Affine Objects, Restrictive Morphisms and Quasi-coherent Sheaves
Master thesis
Thesis advisor: Dr. Owen Biesel
Date master exam: 22 juni 2016
Mathematisch Instituut, Universiteit Leiden
First and foremost I would like to thank my advisor Owen Biesel. He proposed the initial research question and provided great guidance during the whole process.
Furthermore I would like to mention all the people who endured my complaining and encouraged me: Laura Schubert, Fatima Botan, Alexander Gietelink, Kyra Orbons, Se- bastiaan Dielemans, Mathijs Henquet, Maria Heuß, Francesca Ali, Raoul Wols, Vincent Willems, Charlotte Willems and Jasmin Zahirovic. Thank you!
1 Introduction 5
1.1 Motivation . . . 5
1.2 Aim . . . 5
1.3 Outline . . . 6
1.4 Assumptions . . . 6
2 Preliminaries 7 2.1 Basic Category Theory . . . 7
2.2 Topology . . . 9
2.2.1 Basic . . . 9
2.2.2 Sites . . . 10
2.2.3 Sheaves . . . 11
2.2.4 Relative topology . . . 13
2.3 Modules . . . 14
2.4 Restricting and module functors . . . 17
2.5 Schemes . . . 19
2.6 Small Zariski site . . . 20
2.7 Big Zariski Site . . . 22
3 Restrictive morphisms 25 4 Caffine objects 30 4.1 Caffine objects . . . 30
4.2 Caffine schemes . . . 32
4.3 Enough caffines . . . 37
5 Further research 42
1.1 Motivation
Quasi-coherent sheaves, affine schemes and the inclusion of distinguised opens of an affine scheme have a special relationship. A sheaf of modules F on any scheme X is quasi-coherent if and only if for any open affine subset Spec(R) ⊂ X and f ∈ R we have the following coherence property Γ (D(f); F)⊗Γ (Spec(R);O)→ Γ(Spec(R); F). One can find out that a quasi-compact scheme is affine by proving that all higher cohomology of any quasi-coherent sheaf vanishes [Stacks, Tag 01XF].
1.2 Aim
This thesis will investigate, formalize and generalize this trinity. We will generalize affineness to a property that any object of any ringed site may have, which gives us a notion of ‘affine’ object, which we will call an caffine, shorthand for categorically affine, object. This definition will use the functor Λ which is a generalized version of the well- known functor−e that sends modules of the global sections to a quasi-coherent module.
We will define a property of morphisms in a ringed site, to replace the distinguised inclusions in the generalized trinity. Both definitions are defined purely in terms of quasi-coherent sheaves.
Our main results are the following. Any morphism between caffine objects will turn out to be restrictive. If a ringed site has ‘enough caffines’, then a sheaf module is quasi-coherent if and only if it is locally of the form Λ(M) for some module M.
1.3 Outline
In chapter 2 we will build up all the necessary theory that is well-known, but not assumed to be familiar. We will start with some general categorical results and definitions. Then we will introduce the generalisation of topology for categories, the category of sites and the central notion of a sheaf. The induced topology on an over/slice category will get special attention because we will use it extensively.
In chapter 3 we will introduce restrictive morphisms and show how this notion interacts with affine schemes. It will turn out any morphism between affine schemes is restrictive.
In Chapter 4 we will define caffine objects and prove that any morphism between caffine objects is restrictive. Then we will show that the caffine objects in the category of schemes are exactly the affine schemes. Lastly we prove that a sheaf of modules is quasi- coherent if and only if it is locally of the form Λ(M). We will also look at some examples of categories that lack any caffine objects.
In Chapter 5 I will expand on possible avenues of further research.
1.4 Assumptions
The intended audience are my peers, so some mathematical maturity is assumed. I assume familiar with the usual undergraduate curriculum in topology and algebra. Fur- thermore basic categorical notions such as limits and adjunctions are assumed. See the first 3 chapters of [6] for an introduction to these ideas.
This chapter introduces all the well-known results and notions that are needed for the original material. We will start with a discussion of some purely categorical notions like over categories and presheaves. Secondly, we will introduce a notion of a topology on a category and look at some constructions that are relevant for us. Then we will introduce modules on ringed sites and see how they behave with respect to restricting along a functor. Lastly, the notion of a scheme is introduced and we establish some properties in the big and small site of a scheme.
2.1 Basic Category Theory
Some categorical notions like presheaves and over categories will be introduced in this section. See [1] and [4].
Definition 2.1.1 (Presheaf category). Let C be a category. We define the category of presheaves on C as the category of contravariant functors to the category of sets Set.
We will denote it by ^C.
Define the functor h : C→ ^C as follows: for a ∈ C, a7→ Hom(−, a), for f : a0→ a
f7→ f ◦ −.
This functor is fully faithful by the Yoneda lemma.
Notation 2.1.2. Let I, C be categories. Let L : I→ C be a functor. The limit over this functor will be denoted by lim
i∈I L(i). The colimit will be denoted by colim
i∈I L(i).
Definition 2.1.3 (Sections functor). For any a ∈ C define the functor Γ (a; −) : ^C→ Set
by
F→ F(a).
For any presheaf G, we define
Γ (G; −) : ^C→ Set, F→ Hom(G, F).
We will mostly use when G for the terminal presheaf, which will allow us to compute the global sections.
Definition 2.1.4 (Over/Under categories). Let C and C0 be categories. Let F : C→ C0 and z ∈ C0. Define the category Czand Cz as
Obj(Cz,F) :={(a, w) | a ∈ C, w : F(a) → z}, Hom((a, w), (b, v)) :={f : a → b | v ◦ F(f) = w}, and
Obj(Cz,F) :={(a, w) | a ∈ C, w : z → F(a)}, Hom((a, w), (b, v)) :={f : a → b | F(f) ◦ w = v}.
We get faithful functors Cz→ C : (a, w) → a and Cz→ C : (a, w) → a. We will denote them both by u.
Definition 2.1.5 (direct image). Let f : C → D. Define the direct image functor f∗: ^D→ ^C as
f∗= −◦ f.
When u is the forgetful functor Ca → C then we denote u∗F sometimes by F a.
Definition 2.1.6 (inverse image of presheaves). Let C, D be a categories. Let f : C→ D be a functor. Define the inverse image functor f−1 : ^C→ ^D as follows. Let F ∈ ^C. For any d ∈ D
f−1(F)(d) =colim
c∈Cd,fΓ (c; F).
See [Stacks, Tag 00VC].
Lemma 2.1.7. The functor f∗ is left adjoint to f−1. Proof. See [Stacks, Tag 00VE].
Corollary 2.1.8. The functor f∗ commutes with arbitrary colimits.
2.2 Topology
In this section we will define a notion of a topology on a category and look at the related notions of sheaves, sites and restriction of sites.
See [4] for more details.
2.2.1 Basic
Definition 2.2.1 (Sieve). Let C be a category and a ∈ C. Define the maximal sieve max(a) on a to be the collection of all morphisms to a. In formula,
max(a) ={f ∈ C | Codom(f) = a}.
A sieve S on a is a subcollection of max(a) such that f ◦ g ∈ S for any f ∈ S and any g.
Definition 2.2.2 (Sieve category). Let C be a category and a ∈ C. The sieve category Sieves(a) consists of all the sieves on a as objects and inclusions of sieves as morphisms.
Definition 2.2.3 (Pullback of sieve). Let C be a category and a, b ∈ C. Let S be a sieve on a. Let f : b→ a. The sieve f∗S on b is given by f∗S ={g ∈ max b : fg ∈ S}
for any c ∈ C. To show that this is actually a sieve on b, let k : c→ c0 and h ∈ f∗S.
