Sheaves from Scratch Kay Werndli 11Th December, 2010

Sheaves From Scratch Kay Werndli 11th December, 2010

Dieses Werk, sowie alle darin enthaltenen Abbildungen, stehen unter der Creative Commons CC BY:

$\ C Lizenz. Eine Kopie der Lizenz ﬁndet sich unter http://creativecommons.org/licenses/by-nc-sa/2.5/ch/ oder ist brieﬂich bei Creative Commons, 171 2nd Street, Suite 300, San Francisco, California, 94105, USA erhältlich. CONTENTS

Chapter 1 Categorical Preliminaries ...... 4 1. Basic Definitions and Notations...... 4 2. Limits ...... 4 Chapter 2 Kan Extensions ...... 6 1. Definition and First Properties ...... 6 2. Pointwise Kan Extensions ...... 10 3. Tensor-Hom Adjunction...... 16 Chapter 3 Sheaves on a Space ...... 19 1. Presheaves and the Gluing Axiom...... 19 2. Alternative Gluing Axioms...... 21 3. Topological Constructions of Sheaves...... 24 4. Categorical Constructions of Sheaves...... 26 5. Étale Bundles and Sheafification...... 27 6. Sheaves with Algebraic Structure ...... 33 7. More on Étale Bundles...... 34 8. Stalks and Skyscrapers...... 35 9. Image Functors...... 35 Chapter 4 Ringed Spaces ...... 37 1. Definition and First Properties ...... 37 2. Points of a Ringed Space...... 41 3. Sheaves of Modules on a Ringed Space...... 41 4. Zero Sets ...... 44 5. Dimension Theory ...... 45 6. Coherent Sheaves...... 45 7. Sheaf Cohomology ...... 48 8. Čech Cohomology...... 48 9. Flat Morphisms ...... 48 Chapter 5 Topological Preliminaries ...... 53 1. Jacobson Spaces ...... 53 Chapter 6 Algebraic Preliminaries ...... 55 1. Normal Rings ...... 55 2. The Category of Schemes...... 55 Contents 3

3. Graded Rings ...... 56 4. Chow’s Lemma...... 61 Chapter 7 Analytic Preliminaries ...... 63 1. Analytic Spaces ...... 63 2. Weierstraß’ Preparation Theorem...... 64 3. Riemann Extension Theorem...... 64 4. Dimension Theory ...... 64 Chapter 8 GAGA ...... 65 1. Analytiﬁcation of Schemes...... 65 2. GAGA for Projective Spaces ...... 66 Index of Notation ...... 67

Bibliography ...... 68

Index ...... 69 Chapter 1

CATEGORICAL PRELIMINARIES

In this chapter, we review some of the categorical facts we will need later on. The reader is expected to have some knowledge of category theory and should at least be familiar with the basic deﬁnitions of categories, functors, natural transformations, (co-)limits and adjunctions. The facts and deﬁnitions are listed here for completeness’ sake and to introduce the notation used in this text.

1. Basic Deﬁnitions and Notations

2. Limits

Consider a diagram F : J → Sets in the category of sets. Recall that a limit of F is a universal (in this context terminal) cone over F , so how should we construct such a limit? A good starting point would be to take the universal object over the FI, where I runs through Q all objects of J, which is I∈|J| FI and then take the subset of all (xI )i∈|J|, which satisfy the cone condition. That is the set n o a x = (xI )I∈|J| (F a) ◦ prI x = (F a)xI = xJ = prJ x for all I −→ J in J ,

th where prI denotes the standard projection to the I component. I.e. it is the subset of all x such that (F a) ◦ prI and prJ agree for all a: I → J in J, which can obviously be stated as an equalizer condition. This process of constructing a limit is obviously completely categorical and does not depend on the category of Sets, which we only took as an example. All that we need are products and equalizers, so that we arrive at the following, well know theorem.

(2.1) Theorem. Let F : J → C be a diagram in a category C. A limit (E, σ) of F is given by the following equalizer diagram

FJ q8 O pr qq J qq pr qqq a qqq qq p e Y / Y E / / FI / FJ HH q HH I∈|J| (a: I→J)∈J HH σ HH I HH prI pra HH# FI / FJ . F a

That is to say that if (E, e) is an equalizer of p = (prJ )(a:I→J)∈J and q = (F a ◦ prI )(a:I→J)∈J, then E is a limiting object of F with limiting cone (σI = prI ◦e)I∈|J|. Paragraph 2. Limits 5

Proof. If (C, τ) is any cone over F then the arrows τI : C → FI form a cone over the objects Q FI and thus the τI factor uniquely through the product I∈|J| FI. Now, the condition that τ is a cone over F is equivalent to p ◦ (τI )I∈|J| = q ◦ (τI )I∈|J|, by which (τI )I∈|J| factors uniquely through the equalizer (E, e). I.e. there exists a unique arrow t: C → E such that (σ ) ◦ t = (τ ) , so that σ ◦ t = τ , which is what we wanted. I I∈|J| I I∈|J| I I (2.2) Corollary. A category C is (finitely) complete iff it has (finite) products and equalizers.

Of course, all finite products (exept from a terminal object) can be constructed from binary products so that a category is finitely complete iff it has a terminal object, binary products and equalizers. But recall that if 1 ∈ |C| is a terminal object and C,D ∈ |C| any two objects, then a pullback

pr C × D D / D

prC C / 1 is nothing but a product of C and D. Moreover, if a, b: C ⇒ D are two parallel arrows, the product D × D exists, and (a, b): C → D × D is the induced arrow into this product then a pullback

e E / C

(a,b) D / D × D δ along the diagonal δ := (1D, 1D): D → D × D gives us an equalizer (E, e) of a and b. So that we can alternatively state ﬁnite completeness as follows.

(2.3) Corollary. A category C is ﬁnitely complete iﬀ if has a terminal object and pullbacks. Chapter 2

KAN EXTENSIONS

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. Robert Anson Heinlein

1. Deﬁnition and First Properties

Let’s consider the following, so called extension problem: Given the two solid 1-arrows

? • ~ b ~~ c ~~ ~ ~ / • a •

ﬁnd a (dashed) 1-arrow to make the diagram commute (in some sense). If we require commutativity on the nose (i.e. the equality a = c ◦ b), we call c a strict extension of a along b. As one immediately guesses, knowing some bits and pieces about 2-categories, this is usually far too strict to give us a useful tool. One should instead only require commutativity up to isomorphism, i.e. a =∼ c ◦ b, which one would (or at least should) probably call a weak extension of a along b. But we go even one step further and only require a (not necessarily invertible) universal 2-arrow a ⇒ c ◦ b to exist.

(1.1) Deﬁnition. Given an extension problem (i.e. a diagram of solid arrows as below) in a (meta-)2-category, a left Kan extension of a along b is a pair (k, η) of dotted arrows as in the diagram, with η universal from a to b∗ : Hom(B,C) → Hom(A, C) (i.e. initial in a ↓ b∗). The 2-arrow η is sometimes called the unit of the left Kan extension (whose dual for right Kan extensions is then called the counit), while the 1-arrow k is sometimes (by abuse of language) also called the left Kan extension of a along b.

? B ~ b ~~ KS k ~~ η ~ ~ / A a C If we expand this deﬁnition, this means that if we have any other such pair of arrows (k0, η0), then there is a unique 2-arrow α: k ⇒ k0 such that η0 = (α ? b) ◦ η. The notion of a right Kan extension is the co-dual to a left Kan extension (i.e. reverse all 2-arrows but not the 1-arrows). By taking the op- (i.e. reverse only 1-arrows) or coop-dual (i.e. reverse both 1- and 2-arrows), we would get something like a Kan lift. Paragraph 1. Deﬁnition and First Properties 7

As all universal constructions, a left Kan extension us unique up to canonical isomorphism. To wit, if (k, η) and (k0, η0) are two left Kan extensions then there is a unique isomorphism α: k ⇒ k0 such that η0 = (α ? b) ◦ η (and thus also η = (α−1 ? b) ◦ η0). If there is a canonical choice for a left (resp. right) Kan extension of a along b we will usually write “Lanb a” (resp. “Ranb a”) for the 1-arrow of this canonical choice, the 2-arrow being implicit. By the observation above that a Kan extension is unique up ∼ to canonical isomorphism and by abuse of notation, it does no harm to write “k = Lanb a” for the statement “k is the 1-arrow of a left Kan extension of a along b” (and analogously ∼ “k = Ranb a” for a right Kan extension) even if there is no canonical choice for such an extension.

(1.2) Remark. Perhaps we should clarify the use of the adjective “left” in the definiton of a left Kan extensions. One will immediately see the reason if one formulates the external definition of a Kan extension instead of the internal one above. In the same situation as in the last definition, (Lanb a, η) is a left Kan extension of a along b iff ∗ ∗ ∗ η ◦ b : Hom(B,C)(Lanb a, −) → Hom(A, C)(a, b −) is a natural isomorphism of functors Hom(B,C) → Sets (of course to put it that way, we should require each Hom-category to be locally small). One should be aware that b∗ stands ∗ ∗ for two different things here and one should rather write η ◦ 1b instead. Now thas a set it is ` simply the coproduct (i.e. disjoint union) (U,s) PU as a set it is simply the coproduct (i.e. ` 0 disjoint union) (U,s) PU is means exactly that for every k ∈ Hom(B,C) we have a bĳection 0 0 Hom(B,C)(Lanb a, k ) → Hom(A, C)(a, k ◦ b), α 7→ (α ? b) ◦ η, which is natural in k0. This is exactly the definition of a left Kan extension and if we are 0 just given such a natural bĳection, we can recover η by putting k := Lanb a. Then η is just the image of 1Lanb a under the bĳection. Moreover, one immediately deduces from this that if every arrow a has a chosen left Kan extension (Lanb a, η) along b then Lanb is (the object function of) a left adjoint to b∗ (the arrow function of course arising from the universal property of a Kan extension) and η∗ ◦ b∗ is the unit of the adjunction.

(1.3) Example. If J, C are categories and D : J → C a J-diagram in C then D has a colimit if and only if it has a left Kan extension along the unique functor !: J → 1, where 1 is the terminal 1-category. If it has a colimit (L, σ) then L ∈ Ob(C) deﬁnes a functor L: 1 → C and σ is a natural transformation D ⇒ L ◦ ! (which is the constant functor L: J → C) which together satisfy the universal property of a left Kan extension. Conversely if (K, η) is a left Kan extension of D along ! then K : 1 → C is nothing but an object in C and η is a colimiting cone D ⇒ K.

(1.4) Remark. For the following example, recall that if C is a locally small category, R ∈ Ob(C) and F : C → Sets then a natural transformation ϕ: C(R, −) ⇒ F is completely determined by r := ϕR1R. For if we have C ∈ Ob(C) and a ∈ C(R,C), we form the diagram ϕ C(R,R) R / FR

a∗ F a C(R,C) / FC , ϕC which commutes by naturality of ϕ. A quick chase of 1R around the diagram shows that ϕC a = (F a)r. 8 Chapter 2. Kan Extensions

(1.5) Example. Let C be a locally small category and C(R, −): C → Sets a functor, represented by R ∈ Ob(C). Then a left Kan extension of C(R, −) along a functor G: C → D is given by the functor D(GR, −), represented by GR, together with the unit η : C(R, −) ⇒ D(GR, G−), whose component at C ∈ Ob(C) is simply a 7→ Ga (i.e. the nat- 0 ural transformation determined by 1GR). For let K : D → Sets be another functor together 0 0 0 with a natural transformation η : C(R, −) ⇒ K G (which is determined by ηR1R). If there is some α: D(GR, −) ⇒ K0 satisfying η0 = (α ? G) ◦ η, we calculate

0 ηR1R = αGR ◦ ηR1R = αGR1GR, which completely deﬁnes α and gives us the required universal property.

(1.6) Example. Given an adjoint pair f : A B : g (left adjoint on the left) in a 2- category with unit η : 1A → gf and counit ε: fg → 1B then (g, η) is a left Kan extension of 1A along f and moreover f perserves this Kan extension, meaning that after composing everything with f, (fg, fη) is still a left Kan extension of f ◦ 1A = f along f (similarly (f, ε) is a right Kan extension of 1B along g and g preserves it). The converse of this statement is also true: A 1-arrow f : A → B has a right adjoint iﬀ 1A has a left Kan extension (g, η) along f and f preserves this. If that’s the case then g is right adjoint to f and η is the unit of the adjunction.

Proof.“ ⇒”: An even stronger statement is proved in the following remark. “⇐”: As stated above, f preserving the left Kan extension (g, η) means that (fg, f ? η) is a 0 0 left Kan extension of f along itself. Putting k := 1B and η := 1f in the deﬁnition of a left Kan extension gives us a unique 2-arrow ε: fg ⇒ 1B such that (εf) ◦ (fη) = 1f which is one of the triangular equalities. We can derive the other one from the uniqueness properties of our Kan extensions available. For consider the 2-arrow (gεf) ◦ (gfη) ◦ η : 1A ⇒ gf. Because (g, η) is a left Kan extension of 1A along f there is exactly one α: g ⇒ g such that

(α ? f) ◦ η = (g ? ε ? f) ◦ (g ? f ? η) ◦ η ∆=-eq. (g ? f) ◦ η and obviously this must be α = g. But observe that by the swap trick (gfη) ◦ η = (ηgf) ◦ η and thus also α = (gε) ◦ (ηg), which is the second triangular equality.

(1.7) Remark. We can work out this example a little more: If f a g with unit η and counit ε, the Kan extension (g, η) of 1A along f is absolute, meaning that all 1-arrows h: A → C preserve it. To wit, for every 1-arrow h: A → C the composite (hg, hη) is again a left Kan extension of h ◦ 1A = h along f.

? B hg f ~ g ~~ KS ~~ η ~~ A / A / C 1A h

Proof. Let h0 : B → C and η0 : h ⇒ h0f. We are looking for an α: hg ⇒ h0 such that

h?η α?f η0 h /hgf /h0f = h /h0f . Paragraph 1. Deﬁnition and First Properties 9

The only reasonable choice for such an α from the given data is α := (h0 ? ε) ◦ (η0 ? g) and indeed this α satisﬁes the above equation since by the triangle equation (ε ? f) ◦ (f ? η) = 1f we calculate (using the swap trick)

(α ? f) ◦ (h ? η) = (h0 ? ε ? f) ◦ (η0 ? g ? f) ◦ (h ? η) = (h0 ? ε ? f) ◦ (h0 ? f ? η) ◦ η0 = h0 ? (ε ? f) ◦ (f ? η) ◦ η0 = (h0 ? f) ◦ η0 = η0 Moreover, this α is unique. For if any α0 satisﬁes the equation, we ﬁrst apply horizontal composition from the right by g (that’s 1g to be precise) and afterwards apply vertical composition from the left by h0 ? ε, yielding

h?η?g α?fg 0 η0?g 0 hg /hgfg /h0fg h ?ε /h0 = hg /h0fg h ?ε /h0 .

But again by the swap trick and the other triangle equality, we have (h0 ? ε) ◦ (α0 ? f) ◦ (h ? η) ? g = (h0 ? ε) ◦ α0 ? (f ◦ g) ◦ (h ? η ? g) = α0 ◦ (h ? g ? ε) ◦ (h ? η ? g) = α0 ◦ h ? (g ? ε) ◦ (η ? g) = α0 ◦ (h ? g) = α0 thus infering that α0 = (h0 ? ε) ◦ (g ? η0) = α. Motivated by this example, where we viewed adjoint arrows as special cases of Kan extensions, we can ask ourselves on the other hand how Kan extensions behave, when composed with an adjoint arrow.

(1.8) Theorem. Left adjoints preserve left Kan extensions. I.e. if (k, κ) is a left Kan extension of a along b as in the diagram then (fk, fκ) is a left Kan extension of fa along b.

B fk ? @ b ~ KS @ k ~~ @@ ~~ κ @@ ~ @ f ~ / / A a C o _ D g

Proof. Given any k0 : B → D and κ0 : fa ⇒ k0b we consider (g ? κ0) ◦ (η ? a): a ⇒ gfa ⇒ gk0b, where η is the unit of the adjunction. By the universal property of a left Kan extension, we get a unique 2-arrow α: k ⇒ gk0 such that (g ? κ0) ◦ (η ? a) = (α ? b) ◦ κ. This α in turn yields β := (ε ? k0) ◦ (f ? α): fk ⇒ fgk0 ⇒ k0 as required, where ε is the counit of the adjunction. β thus deﬁned has the required property: (β ? b) ◦ (f ? κ) = (ε ? k0) ◦ (f ? α) ? b ◦ (f ? κ) = (ε ? k0 ? b) ◦ (f ? α ? b) ◦ (f ? κ) = (ε ? k0 ? b) ◦ (f ? g ? κ0) ◦ (f ? η ? a) swap= κ0 ◦ (ε ? f ? a) ◦ (f ? η ? a) ∆=-eq. κ0. Moreover, this is the unique β that does so, because α is unique. To wit, if β0 : fk ⇒ k0 is another arrow such that (β0 ? b) ◦ (f ? κ) = κ0, we want to exploit the unicity of α. Thus we consider (g ? β0) ◦ (η ? k): k ⇒ gfk ⇒ gk0 and this 2-arrow satisﬁes that

(g ? β0 ? b) ◦ (η ? k ? b) ◦ κ swap= (g ? β0 ? b) ◦ (g ? f ? κ) ◦ (η ? a) = (g ? κ0) ◦ (η ? a) 10 Chapter 2. Kan Extensions and thus α = (g ? β0) ◦ (η ? k). Because β = (ε ? k0) ◦ (f ? α), we conclude

β = (ε ? k0) ◦ (f ? g ? β0) ◦ (f ? η ? k) swap= β0 ◦ (ε ? f ? k) ◦ (f ? η ? k) ∆=-eq β0

As an immediate consequence of (1.3) and this theorem we get the following well- known fact “for free”.

(1.9) Corollary. Left adjoint functors preserve colimits.

2. Pointwise Kan Extensions

A particularly interesting case of a Kan extension is when we are working in the (strict) (meta-)2-category of all categories, so that our 1-arrows are functors and our 2-arrows are natural transformations. If we assume certain small- and completeness conditions then Kan extensions can be explicitly calculated as a colimit. Moreover, for a very important special case, which we will investigate later, the unit will even be an isomorphism.

(2.1) Remark. For C, D any categories, F : C → D a functor, and D ∈ Ob(D), we always have a projection

Q F ↓ D −→ C, sending an arrow a: C → C0 in C, which makes a diagram

F a FC / FC0 :: Ó :: ÓÓ b :: ÓÓb0 : ÓÑ Ó D

(with b, b0 objects of F ↓ D) commute to itself. This projection is usually not full because nothing guarantees us that for any given a: C → C0 we can ﬁnd such a diagram.

(2.2) Theorem. (Colimit Formula) Given a diagram of solid 1-arrows in CAT

? E ~ G ~~ KS K ~~ η ~~ C / D F such that for each E ∈ Ob(E) we have chosen a colimit

Q F σa colim(G ↓ E −→ C −→ D) = KE, (FC −→ KE)(a: GC→E)∈Ob(G↓E) .

Then the KE constitute the object function of a functor K as in the diagram. Moreover, these colimits determine a left Kan extension (K, η) of F along G, where

η σ1 FC −−→C KGC := FC −−−→GC KGC for σ the chosen colimit for E = GC above. Paragraph 2. Pointwise Kan Extensions 11

Proof. First, let’s extend the family (KE)E∈Ob(E) to a functor. Of course, this is done by exploiting the universal property of the colimit. Any arrow f : E0 → E in E determines a 0 F cocone under G ↓ E → C −→ D, namely (σf◦a : FC → KE)(a: GC→E0)∈Ob(G↓E0), where σ is the colimiting cocone as in the proposition, thus inducing a unique arrow Kf : KE0 → KE. Plainly, this does in fact deﬁne a functor. The next thing we need to check is that η as deﬁned in the theorem is indeed a natural transformation. For this purpose, let C,C0 ∈ Ob(C) and σ, σ0 the chosen colimits from the theorem for E = FC and E = FC0 respectively. We now draw a naturality square and include the obvious diagonal arrow

σ0 1GC0 C0 FC0 / KGC0 HH HH σGf f F f HH KGf HH HH$ CFC / KGC . σ1GC

The upper triangle commutes by definition of K and the lower triangle commutes by the naturality of σ. The last — and hardest — thing to check is the universal property of a left Kan extension being satisfied by η. To do so, let K0 : E → D be another functor and η0 : F ⇒ K0G a natural transformation. We need to find a natural transformation α: K ⇒ K0 such that η0 = (α ? G) ◦ η. The only way to construct α form the given data is to exploit once again the univeresal property of the colimit. So for E ∈ Ob(E) and σ as in the proposition, we have for each a: GC → E in G ↓ E a diagram of solid arrows as shown on the left.

