 2OX;x is not contained in Mx. Set Zb_Xa2(a). This does not contain x. Set V UnZ. Since restriction morphisms reflect localizations (see appendix § 5 for the terminology),aj_Vis invertible, hence ais invertible inO_X;x. This shows thatO_X;xis local, the maximal ideal of which isMx.

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Let f :X!Y be a morphism ofA-schemes. It suffices to show that f^#:OY;f(x)!OX;xis a local homomorphism for anyx2X. Let abe an element ofMy, whereyf(x). Then,

b_Xa2(f^#(a))b_X(a1f)(a2(a)) jfj ¹b_Ya2(a):

Since b_Ya2(a)y, we have jfj ¹b_Xa2(a)x, which shows that My (f^#) ¹Mx.

Hence, we have a functor U:(A-Sch)!(LRCoh). It remains to show that this functor is fully faithful. LetX;Y be twoA-schemes, and f (jfj;f^#):UX!UY be a morphism of locally ringed spaces. It suffices to show that the following diagram

is commutative. Letabe a section ofa1OY. Then

jfj ¹b_Y(a) fxjM_Y;_f(x) ag; fb_Xa₁f^#(a) fxjM_X;xf^#ag;

where MY;f(x) andMX;x are the maximal ideals ofOY;f(x), OX;x, respectively. But the right-hand sides of the both equations coincide, since f^#:O_Y;_f(x) !O_X;xis a local homomorphism for anyx. p In the sequel, we fix a schematizable algebraic type A(s;a₁;a₂;g) ([Tak]).

DEFINITION2.1.2. (1)AnA-scheme X is aQ-scheme, if it is locally isomorphic toSpec^AR, for somes-algebra R.

(2) Let (Q-Sch) be the full subcategory of(A-Sch), consisting of Q- schemes.

Proposition 2.1.1 tells that, if sis the algebraic system of rings, then (Q-Sch) is the category of coherent schemes and quasi-compact morphisms, since a morphism of schemes is a morphism of locally ringed spaces.

PROPOSITION2.1.3. The category(Q-Sch)admits finite patching via quasi-compact opens, namely: let X₁; ;X_n be Q-schemes, U_ijX_i be

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quasi-compact open subsets, andW_ji:Uij!Ujibe isomorphisms satisfying W_kjW_jiW_ki on Uij\Ujk. Then, there exists a co-equalizer X of

`

ijU_ij!!`

iX_i, such that`

iX_i!X is a quasi-compact open covering.

Note that, when we speak of a covering, it must be always surjective.

PROOF. By induction onn, it suffices to prove forn2: letX₁,X₂be twoQ-schemes, andX₁ -U,!X₂ be the intersection quasi-compact open subscheme ofX1andX2. LetXbe the amalgamationX1`

UX2ofX1andX2

alongU. This is well defined, and coincides with the usual topology, sinceX is coherent, thanks to U being quasi-compact. This also shows that

fX_i!Xg_i1;2is indeed a covering. p

The next lemma is peculiar to II-rings.

LEMMA2.1.4. Let C fC^lgbe a projective system of II-rings. Let C be the limit of C, andpl:C!C^lbe the natural morphisms. Then, for any Z2C, the localization C_Zalong Z is naturally isomorphic tolim_l(C^l)_p_l_Z. PROOF. SinceZmaps to 1 by the morphism C!(Cl)_p_l_Z, We have a natural morphism CZ!(C^l)_p_l_Z. This in turn gives a natural morphism C_Z!lim_l(C^l)_p_l_Z.

Next, we see that ifW:C_Z!C_Zis a endomorphism such thatp_lWp_l for any l, then W is the identity.p_lW(x)p_lx in (C^l)_p_l_Z is equivalent to pl(ZW(x))pl(Zx) inC^l. Since this holds for anyl, we haveZW(x)Zx, which is equivalent toW(x)xinCZ.

Finally, we construct lim_l(C^l)_p_l_Z!C_Z. The map f :lim_lC^l!C_Z is already defined, hence we only need to verify thatf(x)f(y) impliesxy in lim_l(C^l)_p_l_Z, but this is obvious.