Hence fh ∈ S and so fhk ∈ S. Conclude that hk ∈ f∗S. This defines a functor f∗: Sieves(a)→ Sieves(b).
Definition 2.2.4 (Grothendieck Topology). A Grothendieck topologyT is a family T(a) of ‘covering’ sieves for every a ∈ C with the following conditions:
1. max(a) ∈T(a)
2. f∗R∈T(a0) if R ∈T(a) for all f : a0→ a
3. if f∗R∈T(a0) for all f ∈ S with S ∈T(a) then R ∈ T(a)
Definition 2.2.5 (Basis). Let C be a category with pullbacks. A Grothendieck pretopol- ogy, or basis, B is a collection B(a) of ‘covering’ families {fi : ai → a} of morphisms for every a ∈ C with the following conditions.
1. every isomorphism is a covering singleton family,
2. (Stability) The pullback of a covering family is a covering family. If {fi : ai → a}
is covering and g : b→ a, then {fi0 : ai×ab→ b} is covering.
3. (Transitivity) If {fi : ai → a} is a covering family and {fij : aij → ai} for every i, then{fij : aij → a} is a covering family.
Generating a topology from a basis: take any sieve containing a covering family to be a covering sieve.
Remark 2.2.6. Sometimes you have a set of families S you would like to generate the topology. So one can take the smallest basis and the topology containing these families.
This will be called ‘the topology generated by S’.
2.2.2 Sites
Definition 2.2.7 (Site). A site (C,T) is a category C with a Grothendieck topology T.
Definition 2.2.8 (continuous functor). Let G : C→ D be a functor between sites. Let c ∈ C and let R be a covering sieve on c. The functor G is said to preserves covers if G(R)generates a covering sieve. It is enough to check that G sends covering families to covering families, if the topology of the sites is defined by a basis. If G also preserves pullbacks then we call it continuous. See [Stacks, Tag 00WU].
Definition 2.2.9 (cocontinuous functor). The functor G is said to lift covers or be cocontinuous if for every R ∈ CovG(c) there is some S ∈ Covc such that G(S) ⊂ R. See [Stacks, Tag 00XJ].
Definition 2.2.10. The category Sites has as objects sites and the morphisms are continuous functors. When no confusion can arise then we will use C to denote the whole site (C,T).
Remark 2.2.11. In most resources a morphism of sites is defined to preserve all finite limits. To get the results that we want we only need preservation of pullbacks and we need the forgetful functor Ca → C to be a morphism of sites, hence this slightly weaker notion than usual.
Definition 2.2.12 (Ringed sites). A ringed site (C,T, O) is a site (C, T) with the sheaf of rings O on it. A morphism of ringed sites is a morphism pair (f, f#) : (C,T, O) → (D,S, U) with f : (C, T) → (D, S) a morphism of sites and f#: f∗U→ O a morphism of sheaf of rings. We will denote the pair (f, f#) by just f when no confusion can arise.
Example 2.2.13 (small site). Let X be a topological space. Let the category Open(X) consist of the opens of X as objects and inclusion as the morphisms. Define a basis on this site consisting of the families {Ui → U} such that SiUi = Ui. A continouous map f : X→ Y induces a morphism of sites Open(Y) → Open(X) by sending U 7→ f−1(U). In this way you can embed the category of topological spaces into the category of sites.
If X is a ringed space we can turn the site Open(X) into a ringed site by setting the sheaf of rings of X to be the sheaf of rings of Open(X).
2.2.3 Sheaves
We will introduce the very important notion of a sheaf. See [4] for a more detailed treatment.
Definition 2.2.14 (Matching family). Let C be a category. Let F be a presheaf on on C.
Let a ∈ C be an object. Let R be a sieve on a. A family{xi}i∈R with xi ∈ Γ (Dom(i); F) indexed by a sieve R and such that xg◦i = F(g)(xi) for any g : b→ Dom(i) and b ∈ C is called a ‘matching family’.
Definition 2.2.15 (Amalgamation). An amalgamation of a matching family{xi}R is an element x ∈ Γ (a; F) such that F(i)(x) = xi.
Definition 2.2.16 (Sheaf). Let (C,T) be a site. Let F ∈ ^C.
A presheaf that admits a unique amalgamation for every matching family is called a sheaf. The category Shv(C) is the full subcategory in ^C of all sheaves.
Definition 2.2.17 (Plus construction). Let (C,T) be a site. Let a, a0∈ C and f : a→ a0. Let F ∈ ^C. Define the functor (−)+: ^C→ ^C as follows.
For all a ∈ C,
F+(a) = {(R, ϕ) | R ∈ T(a), ϕ ∈ Γ(R; F)}
∼ ,
for all morphisms f ∈ C,
F+(f)([(R, ϕ)]) = [(f∗R, ϕh(f))].
The equivalence relation is defined as (R, ϕ) ∼ (S, φ) if ϕ = φ on some covering sieve Q⊂ R ∩ S.
Let L : F→ F0. Then
(L+)a([(R, ϕ)]) = [(R, L ◦ ϕ)]
This functor comes with a natural transformation ω : Id→ (−)+ defined by ωF,a(x) = [(max(a), y]
where
y(i) = F(i)(x).
Lemma 2.2.18. Let F be a presheaf, G a sheaf and g : F→ G a morphism in ^C.
Then g factors through ωF via a unique g0. Proof. See [5, p. 24].
Lemma 2.2.19. For every presheaf F, F+ is separated. If F is separated, then F+ is a sheaf.
Proof. See [5, p. 24].
Definition 2.2.20. Define sh = (−)+◦(−)+. This functor is left adjoint to the inclusion C^ → Shv(C) with unit
ω2F= ωF+◦ ωF.
Lemma 2.2.21. Let f : C → D be a continuous functor. Let O be a sheaf on D Then f∗O is a sheaf.
Proof. See [Stacks, Tag 00WW].
2.2.4 Relative topology
We will look at what the induced topology on a over category looks like and what this implies for restriction of sheaves. See [Stacks, Tag 03A4] for a more detailed treatement.
Remark 2.2.22. Let C be a category and a, b ∈ C. Let f : b → a ∈ Ca. The map max(f)→ max(b) sending a morphism to f to its underlying morphism in C is a bijection.
Moreover composition in C and Ca are the same, so this identification respects pullback of sieves. This observation yields us that Sieves(b) = Sieves(f).
Whenever R is a sieve on b, we will denote the corresponding sieve on f by Rf. Definition 2.2.23 (Relative topology). Let (C,T) be a site. Let a ∈ C.
Define the induced topology Ta on Ca by, for each f ∈ Ca
Ta(f) =T(Dom(f)).
The identification from Remark 2.2.22 implies that Ta is a Grothendieck topology.
Definition 2.2.24 (Oversite). Let Y = (C,T) be a site. Let a ∈ C. Define the site Ya
to be the category Ca with the induced topologyTa. We will denote it by just Ca. Lemma 2.2.25. The functor u : Ca → C is a morphism of sites that is cover lifting.
Proof. Clearly u preserves pullbacks. By Remark 2.2.22 it is also immediate that u preserves covers and lifts covers.
Definition 2.2.26 (Over ringed site). Let Y = (C,T, O) be a ringed site. Let a ∈ C.
Define the ringed site Yato be the site Cawith the structure sheaf u∗Owhere u : Ca→ C is the forgetful functor. This makes sense because u is continuous, see Lemma 2.2.21.
Whenever u is used as a morphism between ringed sites it stands for the morphism of ringed sites (u, Id) as defined in Definition 2.2.12.