0 ηC f F f GC FC / K0GC C0 / C FC0 / FC

σ 0 η0 0 a a K a C0 ηC Gf K0Gf (2.3) EKE / K0E GC0 / GC K0GC0 / K0GC ∃!αE 0 : 0 :: Õ 00 : ÕÕ 0 0 a 0 0 : Õ 0 0 a 0 K a :: ÕÕ K a 0 × : ÕÒ Õ E K0E

0 0 0 0 F By functoriality of K and naturality of η the (K a)◦ηC are a cocone under G ↓ E → C −→ D (cf. the right-hand diagram above) and by the universal property of the colimit there exists 0 a unique arrow αE : KE → K E as in the above diagram, rendering it commutative. These 0 αE do in fact deﬁne a natural transformation α: K ⇒ K as required, which follows once again form the universal property of the colimits σ. To wit, for f : E0 → E in E and σ, σ0 as above and a: GC → E0 in G ↓ E0 arbitrary (so f ◦ a: GC → E0 → E lies in G ↓ E), we get a diagram of solid arrows

FC FC Ø 66 ØØ 0 σ 66 Ø σa f◦a 6 ØØ 66 η0 Ø Kf 6 η0 C ØØ KE0 / KE 66 C (2.4) ØØ 66 ØØ ∃!ψ 66 Ø αE0 αE 6 ØØ 66 ØÓ Ø # 6 K0GC / K0E0 / K0EKo 0GC , K0a K0f K0(f◦a) 12 Chapter 2. Kan Extensions

where the upper square and the two (outer) triangles commute by definition of Kf, αE, 0 and αE0 . Thus the lower square commutes if we precompose it with σa and this holds for 0 0 0 0 all a ∈ Ob(G ↓ E ) meaning that the αE ◦ Kf ◦ σa = K f ◦ αE0 ◦ σa form cocone under G ↓ E0 → C → D and by the universal property of the colimit σ0 there exists a unique 0 0 0 0 diagonal arrow ψ as in the diagram such that ψ ◦ σa = αE ◦ Kf ◦ σa = K f ◦ αE0 ◦ σa and thus by unicity the lower square commutes. So α: K ⇒ K0 is indeed natural and going back 0 to (2.3) it obviously satisfies η = (α ? G) ◦ η, plainly by putting a = 1GC . Finally, what is left to check is the uniqueness of α. To do so let α: K ⇒ K0 be any natural transformation satisfying η0 = (α ? G) ◦ η. Then again (2.3) commutes for any C ∈ Ob(C) and a = 1GC and we need to check that this even holds for any a: GC → E because then our arbitrary α we are considering here would be the same as the one above by unicity of αE in (2.3). To prove our claim, we consider the diagram of solid arrows (2.4) and 0 put E := GC, a := 1GC . In that case the left-hand triangle as well as the both squares still commute and the path around this trapezium is the sameas the right-hand triangle, implying that this commutes, too. But this is exactly (2.3) with a replaced by f : GC → E. (2.5) Remark. As one would expect, there is a similar theorem for right Kan extensions: If in the theorem, we replace G ↓ E by E ↓ G and the colimit by a limit then the KE can be extended to a functor E → D and this together with the limiting cone determines a right Kan extension of F along G, which is then called the limit formula. It is in some sense a dual statement. However, we have to be careful as there are natural transformations involved, which means that we do not have a dual statement in the ordinary (1-categorical) sense. We can still deduce the limit formula from the colimit formula (and vice versa) as “dualising” is really a 2-categorical process. To be precise, the proof boils down to the fact that we have an involution of (meta-)2-categories CATco =∼ CAT, namely the transposition as defined in (2.12) below.

(2.6) Deﬁnition. A left Kan extension, which is given by the colimit formula from the theorem is also called a pointwise left Kan extension. Analogously for a right Kan extension determined by the limit formula.

(2.7) Corollary. Let C be a small category and D cocomplete (i.e. for every small category J the diagonal functor D → DJ has a left adjoint, meaning that we have chosen colimits). Then every functor F : C → D has a (pointwise) left Kan extension along any ∗ E C G: C → E and consequentially (cf. (1.2)) G : D → D has a left adjoint LanG.

As another corollary to the colimit formula, we get the well-known and often used characterisation of an adjunction by the existence of universal arrows. This says that a functor F : C → D has a right adjoint if and only if for every D ∈ Ob(D) there is a universal arrow from F to D, meaning that F ↓ D has a terminal object. The usual elementary proof of this fact involves lots of diagrams but as we have already done this work in the proof of the colimit formula, we get this result “for free”. (2.8) Corollary. For F : C → D the following are equivalent

(a) F has a right adjoint G with unit η : 1C ⇒ GF and counit ε: FG ⇒ 1D.

(b) For every D ∈ Ob(D) we have a chosen universal arrow εD : F GD → D (here GD is just a name for an object of C) from F to D (i.e. (GD, εD) is terminal in F ↓ D).

(c) 1C has a pointwise left Kan extension (G, η) along F , which is absolute (i.e. any functor with domain C preserves it). Paragraph 2. Pointwise Kan Extensions 13

(d) 1C has a left Kan extension (G, η) along F , which is absolute.

(e) 1C has a left Kan extension (G, η) along F and F preserves this.

Proof. “(a) ⇒ (b)”: εD is terminal in F ↓ D by the universal property of the counit ε. “(b) ⇒ (c)”: If H : D → E is any functor and D ∈ Ob(D) there is a canonical choice for a colimit of HFQ: F ↓ D → C → D → E, which is given by evaluating HFQ at the terminal object. To wit, L := HFQ(GD, εD) = HF GD is a colimiting object with colimiting cocone σ : HFG ⇒ L given by σ(C,a: FC→D) := HF !(C,a) : HFQ(C, a) = HFC → L, where !(C,a) : C → GD is the unique arrow from (C, a) into the terminal object (GD, εD). “(c) ⇒ (d) ⇒ (e)”: Trivial. “(e) ⇒ (a)”: This is just (1.6). (2.9) Corollary. In the same situation as in the theorem, if G is fully faithful then η is an isomorphism.

Proof. For C ∈ Ob(C) every object b: GC0 → GC in G ↓ GC is of the form b = Ga for 0 a unique a: C → C because G is fully faithful. Thus 1GC is a terminal object in G ↓ GC and the colimit KGC can be calculated by evaluating G ↓ GC → C → D at this terminal ∼ object, yielding that KGC = FC is this colimit with colimiting cocone ηC = 1FC , which is an isomorphism. (2.10) Corollary. Given adjoint functors F a G with unit η and counit ε then F : C → D is fully faithful if and only if η is an isomorphism.

Proof.“ ⇒”: By the two last corollaries. ∼ “⇐”: η is an isomorphism 1C = GF which is equivalent to each component ηC being an 0 0 0 isomorphism thus rendering (ηC )∗ : C(C ,C) → D(C , GF C) (with C,C ∈ Ob(C)) bĳective. −1 ∼ Furthermore, by deﬁnition of an adjunction ϕC,D : C(C, GD) = D(FC,D), b 7→ εD ◦ F b is also bĳective and thus (using a triangular identity) the composite

0 (ηC )∗ 0 ∼ 0 C(C ,C) −−−→ C(C , GF C) = D(FC ,FC), a 7→ ηC ◦ a 7→ εFC ◦ F ηC ◦ F a = F a is a bĳection, too. But this exactly means that F is fully faithful as we claimed. This corollary can be generalized further and we can ﬁnd necessary and suﬃcient criteria for fullness resp. faithfullness of a left adjoint in terms of the unit.

(2.11) Corollary. In the same situation as in the last corollary.

(a) F is faithfull iﬀ all components ηC of the unit are monic;

(b) F is full iﬀ all components ηC of the unit have a section.

In view of the next proof which might be somewhat confusing because of the unusual practice of viewing a contravariant Hom-functor as a functor with values in Setsop (if we have a contravariant functor from C to D we might either view this a a functor Cop → D or as a functor C → Dop). With the risk of even causing more confusion let us deﬁne the following.

(2.12) Deﬁnition. Let C, D be categories and F : Cop → D. We write F t for the functor C → Dop deﬁned by F and call this the transpose of F . Obviously for F,G: Cop → D,

op DC (F,G) = DopC(Gt,F t) 14 Chapter 2. Kan Extensions and thus transposing itself defines an antiisomorphism from DCop to DopC (i.e. it is an op op isomorphism DC =∼ DopC). In fact, transposing defines a (meta-)1-functor (even an involution) CAT → CAT sending C to Cop and any functor F : C → D to its transpose F t : Cop → Dop. However it does not define a 2-functor CAT → CAT as it is contravariant on Hom-categories (as seen above), i.e. it defines a (meta-)2-functor (even an involution) CATco → CAT.

It is an easy exercise to check that (F ↓ G)op = Gt ↓ F t and with that to deduce the limit formula from the colimit formula. In fact the 2-involution “transpose” allows one to dualise a wider class of statements than “ordinary” dualising because it is a 2-functor and thus we can also dualise statements involving 2-arrows (i.e. natural transformations). Let us adopt the convention that presheaves are of the form Cop → Sets where not stated diﬀerently (in particular any representable presheaf C(−,C) will be viewed as functor of that form). Thus if P,Q are presheaves on C and we write Nat(P,Q) it is always implicitly understood that P and Q take values in Sets and thus the components of a natural transformation between these two as well. The reader might be familiar with the fact that colimits in a locally small category can be deﬁned via limits in Sets using representable presheaves. To wit, for D a locally small category and F : J → D a diagram, a cocone (L, σ) under F is a colimit if and only if ∗ op D(L, D), (σJ )J∈Ob(J) is a limit in Sets (i.e. a colimit in Sets ). So, to check that (L, σ) is a colimit, it is enough to evaluate all representable presheaves at it and see if we get a limit then (we call this the presheaf criterion for colimits). This fact can be generalized to pointwise left Kan extensions and as for colimits we get a presheaf criterion for pointwise Kan extensions. For this, remember that a limit for a diagram F : J → D (with J small and D locally small) is the same as a representation of the functor DJ(∆−,F ): Dop → Sets (where ∆: D → DJ is the diagonal, sending every arrow to the corresponding constant transformation). That is, it’s an object L ∈ Ob(D) and for each D ∈ Ob(D) a bĳection

∼ J ϕD : D(D,L) = D (∆D,F ), which is natural in D. As one would guess, this is a special instance of the Yoneda lemma which gives us a bĳection

D(−,L) ⇒ DJ(∆−,F ) , DJ(∆L, F ) natural in L and F , so that a limiting cone is simply the image of a representation of DJ(∆−,F ) and vice versa.

(2.13) Proposition. Given a diagram E ←−G C −→F D in CAT with D locally small. Then F has a pointwise left Kan extension along G if and only if there is a K : E → D together with an η : F ⇒ K ◦ G such that after composing it with an arbitrary representable presheaf t op t ∗ (viewed as a functor D(−,D) : D → Sets ) the resulting D(K−,D) , (ηC )C∈Ob(C) is a left Kan extension of D(F −,D)t along G.

Proof.“ ⇒”: The representable presheaf D(−,D)t : D → Setsop preserves colimits (because D(−,D) = Dop(D, −): Dop → Sets preserves limits). In particular, it preserves pointwise left Kan extensions. “⇐”: As seen above, a colimit for a diagram H : J → D is nothing but a representation for DJ(H, ∆−): D → Sets, where ∆: D → DJ is the diagonal functor. Thus for our Kan Paragraph 2. Pointwise Kan Extensions 15 extension (K, η) to be pointwise, it is enough to show that for each E ∈ Ob(E), D ∈ Ob(D) there is a bĳection F D(KE,D) =∼ Nat(G ↓ E → C −→ D,D), which is natural in D. To get this, we notice that by the universal property of a left Kan extension (upper row), we have for each K0 : Eop → Sets a bĳection as in the lower row of ∼ Nat(D(K−,D)t,K0t) / Nat(D(F −,D)t,K0tG)

Nat(K0, D(K−,D)) ∼ / Nat(K0G, D(F −,D)), η0 / η0G ◦ (η−)∗ and this bĳection is natural in K0 and D. In particular for K0 := E(−,E) with E ∈ Ob(E), we get bĳections D(KE,D) =∼ NatE(−,E), D(K−,D) =∼ NatE(G−,E), D(F −,D), where the ﬁrst one is just the Yoneda lemma (g 7→ γ with γE0 h := g ◦ Kh). What is left to show is that the natural transformations E(G−,E) ⇒ D(F −,D) are in 1-1-correspondence with cocones (i.e. a natural transformation) from G ↓ E → C → D to D. But this is true; for a natural transformation α: E(G−,E) ⇒ D(F −,D) is completely determined by its components, which is for each C ∈ Ob(C) a map αC : E(GC, E) → D(FC,D). I.e. for each a: GC → E in Ob(G ↓ E) an αC a: FC → D and the naturality of α is the same as the cocone-property of the αC a. Thus we have a bĳection ∼ F Nat E(G−,E), D(F −,D) = Nat(G ↓ E → C −→ D,D), α 7→ (αC a)(C,a)∈Ob(G↓E), which is clearly natural in D and E. To sum this up, we have bĳections F D(KE,D) =∼ NatE(G−,E), D(F −,D) =∼ Nat(G ↓ E → C −→ D,D) all natural in D and E. As a corollary (to the proof really) we get a criterion for deciding if a (K, η) is a pointwise left Kan extension even if D is not cocomplete. (2.14) Corollary. Given a diagram E ←−G C −→F D in CAT with D locally small and K : E → D, η : F ⇒ K ◦ G. Then (K, η) is a pointwise left Kan extension of F along G if and only if for all D ∈ Ob(D), E ∈ Ob(D) the map D(KE,D) → Nat E(G−,E), D(F −,D) , g 7→ γ with γC a := g ◦ Ka ◦ ηC is a bĳection. As another corollary, we get the above mentioned presheaf criterion for colimits. We will state its dual here for we will need it in that form later in the text. Recall that a colimit of a diagram F : J → C is nothing but a left Kan extension along the unique functor !: J → 1 = {0}. Obviously, any such Kan extension is automatically pointwise because the standard projection ! ↓ 0 → J is an isomorphism of categories, so that the colimit formula reads as colim(! ↓ 0 → J → C) = colim(F ) and this colimit exists because we assumed that F has a left Kan extension along !. (2.15) Corollary. (Presheaf Criterion) Let C is a locally small category, F : J → C a small diagram (i.e. J is small) and (L, σ) a cone over F . Then (L, σ) is a limit of F if and only if for every C ∈ Ob(C) the image cone C(C,L), (σJ ∗)J∈Ob(J) of (L, σ) under C(C, −) is a limit of C(C, −) ◦ F : J → C → Sets. In short terms, the family of all representable functors reﬂects limits. 16 Chapter 2. Kan Extensions

3. Tensor-Hom Adjunction

The important special case we mentioned at the beginning of the last section is the case where in theorem (2.2) op F = y: C → Cb = SetsC is the Yoneda embedding. The corollary (2.9) then gives us that a left Kan extension is really a weak extension (i.e. the Kan extension’s unit is an isomorphism). We will study these particular Kan extensions in this and the next paragraph. While we have not introduced ends or the so-called end formula and won’t do that in the rest of the text (simply because it is just another way of writing up the colimit formula using the less comon notion of a end) we will however at least mention the special case of an end, where it reduces to the easy-to-grasp notion of the so-called elements of a presheaf. (3.1) Deﬁnition. Let C be a category and P : Cop → Sets a presheaf. We write Z P := (∗ ↓ P )op = P t ↓ ∗ (where ∗ is a one-point set) C for the category of elements of P (be aware that the same term is also used for R C P := ∗ ↓ P ). Its objects are pairs (C, s) with C ∈ Ob(C) and s ∈ PC and the arrows (C0, s0) → (C, s) are arrows a: C0 → C in C satisfying (P a)s = s0. If C is small, this construction can obviously R R be extended to a functor C : Cb → Cat As in (2.1) the category C P comes with a natural R 0 0 projection πP : C P → C which sends and object (C, s) to C and an arrow (C , s ) → (C, s) to its underlying arrow C0 → C in C. Obviously, this projection is natural in P , i.e. it is a R natural transformation C ⇒ C, where C denotes the constant functor Cb → Cat,P 7→ C. R (3.2) Remark. An alternative description of C P , which we will use is as the comma category y ↓ P . An object of y ↓ P is nothing but a natural transformation α: y C ⇒ P , where C ∈ Ob(C) and y C = C(−,C) and the Yoneda lemma gives us the bĳection ∼ Cb (y C,P ) = Nat C(−,C),P = P C, α 7→ αC 1C , which is natural in C and P . Thus, by identifying these two sets, an object of y ↓ P is nothing but a C ∈ Ob(C) together with an element of PC and (by naturality of the above bĳection in C) an arrow of y ↓ P is what one would expect, indeed yielding an isomorphism y ∼ R ↓ P = C P . But because the Yoneda isomorphism is also natural in P , this is even an y ∼ R isomorphism of functors ↓ − = C and in the following we freely identify these two functors y R with one another. Of course, the standard projection ↓ − → C is precisely C → C under this identiﬁcation. (3.3) Theorem. (Tensor-Hom Adjunction) Given categories C, E with C small and E cocomplete (i.e. we have chosen colimits). Then any functor F : C → E induces an adjunction

−⊗ F C / Cb o _ E , E(F,−) where − ⊗C F := K is the left Kan extension of F along y deﬁned by the colimit formula. Explicitly, this means that for every P ∈ Ob(Cb ) and every E ∈ Ob(E) we have a bĳection Z ∼ πP F E(P ⊗C F,E) = Nat(P, E(F −,E)), where P ⊗C F = colim P −−→ C −→ E C and this bĳection is natural in P and E. Moreover, F factors through − ⊗C F , meaning that ∼ we have an isomorphism of functors η : F = (− ⊗C F ) ◦ y as in the last corollary. Paragraph 3. Tensor-Hom Adjunction 17

Proof. Let P ∈ Cb , E ∈ Ob(E) and θ ∈ Nat(P, E(F −,E)), so that for every f : C → C0 in C, we get a commutative diagram

θ C Nat(y C,P ) =∼ PC C / E(FC,E) O O O f (y f)∗ P f (F f)∗ C0 Nat(y C0,P ) =∼ PC0 / E(FC0,E) . θC0 R So for s ∈ PC ⊆ | C P | we get θC s: FC → E in E and as one sees from the commutative R diagram, all these θC s together form a cocone under C P → C → E thus inducing a unique [ [ arrow θ : P ⊗C F → E satisfying θC s = θ ◦ σ(C,s), where σ is the colimiting cocone from the colimit formula. This deﬁnes a map

Nat(P, E(F −,E)) → E(P ⊗ F,E)

[ [ [ [ as proposed. This map is obviously injective for if θ = ι then θC s = θ ◦σ(C,s) = ι ◦σ(C,s) = ιC s for all s ∈ PC implying that θC = ιC for all C ∈ Ob(C) and thus θ = ι. The map is also surjective for if g : P ⊗ F → E is any arrow then for C ∈ Ob(C), we deﬁne θ : P ⇒ E(F −,E) by θ : PC → E(FC,E), s 7→ g ◦ σ . θ thus deﬁned is obviously natural and θ [ = g. C (C,a)

(3.4) Remark. Let us quickly calculate the unit and counit of this adjunction. The unit

at P ∈ Cb is the unique natural transformation

] ηP = 1P ⊗F : P ⇒ E(F −,P ⊗ F )

[ [ [ such that ηP = 1P ⊗F . But ηP is deﬁned by (ηP )C s = ηP ◦ σ(C,s) = σ(C,s) for all C ∈ |C|, s ∈ PC and this deﬁnes the whole of ηP . Conversely, the counit at E ∈ |E| is

[ εE = 1E(F −,E) : E(F −,E) ⊗ F → E, which is again deﬁned by εE ◦ σ(C,s) = (1E(F −,E))C s = s for all C ∈ |C|, s ∈ E(FC,E).