Combining all the arguments, we see that C_Z coincides with

liml(C^l)_p_l_Z. p

PROPOSITION2.1.5. (1) LetfX_lg be an inductive system of coherent spaces, and Xlim_!_lX_l. Then, there is a natural isomorphism t_X 'lim_li_lt_X_l, wherei_l:X_l!X are the induced morphisms.

(2) Let fX^lg be a filtered projective system of coherent spaces, and Xlim_lX^l. Then, there is a natural isomorphism t_X'lim_!_lp_l¹t_Xl, wherep_l:X!X^l are the induced morphisms.

PROOF. (1) This follows from the above lemma.

(2) First, there is a natural morphismt_X^l!pltX. Taking the adjoint, we obtain f_l:p_l¹t_X^l!t_X. Taking the limit, we obtain lim_!_lp_l¹t_X^l!t_X.

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Next, we show that ifW:tX !tXis an endomorphism withWflflfor any l, then Wis the identity. But this follows from the fact that the projective system is filtered. We can also construct a natural morphism t_X !lim_! p_l¹t_X^l, using the filteredness. Combining all these, we obtain the

required isomorphism. p

PROPOSITION2.1.6. (1)The category(A-Sch)is small co-complete.

(2) The category(A-Sch)admits finite patchings via quasi-compact opens. Moreover, the inclusion functor I:(Q-Sch)!(A-Sch)preserves finite patchings via quasi-compact opens.

PROOF. (1) LetfXlgbe a small inductive system ofA-schemes. First, we will construct the A-schemeX(jXj;O_X;b_X). The underlying space jXjis given by the colimit of the underlying spacesjX_lj. Setji_lj:jXj ! jX_lj be the associated morphisms. The structure sheafO_Xis defined by the limit ofji_ljO_X_l, as a (s-alg)-valued sheaf onX.

We have a morphism

a₁ji_ljO_X_l' ji_lja₁O_X_l^ji^l^j!^b^X^l ji_ljt_X_l;

which extends to givea₁O_X ! ji_ljt_X_l. Taking the limit and using Propo- sition 2.1.5, we obtain

b_X :a1OX !limljiljtXl 'tX:

We will verify that restriction maps reflect localizations. Let OX(U)! O_X(V) be a restriction map, and letZUnV be the closed subset ofU.

LetZ_l be the inverse image ofZbyX_l!X, so that we will denoteZby (Z_l)_l. It suffices to show that ifa(a_l)_l2O_X(U) satisfiesb_Xa₂(a)Z, then a is invertible in O_X(V). Since b_X_la₂(a_l)Z_l, a_l is invertible in OXl(Vl), whereV (Vl)_l. The uniqueness of the inverse element shows thatais also invertible inOX(V). Thus, we have defined anA-schemeX.

We also have natural morphismsi_l:X_l!X.

We will show that X is actually the colimit. Let j_l:X_l!Y be morphisms, compatible with the transition morphisms. There is a unique natural morphism jjj:jXj ! jYj between the underlying spaces. The morphismsOY ! jjljOXlgive

j^#:O_Y !lim_ljj_ljO_X_l'lim_ljjjji_ljO_X_l ' jjjlim_lji_ljO_X_l ' jjjO_X: This is the unique morphism which satisfies jjji^#_l j^#j^#_l. It is easy to see that j(jjj;j^#) commutes with the support morphisms b_X and b_Y.

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Thus, we obtained a unique morphismj:X!YofA-schemes, henceXis indeed the co-limit.

(2) We only have to show that ifXis obtained by patchingX1; ;Xnby quasi-compact opens, thenfX_i!Xgis a covering. By induction, it suffices to prove forn2. There is a surjective morphismX₁`

X₂!X, h ence it remains to show that:

If R1 and R2 are two II-rings, then any element of the spectrum of RR1R2 are in th e image of SpecR1!SpecR or that of SpecR₂!SpecR.

This is easy to prove, so we will skip.