Definition 2.2.27. Let (C,T, O) be a ringed site. Let a, b ∈ C and f : a → b.
Consider the induced functor Ca F
−→ Cb. Let c be the pullback of some pair d→ g ← e.
Then F(c) is the pullback of the pair F(d)→ F(g) ← F(e), so F preserves pullbacks.
Let c → b ∈ Cb. Let R be a covering sieve on c. By Lemma 2.5.9 and the fact that u◦ F = u, we get that F(R) generates a covering sieve, so F is continuous. With the same reasoning one can deduce that F lifts covers.
Let O be the structure sheaf of Ca and U be the structure sheaf of Cb. Note that Γ (c −→ bg −→ a; O) = Γ(c; O) = Γ(g; U). Define the morphism of ringed sites (F, Id) :f (Ca,Ta, O)→ (Cb,Tb, O), which we will denote by F or just f.
2.3 Modules
Presheaf modules and sheaf modules on a ringed site will be introduced in this section.
Next we will introduce two main functors λ and Λ. The functors λ and Λ introduced here will be used extensively. See [Stacks, Tag 03A4] for more detail.
Definition 2.3.1 (Presheaf modules). Let (C,T, O) be a ringed site.
A presheaf module on this ringed site is a presheaf of sets F on C together with a map of presheaves
O× F→ F
such that for every object a ∈ C the map Γ (a; O) × Γ (a; F)→ Γ(a; F) defines a Γ(a; O)- module structure on Γ (a; F).
A morphism
F→ G is a morphism of presheaf modules if
O× F F
O× G G
commutes. The category of presheaf modules on C will be denoted PMod(O). The full subcategory of sheaves in PMod(O) will be denoted Mod(O).
Definition 2.3.2. Let F be a sheaf of modules on (C,T, O). It is called quasi-coherent if the following holds. For any object a ∈ C there exists a covering sieve S such that for any map f : b→ a in S there exists a (global) presentation
O b
L I→ O
b
L J→ F
b → 0
It is enough to have presentation for a generating set of S. Some also call this property that F is locally presentable.
Quasi-coherent modules form a full subcategory of the category of sheaves of modules over (C,T, O) which are denoted by Qcoh(O).
Lemma 2.3.3. Let f : C → D be a morphism of ringed sites. Let F be a quasi- coherent sheaf of modules on D. Then f∗F is quasi-coherent.
Proof. One can view f as a morphism of ringed topoi (f∗, f−1, f#). See [Stacks, Tag 082T].
Definition 2.3.4. Let (C,T, O) be a ringed site. Let R = Γ(1; O). Let M, N be an R-module.
Define
λ : R-Mod→ PMod(O) by for all a ∈ C,
λ(M)(a) = M⊗RΓ (a; O), for all f : b→ a ∈ C,
λ(M)(f) =Id ⊗ O(f), for all g : M→ N ∈ R-Mod,
λ(g) = (a : g⊗ Id).
Lemma 2.3.5. Let (C,T, O) be a ringed site. Let R = Γ(1; O). The functor λ is left adjoint to
Γ (1; −) : PMod(O)→ R-Mod.
Proof. Let M, N be R-modules. Let F, G ∈ PMod(O) be presheaf modules.
Let ϕ : λ(M) → G be a morphism of presheaf modules. Let φ : M → Γ(1; G) be a morphism of R-modules.
Define
α :Hom(λ(M), G)→ Hom(M, Γ(1; G)) :ϕ7→ ϕ1
where ϕ1 is the component of ϕ on the global sections.
Define
β :Hom(M, Γ (1; G))→ Hom(λ(M), G) by, for each a ∈ C
β(φ)a= φ⊗RΓ (a; O).
We will show that β and α are mutually inverse.
Let ϕ ∈ Hom(λ(M), G) and a ∈ C. Let d = β(α(ϕ)). Let m ⊗ g ∈ M ⊗RΓ (a; O). Let p : λ(M)(1)→ λ(M)(a) be the projection map. Let q : G(1) → G(a) be the projection map. Then da(m⊗ g) = ϕ1(m)⊗ g and
ϕa(m⊗ g) = gϕa(m⊗ 1) by linearity
= gϕa(p(m))
= gq(ϕ1(m)) by naturality of ϕ
= g(ϕ1(m)⊗ 1)
= ϕ1(m)⊗ g.
Hence d = ϕ. In words, the natural transformations from presheaves of the from λ(M) are uniquely determined by their global sections component.
Let d = α(β(φ)). Let m ∈ M. Then d(m) = (φ ⊗RR)(m) = φ(m). Hence d = φ, which makes α and β mutual inverses. Naturality is straightforward to check and is omitted.
Definition 2.3.6. Let (C,T, O) be a ringed site. Define Λ : R-Mod→ Mod(O) by sh ◦ λ.
It follows that we have the adjunction Λ a Γ (1; −).
This functor is the generalisation of [Stacks, Tag 01BH] to general sites, so Λ on Open(X) for some ringed space X will coincide with the construction defined in the Stacks Project.
2.4 Restricting and module functors
Lemma 2.4.1. Let f : (D,S, U) → (C, T, O) be a morphism of ringed sites. Suppose f# : f∗O → U is an isomorphism. We get a natural isomorphism f∗ ◦ λ ⇒ λ ◦ (−⊗Γ (1;O)Γ (1; U)).
Proof. Define the natural transformation t : λ ◦ (− ⊗Γ (1;O)Γ (1; U))⇒ u∗◦ λ, by for each Γ (1; O)-module M and for each a ∈ D,
tM,a: M⊗Γ (1;O)Γ (1; U)⊗Γ (1;U)Γ (a; U)→ M ⊗Γ (1;O)Γ (a; O), m⊗ r ⊗ s 7→ m ⊗ rf#,−1(s).
Every component tM,f is an isomorphism by basic commutative algebra and the fact that f# is an isomorphism
Lemma 2.4.2. Let F : (C,T) → (D, S) be a morphism of sites that lift covers. Then F∗◦ (−)+ = (−)+◦ F∗ and hence also F∗◦ sh = sh ◦ F∗.
Proof. Let c ∈ C. Let R ∈ Covc. Let G be a presheaf on D. Let s = {si | i ∈ R} be a matching family in F∗G on c indexed by R. So si ∈ Γ (F(Dom(i)); G). By the cover preserving assumption we know F(R) generates a covering sieve S on F(c). Define the natural transformation T : F∗◦ (−)+→ (−)+◦ F∗ by
s7→ {tj | j ∈ S}
where tj = F(h)si for some factorisation j = F(i)h. Such a factorisation always exists since S is generated by F(R). This assignment is well-defined since F preserves pullbacks, see proof of [3, Lemma 2.3.3].
We will prove that T is injective. Let s0 = {s0i | i ∈ R0}. Assume T(s) = T(s0) then {tj | j ∈ S} = {t0j0 | j0 ∈ S0}, hence there exists covering sieve Q ⊂ S ∩ S0 such that tk= tk0 for every k ∈ Q. Since F lifts covers, there exists covering sieve P ⊂ R ∩ R0 on c such that F(P) generates a sieve P0 ⊂ Q. Hence tk = F(h)sk = F(h)sk0 = tk0 with k ∈ P0 and k = F(k)h. Take h = Id to conclude that s = s0 on P, hence they are in the same equivalence class.
Now comes surjectivity. Let t = {tj | j ∈ S} ∈ .. on F(c). We get a covering sieve R such that F(R) ⊂ S. Consider this matching family now on the covering sieve generated by F(R), which amounts to taking a different representing element from the equivalence class. So now F(R) generates S. For any j ∈ S we get a factorisation j = F(i)h for some i ∈ R. We want to have si = tF(i). This produces a matching family because restriction in F∗G is exactly the same as in G. Note that now T (s) = t. Hence we got surjectivity.