(3.5) Corollary. Every presheaf (of sets) on a small category C is a colimit of repre- sentables. To be more precise, if P : Cop → Sets is a presheaf on a small category C then

Z π y P =∼ colim P −−→P C −→ Cb C

Proof. Consider the left Kan extension − ⊗ y: Cb → Cb of the Yoneda embedding y: C → Cb along itself. By the theorem, this has a right adjoint Cb (y, −): Cb → Cb ,P 7→ Cb (y −,P ) and Cb (y −,P ) =∼ P by the Yoneda lemma (this is indeed an isomorphism of functors since the Yoneda isomorphism Cb (y C,P ) =∼ PC is natural in C) but by naturality of the Yoneda isomorphism in P we do even have an isomorphism of functors Cb (y, −) =∼ 1 and by uniqueness Cb of adjoints we conclude − ⊗ y =∼ 1 , which gives the desired isomorphism as P ⊗ y is just Cb the colimit from the proposition.

The Tensor-Hom adjunction is not only useful because it gives us a pair of adjoint functors (which is already very good) but because it also gives us an equivalence of certain subcategories, as does any adjunction. Because we will need this property time and again, let us state it precisely. 18 Chapter 2. Kan Extensions

(3.6) Theorem. Let F : C → D : G be an adjunction (left adjoint on the left) with unit 0 0 η : 1C ⇒ GF and counit ε: FG ⇒ 1D and let C 6 C, D 6 D be the full subcategories of all objects where η resp. ε are isomorphisms. Then the adjunction restricts to an equivalence 0 0 0 0 ∼ ∼ F : C ' D : G , whose isomorphisms 1C0 = GF , FG = 1D0 are given by η and ε respectively. Proof. We only need to check that the functors F 0, G0 are well-deﬁned (i.e. if C ∈ C0 then FC ∈ D0 and similarly for D ∈ D0). This follows from the triangular identity. Indeed, if we insert C ∈ C0 into one of the triangular identities, we get

1 F η ε FC −−→FC FC = FC −−−→C F GF C −−→FC FC but F ηC is an isomorphism (since ηC is) and thus εFC is its inverse and in particular an ismorphism itself. Similarly for D ∈ D0 using the other triangular identity. Chapter 3

SHEAVES ON A SPACE

I had hoped that [the] hints contained [. . .] would suﬃce for a skilled mathematician to develop the course of the argument. IGOR ROSTISLAVOVICH SHAFAREVICH

1. Presheaves and the Gluing Axiom

Let X be a topological space. The open subsets of X are partially ordered by inclusion and thus form a poset (even a Heyting algebra) and we write L(X) for this poset (viewed as a category) and call it the associated locale to X. In fact, L deﬁnes a functor Top → Catop (or even Top → Loc, where Loc is the category of locales), sending a continuous map f : X → Y to f −1 : L(Y ) → L(X). (1.1) Deﬁnition. A presheaf on a topological space X with values in a category A is a functor P : L(X)op → A and a morphism of presheaves is simply a natural transformation between two such functors. Thus all presheaves on X with values in A form a category, L(X)op namely PShA(X) := A . If U ⊆ X is an open set. then the elements of PU are called (cross-)sections (or U-(cross-)sections to be precise) and if i: V ⊆ U is an arrow in L(X) we write −|V := P i: PU → P V, f 7→ f|V and call this the restriction to V . A presheaf with values in Sets will be simply called a presheaf and we write PSh(X) for the category of presheaves (of sets) on X. (1.2) Example. • If U ⊆ X is open we have the representable presheaf associated to U. This sends and open subset V ⊆ X to a one-point set ∗ if V ⊆ U and to the empty set ∅ otherwise. The restriction maps are obvious.

• The (pre-)sheaf C(−, R) associates to each U ⊆ X the set of C(U, R) of continuous maps to the real number line. The restriction maps are plainly given by the restriction of functions. This is even a (pre-)sheaf of R-algebras. Similarly, if our topological space X is replaced by a smooth resp. complex-analytic manifold or ∞ the like, one can form the (pre-)sheaves C (−, R), H(−, C) etc. of smooth resp. holomorphic functions. • If A has a terminal object 1 (e.g. a one-point set in Sets) then we get for each x ∈ X, A ∈ Ob(A) the so-called skyscraper sheaf over x with stalk A, denoted by op Skyx(A): L(X) → A, which is deﬁned by ( A x ∈ U U 7→ . 1 x∈ / U 20 Chapter 3. Sheaves on a Space

The restriction maps are again obvious (they are either the identity arrow A → A or the unique map to the terminal object).

(1.3) Remark. (Subpresheaves) Recall that a subpresheaf (i.e. a subobject) of P is an equivalence class of monomorphisms S P . Because a morphism of presheaves S → P is a monomorphism iﬀ all the components SU → PU are monomorphisms (i.e. injections), every such monomorphism S P is equivalent to a unique one, where SU ⊆ PU is actually a subset and the components SU PU are inclusions (and so the restriction maps of S are induced by the restriction maps of P ). It follows that the category PSh(X) is well- powered. By abuse of language, we will usually use the term “subpresheaf” interchangeably for a subobject (i.e. a whole equivalence class) or a representative of such a subobject (i.e. a monomorphism S P ) or – even more restrictive – for the unique representative S P , where all the components SU PU are inclusions of subsets. Because the category PSh(X) is complete (since Sets is), every (small) family of T subobjects Pi P has an inﬁmum (i.e. an intersection) S = i∈I Pi, which is nothing but the pullback of all Pi P . In the case where the components of Pi P are inclusions of subsets (which we can always assume), this is particularly easy to describe. It is nothing but T SU := i∈I PiU for U ⊆ X open. It follows that for every family (MU)U∈|L(X)| of subsets MU ⊆ PU there is a smallest subpresheaf S P containing M (i.e. MU ⊆ SU for all U ∈ |L(X)|); it is simply the intersection of all subpresheaves containing M. We call this the subpresheaf generated by M. Explicitly, it is given by

[ 0 SU = MU |U ⊆ PU (U⊆U 0)∈L(X) as one easily checks. Consequentially, every (small) family of subobjects Pi P has a supremum (i.e. a union), too. It is nothing but the subpresheaf generated by the family S i∈I PiU U∈|L(X)|.

The motivation to abstractly deﬁne the notion of a sheaf comes exactly from the second example above. The outstanding property of these (pre-)sheaves is that we can “glue together” a compatible family of functions. To be precise, if U is an open subset of X and S (Ui)i∈I an open cover of U (i.e. U = i∈I Ui) and (fi : Ui → R)i∈I is a family of continuous maps that are compatible (i.e. they agree on intersections, fi|Ui∩Uj = fj|Ui∩Uj ∀i, j ∈ I) then there is a unique continuous map f : U → R such that f|Ui = fi ∀i ∈ I (this condition is sometimes expressed as “matching/compatible functions are uniquely collatable”). This is exactly the deﬁning property and gives one a glimpse of where sheaves might come in handy. They can be used to describe and analyse situations where one has to draw global conclusions from local properties. That is, they can be used almost everywhere.

(1.4) Deﬁnition. Let A be a category with products. A sheaf on a topological space X with values in A is a presheaf F : L(X)op → A (F from the French “faisceau”) on X with values in A that satisﬁes the so-called gluing axiom. This says that every open cover (Ui)i∈I of an open set U ⊆ X yields an equalizer

p e Y / Y FU / FUi / F (Ui ∩ Uj) , q i∈I i,j∈I

th where e := (F (Ui ⊆ U))i∈I , p := (F (Ui ∩ Uj ⊆ Ui) ◦ pri)i,j∈I (for pri the projection to the i component), and q := (F (Ui ∩ Uj ⊆ Uj) ◦ prj)i,j∈I . That is, e, p, and q are the unique arrows Paragraph 2. Alternative Gluing Axioms 21 making the following diagram commute for all i, j ∈ I

F (Ui∩Uj ⊆Ui) FUi / F (Ui ∩ Uj) t9 O O F (U ⊆U) tt i tt pr tt pri i,j tt tt p tt e Y / Y FU / FUi / F (Ui ∩ Uj) J q JJ i∈I i,j∈I JJ JJ pr pr JJ j i,j F (Uj ⊆U) JJ JJ% FUj / F (Ui ∩ Uj) . F (Ui∩Uj ⊆Uj )

If F is a sheaf of sets (or more generally with values in a concrete category) the maps e, p, and q are thus given by

ef = (f|Ui )i∈I , p(fi)i∈I = (fi|Ui∩Uj )i,j∈I , q(fi)i∈I = (fj|Ui∩Uj )i,j∈I and it is an easy exercise to verify that the above diagram being an equalizer is equivalent to the “classical” gluing condition described before the deﬁnition. A morphism between two sheaves is nothing but a natural transformation (i.e. a morphism of presheaves) and thus the sheaves on X (with values in A) form a full subcategory of the category of presheaves on X (with values in A). Because the gluing axiom is obviously invariant under replacing F by an isomorphic presheaf this subcategory is replete (where a subcategory D 6 C is called replete iﬀ whenever D ∈ Ob(D) and a: C → D is an isomorphism in C then C ∈ Ob(D) and a ∈ D). We write ShA(X) or simply A(X) for the full subcategory of PShA(X) with objects all A-valued sheaves. And as for presheaves, a sheaf of sets on X will simply be called a sheaf and we write Sh(X) := Sets(X). Note that since we require A to have products it has in particular a terminal object 1 (being the empty product) and since the empty subset ∅ ⊆ X has an open cover (Ui)i∈I ∼ with I = ∅ it follows from the gluing axiom that F ∅ = 1 is a terminal object.

2. Alternative Gluing Axioms

We will state an alternative description of the gluing axiom which suggests itself do define so-called B-sheaves (although there are more sophisticated methods to do that using Grothendieck topologies). Roughly, a B-sheaf is a sheaf that is only defined on a basis of our space. The problem here is that we cannot state the gluing axiom anymore as a basis is generally not closed under finite intersections. The following alternative gluing axiom (due to Grothendieck) is a typical instance where limits in a category are replaced by limits in Sets using representable functors. For that, remember that representable functors preserve limits and in particular products. Thus if A is a category, (Ai)i∈I a family of objects of A Q ∼ Q having a product and A ∈ Ob(A), we get an isomorphism A(A, i∈I Ai) = i∈I A(A, Ai) Q th given by f 7→ (fi)i∈I , where fi = pri ◦f for pri : j∈I Aj → Ai the i standard projection. (2.1) Proposition. Let A be a locally small category with products and X a topological space. A presheaf P : L(X)op → A is a sheaf if and only if it satisfies the following alternative gluing axiom: For any open cover (Ui)i∈I of an open set U ⊆ X and any A ∈ Ob(A) the map (of small sets) Y ε: A(A, P U) → A(A, P Ui), f 7→ (f|Ui )i∈I i∈I 22 Chapter 3. Sheaves on a Space is injective with image

{(fi)i∈I | fi|V = fj|V ∀i, j ∈ I,V ⊆ Ui ∩ Uj open} .

Proof. This is a special instance of the presheaf criterion (cf. (2.15)), which implies that

p e Y / Y PU / PUi / P (Ui ∩ Uj) , q i∈I i,j∈I is an equalizer diagram (with e, p, q as in the deﬁnition of a sheaf) iﬀ for every A ∈ Ob(A)

p∗ e∗ Q / Q A(A, P U) / A A, i∈I PUi / A A, i,j∈I P (Ui ∩ Uj) q∗ =∼ =∼ p Q ∗ / Q , i∈I A(A, P Ui) / i,j∈I A(A, P (Ui ∩ Uj)) q∗ is an equalizer diagram in Sets, where the vertical isomorphisms are as described above. But this is exactly that e∗ is injective with image

{(fi)i∈I | fi|Ui∩Uj = fj|Ui∩Uj ∀i, j ∈ I} and thus, by noticing that e∗ = ε, all that is left to check is that for a family (fi)i∈I ∈ Q i∈I A(A, P Ui) we have

fi|V = fj|V ∀i, j ∈ I,V ⊆ Ui ∩ Uj open iﬀ fi|Ui∩Uj = fj|Ui∩Uj ∀i, j ∈ I.

But the direction “⇒” is obvious (putting V = Ui ∩ Uj) and for the other one notice that P is a presheaf and thus f | = (f | )| = (f | )| = f | for V ⊆ U ∩ U open. i V i Ui∩Uj V j Ui∩Uj V j V i j By this proposition, we can redefine a sheaf to be a presheaf satisfying this alternative gluing axiom. This has the formal advantage that it makes sense to talk about sheaves with values in a category, that does not necessarily have products and moreover defining a B-sheaf (which is our original motivation) is straightforward. However, this alternative gluing axiom has the logical disadvantage that we have to quantify over a usually large set, namely the objects of A (but then again, to be precise the set of all open covers of an open subset U is also large) and (because of that) it’s less conceptual. Because all categories that we “meet in nature” as codomains of sheaves (like Sets or Ab) are complete the reader may choose her favourite gluing axiom. Still, we will speak of sheaves taking values in a general abelian category which might not be complete and it should be clear from the context which gluing axiom is appropriate. We can again restate the gluing axiom from the last proposition in terms of so-called covering sieves. If we write y: L(X) → PSh(X) for the Yoneda embedding then a sieve S on U ⊆ X open is simply a subfunctor of y U (observe that y U = L(X)(−,U) simply indicates whether an open set V ⊆ X is contained in U or not, i.e. (y U)V = 1 = {∗} if V ⊆ U and 0 = ∅ otherwise). Thus for every V ⊆ X open we have an inclusion SV ⊆ (y U)V (so that again SV ∈ {0, 1}) which is natural in V . This means that a sieve S on U can alternatively be described as a lower set in L(U) meaning that S is a set of open subsets of U such that whenever we have open subsets W ⊆ V ⊆ U with V ∈ S then also W ∈ S and we can regain a functor as the characteristic function of S ⊆ L(U). For the rest of the text, we will freely switch back and forth between these two descriptions and identify the sieve S, viewed as a lower set, with its characteristic function (i.e. S viewed as a functor). We say that a sieve S on U is a covering sieve iff U is the union of the elements of S. Paragraph 2. Alternative Gluing Axioms 23

(2.2) Proposition. Let A be a locally small category and X a topological space. Then a presheaf P : L(X)op → A is a sheaf (in the sense of the last proposition) if and only if for every covering sieve S on an open set U ⊆ X and any A ∈ Ob(A) inclusion ι: S → y U induces a bĳection

ι∗ : Naty U, A(A, P −) =∼ NatS, A(A, P −).

Proof.“ ⇒”: Let us write (Ui)i∈I for the family of elements of S (so one can choose I = S and Ui = i if one wishes to), which is an open cover of U because S is a covering sieve. Observe that a natural transformation α: S ⇒ A(A, P −) is given by its components αV : SV → A(A, P V ) but since SV = ∅ for all V/∈ S, α is completely determined by the components αUi with i ∈ I and since SUi = {∗}, αUi is nothing but an element of A(A, P Ui). This means that we Q have an injection m: Nat S, A(A, P −) → i∈I A(A, P Ui), sending α to (αUi ∗)i∈I and we get a composite map

=∼ ι∗ m Y A(A, P U) −→ Nat y U, A(A, P −) −→ Nat S, A(A, P −) −→ A(A, P Ui), i∈I where the ﬁrst map is the Yoneda isomorphism, given by f 7→ ϕ with ϕV a = (P a)f. A quick calculation shows, that this is the map ε from the last proposition, which is injective. Thus ι∗ is injecitve, too. It is also surjective because the image of ε is

{(αi)i∈I | αi|V = αj|V ∀i, j ∈ I,V ⊆ Ui ∩ Uj open} but since S is a lower set, an open V ⊆ Ui ∩ Uj needs to be in S too, so that (αi)i∈I lies in the image of ε iff the αi define a natural transformation α: S ⇒ A(A, P −) with αUi ∗ = αi. That means that (αi)i∈I lies in the image of ε iff it lies in the image of m and by injectivity of m, ι∗ must be surjective. “⇐”: An open cover (Ui)i∈I of an open set U ⊆ X generates a covering sieve S by V ∈ S ⇔: V ⊆ Ui for some i ∈ I. With the same argument as before an α: S ⇒ A(A, P −) is given by the components αV with V ∈ S but by naturality of α it is already completely determined by its components αUi for if V ⊆ Ui then αV ∗ = αUi ∗ |V so that we again get an injection m and a composite map as above. This time, we already know that ι∗ is a bĳection, so that the composite map ε is indeed injective. (2.3) Corollary. If X is a topological space and P : L(X)op → Sets a presheaf of sets on X the P is a sheaf if and only if for every covering sieve S on an open set U ⊆ X, the inclusion ι: S → y U induces a bĳection

ι∗ : Nat(y U, P ) =∼ Nat(S, P ).

Proof. The ι∗ here is simply the ι∗ from the last proposition with A = {∗}. Thus the condition stated is clearly necessary. It is also suﬃcient, for if ι∗ is a bĳection then (by the proof of the last proposition with A = {∗}) Y ε: PU → PUi, f 7→ (f|Ui )i∈I i∈I is injective with image

{(fi)i∈I | fi|V = fj|V ∀V ⊆ Ui ∩ Uj open} and in particular (putting V = Ui ∩ Uj) the diagram in the deﬁnition of a sheaf is an equalizer. 24 Chapter 3. Sheaves on a Space

3. Topological Constructions of Sheaves

One of the first question that arises after one has understood a definition of a new math- ematical object (especially if that definition is sufficiently complicated) is how to construct new objects from given ones. Obviously, if we have the notion of a morphism between two such objects and all of them form a category, one will look for categorical constructions (like limits and colimits), which we will cover in the next sections but for the time being, we will describe three non-categorical (i.e. topological) constructions to get a sheaf. The first construction is taking subsheaves. As all presheaves form a category we can speak of subpresheaves. A subpresheaf of a presheaf P is just a monomorphism ι: Q P (to be precise, it is an equivalence class of such monomorphisms but this abuse of language is popular practice and usually does no harm). Observe that a natural transformation ι: Q ⇒ P is monic iff all components are monic. Thus for each open subset U ⊆ X, we have a monic iU : QU PU and these respect restriction (i.e. are natural in U). The question now is the following: if F is a sheaf on X and P F a subpresheaf, when is P a sheaf? Fortunatelly, because of the local character of a sheaf, this is a local question and the condition for P to be a sheaf is the one we would expect.

(3.1) Theorem. (Subsheaves) Let F ∈ Sh(X) and ι: P F a subpresheaf. Then P is a sheaf if and only if for every U ⊆ X open, f ∈ FU and every open cover (Ui)i∈I of U we have f ∈ PU ⇔ f|Ui ∈ PUi ∀i ∈ I. Proof. Without loss of generality, we assume PU ⊆ FU for all U ⊆ X open. By the premise, if U ⊆ X is open and (Ui)i∈I an open cover of U, we have a diagram as below, where e, p, and q are as in the deﬁnition of a sheaf and the lefthand square as well as the upper and the lower square on the right obviously commute. p e Y / Y PU / PUi / P (Ui ∩ Uj) q i∈I i,j∈I

p e Y / Y FU / FUi / F (Ui ∩ Uj) q i∈I i,j∈I “⇒”: If P is a sheaf then it does obviously satisfy the stated condition for the upper row is an equalizer and is just a restriction of the lower row. “⇐”: Conversely, if P staisfies the stated condition then the lefthand square is a pullback and a simple diagram chase proves that the upper row is an equalizer. As mentioned in the definition of a sheaf. The condition there says nothing else than that we can glue together a matching family in a unique way. In the same spirit, we can even glue together a matching family of sheaves (up to isomorphism). For this to make sense, one should notice that if F is a sheaf on a space X and U ⊆ X is open, then we can restrict F to U to yield a sheaf F |U on U with the subspace topology, defined in the obvious way. In fact if U ⊆ X is an open subset, we get restriction functors

−|U : PShA(X) → PShA(U) and − |U : ShA(X) → ShA(U), deﬁned in the obvious way. These data do turn PShA and ShA into (meta-)presheaves op op PShA : L(X) → CAT and ShA : L(X) → CAT and the following theorem exactly says that this ShA is almost a (meta-)sheaf because it is so only up to isomorphism (in fact, this eventually leads to the notion of a stack or 2-sheaf ). Paragraph 3. Topological Constructions of Sheaves 25

(3.2) Theorem. (Gluing of Sheaves) Given an open cover (Xk)k∈K of a space X and ∼ for each k ∈ K a sheaf Fk on Xk as well as a family of isomorphisms θk,l : Fk|Xk∩Xl = Fl|Xk∩Xl (k, l ∈ K) satsfying θk,m = θl,m ◦ θk,l on Xk ∩ Xl ∩ Xm for all k, l, m ∈ K. Then there exists a ∼ sheaf F on X and isomorphisms ϕk : F |Xk = Fk (k ∈ K) such that ϕl = θk,l ◦ ϕk on Xk ∩ Xl. Moreover F and the ϕk are unique up to isomorphism.