It is clear from the construction that the functor I preserves finite

patching via quasi-compact opens. p

REMARK2.1.7. Even though small coproducts exist in the category of A-schemes, their behavior is somewhat different from those in schemes.

For example, a point of an infinite coproduct X`

lX_l does not ne- cessarily come from a point of someXl, i.e strictly speaking,fXl!Xgis not a covering ofX.

EXAMPLE2.1.8. The spectrum functor is mal-behaved, when we consider infinite product of rings: the underlying space SpecQ

nR_ndoes not coincide withthe co-product`

n SpecRn, even in the category of coherent spaces.

Here is a typical counterexample: Let k be a field, and set R Q

n2NR_n, where R_nk[x]=(xⁿ). Then, the spectrum ofR_n is a point for any R_n, hence SpecQ

a₁R_n must be the set of all ultrafilters overN, in particular, its Krull dimension is zero.

On the other hand, the Krull dimension of SpecRis not zero: fix a non- principal ultrafilterUonN, and define an idealMofRas

f (f_n)_n2M,f_n2=R_n a:e:U:

Here, P(s) a.e U for a condition P(s) of s means that the set fsjP(s)g belongs toU. This is a maximal ideal ofR(in fact, any maximal ideal ofRis of this form). On the other hand, define an idealpofRas

f (f_n)_n2p,For anyc>0; f_n 2(x^dcne)a:e:U:

This is also a prime ideal, and obviously smaller thanM. Hence, the Krull dimension of R is not 0. In fact, one can prove similarly that the Krull dimension ofRis infinite.

It is obvious thatfSpecR_n!SpecRg_nis not a covering of SpecR.

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2.2 ±Decomposition of morphisms

In this subsection, we prove that there is a functorial decomposition of morphisms in the category of A-schemes. This decomposition plays an important role in the proof of completeness of (A-Sch), since it gives an upper bound of the cardinality of a family of morphisms. Also, note that this decomposition is peculiar to the category of A-schemes.

DEFINITION 2.2.1. A morphism f :X!Y of A-schemes is a P- morphismif:

(1) jfjis epic, i.e. f ¹:C(Y)_cpt!C(X)_cptis injective, (2) f^#:O_Y !fO_X is injective.

Let us mention some trivial facts:

PROPOSITION2.2.2. (1) A P-morphism is epic.

(2) P-morphisms are stable under compositions.

(3) If gf is a P-morphism, then so is g.

These are all obvious, so we will skip the proof.

PROPOSITION2.2.3. Let ff_l:X_l!Y_lg_l be an inductive system of P- morphisms ofA-schemes, and set X1 lim_!lXland Y1lim_!lYl. Then, the natural morphism f :X1!Y1 is also a P-morphism.

PROOF. First, we will see that C(Y1)_cpt!C(X1)_cpt is injective.

Let Z(Z_l)_l and W(W_l)_l be two elements of C(Y₁)_cpt with f ¹(Z)f ¹(W). Since f ¹ is defined by (Z_l)_l7!(f_l ¹Z_l)_l, this implies that f_l ¹Zl f_l¹Wl. Since fl is a P-morphism,Zl andWl must coincide for all l, which is equivalent to ZW. Hence C(Y1)_cpt!C(X1)_cpt is injective. A similar argument shows that O_Y₁!fO_X₁ is also injective,

so that f is a P-morphism. p

DEFINITION 2.2.4. (1) Fix a small index category I. Let Y:I! (A-Sch)be a small projective system ofA-schemes, and f :D(X)!Ybe a morphism in(A-Sch)^I, whereD:(A-Sch)!(A-Sch)^Iis the diagonal functor. (In the sequel, we simply denoteD(X)by X for brevity.) LetS be the set of isomorphism classes of the commutative diagram

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where g is a P-morphism.S is small, from the property of P-morphisms.

Set I(X;Y)lim_!W2SW. Then, by Proposition 2.2.3, X!I(X;Y) becomes a P-morphism.

(2) Let f :X!Y be as above. f is a Q-morphism, if X!I(X;Y)is an isomorphism.