Corollary 2.4.3. We have F∗ω = ω and hence also F∗ω2 = ω2
Lemma 2.4.4 (Λ commutes with restriction). Let (C,T, O) be a ringed site. Let f : (C,T) → (D, S) be a morphism of sites that lift covers. Let f#be an isomorphism.
We have a natural isomorphism
f∗◦ Λ⇒ Λ ◦ (− ⊗Γ (1;O)Γ (1; U)).
Proof. Follows from Lemmas 2.4.1 and 2.4.2.
Lemma 2.4.5. Let f : (D,S, U) → (C, T, O) be a morphism of ringed sites that is cover lifting. Suppose f# be an isomorphism. Let F be a quasi-coherent module on C. Let M = Γ (1; F). Consider : Λ(M)→ F. Then f∗ : Λ(M⊗Γ (1;O)Γ (1; U)))→ f∗F is the counit of the adjunction Λ a Γ (1; −) on D.
Proof. Let a ∈ D. We will show that f∗corresponds to the same morphism λ(M⊗Γ (1;O))→ f∗Fas the counit. By the universal property of the sheafification, this implies that f∗ is the counit.
We have (f∗)a(ω2(m⊗ r)) = rm, which is the same as for the counit.
Example 2.4.6. Let (C,T, O) be a ringed site. Let a ∈ C. Consider the morphism of ringed sites u : Ca → C. Note that u#=Id and by Lemma 2.2.25 it lifts covers. So all the results from this section apply. This is the most important case.
2.5 Schemes
We will recap some definitions in scheme theory that we use later. See [7, 2] for thorough treatments of scheme theory.
Definition 2.5.1 (Spectrum of a ring). Let R be a ring. The spectrum Spec R of R is the locally ringed space defined as follows. The underlying set is the set of prime ideals of R. The (Zariski) topology is generated by the basis of distinguised opens D(f) ={p ⊂ R|f 6∈ p}. The sheaf of rings is given on this basis by
D(f)7→ Rf.
A distinguised open D(f) of Spec(R) viewed as locally ringed space is isomorphic to Spec(Rf), where the inclusion Spec(Rf) → Spec(R) corresponds to the canonical map R→ Rf.
The functor
Spec : Rng→ LRSpaces is left adjoint to
Γ (1; −) : LRSpaces→ Rng, see [Stacks, Tag 01I1].
Definition 2.5.2 (Scheme). We call the locally ringed space Spec(R) an affine scheme.
A scheme S is a locally ringed space that admits a covering of affine schemes. A morphism of schemes is a morphism of locally ringed spaces. The category of schemes we will denote by Sch.
Definition 2.5.3 (Sheaf of algebras). A sheaf of algebras F on a ringed site (C,T, O) is a sheaf of rings that comes with a (structure) morphism of sheaves of rings O→ F.
Definition 2.5.4 (Relative spec). Let X be a scheme. Let S be a sheaf of algebras on Xthat is quasi-coherent as a sheaf of modules.
Define the relative spectrum of S over X to be the scheme Rspec S→ X
that you get by glueing the spectra Spec(Γ (V; S))→ V ⊂ X for every affine open V. See [Stacks, Tag 01LW].
Definition 2.5.5 (Tilde functor). Let Spec(R) be an affine scheme. Let M be a R- module. Define fM to be the unique sheaf on Spec(R) with
fM : D(f)7→ Mf. See [Stacks, Tag 01HR].
Remark 2.5.6. This functor(with this notation) is commonly used in algebraic geometry texts. By [Stacks, Tag 01I7] and uniqueness of left adjoints Λ and −e are canonically isomorphic on spaces, hence on small Zariski sites. We will only use Λ in subsequent sections.
Definition 2.5.7. A morphism f : X → Y of schemes is called affine if for any affine open U ⊂ Ythe open subscheme f−1(U) is affine.
Definition 2.5.8. Let X be a scheme. Let f ∈ Γ (X; O). Define Xf to be the pullback of D(f)→ Spec(Γ(X; O)) ← X.
Lemma 2.5.9. Let X be a scheme. If Xfi is affine for all i and (fi)i∈I= (1) then X is affine.
Proof. Let S
iD(fi) = Spec(Γ (X; O)). Consider the canonical : X → Spec(Γ(X; O)).
By definition −1(D(fi)) = Xfi. By assumption Xfi is affine. By [Stacks, Tag 01S8]] the morphism is affine, hence X is affine.
2.6 Small Zariski site
In this section we will work on the small Zariski site corresponding to a scheme. We will prove that the unit η and of the adjunction Λ a Γ (1; −) are isomorphisms in this case.
Lemma 2.6.1. Let R be a ring. Let M be a R-module. Consider λ(M), Λ(M) as sheaf modules on Open(Spec(R)). Then
ω2λ(M),Spec(R): Γ (Spec(R); λ(M))→ Γ(Spec(R); Λ(M)) is an isomorphism.
Proof. Let p ⊂ R be a prime ideal. As stated in Definition 2.3.6, we may use results from [Stacks, Tag 01BH] in this setting. We will use that Λ(M)p = M⊗RRp. By naturality, localized at p, the map ω2 sends m to m ⊗ 1 ∈ M ⊗RRp, hence is the inverse of the multiplication map which is an isomorphism. Hence globally ω2 is an isomorphism.
Corollary 2.6.2. Let X be a scheme. Consider λ(M), Λ(M) as sheaf modules on Open(X). Let Spec(R) be an open subset of X. Let M be a Γ (X; O)-module. Then ω2λ(M),Spec(R) : Γ (Spec(R); λ(M))→ Γ(Spec(R); Λ(M)) is an isomorphism.
Remark 2.6.3. Let us be in the situation of Definition 2.2.15. Note that ηM is the composition of the unit of λ a Γ (1; −) and sh a i. The first unit is the identity and the second is ω2λ(M),Spec(R).
Corollary 2.6.4. Consider the adjunction Λ(−) a Γ (1; −) on Open(Spec(R)). The unit ηM : M→ Γ(Spec(R); Λ(M)) is an isomorphism.
Lemma 2.6.5. Let F be a sheaf of modules on scheme X. F is quasi-coherent on X if and only if for any open Spec(R) ⊂ X the sheaf F
Spec(R) is isomorphic to Λ(M) with M = Γ (Spec(R); F).
Proof. ⇒: By assumption we get local presentations indexed by a covering. Let S
i∈IUi = X be this covering. Assume without loss of generality that it is an affine open covering. Let Ui =Spec(Ri). Let OL KU
i → OL JUi → F U
i → 0 be one of the given presentations. Taking global sections gives us an exact sequence
RL Ki → RL Ji → Γ(Ui; F)→ 0.
Tensoring it with the localisation Ri,f for any f ∈ Ri yields RL Ki,f → RL Ji,f → Γ(Ui; F)⊗ Ri,f→ 0.
Taking sections at D(f) from the sheaf sequence yields RL Ki,f → RL Ji,f → Γ(D(f); F) → 0.
Hence F
Ui is the unique sheaf with D(f) 7→ Γ(Ui; F)f, which we defined to be Λ(Γ (Ui; F)).
By the affine communication lemma, this property holds for any affine and not just for the affines in this covering.
⇐: Let M = Γ(X; F). Take a presentation RL I → RL J → M and apply Λ(−). Then note that Λ(−) commutes with arbitrary colimits since it is a left adjoint, see lemma ?.