Proof. Let us ﬁx the notation that if U ⊆ X is open, we write Uk := U ∩ Xk. Moreover, for k, l ∈ K we write Fk,l := Fk|Xk∩Xl so that we have a restriction map resk,l : Fk → Fk,l and θk,l ∼ becomes an isomorphism θk,l : Fk,l = Fl,k. If there is a sheaf F as required it must certainly satisfy the gluing condition and thus for U ⊆ X open there must be an equalizer diagram

p Y / Y FU / FkUk / Fk,l(Uk ∩ Ul) q k∈K k,l∈K must be an equalizer diagram, where p := (θk,k(Uk ∩ Ul) ◦ Fk(Uk ∩ Ul ⊆ Uk) ◦ prk)k,l∈K and q := (θl,k(Uk ∩ Ul) ◦ Fl(Uk ∩ Ul ⊆ Ul) ◦ prl)k,l∈K . Since we have equalizers in Sets we can take this as the deﬁnition of FU. If i: V ⊆ U is an arrow in L(X) then the equalizer property gives us a unique arrow FU → FV , which we deﬁne to be F i.

<++>

(3.3) Corollary. Let X be a topological space and P : L(X)op → A a presheaf. Then P is a sheaf iﬀ there is an open cover (Ui)i∈I of X such that each P |Ui is a sheaf. Equivalently (using the axiom of choice) P is a sheaf iﬀ for every x ∈ X there is an open neighborhood U 3 x such that P |U is a sheaf.

Recall that a basis B for the topology of a topological space X is a full subcategory of L(X) (whose objects are called basic open sets) such that every U ⊆ X open can be written as a union of objects of B. This is to say that we get every open subset of X by gluing together basic open sets. Because sheaves behave well under gluing it is reasonable to conjecture that a sheaf is uniquely determined by its values on such a basis B and in fact this is true. To formalise this, we say that a presheaf P : Bop → A is a B-sheaf or a basic sheaf on X for the basis B iff it satisfies the alternative gluing axiom from (2.1) with all open sets replaced by basic open sets. If B is closed under finite intersections (as is the case for the canonical basis of an affine scheme) this is equivalent to the original gluing axiom from (1.4) with all open sets replaced by basic open sets, as we proved in (2.1). A morphism of B-sheaves is again just a morphism of presheaves (i.e. a natural transformation) and we write ShA(B) for the category of all A-valued B-sheaves on X for the basis B.

(3.4) Theorem. (B-Sheaves) Let X be a topological space, A a complete category and B a basis for X. Then the restriction functor

r: ShA(X) → ShA(B)

(with (r F )(V ⊆ U) = F (V ⊆ U)) is an equivalence of categories.

(3.5) Example. (Aﬃne Schemes) For any commutative ring A, we have an associated aﬃne scheme (Spec A, Ae), where X := Spec A is the prime spectrum of A equipped with the Zariski topology (i.e. the closed sets in Spec A are all of the form

VX (a) := V (a) := {p ∈ Spec A | a ⊆ p} 26 Chapter 3. Sheaves on a Space for a A any ideal) and Ae is a sheaf of rings on X (i.e. a scheme is a special instance of a so-calledP ringed space) as deﬁned below. One can show that the so-called special (or basic or distinct) open sets

DX (a) := D(a) := Spec A \ V (Aa) = {p ∈ Spec A | a∈ / p} for a ∈ A form a basis (closed under ﬁnite intersections because D(a) ∩ D(b) = D(ab)) for ∼ the Zariski topology and that D(a) = Spec Aa as topological spaces, where D(a) carries the subspace topology and Aa = A[1/a] is the localisation of A at a (i.e. we formally invert the 2 ∼ multiplicative system Sa := {1, a, a ,... }). This homeomorphism D(a) = Spec Aa is given −1 by p 7→ Aap = Sa p. The only reasonable choice for a structure-ring for the whole of Spec A is AXe := A itself and consequentially, the only reasonable choice of a structure ring for D(a) is ADe (a) := Aa. The restriction maps A → Aa (or more generally Ab → Aa) are of course the universal maps into the localisation. To wit, D(a) ⊆ D(b) means b ∈ p ⇒ a ∈ p for all p ∈ Spec A. Now we form the diagram of solid arrows

λb A / Ab @@ @@ @@ λa @@ , Aa which are the universal morphisms into the localisations and in order for the dotted arrow to exist, λab ∈ Aa must be invertible. If this were not so, then we would ﬁnd q /Aa prime with −1 b ∈ q but then p := λa q /A would be prime, a∈ / p but b ∈ p, which is a contradiction. As one can show, the presheaf Ae thus deﬁned on basic open sets is in fact a B- sheaf for the basis D(a) a∈A, so that it can be uniquely extended to our structure sheaf Ae, which is a sheaf of commutative rings. More generally, if the commutative ring A is even a commutative algebra over a commutative ring R then Ae is even a sheaf of R-algebras because every Aa is again an R-algebra is a canonical way (the R-algebra structure R → Aa is given by composing the R-algebra structure R → A on A with the canonical morphism λa : A → Aa). Finally, a ring homomorphism ϕ: A → B determines a map ϕ← : Spec B → −1 ← −1 Spec A, p 7→ ϕ p, which is continuous since (ϕ ) DSpec A(a) = DSpec B(ϕa) for a ∈ A, so that taking the spectrum is indeed a functor Spec: CRngop → Top or again more generally Spec: R-Algop → Top.

(3.6) Example. If A = k is a ﬁeld then Spec k = {0} is a one-point space and the structure sheaf ke is obviously the constant sheaf k (that is ke{0} = k and ke∅ = 0).

4. Categorical Constructions of Sheaves

(4.1) Proposition. For A an abelian category and X a topological space PShA(X) (the category of A-valued presheaves on X) is abelian, too.

Proof. Limits and colimits of functors are calculated pointwise and A is abelian.

(4.2) Theorem. Let A be an abelian category and X a topological space. Then A(X) (the category of A-valued sheaves on X) is an abelian subcategory of PShA(X). Paragraph 5. Étale Bundles and Sheaﬁﬁcation 27

5. Étale Bundles and Sheaﬁﬁcation

In this section our investment into the abstract machinery of Kan extensions pays off because the ususal sheafification and étalification functors is nothing but a special instance of the Tensor-Hom adjunction we get by taking Kan extensions along the Yoneda embedding. Recall that any functor F : C → E with C small and E cocomplete induces an adjunction − ⊗C F : Cb E : E(F, −) (left adjoint on the left), where Z πP F P ⊗C F = colim P −−→ C −→ E C for any presheaf P ∈ Ob(Cb ). Let’s consider the case where C = L(X) (for X some topological space), E = Top ↓ X the category of bundles over X and F : L(X) → Top ↓ X sending an open set U to the inclusion U,→ X and U ⊆ V to U,→ V (which is evidently a continuous map of spaces over X). For the sake of readability, let us write Λ := − ⊗L(X) F and Γ := (Top ↓ X)(F, −) for the induced adjoints. op R If P : L(X) → Sets is any presheaf then the category of elements L(X) P has as objects pairs (U, s) with U ⊆ X open and s ∈ PU while an arrow (V, t) → (U, s) exists (which R we abreviate by (V, t) 6 (U, s)) iff V ⊆ U and s|V = t, so that L(X) P is a poset. We can now describe the colimit ΛP , which is determined by the coequalizer diagram p a / a V / U / / ΛP , q (V,t)6(U,s) (U,s) where for (V, t) (U, s) the maps p and q are defined on ι V as the inclusions into 6 (V,t)6(U,s) ι(V,t)V and ι(U,s)U respectively (where the ιs denote standard inclusions into the coproduct). ` In a more set-theoretic manner, if we write x(U,s) for the element x ∈ ι(U,s)U ⊆ (V,t) V , ` we may say that ΛP is the quotient of (U,s) U by the equivalence relation ∼, which is the smallest equivalence relation such that x(V,t) ∼ x(U,s)U for all (V, t) 6 (U, s) and x ∈ V . Plainly, this gives the equivalence relation y(V,t) ∼ x(U,s) iff x = y and there is some W ⊆ U ∩V open such that x ∈ W and t|W = s|W . If for U ⊆ X open, x ∈ U and s ∈ PU, we write germx s for the equivalence class of x(U,s) and for a point x ∈ X

Px := {germx s | U ⊆ X open, x ∈ U and s ∈ PU} , ∼ ` we immediately see that ΛP = x∈X Px (as a set). As this colimit of the above diagram is as good as any other, we take this as the definition of ΛP , together the standard projection ΛP → X (which is really induced by the universal property of the colimit), turning ΛP into an object of Top ↓ X. Let us recall that the topology on the colimit ΛP is nothing but the final topology on it with respect to the colimiting cocone, having as components the cross- R sections s˙ : U → ΛP, x 7→ germx s (for (U, s) ∈ | L(X) P |) of the bundle ΛP → X. Finally, if α: P → Q is a morphism of presheaves on X, the map Λα:ΛP → ΛQ (given by the universal property of the colimit) is nothing but ΛP → ΛQ, germx s 7→ germx αU s. Let’s sum this all up again with the following definition. (5.1) Definition. Let X be a topological space and P ∈ |PSh(X)|. For x ∈ X an equivalence relation ∼ is defined on S {PU | U ⊆ X open and x ∈ U} by s ∼ t iff there is some W ⊆ U ∩ V open with x ∈ W and s|W = t|W . The equivalence class of s ∈ PU is then denoted by germx s and called the germ of s at x. We write Px for the resulting quotient and call this the stalk of P at x. The disjoint unionbecause limits of functors are calculated pointwise, they also carry the same topology. F a ΛP := Px, x∈X 28 Chapter 3. Sheaves on a Space

equipped with the final topology with respect to all s˙ : U → ΛP, x 7→ germx s (U ⊆ X open, s ∈ PU) is called the stalk-space or the associated étale bundle of P (strictly speaking, the associated bundle is ΛP equipped with the standard projection ΛP → X, germx s 7→ x but we usually won’t be pedantic about it). Because all the cross-sections s˙ of our bundle ΛP are injective, they are open maps and so the sU˙ = {germx s | x ∈ U} with U ⊆ X open and s ∈ PU form a subbasis for the topology of ΛP . Λ thus defined yields a functor PSh(X) → Top ↓ X by sending α: P → Q in PSh(X) to Λα:ΛP → ΛQ, germx s 7→ germx αU s. (5.2) Lemma. Let X be a topological space, F ∈ |Sh(X)|, U, V ⊆ X open and s ∈ FU, t ∈ FV . Then s|U∩V = t|U∩V iff s˙|U∩V = t˙|U∩V .

Proof. The direction “⇒” is obvious. For the other direction, observe that for each x ∈ U ∩V , ˙ we have germx s =sx ˙ = tx = germx t, so that there is some open neighborhood Wx ⊆ U ∩ V of x, satisfying s|Wx = t|Wx . But now (Wx)x∈X is an open cover of U ∩ V and the s|Wx form a matching family for this cover. Because F is a sheaf, if follows that we can collate them uniquely and thus s| = t| . U∩V U∩V (5.3) Remark. Observe that we do not need the full strength of the gluing axiom in the above proof. It suﬃces for F to be a so-called separated presheaf, meaning that the map e in the gluing axiom is a monomorphism but not necessarily an equalizer. This means that if

U ⊆ X is open, (Ui)i∈I an open cover of U and s, t ∈ PU such that s|Ui = t|Ui for all i ∈ I, then s = t. I.e. a presheaf is separated iﬀ any matching family which can be glued together can be done so uniquely.

While the “germ maps” germx : PU → Px are now deﬁned, they were done so quite nastily, ﬁrst forming the union S {PU | U ⊆ X open and x ∈ U} and then imposing an equivalence relation on the resulting set, making germx the standard projection into the quotient. There is in fact a much neater way to do this, not depending on set-theoretic notions such as the union of some sets. The intuition behind it remains the same; the germ of a section at a point x should capture its behaviour near that point. One way to do this is the one described above; another one is to observe that the best approximation of the point x in L(X) would be the limit (i.e. the intersection) of all open neighborhoods of x. This limit need not exist in L(X) but after applying the given presheaf P , we land in a bicomplete category and our imaginary limit can be calculated there as a colimit. (5.4) Proposition. Let X be a topological space and P ∈ |PSh(X)|. For x ∈ X, let L(X, x) be the full subcategory of L(X) having as objects all U ⊆ X open such that x ∈ U. We then have ∼ Px = colim P |L(X,x)op = colimx∈U PU with germx : L(X, x) → Px as the colimiting cocone, which has components germx : U → Px, s 7→ germx s. Moreover, germx is natural in the sense that for α: P → Q any morphism in PSh(X) there is a unique map αx : Px → Qx such that the following square is commutative

germx PU / Px

αU αx QU / Qx germx for all U ⊆ X open with x ∈ U. So taking the germ at x is a functor −x : PSh(X) → Sets, (α: P → Q) 7→ (αx : Px → Qx) and for U ⊆ X open with x ∈ U the maps germx as in Paragraph 5. Étale Bundles and Sheaﬁﬁcation 29

the above square determine a natural transformation evU ⇒ −x, where evU is the evaluation functor (α: P → Q) 7→ (αU : PU → QU).

Proof. Let M be any set, together with a cocone θU : PU → M (U ⊆ X open with x ∈ U). If there is any map t: Px → M such that θ = germx ◦t, it must be given by germx s 7→ θU s for s ∈ PU (for the maps germx form a surjective family, i.e. every element of Px lies in the image of some germx). This is a well defined map, for if we have open neighborhoods U, V of x and s ∈ PU, t ∈ PV with germx s = germx t then there is some W ⊆ U ∩ V open with x ∈ W and s|W = t|W , so that θU s = θW s|W = θW t|W = θV t. This proves (P, germx) the stated colimit. As for the induced map αx, observe that the germx ◦αU : PU → Qx form a cocone under P | and α is thus given by the universal property of the colimit. L(X,x) x The second functor Γ: Top ↓ X → PSh(X) is much easier to describe (because it is not defined by a universal property). It sends a bundle E → X (which, we will usually identify with E and whose fibre over x ∈ X will be denoted by Ex) to (Top ↓ X)(−,E), so that for U ⊆ X open n o s (ΓE)U = s: U → E U −→ E → X = U,→ X = {s: U → E | sx ∈ Ex ∀x ∈ U} is the set of all U-cross-sections of E and if V ⊆ U are open sets then the corresponding restriction map is simply the restriction of maps, i.e.

−|V := (ΓE)(V ⊆ U): s 7→ s|V . (5.5) Lemma. Λ: PSh(X) → Top ↓ X preserves ﬁnite limits.

Proof. Let D : J → PSh(X) be a finite diagram, together with a limit (P, λ) and let (E, µ) be a limit of Λ◦D. Observe that for any x ∈ X we get a bifunctor J×L(X, x)op → Sets, (J, U) 7→ (DJ)U. But as J is finite and L(X, x)op is filtered (that is to say that for U, V ∈ |L(X, x)| he also have U ∩ V ∈ |L(X, x)|) we have a canonical isomorphism ∼ ∼ ∼ ∼ ϕx : Px = colimU PU = colimU limJ (DJ)U = limJ colimU (DJ)U = limJ (DJ)x = Ex, which is simply germx s 7→ e, where e ∈ E is the unique element such that µJ e = germx λJ s ∈ (DJ)x for all J ∈ |J|. By the universal property of the disjoint union these maps induce a bĳection ϕ:ΛP → E, which is even a homeomorphism. The inverse E → ΛP is continuous by the universal property of the limit E. For ϕ itself to be continuous, recall that ΛP carries the final topology with respect to all cross-sections s˙ : U → ΛP with U ⊆ X open ∼ and s ∈ PU = limJ (DJ)U and moreover, E carries the initial topology with respect to all µJ : E → ΛDJ. Thus all we need to check is that all composites µJ ◦ ϕ ◦ s˙ are continuous. But this is trivial, for if we write t := λJ s ∈ (DJ)U then µJ ◦ ϕ ◦ s˙ = t˙: U → ΛDJ, which is continuos by definition of ΛDJ. (5.6) Remark. The converse statement, that Γ preserves finite colimits is not true in general and Γ does usually not even preserve finite coproducts, let alone arbitrary coproducts. However, it is possible to classify those spaces X for which this holds. In fact, the following are equivalent

(a) Γ preserves coproducts;

(b) Γ preserves ﬁnite coproducts;

Recall that X is irreducible (also called hyperconnected) iﬀ any open subset of X is connected.

Proof. If X is irreducible and (Ei)i∈I is a family of bundles over X then their coproduct is the ` disjoint union E := i∈I Ei carrying the final topology with respect to all inclusion Ei ,→ E. If now, U ⊆ X is open then U must be connected and in particular, any cross-section s: U → ` E must map all of U into one of the summands Ei. Consequentially (ΓE)U = i∈I (ΓEi)U as claimed. Conversely, if Γ preserves finite coproducts, let U ⊆ X open and consider the bundles E = E0 = U (with the inclusion map U,→ X) over X. The only cross-section U → E (resp. U → E0) is obviously the identity map and so Γ(E q E0)U =∼ (ΓE)U q (ΓE0)U is a two-point set, meaning that the only cross-sections U → E q E0 are the inclusions into 0 either E or E . This implies that U is connected for if U1, U2 ⊆ U are disjoint open subsets 0 with U = U1 ∪ U2 then the two inclusions U1 ,→ E and U2 ,→ E define a cross-section s: U → E q E0. Because sU = E or sU = E0 we conclude that U = or U = . 1 ∅ 2 ∅ Before putting all the above together and stating it as a theorem, let us analyze what properties are enjoyed by the images of objects under either of the functors Λ and Γ. By observing that the image ΓE of a space E over X is a presheaf of continuous maps into E, with the additional (but local) requirement that they should form a cross-section of E → X it is obvious that ΓE is always a sheaf. (5.7) Lemma. The functor Γ: Top ↓ X → PSh(X) has its values in Sh(X). Just as the local nature of being a cross-section of a bundle E is encaptured by the fact that the presheaf ΓE is in fact a sheaf, there is a corresponding notion on the topological side that accounts for the local nature of the germ of a section. This notion is that of a local homeomorphism or étale bundle. (5.8) Definition. A continuous map p: E → X is called étale or an étale bundle or étalé space over X iff it is a local homeomorphism in the sense that for every e ∈ E there is some open neighborhood U 3 e such that pU ⊆ X is open and p restricts to a homeomorphism U =∼ pU. We write Etale´ (X) for the full subcategory of Top ↓ X having as objects all étale bundles over X. (5.9) Example. Every covering space is an étale bundle.

(5.10) Example. If X = {∗} is a one-point space and E any topological space, then the unique bundle E → X is étale iﬀ E is discrete.

(5.11) Remark. It follows immediately from the definition that étale bundles have discrete fibres, that every étale map is open and that any morphism of étale bundles (i.e. any continuous map of bundles over X, which happen to be étale) is again an étale map. More- 0 0 over, if E, E are two étale bundles over X then their fibred product E ×X E is again an étale bundle over X. Finally, if E is an étale bundle over a space X, then a basis for the topology of E is given by the images of all cross-sections s: U → E with U ⊆ X open (and consequentially any such cross-section is an open map), so that E carries the final topology with respect to all such cross-sections.

Proof. Only the last claim needs some justification. Let V ⊆ E open. By definition of an étale bundle, we may cover V by open sets (Vi)i∈I such that all the subsets Ui := pVi ⊆ X −1 Vi ∼ −1 are open and p induces a homeomorphism si := p|U : Vi = Ui. Because the si are all i S homeomorphisms, their inverses si : Ui → Vi are cross-sections of E and V = i∈I siVi, which proves our claim. Paragraph 5. Étale Bundles and Sheafification 31

(5.12) Lemma. Any two cross-sections of an étale bundle E over a space X, which agree on a point x ∈ X, agree locally around x. To wit, if U, U 0 ⊆ X are open neighborhoods of x ∈ X and s: U → E, t: U 0 → E cross-sections of E satisfying sx = tx then there is some 0 open neighborhood W ⊆ U ∩ U of x such that s|W = t|W .