Roughly speaking, a P-morphism can be regarded as a schematic surjection and a Q-morphism as a schematic immersion. Thus, if X!I(X;Y)!Y is the PQ-decomposition of a morphism f :X!Y, I(X;Y) can be regarded as the `image scheme' of f.

The next proposition is purely category-theoretical.

PROPOSITION2.2.5. (1) Let f :X!Y be a morphism from an A-scheme X to a projective system Y of A-schemes. Then, the morphism h:I(X;Y)!Y is a Q-morphism.

(2) A morphism f :X!Y of A-schemes is an isomorphism if and only if f is a P-morphism and Q-morphism.

PROOF. (1) Let g:X!I(X;Y) be the induced P-morphism. Set WI(I(X;Y);Y), and let ~h:W!Y be the induced morphism. Since p:I(X;Y)!W is a P-morphism, X!W is also a P-morphism by (1).

Hence, there is a morphismi:W!I(X;Y) suchthat ipgg. Th is implies thatipis the identity, sincegis epic. Hence,pipp, and this shows that pi is the identity since p is also epic. This shows that I(X;Y)!Wis an isomorphism.

(2) The proof is similar to (1). p

COROLLARY2.2.6. (1)LetX!^f Y !^g Zbe a series of morphisms ofA- schemes. Ifgfis a Q-morphism, then so is f.

(2) Let f :X!Y be a morphism of A-schemes. Then, the decomposition f hg of f into a P-morphism g and a Q-morphism h is unique up to unique isomorphism.

We refer to the decomposition of (2) asPQ-decomposition.

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PROOF. (1) Let h:X!I(X;Y) and ~f :X!I(X;Z) be the induced morphisms. By universality, we have a morphism p:I(X;Y)!I(X;Z) suchthat ph~f.~f is an isomorphism sincegf is a Q-morphism. Hence,

~f ¹phis the identity. Also,h~f ¹phhandhepic implies thath~f ¹pis also the identity. Hence,his an isomorphism.

(2) LetX!^g W!^h Ybe a decomposition off into a P-morphism and a Q-morphism. Then, there is a morphismp:W!I(X;Y) by universality.

SinceX!I(X;Y) is a P-morphism,pis also a P-morphism. Sincehis a Q- morphism, (1) tells thatpis a Q-morphism, hence an isomorphism. p On the other hand, it seems to be impossible to prove that Q-morphisms are stable under compositions, using only categorical operations.

LEMMA2.2.7. LetA(s;a₁;a₂;g)be the schematizable algebraic type.

Then:

(1) a1 preserves filtered colimits.

(2) a₁preserves images: namely, if f :A!B is a homomorphism ofs- algebras, thena₁(Imf)Im (a₁f).

PROOF. (1) LetfR_lg_lbe a filtered inductive system ofs-algebras, and setR1 lim_!lRl. Then we have a natural homomorphismW:lim_!la1Rl! a1(R1). We will show thatWis bijective.

First, we will prove the surjectivity. For anya2a1R1,acan be written asP_n

i a₂(a_i) for somea_i2R₁, sincea₂(R)a₁Rgeneratesa₁R. Since the inductive system is filtered, there existsl₀suchthatfa_ig_iR_l₀. Then,ais contained in the image of

a₁R_l₀ !lim_!_la₁R_l!^W a₁R₁; hence in the image ofW.

Next, we prove the injectivity. Suppose W(a)W(b) for some aP

ia2(ai);bP

ja2(bj)2lim_!la1Rl. Then, a and b must coincide in a₁R_l₀ for somel₀, which shows thatabin lim_!_la₁R_l.

(2) Letabe an element ofa1(Imf). Then, aX

i

a2f(ai)(a1f)X

i

a2(ai)

for some ai2A. This shows that a1(Imf)Im (a1f). The converse is

similar. p

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PROPOSITION 2.2.8. Let f :X!Y be a morphism of A-schemes.