We have Λ(R) = OSpec(R) so we get a presentation OL ISpec(R) → OL JSpec(R) → F on every affine open subset Spec(R) ⊂ X, hence F is quasi-coherent.
Corollary 2.6.6. Consider the adjunction Λ(−) a Γ (1; −). For quasi-coherent sheaf F the counit F: Λ(M)→ F is an isomorphism.
2.7 Big Zariski Site
In this section we will introduce the big Zariski ringed site and look at how quasi- coherence and Λ behave on this site. We will prove that the category of quasi-coherent sheaves on an affine scheme Spec(R) ∈ Sch is equivalent to the category of R-modules.
Definition 2.7.1 (Big Zariski site). Define the big Zariski site to be (Sch,T, O) with the following components. The underlying category is Sch. The topology T is generated by the basis cosisting of the covering families{Xi
fi
−→ X} where fi is an open immersion and S
ifi(Xi) = X. The sheaf of rings O sends (U, Q)→ (X, O) to Γ(U; Q).
We will mostly be interested in the site SchX for a scheme X.
Definition 2.7.2 (k). Let X be a scheme. Define the functor k : Open(X) → SchX by U7→ ((U, OU), i) where i : U→ X is the inclusion of the open subscheme into X.
We will show that it preserves limits and covers. The terminal X ∈ Open(X) is sent to the terminal X→ X. Let U → V and W → V be two morphism in Open(X). We have k(U∩ W) = U ∩ W → X which is the pullback of k(U) → k(V) and k(W) → k(V).
Let S = {D(fi) → Spec(R)} be one of the generating family in Open(X). Note that k(D(fi)) is isomorphic to the object Spec(Rfi) → Spec(R). Hence k(S) generates a covering sieve on Sch and hence on SchX. We established that k is a morphism of sites.
Consider (Open(X),T, O) and (SchX,S, U). Note that k∗U = Oby construction. Define the morphism of ringed sites (k, Id) : (Open(X),T, O) → (SchX,S, U). We will denote this by k.
Lemma 2.7.3. Let X be a scheme. The morphism of sites k : Open(X) → SchX is cover lifting.
Proof. Let U ∈ Open(X). Let W ={Ui fi
−→ k(U)} be a covering family on k(U). Then V ={fi(Ui)→ U} is a covering family on U such that k(V) generates the same sieve as W. Hence k is cover lifting.
Lemma 2.7.4. Let X be a scheme. Consider k : Open(X)→ SchX. Then k∗Λ = Λ.
Proof. By definition k# is an isomorphism. By Lemma 2.7.3 we get that k is cover lifting. Then we can apply Lemma 2.4.4. Let U be the structure sheaf of SchX. Let O be the structure sheaf of Open(X). Note that the structure morphism Γ (1; k∗U)→ Γ(1; O) is an isomorphism, so Lemma 2.4.4 implies this result.
Lemma 2.7.5. Let X = Spec(R) be a scheme. Let F be a quasi-coherent sheaf on SchX. Let M = Γ (X; F). Then F: Λ(M)→ F is an isomorphism.
Proof. By Lemma 2.3.3, k∗ sends quasi-coherent sheaves to quasi-coherent sheaves. so k∗F = Λ(M), where k : Open(X) → SchX. Hence Γ (D(f); F) = Γ (D(f); ΛM) = Mf for any f ∈ R.
Let D(fi) O
L J α−→ D(fi i) O
L K → D(fi)
F→ 0 be a presentation with (fi) = (1). Note that the presheaf cokernel Coker(αi) of the sheaf morphism is λ(Coker(αi,D(fi)))where αi,D(fi) is the component at D(fi)of αi. So D(fi)
F= Λ(Mf) since ω2 is iso for affines.
Hence
Γ (D(f)×XSpec(S); F) = Γ (D(f) ×XSpec(S); ΛF)
= Γ (Spec(Sf); ΛMf)
= Mf⊗ Sf
= M⊗ Sf.
By the sheaf property it follows that Γ (Spec(S); F) = Γ (X; F) ⊗ S, hence the counit Λ(M)→ F is an isomorphism.
Lemma 2.7.6. Let X = Spec(R) be a scheme. Let M be a R-module. We work on the ringed site SchX. The unit η : M→ Γ(X; Λ(M)) is an isomorphism
Proof. Let Λa(M) be Λ(M) over Open(X). By Lemma 2.7.4, we have Λa(M) = k∗Λ(M). Hence Γ (X; Λ(M)) = Γ (X; Λa(M)). Moreover Γ (X; Λ(M)) = M by Corol- lary 2.6.4.
This section will introduce the notion of a restrictive morphism. We will see some examples, non-examples and results in the category of schemes and see that this notion is closely related to affinenes. We will always work over small sites in this section. For some of the examples and results see the chapter on quasi-coherent modules in [7].
Definition 3.0.1 (Restrictive morphism). Let (C,T, O). A morphism f : a → b ∈ C is called restrictive if for every quasi-coherent module G on Cb the morphism
bf : Γ (b; G)⊗Γ (b;O)Γ (a; O)→ Γ(a; G) (3.1) is an isomorphism.
Remark 3.0.2. Let us be in the setting of Definition 3.0.1. Assume G = Λ(M) where M = Γ (1; G). Consider ω2G,a: λ(M)(a)→ Λ(M)(a). This is map is equal to bf.
Example 3.0.3. In SchSpec(A)the morphism Spec(Af)→ Spec(A) is restrictive. Let G be a quasi-coherent sheaf on SchSpec(A). This implies that G = Λ(Γ (Spec(A); G)), see ?.
The morphism
Γ (Spec(A); G) ⊗Γ (1;O)Γ (Spec(Af); O)→ Γ(Spec(Af); G) = Γ (A; G)f, m⊗ r→ rm
is an isomorphism by basic commutative algebra.
Example 3.0.4. Let R be a ring. Consider the open immersion U = Spec(R[x, y])\ {(x, y)} → Spec(R[x, y]) and the quasi-coherent sheaf G = Λ(R[x,y]xy ). The global sections of this sheaf are R[x,y]xy , as shown in Lemma 2.6.1.
Define U1 = D(x) → U and U2 = D(y) → U. Note that these cover U together. We have Γ (U1; G) = 0 and Γ (U1; G) = 0, since R[x,y]xy
x = 0 and R[x,y]xy
y = 0. Hence since G is a sheaf, we get Γ (U; G) = 0.
The sections over U of Λ(R[x, y]) are (also) R[x, y]. See [7, p. 4.4.1]. We conclude that Γ (1; G)⊗Γ (1;O)Γ (U; O)→ Γ(U; G) is not an isomorphism.
Lemma 3.0.5. Let (C,T, O) be a ringed site. Let f : b → a ∈ C and g : c → b ∈ C be morphisms.
1. If fg and f are restrictive, then g is.
2. if f and g are restrictive, then fg.
If Γ (b; O)→ Γ(c; O) is faithfully flat then fg and g restrictive implies f is restrictive.
Proof. Consider the following diagram
Γ (a; G)⊗Γ (a;O)Γ (c; O) Γ (b; G)⊗Γ (b;O)Γ (c; O)
Γ (a; G)⊗Γ (a;O)Γ (c; O) Γ (c; G),
f⊗b Γ (b;O)Γ (c;O)
bg fgb
where−b is as in lemma ?.
This diagram commutes: going either direction sends g ⊗ r to rg. The results follows from commutativity.
Lemma 3.0.6 (coproduct). Let X1, X2, Y be a schemes. X1 −f→ Y and X1 2 f2
−→ Y are restrictive morphisms if and only if the corresponding morphism X1t X2−−−f1tf→ Y is2 restrictive.