Proof. We write e := sx = tx and choose V 3 x open, such that the standard projection p: E → X induces a homeomorphism p0 : V =∼ pV . By the above remark, pV ⊆ X is open 0 and so W := U ∩ U ∩ pV is an open neighborhood of x. But s|W and t|W are both inverses of the homeomorphism p0−1W → W we get by restricting p0, so that they must agree. (5.13) Lemma. The functor Λ: PSh(X) → Top ↓ X has its values in Etale´ (X).

Proof. Let P ∈ |PSh(X)| and germx s ∈ ΛP for some x ∈ X, U ⊆ X an open neighborhood of x and s ∈ PU. The map s˙ : U → ΛP, y 7→ germy s is open, so that V :=sU ˙ is an open neighborhood of germx s. Moreover, because s˙ is a section of the standard projection p:ΛP → X, pV = U ⊆ X is open and p restricts to a homeomorphism V =∼ pV = U having s˙ as its inverse. (5.14) Main Theorem. For any topological space X, we have adjoint functors as above

Λ / PSh(X)o _ Top ↓ X , Γ which both preserve ﬁnite limits and factor through the inclusions of full subcategories as

PSh(X) −→Λ Etale´ (X) ,→ Top ↓ X and Top ↓ X −→Γ Sh(X) ,→ PSh(X), yielding and adjoint equivalence Λ: Sh(X) ' Etale´ (X):Γ.

(5.15) Remark. Before proceeding to the proof, let us quickly calculate the unit and counit of the adjunction. As this arises as a Tensor-Hom adjunction, we may do so by using (3.4) and the fact that for P ∈ |PSh(X)| the colimiting cocone to ΛP has components R s˙ : U → ΛP, x 7→ germx s for (U, s) ∈ | L(X) P |. By this, the unit ηP : P → ΓΛP at a presheaf P is given by (ηP )U s =s ˙ for U ⊆ X open and s ∈ PU, while the counit εE : ΛΓE → E is determined by εE ◦ s˙ = s for all U ⊆ X open, s ∈ (ΓE)U = {s: U → E | sx ∈ Ex ∀x ∈ U} (this does determine εE by the universal property of the colimit ΛΓE).

Proof. The only thing we have not shown yet is the very last claim. By (3.6) it suffices to check that the unit ηP : P → ΓΛP at a presheaf P on X is an isomorphism iff P is a sheaf, while the counit εE : ΛΓE → E at a bundle E over X is an isomorphism iff E is étale. The “only if” parts have already been stated as lemmas (5.7) and (5.13), so that we only need to check the “if” parts. For this, let first P be a sheaf and we need to check that each

(ηP )U : PU → (ΓΛP )U = {t: U → ΛP | tx ∈ (ΛP )x = Px ∀x ∈ X} , s 7→ s˙ (U ⊆ X open) is bĳective. This is so, for if U ⊆ X is open and t: U → ΛP is a cross-section of ΛP , then tU ⊆ ΛP is open and we can choose for each e ∈ tU an open neighborhood Ve ⊆ tU, ∼ such that the standard projection p:ΛP → X induces a homeomorphism Ve = pVe ⊆ U, −1 having the restriction t: t Ve → Ve as its inverse. By deﬁnition of the topology of ΛP , we may shrink each Ve, so that is still an open neighborhood of e but is now given as the image −1 of some s˙e : Ue → ΛP with Ue ⊆ t Ve open and se ∈ PUe. But then s˙e is nothing but the 0 restriction of t, so that s˙ | =s ˙ 0 | = t| for all e, e ∈ pU. By (5.2), this e Ue∩Ue0 e Ue∩Ue0 Ue∩Ue0 32 Chapter 3. Sheaves on a Space

implies that the se form a matching family for the open cover (Ue)e∈pU of U and we may thus uniquely collate them to s ∈ PU. So for t ∈ (ΓΛP )U, we have found a unique s ∈ PU such that s˙ = t, meaning that (ηP )U is bĳective. If now, p: E → X is étale, we need to check that εE : ΛΓE → E, determined by εE ◦ s˙ = s for all U ⊆ X open and s ∈ (ΓE)U, is a homeomorphism. It is surjective, for if e ∈ E, we choose V 3 e open such that the standard projection p: E → X induces a homeomorphism V =∼ pV =: U and we write s: U → V,→ E for the inverse. This is a cross- section of E (i.e. s ∈ (ΓE)U) and so εE ◦s˙ = s and in particular εE ◦spe˙ = spe = e. It is also 0 0 0 0 injective, for if εEf = εEf for some f, f ∈ ΛΓE, we have f = germx s and f = germx0 s for 0 0 0 0 0 some x, x ∈ X, U 3 x, U 3 x open and s ∈ (ΓE)U, s ∈ (ΓE)U . Because εE is a map of 0 0 0 spaces over x, it follows that x = x and by definition of εE, we have sx = εEf = εEf = s x. So s and s0 are two cross-sections of the étale bundle E, which agree at x and thence, by 0 0 0 (5.12), s|W = s |W for some W 3 x open, meaning that f = germx s = germx s = f . Obviously, εE is continuous, because all the εE ◦ s˙ = s are continous and ΓΛE carries the −1 final topology with respect to all the s˙ and finally the inverse εE is continous, too, because −1 if s: U → E is a cross-section of E with U ⊆ X open, then εE ◦ s =s ˙ is continuous and E carries the final topology with respect to all cross-sections s: U → E.

(5.16) Remark. Observe that for the unit ηP at P to be injective, it is enough for the presheaf P to be separated.

(5.17) Corollary. (Sheafification) For any topological space X, Sh(X) is a reflective full subcategory of PSh(X), i.e. we have an adjunction

a / PSh(X)o _ Sh(X), i where i: Sh(X) ,→ PSh(X) is the inclusion and a is the so-called sheafification (or associated sheaf ) functor. The counit ε: a i ⇒ 1Sh(X) is an isomorphism and moreover, a preserves and reflects finite limits and consequentially, a preserves and reflects monomorphisms, too. (5.18) Remark. Notice that in general, if C is a full reflective subcategory of a category D (i.e. C is a full subcategory of D and the inclusion i: C → D has a left adjoint F : D → C) then by (2.10) of the last chapter the counit ε: F i ⇒ 1C is always an isomorphism. ∼ (5.19) Corollary. If X is a topological space, x ∈ X and P ∈ |PSh(X)| then Px = (a P )x, i.e. a presheaf and its sheafification have isomorphic stalks. If P is a presheaf, it is sometimes convenient to work with the associated étale bundle ΛP instead of the sheafification a P because ΛP already contains all information about a P but is usually a less complicated object to work with. To illustrate this claim, let us analyse the situation for constant presheaves, thereby introducing constant sheaves.

(5.20) Definition. (Constant Sheaves) Let X be a topological space and M any set. op The constant presheaf PM on X is simply the presheaf PM : L(X) → X, which has PM U := U for all U ⊆ X open and whose restriction maps are all identities. If there is no risk of confusion, we will usually identify M with PM . The constant sheaf FM on X is now the sheafification FM := a M of the constant presheaf M. Let us quickly analyze constant presheaves and their sheafification. For M a set (identified with he corresponding constant presheaf on X), let U, V be open neighborhoods Paragraph 6. Sheaves with Algebraic Structure 33

0 0 of a point x ∈ X and m ∈ MU = M, m ∈ MV = M. Then obviously germx m = germx m 0 iff m = m because all restriction maps are identities and it follows that Mx = M, thus ` ∼ ΛM = x∈X M = X × M as a sets over X (where the projection X × M → X is the obvious ` ∼ one). The topology on ΛM has as a subbasis all sets mU˙ = {germx m | x ∈ U} = x∈U {m} = X × {m} with U ⊆ X open (which is even a basis in this case) and in particular all fibres are descrete. Thus for the constant presheaf M we have a particularly easy description of ΛM; it is simply the product space X × M, where M carries the discrete topology. By definition a M = ΓΛM is the sheaf of all cross-section of ΛM and we arrive at the following, more elementary description: The constant sheaf a M associated to the set M is given by

(a M)U =∼ {f : U → M | f is continuous} with the usual restriction of maps, where M carries the discrete topology. Observe that if U is connected, then f : U → M must be constant. More generally, if we write c(U) for the set of connected components of U, we get an injective map

c(U) {f : U → M | f is continuous} → M , f 7→ (fC)C∈c(U) and if the connected components of X are open (e.g. if X is locally connected or if X has only ﬁnitely many connected components) then it is even surjective. Thus in that case we can identify

(a M)U =∼ M c(U) but the restriction maps get a little more complicated. To wit, if V ⊆ U is open and D ∈ c(V ) then D is a connected subset of U and thus there is some (necessarily unique) c(U) c(V ) connected component CD ∈ c(U) such that D ⊆ CD and the restriction map M → M is given by (mC )C∈c(U) 7→ (mCD )D∈c(V ).

6. Sheaves with Algebraic Structure

In most cases, the sheaves occurring in geometric contexts are really sheaves carrying an algebraic structure (e.g. the structure sheaf of a manifold, which is a sheaf of R-algebras). As an example, we consider the case of sheaves of R-modules, for R any fixed commutative ring. But the reader should carry in mind that all what follows is mutatis mutandis true for any algebraic structure (in the sense of universal algebra), e.g. sheaves of rings or algebras. A presheaf of R-modules on a topological space is usually defined as a presheaf P : L(X)op → R-Mod with values in the category of R-modules. Obviously, because R-Mod is a subcategory of Sets and we can always compose P with the standard inclusion functor i: R-Mod ,→ Sets, we could as well say that a presheaf of R-modules is a presheaf P : L(X)op → Sets of sets such that each FU carries the structure of an R-module and all restriction maps are R-linear. An important observation at this point is that the sheaf condition (i.e. the gluing axiom) is not ambiguous in this context. I.e. P : L(X)op → R-Mod satisfies the gluing axiom (in the category R-Mod that is) iff the composite i ◦ P : L(X)op → R-Mod ,→ Sets satisfies the gluing axiom (in the category Sets). The reason for this is simply that i creates and preserves limits (which comes from the fact that i has a left adjoint, namely the free construction). Thus it does no harm to define a sheaf of R-modules to be a sheaf F : L(X)op → Sets of sets such that each FU carries the structure of an R-module and all restriction maps are R-linear. Observe now that an R-module is nothing but a set M, equipped with maps M × M → M, 1 → M and R × M → M (usually called “addition”, “zero” and “scalar 34 Chapter 3. Sheaves on a Space multiplication”), such that the obvious diagrams commute. Because we can identify R with the constant presheaf PR and because products of functors are calculated pointwise, we may equally well define a presheaf of R-modules to be an R-module-object (read “PR-module- object”) in PSh(X). As noted in the last section, the functors Λ and Γ preserve products, so that for P an R-module-object in PSh(X) the corresponding étale bundle ΛP is a ΛR- module-object but ΛR =∼ X × R (where R carries the discrete topology)

7. More on Étale Bundles

As always, one of the questions one is concerned with sonner or later if one tries to work in a newly constructed category is the question of subobjects, i.e. describing the monomorphisms. This is easy in categories of algebraic structures because there a morphism is monic iff it is 0 injective. The same is true in Top and Top ↓ X because if ϕ: E → E is monic and e1, e2 ∈ E with ϕe1 = ϕe2 then we precompose ϕ with the two maps {∗} → E mapping ∗ to e1 and e2 respectively and conclude that e1 = e2. This trick no longer works in Etale´ (X) because {∗} → X is only étale iff the image of ∗ is an open point in X. This seems disappointing at first but as it turns out, monomorphisms in Etale´ (X) are even nicer than in Top ↓ X. Because observe that if ϕ: E → E0 is a monomorphism (i.e. an injective map) in Top or Top ↓ X, so that E is a subspace of E0 in the categorical sense, then this does not imply that E is a subspace of E0 in the topological sense, i.e. E =∼ ϕE, so that E can be identified with the subset ϕE of E0, carrying the subspace topology induced by that of E0. The reason for this is that the topology of E can be finer than that of ϕE. However, in the étale case, this cannot happen and even better, E is not only a subspace (in the topological sense) but even an open subspace. (7.1) Proposition. If ϕ: E → E0 is a morphism of étale bundles over X, then ϕ is monic iff it is an open embedding. Proof. Because ϕ is a morphism of étale bundles, it is itself étale and in particular open. If ϕ is an open embedding, we identify E with the open subset ϕE ⊆ E0, carrying the subspace topology. If e ∈ E ⊆ E0 then there is some open neighborhood U 0 ⊆ E0 of e such that pU 0 ⊆ X is open and U 0 =∼ pU 0 via p: E0 → X. Consequentially, U := U 0 ∩ E is an open neighborhood of e, pU is open in pU 0 and thus also in X and p restricts to U =∼ pU, which is a homeomorphism because the restriction U 0 =∼ pU 0 is so. Conversely, if ϕ is monic then it is 0 injective, for if we have e1, e2 ∈ E with ϕe1 = ϕe2 =: e , then we find open neighborhoods U1, 0 U2 ⊆ E of e1 and e2 respectively, such that ϕU1 = ϕU2 =: V ⊆ E is open and ϕ restricts to ∼ ∼ 0 homeomorphisms U1 = V and U2 = V mapping e1 resp. e2 to e . As shown above, V together ∼ with the restricted projection p|V : V → X is étale and the two maps f1 : V = U1 ,→ E and ∼ f2 : V = U2 ,→ E are morphisms of étale bundles, satisfying ϕ ◦ f1 = ϕ ◦ f2 and thus f1 = f2. 0 0 In particular, e1 = f1e = f2e = e2, so that ϕ is indeed injective, as well as continuous and open. Consequentially, ϕ is an open embedding as claimed. (7.2) Remark. (Subsheaves) Using the sheafification functor a, we can discuss subsheaves, analogous to (1.3) where we discussed subpresheaves. To wit, because a as well as i preserve (and reflect) monomorphisms, we have for any sheaf F a Galois connection

a / SubPSh(X)(F )o _ SubSh(X)(F ), i ∼ meaning S 6 i a S and a iE 6 E (here even a iE = E) or equivalently a S 6 E ⇔ E 6 iS for all subpresheaves S F and all subsheaves E F . Paragraph 8. Stalks and Skyscrapers 35

Describing the intersection of subsheaves is easy and we don’t need sheafification there. If F is a sheaf on X and (Fi F )i∈I a (small) family of subsheaves (without loss of generality the components FiU FU for U ⊆ X open are inclusions of subsets) then their T intersection E := i∈I Fi as subsheaves is the same as their intersection as subpresheaves. T That is EU = i∈I FiU for U ⊆ X open with restriction maps induced by those of F . Using (3.1) and the fact that the Fi are sheaves, one readily checks that E thus defined is indeed a sheaf. Consequentially, if (MU)U∈|L(X)| is a family of subsets MU ⊆ FU there is a smallest subsheaf containing M, called the subsheaf generated by M. However, its description is not as simple as in the case of subpresheaves, for if S is the subpresheaf generated by M (i.e. S 0 SU = (U⊆U 0)∈L(X) MU |U ) then a S is the subsheaf generated by M. Indeed, if E F is any subsheaf containing M then E = iE is a subpresheaf containing M and so S 6 iE, ∼ implying a S 6 a iE = E. As usually for sheafification, the subsheaf a S generated by M is best described in terms of the corresponding étale bundle ΛS. The topology on ΛS ⊆ ΛF is obviously just the subspace topology (because Λ preserves monomorphisms) and a short inspection of the definition of the stalk and of S yields that

(a S)x = Sx = Mx := {germx s | U ⊆ X open, x ∈ U and s ∈ MU} . Now, as in the case of subpresheaves, it follows that we have (small) unions of subsheaves, too and they are obviously nothing but the sheafification of the unions as subpresheaves (because a is left adjoint and thus preserves unions). Finally, we observe that by the above proposition (7.1), subsheaves of a sheaf F correspond to open subsets of ΛF under the equivalence Λ: Sh(X) ' Etale´ (X) and we can equally well define a subsheaf by specifying an open subset of the stalk space ΛF . This is in fact very convenient and will come in handy when describing ideal sheaves or more generally submodules of OX -modules (for X a locally ringed space) in the next chapter.

8. Stalks and Skyscrapers

9. Image Functors

One of the major features of étale bundles is that they are stable under base change. Thereby, if E → X is a bundle over X and Y → X a continuous map (called the base change map), ∗ then the projection ϕ E := E ×X Y → Y in the pullback diagram ϕ∗E / E

Y / X is called the pullback bundle of E along Y → X and we say that we obtained ϕ∗E → Y by base change along Y → X. Observe that while the roles of E and Y can of course be interchanged, leading to the same result, the speciﬁc terms “bundle over X” and “base change map” (both meaning nothing but “continuous map into X”) reveal the underlying line of thought, in which the roles of E and Y are not interchangeable. (9.1) Theorem. (Base change) Étality of bundles is stable under base change. I.e. if p: E → X is étale and ϕ: Y → X is continuous then the pullback bundle q : ϕ∗E → Y along ϕ is étale, too. Thus any continuous map ϕ: Y → X induces a so-called base change functor ∗ ∗ 0 ϕ : Etale´ (X) → Etale´ (Y ), mapping E to ϕ E = E ×X Y and ψ : E → E to the unique map ϕ∗E → ϕ∗E0, (e, y) 7→ (ψe, y) deﬁned by the universal property of the pullback. 36 Chapter 3. Sheaves on a Space

Proof. Let (e, y) ∈ ϕ∗E. Because p is étale there is some open neighborhood U ⊆ E of e, which is mapped homeomorphically by p to the open set pU ⊆ X. Now, U × ϕ−1pU is open in E ×Y and thus the intersection V := (U ×ϕ−1pU)∩ϕ∗E is an open neighborhood of (e, y) ∗ −1 U −1 in ϕ E. If y ∈ ϕ pU with x := ϕy ∈ pU, there is exactly one point u = p|pU x ∈ U, which is mapped to x by p and the map

U −1 U −1 p|pU ◦ ϕ, 1ϕ−1pU : ϕ pU → V, y 7→ u = p|pU ϕy, y

V −1 thus deﬁned is the inverse to the restriction q|qV : V → qV = ϕ pU, so that this is indeed a homeomorphism and ϕ∗E is étale. Chapter 4

RINGED SPACES

1. Deﬁnition and First Properties

For the rest of the text let us fix the convention that all rings considered are commutative and have a unit and in the same vein all algebras over a ring are required to be associative, commutative and unital. Observe, that many of the definitions still make sense in the non- commutative case but as we are going to study the connection of algebraic and analytic geometry over C all rings and algebras of our interest will be commutative. (1.1) Definition. For R a ring, a ringed space over R (or R-ringed space for short) is a topological space X together with a sheaf (called the structure sheaf of the ringed space) O : L(X)op → R-Alg of R-algebras on X. Equivalently, if we identify the ring R with the corresponding constant sheaf a R on X then we could require O to be an a R-algebra object in the category Sh(X). As an immediate consequence all stalks Ox are again R-algebras and we say that (X, O) is a locally ringed space iff all the stalks Ox are local rings, whose maximal ideal we will denote by mx(OX ) or simply mx if there is only one structure sheaf O involved. If there is no risk of confusion, we usually identify a ringed space with its underlying topological space X and the structure sheaf whose existence is implicitly assumed will then be called OX . Consequentially the stalk at x ∈ X will then by denoted by OX,x and in the case of a locally ringed space mX,x/OX,x is the maximal ideal of OX,x. If (X, OX ) and (Y, OY ) are ringed spaces # then a morphism of ringed spaces (ϕ, ϕ ):(X, OX ) → (Y, OY ) consists of a continuous map # −1 ϕ: X → Y together with a morphism of R-Alg-valued sheaves ϕ : OY → ϕ∗OX = OX ◦ϕ . Notice that for every x ∈ X and every V ⊆ Y open with ϕx ∈ V (and thus x ∈ ϕ−1V ), we have an arrow ϕ# V −1 germx OY V −−→ OX ϕ V −−−−→ OX,x # and by naturality of ϕ and germx these together constitute a cocone under the restricted op functor OY : L(Y, ϕx) → R-Alg and thus by the universal property of the colimit induce a unique morphism # # # ϕx : OY,ϕx → OX,x such that germx ◦ϕV = ϕx ◦ germϕx ∀V 3 ϕx open. # By this it makes sense to define that for (X, OX ) and (Y, OY ) locally ringed spaces and (ϕ, ϕ ) a morphism of ringed spaces we say that (ϕ, ϕ#) is a morphism of locally ringed spaces iff for # #−1 every x ∈ X the morphism ϕx is local in the sense that ϕx mX,x = mY,ϕx (or equivalently # # ϕx mY,ϕx ⊆ mX,x). As for ringed spaces, we usually identify (ϕ, ϕ ) with ϕ. We write SpcR for the category of R-ringed spaces R and LSpcR for the category of locally ringed spaces over R. 38 Chapter 4. Ringed Spaces

(1.2) Remark. In most texts, a ringed space is defined as topological space X together with a sheaf of rings on X but as Z is initial in the category of rings, every ring is a Z-algebra in a unique way and the category of Z-algebras is the category of rings. Thus our definition encompasses this case and we write Spc and LSpc as a shorthand for Spc and LSpc . Z Z Moreover, as most geometry (algebraic or analytic) is done over a fixed base field k all ringed spaces occurring in that context are naturally sheaves of k-algebras.