Then, there exists a decomposition X!^g W!^h Y of f , where g is a P- morphism, and h satisfies

(i) h ¹:C(Y)_cpt!C(W)_cptis surjective, and

(ii) h^#:OY!hOW is stalkwise surjective, that is, O_Y;h(w)! O_W;w is surjective for any w2W.

PROOF. Let R be the image of f ¹:C(Y)_cpt!C(X)_cpt, and set jWj SpecR^y. The structure sheaf OW:R'C(W)_cpt!(s alg) is defined by the sheafification of

R3Z7!Im [f ¹O_Y(Z)!O_X(Z)]_S;

where S fajb_Xa2(a)1 inRZg is a multiplicative system of Im [f ¹O_Y(Z)!O_X(Z)].

The support morphismb_W :a₁O_W!t_W is defined as follows: for any Z2C(W)_cpt,a₁O_W(Z) is locally isomorphic to

a₁Im [f ¹O_Y(Z)!O_X(Z)]_S'a₁lim_!_f ¹_VZIm [O_Y(V)!O_X(Z)]_S 'lim_!_f ¹_VZIm [a₁O_Y(V)!a₁O_X(Z)]_S by Lemma 2.2.7. Since we have a commutative square

and the lower horizontal arrow factors throughR_Zt_W(Z), we obtain a homomorphisma₁O_W(Z)!t_W(Z). Note that the localization bySdoes not affect. It is obvious that the restrictions reflect localizations, hence W(jWj;O_W;b_W) is well defined as anA-scheme.g:X!Wis defined by the injectionsg ¹:R!C(X)_cptand

g^#:O_W(Z)Im [f ¹O_Y(Z)!O_X(Z)]_S!O_X(Z):

It is obvious that g is a P-morphism. h:W !Y is defined by h ¹:C(Y)_cpt!Rand

h^#:O_Y(Z)!O_W(g ¹Z)Im [O_Y(Z)!O_X(f ¹Z)]_S:

Let us verify that h^# is stalkwise surjective, namely, OY;h(w)!OW;w is surjective for anyw2W. LethU;aibe any element ofOW;w. Sinceh ¹is

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surjective,Uh ¹Vfor some quasi-compact openV Y. The germacan be expressed as b=c, where b_Xa2(c)1, and b;c is in the image of O_Y(V)!O_W(U). This implies thatcis a unit inO_W;w, hence also a unit in O_Y;f_(w). HenceO_Y;h(w)!O_W;wis surjective. p COROLLARY 2.2.9. (1) Let f :X!Y be a morphism of A-schemes.

Then, the W constructed in 2.2.8 is naturally isomorphic to I(X; Y). In particular, the followings are equivalent:

(i) f is a Q-morphism.

(ii) f ¹:C(Y)_cpt!C(X)_cpt is surjective, and f^#:O_Y !hO_X is stalkwise surjective.

(2) Q-morphisms are stable under compositions.

(3) Q-morphisms are monic.

PROOF. (1) By Proposition 2.2.8, there exists a decomposition X!^g W!^h Y of f, wh ere g is a P-morphism, h ¹:C(Y)_cpt!C(W)_cpt is surjective, andh^#:OY !hOW is stalkwise surjective. By the universal property, there is a P-morphism u:W!I(X;Y). Note that I(X;Y) is isomorphic to X, since f is a Q-morphism. Since h ¹ is surjective, u ¹:C(X)_cpt!C(W)_cptis an isomorphism. Also, the stalkwise surjectivity ofh^#implies thatu^#is also an isomorphism.

(2) It is clear from (1).

(3) Obvious. p

LEMMA2.2.10. If X!Y!Z is a series of morphisms ofA-schemes, then we have a natural isomorphism

I(X;Z)'I(I(X;Y);I(Y;Z)):

PROOF. Since WI(I(X;Y);I(Y;Z))!I(Y;Z) and I(Y;Z)!Z are Q-morphisms, the composition W!Z is also a Q-morphism. Hence,

X!W!Zis the PQ-decomposition ofX!Z. p

PROPOSITION2.2.11. The PQ-decomposition is functorial.