Proof. ⇒:
Note that Γ (X1t X2; −) = Γ (X1; −)× Γ (X2; −) by the sheaf property. We will show that f\1t f2: Γ (Y; G)⊗Γ (Y;O)Γ (X1t X2; O)→ Γ(X1t X2; G)
is an isomorphism. Tensor commutes over products, so this becomes
fb1× bf2: (Γ (Y; G)⊗Γ (Y;O)Γ (X1; O))× (Γ (Y; G) ⊗Γ (Y;O)Γ (X2; O))→ Γ(X1; G)× Γ (X2; G).
By assumption bf1 and bf2 are isos, so their product is.
⇐: By Lemma 3.0.7, the canonical morphism Xi→ X1tX2is restrictive. By assumption X1t X2 → Y is restrictive. Composing these morphisms yields fi, by Lemma 3.0.5 this yields a restrictive morphism.
Lemma 3.0.7. Let X, Y be a schemes. The canonical morphism X → X t Y is restrictive.
Proof. Let G be a quasi-coherent sheaf on X t Y. By the sheaf property Γ (X t Y; G) = Γ (X; G)× Γ (Y; G). The same holds also for O.
We are considering the morphism
Γ (Xt Y; G) ⊗Γ (XtY;O)Γ (X; O)→ Γ(X; G).
By the previous remark about disjoint unions and the sheaf property and some basic commutative algebra one sees that this becomes
(Γ (X; G)⊗Γ (XtY;O)Γ (X; O))× (Γ (Y; G) ⊗Γ (XtY;O)Γ (X; O))→ Γ(X; G).
Since Γ (Y; G) ⊗Γ (XtY;O)Γ (X; O) = 0, we are left with
(Γ (X; G)⊗Γ (XtY;O)Γ (X; O))→ Γ(X; G) g⊗ r→ rg.
Note that Γ (X; G) already is an Γ (X; O)-module and conclude that hence this morphism is an isomorphism.
Lemma 3.0.8 (Restrictive to affines). If f : X→ Spec(R) is a restrictive open immer- sion, then X is affine.
Proof. Since X is an open in Spec(R), we get a distinguised covering S
iD(fi) = Xwith fi∈ R and i ∈ I. We will prove that (fi
X)i∈I = (1)in S = Γ (X; O).
Then we invoke the result in Lemma 2.5.9 that states the following. For a scheme Y let Yf be the support of f ∈ Γ (Y; O) as in lemma ?. if Ygj are affine and (gj)j∈J = (1) then Y is affine.
Note that D(fi) = X
fi
X
. Consider M = (fR
i) as an R-module and look at Λ(M). By restrictiveness we get M ⊗RS = Λ(M)(S) and by M ⊗RRfi = Λ(M)(D(fi)) = Mfi = 0.
Hence Λ(M)(S) = 0 by the sheaf axiom. This implies that (fi
X)i∈I = (1)in S. So X is affine.
Lemma 3.0.9. Any morphism Spec(S)−→ Spec(R) ∈ Schf Spec(R)between affine schemes is restrictive.
Proof. Let G be a quasi-coherent module on SchSpec(R). Set M = Γ (Spec(R); G). We want to prove that
f : Mb ⊗RS→ Γ(Spec(S); G) is an isomorphism.
By Lemma 2.7.5, we get G = Λ(M). As said in Remark 3.0.2, in this case bf = ω2Spec(S). By Definition 2.2.15, we know that ω2 is an isomorphism at affine schemes.
Example 3.0.10 (Affine non-restrictive map). One might expect(or want) that any property of all maps between affine schemes also hold for affine maps between any schemes. This is not the case for restrictiveness, so it is not local on the target.
Consider the canonical inclusions A1→ P1and the shifted quasi-coherent module O(−1).
This module is locally free of degree 1, this is often called an invertible module.
The global sections of the module O(−1) are the elements of degree −1 in the global sections of O. There are no such elements, hence the global sections are the zero module.
On A1 all invertible modules are isomorphic to the structure sheaf. See [7, p. 14.2.8].
Similary any inclusion Spec(κ(p))→ P1 of a point is not restrictive which can be shown with the same argument.
This is a (more opaque) way of saying that on projective space not every quasi-coherent sheaf is generated by global sections.
In this chapter we will define the property affineness for an object in any ringed sites.
This will be called ‘caffine’. We will prove that caffine and affine are equivalent in the ringed site of schemes. Lastly, we introduce the notion of enough affines. If this conditions holds then a sheaf is quasi-coherent if and only if it is locally Λ(M) for any caffine object and some (non-)examples.
4.1 Caffine objects
In this section we will introduce caffine objects, see some examples and non-examples and prove some properties. Most of these results will be generalisations of their counterpart for affine schemes.
Definition 4.1.1 (Caffine object). Let (C,T, O) be a ringed site. Let a ∈ C be an object. We call a caffine if the unit η and co-unit of the adjunction Λ a Γ (a; −) on Ca are natural isomorphisms for any Γ (a; O)-module M and any quasi-coherent F on Ca
In other words, that Λ : Γ (a; O)-Mod → Qcoh(O) is an equivalence of categories with Γ (a; −)as pseudo-inverse and the unit and co-unit as witnessing natural isomorphisms.
Example 4.1.2 (Examples of caffine objects). The main example to keep in mind is Spec(R) ∈ Sch. See Lemmas 2.7.5 and 2.7.6 for proofs that the unit and co-unit are isomorphisms.
Example 4.1.3. The scheme P1 is not caffine in Sch. The counit at the quasi-coherent sheaf of modules O(−1) has signature : Λ(0)→ O(−1). Since O(−1) is locally free of degree 1, this cannot be an isomorphism.
Definition 4.1.4 (caffine cover). Let (C,T, O) be a ringed site. A family of maps {ai → a} is called a caffine covering of a if every ai is caffine and the family is a covering family.
Definition 4.1.5. We say that a ringed site (C,T, O) has enough affines if any object admits a caffine covering.
Lemma 4.1.6. Let (C,T, O) be a ringed site. Let a ∈ C. Let {bi → a} be a caffine covering on a. Assume every map bi −→ a is restrictive. Then the counit of thei adjunction Λa(−)a Γ (a; −) is a natural isomorphism.
Proof. Let F be a quasi-coherent sheaf module. Set M = Γ (a; F). Set Mi = Γ (bi; F).
Set βi = i∗F,a. By Definition 2.2.27 the induced functor of i satisfies all requirements of Lemma 2.4.5, so we have i∗F,a = ∼ i∗F,bi. Since bi is caffine βi is an isomorphism.
Hence is an isomorphism between sheaves, since it is an isomorphism restricted along every morphism from a covering.
Lemma 4.1.7. Let (C,T, O) be a ringed site. Let a ∈ C. Let M be a Γ(a; O)-module.
The component
ω2λ(M),a: λ(M)(a)→ Λ(M)(a) at Ida of the sheafification morphism
ω2λ(M) : λ(M)→ Λ(M) is equal to the unit of Λ a Γ (1; −) in Ca.
Proof. Consider the following maps, which you get by repeatedly calling on an adjunc- tion.
Id : Λ(M)→ Λ(M)
ω2Λ(M): λ(M)→ Λ(M) use sheafification adjunction ω2λ(M),aM→ Γ(a; Λ(M)) take sections at a
We took the adjunct of Id with respect to the sheafification adjunction as the first step.
Then we took the adjunct of the result wrt the λ ajunction. Hence we get the adjunct of Id wrt the Λ adjunction. so the last map is actually the unit of the Λ adjunction.