(1.3) Remark. If (X, OX ) is a (locally) ringed space and U ⊆ X open then (U, OX |U ) is again a (locally) ringed space. Because the notation OX |U is rather cumbersome, we will write OU := OX |U where there is no risk of confusion.

(1.4) Example. (Affine Schemes [bis]) In (3.5) of the last chapter we already defined the affine scheme (Spec A, Ae) associated to an R-algebra A (R any ring), which is a ringed space over R and in fact a locally ringed one for if p ∈ Spec A then the stalk Aep is isomorphic to Ap, the localisation of A at p (i.e. we formally invert all elements of the multiplicative system Sp := A \ p), while the colimiting cocone germp : AUe → Ap for U ⊆ Spec A open with p ∈ U is given by the canonical maps into the localisation. This Ap is obviously a local ring, −1 having Sp p (the image of p under the canonical morphism A → Ap) as its unique maximal ideal. Now, one could take the full subcategory of LSpcR containing all (Spec A, Ae) as the category of affine schemes over R but the requirement that an affine scheme is (rather than is like) a (Spec A, Ae) is usually too restrictive and one instead takes the repletion of this full subcategory. So that an affine scheme over R is a locally ringed space (X, OX ) over R, isomorphic to (Spec A, Ae) for some R-algebra A and a morphism between two affine schemes is simply a morphism of locally ringed spaces over R. We write AffSchR < LSpcR for the full subcategory of all affine schemes over R and again AffSch as a shorthand for AffSchZ; In the same spirit an affine scheme (with no base ring mentioned) is always understood to be an affine scheme over Z. In (3.5) of the last chapter we already defined Spec as a functor R-Algop → Top and one can extend this to a functor

ϕ (ϕ←,ϕ) Spec: R-Algop → AﬀSch,A −→ B 7→ (Spec B, Be) −−−−→e (Spec A, Ae), where ϕ is given on the basic sets D (a) of X := Spec A by ϕ := ϕ : A → B , which e X eDX (a) a a ϕa is the unique map induced by the composite of ϕ: A → B with the canonical map B → Bϕa. ← −1 Here we used that ADe X (a) = Aa and that (ϕ ) DX (a) = DSpec B(ϕa). This functor Spec is obviously an equivalence of categories and a quasi-inverse is given by

# # (ϕ,ϕ ) ϕY (X, OX ) −−−−→ (Y, OY ) 7→ OY Y −−→ OX X.

(1.5) Remark. One could think (possible encouraged by (1.2)) that our approach, using R-algebras gives a more general notion of an affine scheme but it really doesn’t. Because an R algebra A is nothing but a ring A, equipped with a morphism of rings R → A and a morphism of R-algebras is nothing but a morphism “under R”, so that R-Alg = R ↓ CRng. And by op op the above example, we know that R-Alg ' AffSchR (and in particular CRng ' AffSch for R = Z) via the spectrum functor, so that we have a series of equivalences of categories op op op AffSchR ' R-Alg = (R ↓ CRng) = CRng ↓ R ' AffSch ↓ Spec R. Thus an affine scheme over R is nothing but an object over Spec R in the category of affine schemes (over Z). As indicated in the corollaries below, such an equivalence of categories holds in a much more general context. Paragraph 1. Definition and First Properties 39

While the spectrum of a ring (as a locally ringed space) is a basic concept in algebraic geometry nowadays, it is really of a much more general nature and naturally arises in a wider context as the next theorem shall illustrate. For this let R be any ring and let us op # write Γ: LSpcR → R-Alg for the global sections functor, which sends (ϕ, ϕ ):(X, OX ) → # #op (Y, OY ) to the R-algebra morphism ϕX : OY Y → OX X (that is really to ϕX ). Γ / op (1.6) Theorem. There is an adjunction LSpcR o _ R-Alg and Spec is fully faithful. Spec

Proof. Let us check the universal property of the counit. If A is an R-algebra and Y := Spec A, the counit εA at an R-algebra A must be a ring morphism A → AYe = A and the natural candidate is εA := 1A. If now (X, OX ) is a locally ringed space and ϕ: A → OX X a morphism # of rings, we need to ﬁnd a unique morphism of locally ringed spaces (ϕ,b ϕb ):(X, OX ) → (Y, Ae) making the following diagram commute.

(X, OX ) OX X B BB ϕop (ϕ,ϕ#) ϕ#op BB b b bY BB BB! (Y, Ae) A / A 1A

If there is such a morphism then for x ∈ X and p := ϕxb we get a commutative square

ϕ# = ϕ bY A / OX X

germp germx A / , p # OX,x ϕbx where germp is the canonical morphism into the localisation. Thus −1 −1 #−1 #−1 −1 −1 ϕxb = p = germp mp = germp ϕbx mx = ϕbSpec A germx mx = (germx ◦ϕ) mx, # where in the third equality, we used that ϕbx must be local. We take this as the deﬁnition of ϕb and this is indeed a continuous map for if a ∈ A and D(a) = Spec A \ V (Aa) is a basic open set then −1 ϕb D(a) = {x ∈ X | ϕxb ∈ D(a)} −1 = x ∈ X a∈ / (germx ◦ϕ) mx = {x ∈ X | germx ϕa∈ / mx} = {x ∈ X | ∃U 3 x open, ψ ∈ OX U : ϕa|U · ψ = 1} −1 so that if x ∈ ϕb D(a) =: Ua there is an open neighborhood U 3 x contained in Ua. Observe that since OX is a sheaf, Ua is the set of all x ∈ X, where ϕa is locally invertible at x and inverses are unique, we can collate the local inverses of ϕa to derive that ϕa|Ua is invertible in # OX Ua (and Ua is the biggest such open set). Now, we can describe ϕb : Ae → ϕb∗OX . If there is such a transformation then (with the same notation as before) we get another commutative square, similar to the last one

ϕ# = ϕ bY A / OX X

, Aa / OX Ua ϕ# bD(a) 40 Chapter 4. Ringed Spaces where the vertical arrows are the restriction maps of the corresponding structure sheafs. The upper leg A → OX X → OX Ua, b 7→ ϕb|Ua sends a to an invertible element and thus, by the # universal property of the localisation, ϕbD(a) must be the unique morphism

# b ϕb ϕ : Aa → OX Ua, 7→ bD(a) an ϕan

# which renders this diagram commutative and we again take this as the definition of ϕbD(a). With (ϕ, ϕ#) thus defined, the first commutative square gives us that this does indeed induce local maps between the stalks, for if x ∈ X and p := ϕxb , we have

# # −1 ϕ m = (ϕ ◦ germ )p = (germ ◦ϕ)p = (germ ◦ϕ)(germ ◦ϕ) mx ⊆ mx. bx ϕxb bx p x x x The uniqueness of (ϕ, ϕ#) follows by the above construction. b b

(1.7) Corollary. For A an R-algebra and (X, OX ) a locally ringed space over R, the map

∼ # # # LSpcR (X, OX ), (Spec A, Ae) = R-Alg(A, OX X), (ϕ, ϕ ) 7→ Γ(ϕ, ϕ ) = ϕSpec A, induced by the global sections functor is a bĳection, which is natural in (X, ) and A. OX

(1.8) Corollary. (Spec R, Re) is terminal in LSpcR and in particular, (Spec Z, Ze) is terminal in LSpc.

(1.9) Corollary. For R any ring the functor !: LSpcR → LSpc ↓ Spec R, which maps an object X to the unique morphism X → Spec R of locally ringed spaces over R, is an isomorphism of categories. The inverse is determined by the standard projection Q: LSpc ↓ Spec R → LSpc.

Proof. The standard projection Q: LSpc ↓ Spec R → LSpc does indeed deﬁne a functor Q: LSpc ↓ Spec R → LSpcR. For if a: X → Spec R is an object in LSpc ↓ Spec R and U ⊆ X open, the composition

Γa −|U R −→ OX X −−→ OX U deﬁnes the structure of an R-algebra on OX U. By functoriality of Γ and the structure sheaves this R-algebra structure is preserved under morphisms X → Y in LSpc ↓ Spec R. We have natural isomorphisms η : 1 ∼ Q ◦ ! and ε:! ◦ Q ∼ 1 whose components = =

(1.10) Example. (Schemes) Just like manifolds are spaces that locally look like an affine space, a scheme is a space that locally looks like an affine scheme. That is, a scheme over R (R any ring) is a locally ringed space (X, OX ) over R such that there is an open cover (Ui)i∈I of X and for every i ∈ I an R-algebra Ai as well as an isomorphism of locally R-ringed spaces ϕi :(Ui, OU ) ' (Spec Ai, Aei). A scheme (X, OX ) over R is locally finitely generated (or locally of finite type for short) iff all the Ai are finitely generated R-algebras. Although not a local property (but still stated here), we say that a scheme (X, OX ) is of finite type iff it is locally finitely generated and in addition the index set I is finite. Paragraph 2. Points of a Ringed Space 41

2. Points of a Ringed Space

In this section we discuss some connections of a point x ∈ X with its stalk OX,x. Speciﬁcally, we are interested in how good or how bad a point can behave.

(2.1) Deﬁnition. Let X be a ringed space. A point x ∈ X is called

• normal iﬀ OX,x is normal (i.e. OX,x is an integral domain and is integrally closed in its quotient ﬁeld Quot OX,x);

• irreducible iff OX,x is an integral domain, otherwise it is called reducible; √ • reduced iff OX,x is a reduced ring (i.e. 0 = 0, meaning that the only nilpotent element of OX,x is 0). We call X normal (resp. reduced) iff it is normal (resp. reduced) at every point x ∈ X. We call X locally irreducible iff it is irreducible at every point x ∈ X. Note that being irreducible is defined differently and is not a local property.

(2.2) Example. A scheme X is reduced iﬀ all OX U with U ⊆ X open are reduced rings.

(2.3) Deﬁnition. A scheme X is called integral iﬀ all the OX U with U ⊆ X open are integral domains. Although this is not a local property (in the sense that it cannot be checked on stalks) it still has some connection with it as the following proposition shall show.

(2.4) Proposition. A scheme X is integral iﬀ it is irreducible and reduced.

3. Sheaves of Modules on a Ringed Space

(3.1) Deﬁnition. Let (X, O) be a ringed space over a ring R.A sheaf of O-modules (or simply O-module for short) is an O-module-object in the category Sh(X). That is to say it’s a sheaf F : L(X)op → Sets of such that every FU is additionally equipped with the structure of an OU-module. The restriction maps of F are required to be compatible with the module structures in the sense that if i: V ⊆ U are open sets of X the following diagram commutes

OU × FU / FU

Oi×F i F i OV × FV / FV , where the horizontal arrows are the scalar multiplications. To explain this a little more, O gives us a morphism Oi = −|V : OU → OV of R-algebras and via this, every OV -module M is also an OU-module by λ · m := λ|V m for λ ∈ OU and m ∈ M. Moreover, we have a morphism of abelian groups F i = −|V : FU → FV and the above condition on F now means that this restriction map is OU-linear, i.e. λ · m|V = λ|V m|V = (Oi)λ (F i)m = (F i)(λm) = (λm)|V ∀λ ∈ OU, m ∈ FU

(one should be weary that in the second expression, −|V stands for two diﬀerent things, as indicated). A morphism α: F → G between sheaves of O-modules is simply a morphism of OX -module-objects in Sh(X). That is to say, it’s a natural transformation α: F ⇒ G such that for every U ⊆ X open αU : FU → GU is OU-linear. With this deﬁnition of a morphism we get for a ringed space (X, OX ) the category of OX -modules OX -Mod. 42 Chapter 4. Ringed Spaces

(3.2) Remark.

• Of course if F is an O-module, then every FU is not only a set but an abelian group or even better an R-module or better still an OX-module. So, in the above deﬁnition, one can replace the category Sets by Ab or R-Mod or OX-Mod if one wishes to. Consequentially, OX-Mod is a subcategory of the abelian category Ab(X) (resp. R-Mod(X) or OX-Mod(X)).

• The theory of OX -modules encompasses (and in fact shares most of the properties with) the theory of sheaves of modules on a space X. For if X is any topological space and R a (commutative) ring, we may turn X into a ringed space by giving it the constant structure sheaf OX := a R. Then an OX -module is nothing but a sheaf of R-modules over X.

(3.3) Example. If X is a ringed space then OX is obviously a module over itself.

(3.4) Remark. (Submodules) Obviously, if F is an OX -module and (Fi F )i∈I a T (small) family of submodules then their intersection i∈I Fi as subsheaves is again a submodule and thus the poset SubOX -Mod(F ) has arbitrary intersections. Consequentially, if (MU ⊆ FU)U∈|L(X)| is a family of subsets, there is a smallest submodule of F , containing

M. It is called the submodule generated by M and denoted by hMiOX or OX M. It can be described explicitly in the following manner: Let S be the subpresheaf of F generated by

M and define PU := hSUiOX U . This is a submodule of F in the category PSh(X) and it is obviously the smallest one containing M. Alas, it need not be a sheaf and so we need to apply sheafification, yielding finally OX M = a P . This is indeed an OX -module because a preserves products (and this OX -module-objects), it is a submodule of F because a preserves monomorphisms and it is the smallest such because if E F is another one then it also a ∼ submodule in PSh(X) and so P 6 E, yielding a P 6 a E = E. The description of OX M in terms of the associated étale bundle Λ(OX M) is particularly easy. The stalks (i.e. fibres) are given by

(OX M)x = {germx m | U ⊆ X open, x ∈ U and s ∈ hMUiOX U } = OX,xMx Fx P (where Mx is as in (7.2) of the last chapter) and the topology is obviously just the subspace topology of Λ(OX M) ⊆ ΛF . One could alternatively have the idea to interchange the first two steps. I.e. first replace MU by hMUiOX U and then take the presheaf S generated by this family. However, this is in general not a submodule and one would need to define P as above which gives the same presheaf in both cases. Another alternative would be to first constructs the subsheaf E generated by M ∼ (i.e. E = a S for the S from above) and then PU := hEUiOX U , which is again a presheaf but not necessarily a sheaf. So one would have to apply a again, which gives us the same result.

(3.5) Deﬁnition. Consider OX as a module over itself. A submodule I of OX is called an ideal sheaf or an ideal of OX . Explicitly, this means that I is an ideal and I U OX U P for all U ⊆ X open or equivalently, Ix OX,x for all x ∈ X. P

There is an analogue of theorem (1.6), which established an adjunction Γ: LSpcR R-Alg : Spec, for modules. For this, we ﬁrst need to generalise the construction of the structure sheaf Ae of an aﬃne scheme to modules. Paragraph 3. Sheaves of Modules on a Ringed Space 43

(3.6) Definition. For A an R-algebra, X := Spec A and M a module over A, we write op Mf: L(X) → R-Mod for the sheaf, that is defined on basic open sets Xa (for a ∈ A) by −1 2 MXf a := Ma := Sa M where Sa := {1, a, a ,... }. This is indeed a sheaf and in fact even an Ae-module. Moreover the stalks at a point p ∈ Spec A is Mfp = Mp, as one would expect. If we again write Γ: Ae-Mod → A-Mod for the global sections functor, defined by F 7→ FX = F Spec A and f− : A-Mod → Ae-Mod for the functor that sends an A-module M to its assocaited Ae-module Mf, we get the above mentioned analogue. (3.7) Theorem. For A an R-algebra and X := Spec A, we have an adjunction

Γ / Ae-Modo _ A-Mod. f− Moreover, f− is fully faithful, preserves tensor products and (possibly inﬁnite) direct sums. (3.8) Theorem. Let Γϕ: A → B be a morphism of R-algebras and ϕ: Spec B → Spec A the induced morphism of schemes over R. If M is an A-module and N a B-module, then ∼ ∗ ∼ ϕ∗Ne = AgN and ϕ Mf = M^⊗A B, where AN is N viewed as an A-module (via Γϕ). Thus with the last theorem, every morphism Γϕ: A → B of R-algebras gives us a square

−⊗ B A / A-ModO o _ B-ModO A− f− Γ Γ f− ϕ o ∗ Ae-Mod _ / Be-Mod ϕ∗ and the above exactly says that the square of all right adjoints as well as the outer square both commute up to isomorphism.

(3.9) Theorem. For (X, OX ) a ringed space, the category OX -Mod is an abelian subcategory of R-Mod(X) (and thus also of PShR-Mod(X)), which is closed under taking limits and colimits. (3.10) Deﬁnition. Just as for any commutative ring R and R-modules M,N the Hom- set HomR(M,N) is naturally an R-module (via pointwise addition and scalar multiplication). An analogous result is true for OX -modules. If X is a ringed space and F , G are OX -modules then HomOX (F,G), deﬁned by

HomOX (F,G)U := HomOX U (F U, GU) with the obvious restriction maps is an OX -module, too. It is called the sheaf of local morphisms from F to G or sheaf Hom for short. Just as for R-modules, where for M ﬁxed there is a left adjoint − ⊗R M to HomR(M, −) the same is true for OX -modules. To wit, if F and

G are OX -modules, the componentswise tensor product P defined by PU := FU ⊗OX U GU with the obvious restriction maps is a presheaf, called the presheaf tensor product of F and G. It’s sheafification

F ⊗OX G := a P = a(U 7→ FU ⊗OX U GU) is called the sheaf tensor product or simply tensor product of F and G. 44 Chapter 4. Ringed Spaces

4. Zero Sets

The motivation for considering (locally) ringed spaces came from classical algebraic and differential geometry, where the considered (locally) ringed spaces are varieties and manifolds respectively. The structure sheaves in these cases was given by the sheaf of rational resp. smooth functions on the space considered. Thus the structure sheaves were always algebras of functions into a field and in particular, every point was reduced and we can apply elements f ∈ OX U to points x ∈ U. The maximal ideal mx / OX,x then corresponds to functions which are 0 at x. (to wit mx is the set of all germx f with fx = 0). In the abstract setting this is no longer possible, at least not directly. But using a little trick we can still treat elements of OX U like functions on U. The idea behind it is that ` any s ∈ OX U can be identified with the cross section s˙ : U → ΛOX = x∈X OX,x, which is defined by sx˙ = germx s. As germs in mx should correspond to germs of functions which are 0 at x, we need to mod out mx to finally arrive at the following definition.

(4.1) Definition. Let X be a locally R-ringed space. For x ∈ X the stalk OX,x is a local ring with maximal ideal mx and so OX,x/mx is a field, which is called the residue field at x and is denoted by κx (resp. κX,x). If now U ⊆ X is open, s ∈ OX U and x ∈ U we write

[s]x := [germx s] = germx s + mx ∈ OX,x/mx = κx.

Thus, ﬁxing x ∈ U, the map OX U → κx, s 7→ [s]x is nothing but the composition

germx OX U → κx = OX U −−−−→ OX,x OX,x/mx = κx, where the last arrow is the standard projection. As these are both homomorphisms of R- algebras, the whole map is so. If all the residue fields κx for x ∈ X (or at least x ∈ U) are embedded into a common field k over R (e.g. for schemes of finite type over C or complex analytic spaces or smooth manifolds) then obviously, this determines a function [s]: U → k and we get an R-algebra homomorphism OX U → Map(U, k), where Map(U, k) = Sets(U, k) = kU is the set of all k-valued functions (which is an R-algebra by pointwise addition and multiplication). Observe that in the above construction, we could replace OX by an arbitrary sheaf of local R-algebras.

Obviously, using this construction, we can generally not capture the whole structure of OX in this way, e.g. we cannot treat the issue of non-reducedness because Map(U, k) is always reduced. To be more precise, we have the following result.