PROOF. Suppose given a commutative square

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Then, we will show that there is a natural morphismI(X1;Y1)!I(X2;Y2).

By the universality, we have a unique morphismI(X1;Y1)!I(X1;Y2). On the other hand, we have again a unique morphismI(X1;Y2)!I(X2;Y2) by Lemma 2.2.10, hence combining them gives the required morphism

I(X₁;Y₁)!I(X₂;Y₂). p

Let us summarize what we have obtained in this subsection:

THEOREM 2.2.12. For any morphism f :X!Y of A-schemes, we have a functorial decomposition X!I(X;Y)!Y of f , where

(1) X!I(X;Y)is a P-morphism (in particular, epic), and (2) I(X;Y)!Y is a Q-morphism (in particular, monic).

Moreover, the decomposition of the given morphismfinto a P-morphism and a Q-morphism is unique up to unique isomorphism. Also, this decomposition is universal: if f is factors asX!W!Y whereX!W is a P- morphism (resp.W!Yis a Q-morphism), then there is a unique morphism W!I(X;Y) (resp.I(X;Y)!W) making the whole diagram commutative.

REMARK2.2.13. We know that this decomposition is impossible in the category of schemes. For example, let k be a field, XSpeck[x;y=x], YSpeck[x;y] wh ere x and y are indeterminants. The image of the natural morphism f :X!Ycannot be a scheme: the origin has no affine neighborhood in the image.

2.3 ±Completeness

PROPOSITION2.3.1. The category(Q-Sch)is finite complete, i.e. there are fiber products.

PROOF. The construction of fiber products are similar to that of gen- eral schemes. Note that we use the fact that quasi-compact open immer- sions are stable under base changes by quasi-compact morphisms. p REMARK 2.3.2. Let s be the algebraic system of rings. Then, the natural inclusion functor (Q-Sch)!(Sch) preserves fiber products. This is clear from the construction.

We already know that the category of ordinary schemes is not complete. However:

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PROPOSITION2.3.3. The category(A-Sch)is small complete.

PROOF. LetXbe a small projective system ofA-schemes. LetS be the set of all isomorphism classes of Q-morphismsY !X.S is indeed a small set: ifY!Xis a Q-morphism, then the underlying space ofYand its structure sheaf is generated by those ofX, henceS is small.

LetXbe the co-limit ofS . Then,Xis the limit ofX. p COROLLARY2.3.4. The natural inclusion functor(Q-Sch)!(A-Sch) preserves fiber products.

PROOF. LetX;YbeQ-schemes over aQ-schemeS. We will show that the fiber productV X_SYin the category ofQ-schemes is indeed that in the category ofA-schemes.

STEP1. IfX;Y;Sare all affine, thenVis the fiber product in (A-Sch), by the adjunction Spec^A:(s alg)!(A-Sch)^op:G.

STEP2. SupposeY;Sis affine. LetX [_iX_ibe an open affine cover of X. Suppose given the following commutative square:

Then, there is a unique morphismf ¹(X_i)!X_i_SYby Step 1 for eachi, which patches up to give the morphismZ!X_SY.

STEP3. Same arguments as in Step 2 shows that ifSis affine, thenVis the fiber product in (A-Sch).

STEP 4. Suppose S [_iS_i is an open affine cover of S. Set X_iX_SS_i,Y_iY_SS_i. Then,V_iX_i_S_iY_iis also the fiber product in (A-Sch) by Step 3. Note thatV [_iV_iis an quasi-compact open covering. Suppose given a commutative diagram as in Step 2, and set Zif ¹(Xi). This coincides withg ¹(Yi). Then we have a unique morphism Z_i!V_ifor eachi, which patches up to give a morphismZ!V. p

The next proposition is purely category-theoretical.

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PROPOSITION2.3.5. (1)Q-morphisms are stable under pullbacks.

(2) P-morphisms are stable under pushouts.