Corollary 4.1.8. Let (C,T, O) be a ringed site. Let a ∈ C be caffine. Then ω2λ(M),a is an isomorphism for any O(a)-module M.
Theorem 4.1.9 (Morphism between caffines is restrictive). Let Y = (C,T, O). Let f : b→ a ∈ C be a morphism between caffine objects, then f is restrictive.
Proof. Let F be a quasi-coherent module on Ya. Let M = Γ (a; F). Since a is caffine, we have F = Λ(M).
We have to show that
f : Γ (a; F)b ⊗Γ (a;O)Γ (b; O)→ Γ(b; F) is an isomorphism, where bf as defined in Definition 3.0.1.
Note that bf is the component at b of the natural transformation ω2λ(Γ (1;F)), see Re- mark 3.0.2. Since b is caffine, this component is an isomorphism by Corollary 4.1.8.
4.2 Caffine schemes
Let X ∈ Sch.
Set R = Γ (X; O). We will prove that the canonical morphism
X: X→ Spec(R)
is an isomorphism if X is a caffine object. This morphism sends x ∈ X to ker x as defined later.
We will start this section with some general results that work for any scheme.
Definition 4.2.1 (Support of module). Let M be a O-module. The support Supp(M) of this module is the subspace
{x ∈ X | Mx = 0} ⊂ X.
Definition 4.2.2 (Set cut out by ideal). Define the set cut out by an ideal I of R to be VX(I) =Supp(ΛX(R
I)).
Definition 4.2.3. For a global section a ∈ R define DX(a) ={x ∈ X | a 6∈ ker(x)}.
The following lemma will be the tool we use to prove that X is affine if it is caffine.
Lemma 4.2.4. The following are equivalent:
1. X is affine
2. X is an isomorphism 3. X is a homeomorphism
Proof. 1⇒ 2: Assume that X = Spec R. Then ker p = p for p ∈ X, so the topological part of Xis an homeomorphism. The sheaf part of Xsends the global section 1 ∈ O(X) to 1 ∈ R, since the sheaf morphism is a OSpec(R)-algebra map, it has to be the identity.
2⇒ 3: By definition.
3 ⇒ 1: Let Spec Ai = Ui ⊂ X be open affines and suppose S
iUi = X. Assume it is a finite affine cover, which can be done since X is quasi-compact. Using our base, we get a cover of Ui = S
jDX(aij) with aij global sections. Observe that DX(aij) ⊂ Ui, hence DUi(aij
Ui) = DX(aij) which makes them affine. Continuing like this, we get a finite cover of affines DX(aij)of X. Since
X(X) = X([
ij
DX(aij)) =[
ij
DSpec(R)(aij) =Spec R,
we have (aij) = (1). Now both requirements of Lemma 2.5.9 are satisfied, hence X is affine.
Lemma 4.2.5 (Stalks). Let M be a Γ (X; O)-module. Let x ∈ X. Then Λ(M)x = MOx. Proof. See [Stacks, Tag 01BH].
Lemma 4.2.6. For x ∈ X TFAE:
1. x ∈ VX(I) 2. IOx6= Ox 3. I ⊂ ker(x).
Proof. 1⇒ 2:
Assume x ∈ VX(I). We have ΛX(RI) = ΛΛX(R)
X(I) and taking stalks is exact soΛΛX(R)
X(I) x= ΛΛX(R)x
X(I)x. Lastly by Lemma 4.2.5 we get ΛX(O(X)I )x = IOOx
x. Since x ∈ VX(I) we know IOOx
x 6= 0, hence IOx6= Ox.
2 ⇒ 3:
Assume IOx 6= Ox. Then IOx is proper hence contained in the unique maximal ideal of the local ring Ox, therefore I 7→ 0 in k(x) or equivalently I ⊂ ker(x).
3 ⇒ 1:
Assume I ⊂ ker(x). Then I maps into mx, hence IOx ⊂ mx. Therefore Ox
ΛX(I)x = Ox IOx 6= 0.
Corollary 4.2.7. If y ∈ I then DX(y)∩ VX(I) =∅.
Proof. Assume y ∈ I. Let z ∈ VX(I), then y ∈ ker(z) by Lemma 4.2.6. This implies z6∈ DX(y).
Corollary 4.2.8. VX(I)∪ VX(J) = VX(IJ)
Proof. Let z ∈ VX(I)∪ VX(J). Then I ⊂ ker(z) and J ⊂ ker(z) by Lemma 4.2.6, hence IJ ⊂ ker(z). Apply Lemma 4.2.6 again to get z ∈ VX(IJ). Let z ∈ VX(IJ). Then IJ ⊂ ker(z) by Lemma 4.2.6. The ideal ker(z) is prime, so I ⊂ ker(z) or J ⊂ ker(z).
Invoke Lemma 4.2.6 again to get z ∈ VX(I)∪ VX(J).
Lemma 4.2.9. Let I ⊂ Γ (X; O) be an ideal. The set VX(I) is closed.
Proof. Consider the exact sequence
O(X)→ O(X) I → 0.
The functor Λ is a left adjoint hence right exact so O−→ Λ(f O(X)
I )→ 0
is exact. Hence the sequence
Ox −f→ Λ(x O(X) I )x → 0 is exact. So we get that fx(1x) generates Λ(O(X)I )x.
Note that fx(1x) = f(1)x by definition of fx, hence f(1)x is a generating element. Hence Λ(O(X)I )x6= 0 if and only if f(1)x6= 0.
This implies VX(I) = Supp(f(1)) which makes VX(I) closed as the support of a global section.
From now on assume that X is a caffine object in Sch.
Lemma 4.2.10. Every closed set W ⊂ X can be written as VX(I) for some ideal I⊂ Γ (x; O).
Proof. Let I be some ideal sheaf inducing a closed subscheme structure on W. This is by definition a quasi-coherent module. Let I = Γ (x; I). Since X is caffine, we get I = Λ(I). Let OW be the structure sheaf of this closed subscheme. Consider the closed immersion W −→ X. By construction I → O → ii ∗OW → 0 is exact, hence i∗OW = OI. Hence VX(I) =Supp (i∗OW) = W.
Lemma 4.2.11. The sets DX(a) form a basis for the topology of X, with a ∈ R.
Proof. Let x ∈ DX(a). Then a 6∈ ker x, hence 1a exists in Ox. We get an open neigh- bourhood V of x such that a1 ∈ Γ (V; O). Let y ∈ V, then clearly a 6∈ ker y. We have shown that x ∈ V ⊂ DX(a), hence DX(a) is open.
Let U ⊂ X be any open. Let x ∈ U. By Lemma 4.2.10 we get I such that VX(I) = Uc. It follows that x 6∈ VX(I)and I 6⊂ ker(x) by Lemma 4.2.6. So we get a g ∈ I with g 6∈ ker(x).
We get x ∈ DX(g) and DX(g)⊂ U, because DX(g)∩ VX(I) =∅ by Corollary 4.2.7.
Let a, b ∈ R. Note that DX(ab) = DX(a)∩ DX(b) since ker(x) is a prime ideal. So the opens DX(a) form a basis.
Lemma 4.2.12. The map X : X→ Spec(R) is surjective.
Proof. Let p ∈ Spec R be a point in the target of X. Then ΛX(κ(p))is a quasi-coherent sheaf of modules. In fact κ(p) ⊗Γ (X;O)O(U)is a O(U) algebra for any open U ⊂ X, hence ΛX(κ(p))is a quasi-coherent sheaf of algebras. Hence we can compute the relative spec Rspec (ΛX(κ(p)))−→ X.h
Since X is caffine we have ΛX(κ(p))(X) = κ(p). So
Rspec (ΛX(κ(p)))−−−X◦h→ Spec(R) factors as
Rspec (ΛX(κ(p)))→ Spec(κ(p)) → Spec(R).