(4.2) Remark. Let X be a locally ringed space, U ⊆ X open, x ∈ U and s ∈ OX U with germx s ∈ OX,x nilpotent. Then there is some open neightborhood V ⊆ U of x such that [s]y = 0 for all y ∈ V .

n Proof. There is some n ∈ N such that (germx s) = 0 ∈ OX,x and thus we ﬁnd some V ⊆ U n n n n open with x ∈ V and s |V = (s|V ) = 0 but then [s ]y = ([s]y) = 0 ∈ κy for all y ∈ V so that indeed [s]y = 0 ∈ κ for all y ∈ V . y

(4.3) Deﬁnition. Let X be a locally ringed space, U ⊆ X open and S ⊆ OX U. The zero locus of S is the set

NU (S) := N(S) := {x ∈ U | [s]x = 0 ∀s ∈ S} .

If S = {f1, . . . , fk} is ﬁnite we simply write N(f1, . . . , fk) for N({f1, . . . , fk}). Obviously, for S, T ⊆ OX U, we have N(S ∪ T ) = N(S) ∩ N(T ), N(S ∩ T ) = N(S) ∪ N(T ) and moreover Paragraph 5. Dimension Theory 45

if s := hSiOX U OX U is the ideal generated by S then N(s) = N(S). Consequentially, we P can generalize this construction to the following: For I OX an ideal sheaf the zero locus of I is the set P

N(I ) := Supp(OX /I ) = {x ∈ X | (OX /I )x 6= 0} = {x ∈ X | OX,x 6= Ix} ∼ 0 (where in the last equation, we used that (OX /I )x = OX,x/Ix). Obviously, if I , I OX are two ideal sheaves then P

N(I + I 0) = N(I ) ∩ N(I 0) and N(II 0) = N(I ∩ I 0) = N(I ) ∪ N(I 0).

Consequentially, if I 0 ⊆ I then N(I 0) ⊇ N(I ). A subset A ⊆ X is called a zero set iﬀ for each x ∈ X there is some open U 3 x open and ﬁnitely many f1, . . . , fk ∈ OX U such that

<++>

5. Dimension Theory

(5.1) Deﬁnition. Let X be a ringed space and x ∈ X we deﬁne the (local) dimension of X at x as the Krull-dimension dimx X := dim OX,x. The dimension of X is now simply dim X := supx∈X dimx X ∈ N ∪ {∞}.

While this deﬁnition makes sense for an arbitrary ringed space, it is mostly of interest in the case where X is a locally ringed space and all the stalks Ox are Noetherian. There is a rich theory for the dimension of Noetherian local rings and we may refer the reader to Atiyah & MacDonald, Chapter 11. E.g. we can say that the dimension of a Noetherian local ring is always ﬁnite.

x 2 (5.2) Deﬁnition. The cotangent space of X at x is T X := mx/mx, viewed as a vector space over κx (of course this also carries the structure of an Ox-module). The tangent space x TxX of X at x is simply the dual vector space of T X. That is it’s the κx-vector space 2 2 ∗ TxX := κx-Lin(mx/mx, κx) = (mx/mx) .

(5.3) Definition. Let X be a locally ringed space with Noetherian stalks. A point x ∈ X x is called smooth or regular iff dimx X = dimκx T X (or equivalently dimx X = dimκx TxX because dimx X is always finite).

6. Coherent Sheaves

op (6.1) Definition. Let (X, OX ) be a ringed space. An OX -module F : L(X) → R-Mod is said to be of finite type at x ∈ X or finitely generated at x ∈ X iff there is an open n neighborhood U 3 x together with an epimorphism of OU -modules ε: OU F |U for some n n ∈ N. Because OU is free and ε natural, ε is completely determined by the values s1, . . . , sn ∈ n FU of the unit vectors e1, . . . , en ∈ OU U under εU and one thus says that F |U is generated by the sections s1, ··· , sn. In that case, for V ⊆ U open, every section s ∈ FV can be written Pn as s = i=1 λisi|V for some λi ∈ OU V . We say that F is of finite type or locally finitely generated iff it is finitely generated at every x ∈ X.

Because a morphism of sheaves is epi iff all induced stalk-maps are epi, one deduces that if an O-module F is finitely generated at x then Fx is a finitely generated Ox-module. However, being finitely generated at x is of course much more than that because there need 46 Chapter 4. Ringed Spaces

to be sections, defined at a vicinity U of x, that generate Fx and do so at the whole of U. Still, if we know that F is finitely generated at x, being defined at U is really enough and it automatically follows that such sections generate F at a (possibly smaller) vicinity of x, as the follwoing proposition illustrates.

(6.2) Proposition. Let X be a ringed space, x ∈ X and F an OX -module, which is of ﬁnite type at x. If U 3 x is open and s1, . . . , sn ∈ FU such that their germs germx si generate Fx as an OX,x-module. Then there exists V ⊆ U open with x ∈ V , such that s1|V , . . . , sn|V generate F |V as an OV -module. By a direct application of the famous Nakayama Lemma we can gain a stronger result, where the germx si from the last proposition are not directly required to generate the whole stalk Fx. It is enough for their residue classes [germx si] ∈ Fx/mxFx to span Fx/mxFx as a κx-vector space. (6.3) Lemma. (Nakayama Lemma) Let A be a commutative ring, a A and M a ﬁnitely generated A-module such that aM = M. Then there is some a ∈Pa such that (1 + a)M = 0. In particular, if A is local an a = m its maximal ideal then M = 0.

Proof. By induction on n, the number of generators of M. If n = 0 then M = 0 and there is nothing to show. If n > 1 and M = Am1 + ··· + Amn for some m1, . . . , mn ∈ M then mn = a1m1 + ··· + anmn for some a1, . . . , an ∈ a and so (1 − an)mn = a1m1 + 0 ··· + an−1mn−1 ∈ Am1 + ··· + Amn−1. By induction hypothesis there is some a ∈ a such 0 0 that (1 + a )(Am1 + ··· + Amn−1) = 0 and in particular (1 + a )(1 − an)mn = 0. Putting 0 0 a := a − an − a an ∈ a gives the desired result because (1 + a)mn = 0 by definition of a and (1 + a)m = (1 − a )(1 + a0)m = 0 for all i ∈ {1, . . . , n − 1} by definition of a0. i n i (6.4) Corollary. Let A be a local ring with maximal ideal m, M a finitely generated A-module and m1, . . . , mn ∈ M such that their residue classes [m1],..., [mn] ∈ M/mM generate M/mM as a vector space over A/m. Then m1, . . . , mn generate M as an A-module. Moreover, the [mi] form a basis of M/mM (as an A/m-vector space) iff m1, . . . , mn is a minimal set of generators of M (as an A-module).

Proof. First observe that plainly [m1],..., [mn] also generate M/mM as an A-module. Let 0 0 M := Am1 +···+Amn and consider N := M/M . This is a finitely generated A-module and the quotient map M → N, which maps every mi to 0 by definition of N, yields a surjection M/mM → N/mN. But as [mi] 7→ 0 under this surjection and the [mi] span M/mM, we conclude that N/mN = 0 and thus by Nakayama’s Lemma N = 0, which finishes the proof that [m1],..., [mn] span M/mM iff m1, . . . , mn generate M. As for the last claim, the direction “⇐” is obvious because if mn ∈ Am1 + ··· + Amn−1 then [mn] ∈ A[m1] + ··· + A[mn−1] = (A/m)[m1] + ··· + (A/m)[mn−1]. Conversely, assume that [mn] lies in the span of [m1],..., [mn−1], so that there are a1, . . . , an−1 ∈ A and b1, . . . , bn ∈ m such that mn = a1m1 + ··· + an−1mn−1 + b1m1 + ··· + bnmn but then (1 − b )m ∈ Am + ··· + Am and 1 − b is a unit. n n 1 n−1 n

(6.5) Proposition. Let X be a locally ringed space, x ∈ X and F an OX -module, which is of ﬁnite type at x. If U 3 x is open and s1, . . . , sn ∈ FU such that the residue classes of their germs [germx si] ∈ Fx/mxFx span the whole κx-vector space Fx/mxFx, then there is some open neighborhood V ⊆ U of x such that s1|V , . . . , sn|V generate the OV -module F |V . Moreover, the [germx si] form a basis of Fx/mxFx (as a κx-vector space) iﬀ the si|V form a minimal set of generators of the F |V (as an OV -module) Paragraph 6. Coherent Sheaves 47

Proof. Because F is of finite type at x the stalk Fx is a finitely generated OX,x-module we can apply the above corollary to conclude that the germx si generate the whole of OX,x. By the above proposition, there is thus some V ⊆ U open with x ∈ V and the si|V generate F | . V op (6.6) Definition. An OX -module F : L(X) → R-Mod on a ringed space (X, OX ) is said to be of relation finite type at x ∈ X iff for every open neighborhood U 3 x, every n ∈ N n and every morphism ϕ: OU → F |U of OU -modules, the kernel Ker ϕ is of finite type at x (as an OU -module). F is said to be of relation finite type iff it is so at every x ∈ X. Finally, we call F coherent iff it is of finite type and of relation finite type.

(6.7) Proposition. If X is a ringed space and F is a coherent OX -module then F is locally ﬁnitely presentable. That is, for every x ∈ X there is an open neighborhood U 3 x and an exact sequence of OU -modules m n OU → OU F |U m n for some m, n ∈ N (i.e. F |U is a cokernel of some morphism OU → OU ). Conversely, if OX is coherent (as a module over itself) then any locally ﬁnitely presentable OX -module is coherent.

(6.8) Definition. Let X be a ringed space and F an OX -module. We say F is quasi- coherent at x ∈ X iff there is some open neighborhood U 3 x, cardinal numbers κ, λ and an exact sequence of OU -modules ⊕κ ⊕λ OU → OU F |U → 0, where the first two objects are κ- resp. λ-fold direct sums of OU . F is simply called quasi- coherent iff it is quasi-coherent at every x ∈ X.

Many authors in algebraic geometry define (quasi-)coherent sheaves on schemes a little differently. They call a sheaf F of OX -modules on a scheme (X, OX ) quasi-coherent iff it is locally isomorphic to some Mf. I.e. iff there is an open cover (Ui)i∈I of X, for every i ∈ I ∼ an R-algebra Ai together with an isomorphism of locally R-ringed spaces ϕi : Ui = Spec Ai ∼ as well as an Ai-module Mi and an isomorphism ϕi∗F |Ui = Mfi of Aei-modules. F is then said to be coherent iff all the Mi are finitely generated Ai-modules. These two definitions are not equivalent! But they are so for locally noetherian schemes as is asserted by the following theorem. (6.9) Theorem. Let A be a noetherian ring, X := Spec A and U ⊆ X open and F an OU -module (where OU := Ae|U ) then the following are equivalent: (a) F is coherent; (b) F is of finite type and quasi-coherent; ∼ (c) F = Mf|U for some finitely generated A-module M. Proof. EGA 1, Théorème 1.5.1. (6.10) Theorem. Given a scheme X and a short exact sequence 0 00 F F F 0 00 of quasi-coherent OX -modules. Then if any two of F , F , F are coherent then so is the third. 48 Chapter 4. Ringed Spaces

7. Sheaf Cohomology

(7.1) Lemma. Let ϕ: X → Y be a closed immersion of locally ringed spaces and F ∈ i i Ab(X) a sheaf of abelian groups on X. Then the induced morphisms H (X,F ) → H (Y, f∗F ) are isomorphisms for all i ∈ N.

8. Čech Cohomology

(8.1) Deﬁnition. Let X be a topological space and U = (Ui)i∈I an open cover of X and p+1 for a (p + 1)-tuple (i0, . . . , ip) ∈ I let us write Ui0,...,ip := Ui0 ∩ · · · ∩ Uip . If F ∈ Ab(X) is a sheaf of abelian groups on X, we deﬁne the Čech complex Cˇ•(U,F ) as the cochain complex, which is 0 in negative degrees, ˇp Y C (U,F ) := FUi0,...,ip p (i0,...,ip)∈I

ˇp in degree p ∈ N (so that an element α ∈ C (U,F ) is a family (αi0,...,ip )i0,...,ip∈I with αi0,...,ip ∈

FUi0,...,ip ) and has coboundary operators

p+1 ˇp ˇp+1 X k d: C (U,F ) → C (U,F ) with (dα)i0,...,ip+1 := (−1) α |U i0,...,ibk,...,ip i0,...,ip+1 k=0

p+2 for (i0, . . . , ip+1) ∈ I (where i0,..., ibk, . . . , in := i0, . . . , ik−1, ik+1, . . . , in is the tuple we get by leaving out ik). It is a standard exercise (which – together with the Jacobi identity – is one of those algebraic identities, which every mathematician should have proved at least once) that with this d2 = 0 so that Cˇ•(U,F ) is indeed a cochain complex. Now the Čech cohomology

Hˇ •(U,F ) := H •(Cˇ•(U,F )) of X with respect to U and values in F is simply deﬁned as the cohomology of the Čech complex.

(8.2) Remark. Some authors define the Čech complex a little differently. They require the index set I to be well-ordered (or at least totally ordered) and define the Čech complex in degree p ∈ N as p Y C (U,F ) := FUi0,...,ip i0

<++>

9. Flat Morphisms

(9.1) Definition. Recall that if A is a ring and M an A-module then tensoring with M is a functor − ⊗A M : A-Mod → A-Mod. It has a right adjoint HomA(M, −) and is thus right-exact. M is said to be flat iff the functor − ⊗A M is even exact; M is said to be Paragraph 9. Flat Morphisms 49

0 00 faithfully flat iff it is flat and in addition −⊗A M reflects exactness, i.e. if N → N → N are 0 00 morphisms of A-modules and the image N ⊗A M → N ⊗A M → N ⊗A M under − ⊗A M is a short exact sequence then N 0 → N → N 00 must be a short exact sequence. If ϕ: A → B is a morphism of rings (i.e. B is an A-algebra and in particular an A-module) then ϕ is said to be flat (resp. faithfully flat) iff B is so as an A module. Observe that − ⊗A B is now even a functor A-Mod → B-Mod (called extension of scalars), whose right adjoint HomA(B, −) is naturally isomorphic to the functor A− (called restriction of scalars), which sends a B-linear map M → N to itself, where the modules M and N are now regarded as A-modules via a · m := ϕa · m. It is now obvious that if A → B → C are morphisms of rings, which are both flat (resp. faithfully flat) then their composition is so as well.

(9.2) Remark. More generally (but not important in this text) if A, B are (not necessarily commutative) rings, any A-B-bimodule M (i.e. M is a left A-module and a right B-module such that (am)b = a(mb) for all m ∈ M, a ∈ A, b ∈ B) yields two functors − ⊗A M : Mod-A → Mod-B and M ⊗B −: B-Mod → A-Mod, which both have a right adjoint HomB(M, −) and HomA(M, −) respectively. That these functors occur in pairs is of course no conincidence, for if A is a (not necessarily commutative) ring, we can deﬁne the dual ring Aop, which has the same elements as A but with multiplication reversed, i.e. a ·op b := b · a for a, b ∈ A (this is only a special case of forming the dual category because a ring is nothing but an Ab-category with just one object). Now any left A-module may equally well be viewed as a right Aop-module and vice versa, giving us an isomorphism of categories A-Mod =∼ Mod-Aop. Thus all functors between module-categories have a “left” and a “right” version in the non-commutative case.

(9.3) Lemma. Let ϕ: A → B be a morphism of rings.

(a) p ∈ Spec A lies in the image of ϕ← : Spec B → Spec A iﬀ ϕ−1(Bϕp) = p.

(b) If N is any B-module then sN : N → AN ⊗A B, n 7→ n ⊗ 1 is a section of the counit εN of − ⊗A B a A−. In particular, it is injective and N is a direct summand of AN ⊗A B ∼ ∼ via N ⊕ Ker εN = sN N ⊕ Ker εN = AN ⊗A B.

Proof. Ad (a): If p = ϕ−1q for some q ∈ Spec B then ϕ−1(Bϕp) = ϕ−1(Bϕϕ−1q) ⊆ ϕ−1q = p and the other inclusion is obvious. Conversely, if ϕ−1(Bϕp) = p, we put S := ϕ(A \ p) ⊆ B, which is multiplicatively closed. Now b := S−1(Bϕp) /S−1B is a proper ideal because −1 Bϕp ∩ S = ∅ and thus b ⊆ m for some maximal ideal m /S B. But then q := (B → −1 −1 −1 S B) m is prime, ϕp ⊆ q by which p ⊆ ϕ q and conversely q ∩ S = ∅ so that also ϕ−1q ⊆ p and thus p = ϕ−1q. Ad (b): The counit is given by εN : AN ⊗A B → N, n ⊗ b 7→ bn and one easily checks that sN is a section.

(9.4) Proposition. Let ϕ: A → B be a ﬂat morphism of rings. The following are equivalent:

(a) ϕ is faithfully ﬂat;

(b) ϕ−1(Bϕa) = a for all a A; P (c) ϕ← : Spec B → Spec A is surjective;

(d) Bϕm 6= B for all m /A maximal; 50 Chapter 4. Ringed Spaces

(e) M ⊗A B 6= 0 for all A-modules M 6= 0;

(f) The A-linear map M → M ⊗A B, m 7→ m ⊗ 1 is monic for all A-modules M;

(g) − ⊗A B : A-Mod → B-Mod is faithful.

−1 Proof. “(a) ⇒ (b)”: Consider the diagram ϕ (Bϕa) A A/a, whose exactness we need to check. Tensoring with B yields a commutative diagram

−1 ϕ (Bϕa) ⊗A B / / A ⊗A B / / A/a ⊗A B

=∼ =∼ =∼ Bϕϕ−1(Bϕa) = Bϕa / / B / / B/Bϕa , where the last vertical isomorphism is by flatness of ϕ. Because ϕ is faithfully flat, we deduce that the original diagram is exact. “(b) ⇒ (c)”: By the previous lemma. “(c) ⇒ (d)”: Obvious, if m/A is maximal, then by surjectivity of ϕ← there is some q ∈ Spec B such that m = ϕ−1q, so that ϕm ⊆ q and thus Bϕm ⊆ q 6= B. ∼ “(d) ⇒ (e)” For M any A-module, we have M = F/K for some free A-module F and K 6 F . 0 ∼ Now choose any free summand A A ⊕ F = F such that K ∩ A =: a 6= A, yielding 0 6= A/a M. By flatness of B, A/a ⊗A B M ⊗A B is monic and it thus suffices to check ∼ that A/a ⊗A B = B/Bϕa 6= 0, which is true by (d) because a /A is proper and we may thus find m /A maximal with a ⊆ m. “(e) ⇒ (f)”: Let M be any A-module and tensor the short exact sequence K M M ⊗AB, yielding another short exact sequence K ⊗A B M ⊗A B M ⊗A B ⊗A B. By the previous lemma, M ⊗A B → M ⊗A B ⊗A B is injective, so that K ⊗A B = 0 and thus K = 0. “(f) ⇔ (g)”: The map in (f) is the unit of − ⊗A B a A−. Now use (2.11). “(f) ⇒ (a)”: If M 0 → M → M 00 is a sequence of A-modules (not necessarily exact), such that its image under − ⊗A B is exact, we form the commutative diagram

M 0 / M / M 00

0 00 , M ⊗A B / / M ⊗A B / / M ⊗A B where all the vertical maps are as in (f). And a quick diagram chase gives the exactness of the upper row.

(9.5) Lemma. (Descent for Flatness) Let B be an A-algebra and M a faithfully flat B-module. If M is (faithfully) flat over A then B is (faithfully) flat over A. In particular, if

ϕ ψ A −→ B −→ C are morphisms of rings with ψ (faithfully) flat and ψ ◦ ϕ faithfully flat then ϕ is (faithfully) flat.

(9.6) Remark. Recall that if A is a ring and a A an ideal, the a-adic completion Ab of A is by deﬁnition the limit of the diagram P

n+1 n ··· A/a A/a ··· A/a 0 Paragraph 9. Flat Morphisms 51 in A-Alg. Thus if A, B are rings a A, b B ideals and ϕ: A → B a morphism of rings such that ϕa ⊆ b then, by the universalP propertyP of the limit, ϕ induces a unique morphism ϕ: Ab → Bb form the a-adic completion of A to the b-adic completion of B, such that

ϕ n n n Ab −→b Bb → B/b = Ab → A/a → B/b for all n ∈ N, where the maps Ab → A/an and Bb → B/bn are the limiting cones and A/an → B/bn is induced by ϕ.