PROOF. We will only prove (1): the proof of (2) can be proven by the dual argument. Consider the following pullback diagram:

Supposef is a Q-morphism. Then the PQ-decomposition XT^a(!^~^f⁾Im~f I(XT;X)^b(!^~^f⁾X

of ~f gives a morphism w:Im~f !T suchthat wa(~f)~g and f w gb(~f). By the universal property of the pullback, there exists a unique morphismu:Im~f !X_Tsuchthat~guwand~f ub(~f). The second equality shows thatuis a Q-morphism, hence it suffices to show thatuis a P-morphism. To prove this, we will show thatua(~f) is the identity. Since

~gua(~f)wa(~f)~g; and ~f ua(~f)b(~f)a(~f)~f; the universal property of the pullback shows that ua(~f) is th e

identity. p

2.4 ±Filtered limits

PROPOSITION2.4.1. Let X fX^lgbe a small filtered projective system ofA-schemes, and Y be the limit of X. Then, the underlying space of Y coincides with the limitlimX^l in the category of coherent spaces. The structure sheaf O_Y coincides with the colimit lim_!p_l¹O_X^l, where p_l:Y!X^lare the natural morphisms.

PROOF. Let X¹ be the limit of the X in the category of coherent spaces, and set OX¹ lim_!p_l¹O_X^l, where pl:X¹!X^l is the natural morphism of coherent spaces. First, we will construct the support morph- ismb_X1. By Proposition 2.1.5, we have a natural map

a1p_l¹O_X^l!p_l¹t_X^l!lim_!_lp_l¹t_X^l't_X¹:

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This gives a natural map lim_!l(a1p_l¹O_X^l)!tX¹. The left-hand side is isomorphic to a1OX¹ since a1 is filtered co-continuous by Lemma 2.2.7.

Therefore, we obtain the required morphism b_X1 :a1O_X¹ !t_X¹. It is obvious that restrictions reflects localizations. Hence, we have constructed an A-scheme X¹(X¹;O_X¹;b_X1). There are also natural morphisms p_l:X¹!X^lofA-schemes, compatible with the transitions.

We will show that X¹ is naturally isomorphic to Y. Since we already have a morphism X¹!Y by the universal property ofY, it suffices to show that:

(i) IfW:X¹!X¹is a endomorphism withplWplfor anyl, thenW is the identity.

(ii) There exists a morphismc:Y!X¹withp_lcp_l for anyl.

First, we prove (i). It is obvious thatWis the identity on the underlying space. For the structure sheaves, we have the commutative diagram:

which shows thatW^#is the identity, sinceOX¹ lim_!lp_l¹O_X^l.

It remains to prove (ii). There is a natural morphismjcj:jYj ! jX¹j between the underlying spaces. Since there are morphisms

p^#_l :O_X^l!p_lO_Y 'p_ljcjO_Y;

these give morphisms p_l¹O_Xl ! jcjO_Y. It is obvious that these are compatible with the transition morphisms, hence we obtainOX¹ !cOY. Also, this morphism commutes withb_X¹andb_Y, hence we have a morphism ofA-schemes. It is obvious thatp_lcp_l. p PROPOSITION2.4.2. Letfg_l:X^l!Y^lg_lbe a filtered projective system of Q-morphisms ofA-schemes, and set X¹limlX^l and Y¹limlY^l. Then, the natural morphism g:X¹ !Y¹ is also a Q-morphism.

PROOF. First, we will see thatg ¹:C(Y¹)cpt!C(X¹)cptis surjective.

SinceC(X¹)_cptlim_!lC(X^l) is a filtered colimit, any elementZofC(X¹)_cpt is in the image ofC(X^l)_cptfor somel. SinceC(Y^l)_cpt!C(X^l)_cptis surjective, there is an element W2C(Y^l)_cpt suchthat g_l¹WZ. Hence, g ¹p_l¹W g_l¹WZ. This shows thatg ¹ is surjective. A similar argument shows thatO_Y¹ !gO_X¹ is also stalkwise surjective. p

<span class="mathjax-formula">$\mathcal {A}$</span>-Schemes and Zariski-Riemann Spaces

A-Schemes and Zariski-Riemann Spaces

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