This gives us a commutative square
X Spec R
Rspec (ΛX(κ(p))) Spec κ(p)
X
Since ΛX(κ(p)) is not the zero sheaf hence the structure sheaf of Rspec (ΛX(κ(p))) is non-zero. This implies that Rspec (ΛX(κ(p))) is not the empty scheme. Therefore the point p is in the image of Rspec (ΛX(κ(p)))→ Spec(κ(p)) → Spec(R), hence also in the image of X.
Lemma 4.2.13. Let p ⊂ R be a prime ideal. The closed set VX(p) is irreducible.
This implies that X is injective.
Proof. Let z ∈ X. Assume that X(z) = p. So ker z = p. Let y ∈ VX(p). Then ker(z) ⊂ ker(y), hence if y ∈ DX(a) then z ∈ DX(a). Therefore z is contained in any open subset of VX(p), hence it is irreducible. Note that z is its unique generic point since the biggest closed subset of VX(p) not containing z is the empty set. This shows that X is injective.
Lemma 4.2.14. The counit X is open, hence a homeomorphism.
Proof. Let a ∈ R. Let x ∈ X. Let p = X(x). Since X is a morphism of locally ringed spaces, if a 6∈ ker x then a 6∈ ker p = p. Note that ker − is defined for arbitrary schemes so also works on Spec(R). Hence X(DX(a)) ={X(x)| a 6∈ ker(x)} = X(X)∩ DSpec R(a) = DSpec R(a). Our map X is continuous and open, so a homeomorphism.
Proposition 4.2.15. The caffine scheme X is affine.
Proof. By Lemmas 4.2.12 to 4.2.14 is an homeomorphism and so by Lemma 4.2.4 it follows that X is affine and is an isomorphism.
4.3 Enough caffines
For schemes a quasi-coherent module often is defined by a module that over affine schemes is of the form Λ(M) for some module M. We will look how often this generalizes, which we will do through the notion of ‘enough caffines’. We will see some examples where it does not work as in Sch and some results about when it does.
Definition 4.3.1. We say that a ringed site (C,T, O) has enough affines if any object admits a caffine covering.
Lemma 4.3.2. Let C be a ringed site that has enough affines. Let F be a sheaf of modules on C. Then F is quasi-coherent if and only if for each caffine a ∈ C we have F
a= Λ(M) for some Γ (a; O)-module M.∼ Proof. ⇒: This is Lemma 2.3.3.
⇐: Since C has enough affines, every object has a cover of caffine objects. For any caffine a ∈ C, we have F
a = Λ(M) so F∼
a has a global presentation. Hence F is quasi-coherent.
Lemma 4.3.3. Any ringed site (C,T, O) that has a finite poset as underlying cate- gory, has enough caffines.
Proof. Let x0∈ C. If x0 is covered by the maximal sieve only or the maximal sieve and the empty sieve, it is caffine and we are done. Assume otherwise. Let S ={yi → x0} be a non-maximal, non-empty cover of x0. Then S does not contain isomorphisms.
We can associate to any non-maximal non-empty covering sieve S of an element x0, the set of all NA-chains x0 ← x1 ← . . . ← xn. An NA-chain, associated to R, is a chain of maps ending in x0 such that xi ← xi+1 is contained in a non-maximal, non-empty cover of xi, where x0 ← x1 is contained in R.
By finiteness of C, any chain of maps is bounded by the size of C or contains a cycle.
If a chain contains a cycle, it contains isomorphisms. By construction, no isomorphism can be present in a NA-chain. Therefore the length of any NA-chain is bounded by kCk.
Let H be a NA-chain associated to S of maximal length m. Then the last map . . . ← h← g in H has a caffine object g as domain, because H cannot be increased and so g has no non-maximal, non-empty coverings which makes it caffine. Also the non-maximal, non-empty covering of h where this map appears must be a caffine covering by applying the same reasoning to the other objects occurring in it. Hence all objects occurring at the (m − 1)th place in any NA-chain admits a caffine cover. Let i 6 m − 1. Assume all elements at the (i − 1)th place admit a caffine cover. Let b be a object occurring at the (i − 1)th place in a chain. It is either caffine or all objects in any non-maximal, non- empty cover occur at the ith place in some chain hence admit a caffine cover. Therefore any non maximal, non empty cover on b can be refined to a caffine cover. This provides us with a caffine cover of b. By reversed induction, x0 admits a caffine cover.
Example 4.3.4. The following will be an example of a category that does not have enough caffines.
The category C is Z×Z as a product partial order. An element (i, j) is only non-trivially covered by{(i, j − 1) → (i, j), (i − 1, j) → (i, j)}. Let k be any field. Let R = k[xij|i, j ∈ Z].
Define the structure sheaf as O(i, j) = R[x−1kl |i 6 k & j 6 l].
Fix (a, b) ∈ C. Consider the over category C(a,b) at this point. Let (i, j) → (a, b) be an object of C(a,b). So i 6 a and j 6 b. Define the presheaf of modules F(i, j) = O(i, j)/(xa−1,bxa,b−1) on X. We have a > i or b > j or (i = a and j = b). If a > i or b > j, then xa−1,b or xa,b−1 is invertible in O(i, j), hence F(i, j) = 0 in both cases.
This presheaf is zero everywhere except at (a, b), because both xa−1,band xa,b−1are not invertible in O(a, b) hence sheafifies to the zero sheaf. In other words: Λ((x O(a,b)
a−1,bxa,b−1)) =
0, Hence (a, b) is not affine, which shows that C has no affine objects.
Example 4.3.5. Let C be as in Example 4.3.4. Consider G = O(i, j)[yk,l|k 6 i & l 6 j].
Let L
k∈IO −→ G be any sheaf map. Let α(eα k) be the image of the generators ek ∈ L
k∈IO in the global sections. The section y1,1∈ G(1, 1) cannot be written as a finite sumP
kλkα(ek)for scalars λk∈ O(i, j) for any (i, j). This shows that α is not surjective hence G is not quasi-coherent(locally presentable).
Example 4.3.6. This is the example in [Stacks, Tag 01BL]. We state it here because we will categorify it to yield a category that has not enough affines.
Let L = (R, OR)be the real line with the euclidean topology and the sheaf of continuous real valued functions as structure sheaf. Let
X = S∞
i=0Li
∼
with [i, x]∼ [j, y] if and only if i = j and x = y or y = x = 0. The real lines are glued to each other at zero. Define the open Un ⊂ X as Un∩ Li= (−n1,n1). These opens form a basis of neighbourhoods of 0. Let f : R → R be any continuous function such that f(x) = 0if x ∈ (−1, 1) and f(x) = 1 if x ∈ (−∞, −2) ∪ (2, ∞). Let fn(x) = f(nx).
Define the sheaf map
M
i
OR −→α M
ij
OR,
ei 7→X
j
fi1Ljeij.
To proof that this is well-defined, we need to show that the sum P
jfi1Ljeij is locally finite for every i. Let [k, y] ∈ X. If y 6= 0, then
W[k,y]={[k, z] ∈ X | z ∈ (y − δ, y + δ)} ⊂ Lk
is open in X and αW[k,y](ei) = fieik for any δ <|y2|. If y = 0, then αUn(ei) = 0if n > i because fi is zero on Un. Hence we found a cover on which our sum is locally finite, which makes α well-defined.
The adjunction Λ a Γ (X; −) implies that Λ commutes with arbitrary colimits. Moreover OX= Λ(Γ (X; O)),∼