(9.7) Notation. For the rest of the text, if A is a local ring we write Ab and speak of the completion of A when we mean m-adic completion, where m /A is the unique maximal ideal. Let us also ﬁx the notation that if A is any ring, a A, Ab the a-adic completion and b A any ideal, we write b for the extended ideal of b in PAb. P

(9.8) Lemma. Let A be a Noetherian ring, a A an ideal and Ab the a-adic completion of A. Then the canonical morphism A → Ab is flat.P It is even faithfully flat iff a is contained in the Jacobson radical JA := T Max A of A. In particular, if A is a Noetherian local ring then A → Ab is faithfully flat.

Proof. For A Noetherian, A → Ab is always flat by Atiyah-MacDonald, Proposition 10.14. “⇒”: If A → Ab is faithfully flat and m /A is any maximal ideal then mb 6= Ab by (9.4) −1 and we find nm / Ab maximal such that mb ⊆ nm. Because m = ϕ (mb ) (again by (9.4)) and −1 −1 m ⊆ ϕ nm 6= A it follows by maximality of m that even m = ϕ nm. We thus deduce that

−1 −1 \ \ −1 \ ϕ JAb ⊆ ϕ nm = ϕ nm = m = JA. m∈Max A m∈Max A m∈Max A

By Atiyah-MacDonald, Proposition 10.15 iv), ba ⊆ JAb and thus (using (9.4) a last time) −1 −1 a = ϕ (ba) ⊆ ϕ JAb ⊆ JA. −1 e “⇐”: For m /A maximal, Am is a local ring with maximal ideal Sm m =: m . By Atiyah- e MacDonald, Proposition 10.16, the m -adic completion Adm is again a local ring with maximal e −1 −1 e ideal mc and because a ⊆ m it follows that Sm a ⊆ Sm m = m yielding a commutative square

A / Ab

/ . Am Adm

Chasing an element m ∈ m around the lower leg gives us that its image in Adm is not a unit and thus (going around the upper leg), its image in Ab can’t be a unit either, yielding that m 6= A. Faithfull ﬂatness now follows from (9.4). b b

(9.9) Proposition. Let A, B be Noetherian rings a A, b B with a ⊆ JA and b ⊆ JB. P P Furthermore, let ϕ: A → B be a morphism of rings with ϕa ⊆ b such that ϕb: Ab → Bb is faithfully flat. Then ϕ is flat. In particular, if A, B are Noetherian local rings and ϕ: A → B is local with ϕb faithfully flat, then ϕ is faithfully flat.

ϕ ϕ Proof. By the last lemma, B → Bb and A −→ B → Bb = A → Ab −→b Bb are faithfully ﬂat. Now ϕ apply (9.5) to A −→ B → B. b 52 Chapter 4. Ringed Spaces

(9.10) Definition. Let f : X → Y be a morphism of ringed spaces and F an OX -module. We say that F is f-flat (or flat over Y if there is no risk of confusion) at a point x ∈ X iff Fx is a flat OY,fx-module. The OY,fx-module structure on Fx is given by restriction of scalars # via the stalk map fx : OY,fx → OX,x. F is said to be f-flat iff it is f-flat at every point x ∈ X. Finally, the morphism f is said to be flat iff OX (viewed as a module over itself) is f-flat. I.e. f is flat iff for every point x ∈ X the stalk OX,x is a flat OY,fx-module. Put # in a more concise manner: f is flat iff for every x ∈ X the stalk map fx : OY,fx → OX,x is flat. f is said to be faithfully flat iff f is flat and surjective. This definition sensible, for if A → B is a morphism of rings then it is flat (resp. faithfully flat) iff the induced morphism Spec B → Spec A is flat (resp. faithfully flat).

A geometric equivalent of our proposition (9.9) is the following corollary, which we will need later in the text.

(9.11) Corollary. If X,Y are locally ringed spaces, all of whose stalks are Noetherian (e.g. locally Noetherian schemes) and ϕ: X → Y a morphism which induces a faithfully # flat map between the completed stalks (i.e. for every x ∈ X the map ϕbx : ObY,ϕx → ObX,x is faithfully flat), then ϕ is flat. (9.12) Theorem. Let X,Y be locally ringed spaces and ϕ: X → Y a morphism. Then ∗ ϕ is flat iff ϕ : OY -Mod → OX -Mod is exact. Chapter 5

TOPOLOGICAL PRELIMINARIES

1. Jacobson Spaces

Recall that a subset Y of a topological space is said to be locally closed (or relatively open) iff Y = U ∩ A for some U ⊆ X open, A ⊆ X closed. Equivalently, Y is locally closed iff Y is open in Y . A subset Y of X is said to be constructible iff it is a finite union of locally closed subsets. Finally, Y is said to be locally constructible iff for each x ∈ X, there is some open neighborhood U 3 x such that Y ∩ U is a constructible subset of U. For the following proposition, if X is a topological space, let L(X), Cl(X), Cstr(X) and LCstr(X) denote the set of open, closed, constructible and locally constructible subsets of X respectively. Obviously, we have inclusions L(X), Cl(X) ⊆ Cstr(X) ⊆ LCstr(X).

(1.1) Proposition. Let X be a topological subspace and Y ⊆ X. Then the following are equivalent:

(a) Y ∩ Z 6= ∅ for each non-empty locally closed subset Z ⊆ X;

(b) A = A ∩ Y for all A ⊆ X closed;

(d) Z ⊆ Y ∩ Z for all Z ⊆ X locally constructible;

(e) L(X) → L(Y ),U 7→ U ∩ Y is injective (thus bĳective);

(f) Cl(X) → Cl(Y ),A 7→ A ∩ Y is injective (thus bĳective);

(g) Cstr(X) → Cstr(Y ),Z 7→ Y ∩ Z is injective (thus bĳective);

(h) LCstr(X) → LCstr(Y ),Z 7→ Y ∩ Z is injective (thus bĳective).

Proof. EGA 4, Proposition 10.1.2

(1.2) Definition. For X a topological space, a subset Y ⊆ X is called very dense iff it satisfies one (and thus all) of the above properties.

(1.3) Proposition. Let X be a topological space, (Ui)i∈I an open cover of X and Y ⊆ X. Then Y lies very dense in X if and only if Ui∩Y lies very dense in Ui for all i ∈ I. In particular, if Y ⊆ X is very dense and U ⊆ X open then U ∩ Y lies very dense in U. 54 Chapter 5. Topological Preliminaries

Proof. “⇒” is obvious because if U ⊆ X is open and Z ⊆ U is locally closed in U then it is locally closed in X, too. Conversely, if Z ⊆ X is locally closed then Z ∩ Ui ⊆ Ui is locally closed for all i ∈ I. Because (Ui)i∈I is a cover of X there is at least one i ∈ I such that Ui ∩ Z 6= ∅, so that Ui ∩ Y ∩ Z 6= ∅ because Ui ∩ Y lies very dense in Ui. In particular Y ∩ Z 6= , which proves “⇐”. ∅ (1.4) Definition. Let X, Y be topological spaces. A continuous map ϕ: X → Y is called a quasi-homeomorphism iff it induces a bĳection L(Y ) → L(X),V 7→ ϕ−1V . Put differently a quasi-homeomorphism is a continuous map, whose image under the functor L: Top → Loc is an isomorphism.

With this definition one can restate the definition of very dense in the following way: If X is a topological space and Y ⊆ X then Y lies very dense in X iff the inclusion Y,→ X is a quasi-homeomorphism. We can “reverse” this result in the sense of the following proposition.

(1.5) Proposition. If ϕ: X → Y is a quasi-homeomorphism then ϕX ⊆ Y is very dense.

−1 −1 Proof. Let V1, V2 ⊆ Y open with V1 ∩ ϕX = V2 ∩ ϕX. Then ϕ V1 = ϕ (V1 ∩ ϕX) = ϕ−1(V ∩ ϕX) = ϕ−1V , so that V = V because ϕ is a quasi-homeomorphism. 2 2 1 2 (1.6) Deﬁnition. A topological space X is called a Jacobson space (or simply Jacobson) if the set Max(X) of closed points of X lies very dense in X. That is to say that every locally closed subset Y ⊆ X contains at least one closed point of X.

As a corollary to the above proposition, we can immediately determine how the property of being a Jacobson space behaves under taking “good” (i.e. locally constructible) subspaces and under collation.

(1.7) Corollary. If X is a Jacobson space and Y ⊆ X locally constructible, then a point y ∈ Y is closed in Y if and only if it is closed in X. Consequentially, Y is a Jacobson space, too.

(1.8) Proposition. Let X be a topological space and (Ui)i∈I an open cover of X. Then X is Jacobson if and only if Ui is Jacobson for all i ∈ I.

Proof. “⇒” is obvious by the last corollary. For “⇐” observe that if x ∈ Ui is closed in Ui for some i ∈ I then x is closed in X for there is some A ⊆ X closed such that A ∩ U = {x} i <++> Chapter 6

ALGEBRAIC PRELIMINARIES

Tu as voulu de l’algèbre, et tu en auras jusqu’au menton! Jules Verne

1. Normal Rings

(1.1) Definition. Let A be a ring and p /A prime. We define the height height p as the supremum of the length n over all chains p0 ( p1 ( ··· ( pn = p of prime ideals, which end at p. We define its depth depth p as the supremum of the length n over all chains p = p0 ( p1 ( ··· ( pn of prime ideals which start at p.

(1.2) Remark. One readily checks that height p = dim Ap and depth p = dim A/p. In particular, if A is Noetherian every prime ideal has ﬁnite height, which follows from the fact that for A Noetherian Ap is again Noetherian, as it is also local it must have ﬁnite dimension.

2. The Category of Schemes

(2.1) Lemma. Given a pullback diagram on the left

π π Y / −1 −1 Y / X ×Z Y Y πX U ∩ πY V V

πX ψ πX ψ X / Z / ϕ U ϕ W in SchR of schemes over a ring R as well as open subschemes U ⊆ X, V ⊆ Y and W ⊆ Z with ϕU, ϕV ⊆ W . Then the diagram on the right (which is a restriction of the left one) is a pullback, too. In particular, this pullback U ×W V does not depend on W ⊆ Z.

Proof. Let S be a scheme over R, together with morphisms f : S → U, g : S → V such that

f ϕ g ψ S −→ U −→ W = S −→ V −→ W.

Composing f and g the inclusions U,→ X and V,→ Y respectively gives a unique h: S → X ×Z Y such that

h πX f h πY g S −→ X ×Z Y −−→ X = S −→ U,→ X and S −→ X ×Z Y −−→ Y = S −→ V,→ Y

<++>

(2.2) Theorem. The category SchR of schemes over a ring R is ﬁnitely complete. 56 Chapter 6. Algebraic Preliminaries

Proof. Spec R is terminal in SchR and it suﬃces to check that SchR has pullbacks. For this consider a diagram of solid arrows in SchR

πY X ×Z Y / Y

πX ψ / . X ϕ Z

In the case where X, Y and Z are aﬃne, it the pullback exists because Γ a Spec and thus Spec preserves limits.

<++>

3. Graded Rings

(3.1) Definition. Let R be a ring. A graded R-algebra or graded ring over R is an R- algebra A, together with a family (An)n∈N of R-submodules of A such that A is the internal direct sum A = L A (i.e. A = P A and A ∩ A = 0 for all m 6= n) and n∈N n n∈N n m n A A ⊆ A for all m, n ∈ . In the same vein, if A = L A is a graded R-algebra, a m n m+n N n∈N n graded A-module over R is an A-module M, together with a family (Mn)n∈N of R-submodules such that M is the internal direct sum M = L M and A N ⊆ M for all m, n ∈ . n∈N n m n m+n N As usually, we will identify a graded module (resp. algebra) with its underlying module (resp. algebra) and assume the corresponding family of submodules being implicitly given. An element m of a graded module (resp. algebra) M is called homogeneous iff m ∈ Mn for some n ∈ N. For m 6= 0, this n ∈ N is by definition unique and is called the degree of m, which we denote by deg m. Any element m ∈ M can be uniquely written as m = P m n∈N n with mn ∈ Mn for all n ∈ N and only finitely many non-zero mn. These non-zero mn are called the homogeneous components of m and we define the degree deg m of m 6= 0 as the maximal n ∈ N such that mn 6= 0. deg thus defined obviously satisfies deg(ab) = deg a+deg b for all a, b ∈ A with ab 6= 0 and in particular deg 1 = 0, i.e. 1 ∈ A0. Finally, a submodule N 6 M of a graded module M is called homogeneous iff N is the internal (direct) sum N = P (N ∩ M ) = L (N ∩ M ). In particular, an ideal a A of a graded ring is n∈N n n∈N n homogeneous iff a is the internal direct sum a = L (a ∩ A ). P n∈N n

(3.2) Remark. It is sometimes usefull to consider graded modules (resp. ring) of the form M = L M (i.e. the module is -graded rather than -graded) and the deﬁnition n∈Z n Z N is completely analogous. If we have an -graded module (resp. ring) M = L M then N n∈N n conventionally we put Mn := 0 for n ∈ Z, n < 0 and thus unspokenly think of M as a Z-graded module.

Obviously, if A = L A is a graded R-algebra, A is an R-subalgebra of A (in n∈N n 0 particular, A is an A0-algebra) and every An is an A0-module. More generally, if M = L M is a graded A-module, then every M is an A -module. It follows that a graded n∈N n n 0 rings A is an R-algebra iﬀ A0 is an R-algebra; In other terms, the R-algebra structure R → A factors through the inclusion A0 ,→ A and it follows that the image of R in A must lie entirely in A0. For the rest of the section, R is a ﬁxed base ring and all (graded) rings are assumed to be (graded) R-algebras.

(3.3) Example. Paragraph 3. Graded Rings 57

• Any R-algebra A can be given a structure of a graded A-algebra by declaring A0 := A and An := 0 for all n 6= 0. A graded algebra of this form is called concentrated in degree 0.

• A := R[X1,...,Xn] can be turned into a graded ring by deﬁning An as the set of all homogeneous polynomials of degree n.

• If A is a graded ring, we write A := L A . If we let A operate on A by + n∈N>0 n + the multiplication given by the ring structure of A, then A+ is a graded A-module (with (A+)0 = 0). • If A is a graded ring and a A a homogeneous ideal then a is a graded A-module P by putting an := a ∩ An. More generally for N 6 M a homogeneous submodule of a graded A-module M, then N becomes a graded A-module by deﬁning Nn := M ∩N.

(3.4) Proposition. Let A be a graded ring and x1, . . . , xs ∈ A homogeneous of positive degree (i.e. xi ∈ A+). Then the following are equivalent:

(a) A = A0[x1, . . . , xs];

(b) A+ = hx1, . . . , xsiA = Ax1 + ··· + Axs;

2 Ps 2 (c) A+/A+ = i=1 A0xi, where xi := xi + A+; n n+1 P α s (d) For n ∈ N>0 any positive integer A+/A+ = |α|=n A0x , where α ∈ N is a multi- n+1 index and x := (x1,..., xs) with xi := xi + A+ .

Proof. “(a) ⇔ (b)”: The direction “⇒” is obvious and we only need to prove “⇐”. For 0 0 this, we write A := A0[x1, . . . , xs] and it suﬃces to show that An ⊆ A for all n ∈ N, which we prove by induction on n. The case n = 0 is trivial. As for the inductive step, let n ∈ N>0, a ∈ An ⊆ A+ and let us write d1, . . . , ds for the degrees of x1, . . . , xs. By P assumption a = i=0 cixi for some ci ∈ An−di (recall that An = 0 for n < 0). But as di > 0 for all i ∈ {1, . . . , s}, it follows that ci ∈ A0[x1, . . . , xs] by induction hypothesis and thence a ∈ A0[x1, . . . , xs]. “(a) ⇒ (c)”: Any a ∈ A+ is of the form a = p(x1, . . . , xs) for some polynomial p(X1,...,Xs) ∈ A0[X1,...,Xs] in s indeterminates. Putting X := (X1,...,Xs) (and x := (x1, . . . , xs)) P α and using multi-index notation, we have p(X) = s c X for some c ∈ A and since α∈N α α 0 2 a = p(x) ∈ A+, p(X) must not have a constant term. But since xixj ∈ A+ for all i, P α P α j ∈ {1, . . . , s}, we conclude that a = p(x) = p(x) = s c x = c x , where α∈N α |α|=1 α x := (x1,..., xs). “(c) ⇒ (d)”: By induction on n. The induction basis n = 1 is our assumption (c). As for n−1 n > 1, let a ∈ A+ and b ∈ A+. By induction hypothesis

X α 00 0 00 Ps 00 0 00 a = cαx + a =: a + a and b = i=0 dixi + b =: b + b |α|=n−1

00 n 00 2 0 00 0 00 for some cα, di ∈ A0, a ∈ A+ and b ∈ A+. Consequentially ab = (a + a )(b + b ) ≡ 0 0 n+1 0 0 a b mod A+ and a b is of the required form. 0 “(d) ⇒ (a)”: Any a ∈ A is uniquely of the form a = a0 + a1 + ··· + an =: a0 + a with ai ∈ Ai and by assumption, we can expand a as

0 X α 00 X α 000 X α a = a0 + a = a0 + cαx + a = a0 + cαx + a = ··· = a0 + cαx |α|=1 |α|62 |α|6n 58 Chapter 6. Algebraic Preliminaries

s 0 00 2 000 3 with cα ∈ A0 for all multi-indices α ∈ N , a ∈ A+, a ∈ A+, a ∈ A+ and so on. As indicated, this process terminates at the n + 1st step.

(3.5) Corollary. A graded ring A is Noetherian if and only if A0 is Noetherian and A is ﬁnitely generated as an A0-algebra.

Proof. “⇐” follows from Hilbert’s basis theorem and we only need to check “⇒”. If A is Noetherian, the ideal A+ is finitely generated by x1, . . . , xs ∈ A+, say. Without loss of generality, we can assume that each xi is homogeneous for otherwise we replace it by its homogeneous components. Now the claim follows by (b) ⇒ (a) in the above proposition. (3.6) Definition. Let A be a graded ring, M, N graded A-modules. An A-linear map ϕ: M → N is called a morphism of graded A-modules iff ϕMn ⊆ Nn for all n ∈ N. Similiarly, if A, B are graded rings a morphism (of R-algebras that is) ϕ: A → B is called a morphism of graded rings (resp. R-algebras) iff ϕAn ⊆ Bn for all n ∈ N. With this definition, we have a category R-gAlg of graded R-algebras (and GRng := Z-gAlg the category of all graded rings) and for A any graded ring a category A-gMod of graded modules over A.

The next thing (and really the only reason we introduced graded rings) is the so- called Proj construction, which is a graded analogue of Spec. For this, we need some facts about homogeneous ideals.

(3.7) Proposition. Let A be a graded ring, M a graded A-module.

(a) A submodule N 6 M is homogeneous iﬀ for all m ∈ N the homogeneous components of m (as an element of M) lie in N, too.

(b) A submodule N 6 M is homogeneous iﬀ it has a set of homogeneous generators. (c) The set of all homogeneous submodules of M is closed under taking sums and intersections.

(d) If N 6 M is a homogeneous submodule and a A a homogeneous ideal, then aN 6 M is a homogeneous submodule of M. P

(e) The set of all homogeneous ideals of A is closed under taking sums, intersections, products and radicals.

(f) Let a, b A be homogeneous ideals. Then a = b if and only if for all a ∈ A homogeneous a ∈ a ⇔Pa ∈ b.

(g) A homogeneous ideal a A is radical if and only if for all a ∈ A homogeneous ak ∈ a P for some k ∈ N implies a ∈ a. (h) A proper homogeneous ideal a /A is prime if and only if for all a, b ∈ A homogeneous ab ∈ a implies a ∈ a or b ∈ a.

Proof. Ad (a): If N = P (N ∩ M ) is homogeneous and m ∈ N, we can write m = n∈N n P m with m ∈ N ∩ M . Because m ∈ M and m = P m ∈ M the m are the n∈N n n n n n n∈N n n homogeneous components of m and by deﬁnition mn ∈ N. Conversely if m ∈ N and N contains all homogeneous components m ∈ M of m, we have m = P m ∈ P (N ∩M ). n n n∈N n n∈N n Ad (b): Let S ⊆ N be a set of generators and deﬁne S0 as the set of all homogeneous components of all elements of S. By (a) S0 ⊆ N and this set still generates the whole of N. Paragraph 3. Graded Rings 59

Ad (c): Using (a), the case of intersections is trivial. In the case of sums, observe that if m, 0 0 0 m ∈ M have homogeneous components mn and mn respectively, then m + m has homoge- 0 neous components mn + mn and by (a) a sum of homogeneous submodules is homogeneous. Ad (d): By the same argument from the proof of (c) it suffices to consider the case am ∈ aN where a ∈ a, m ∈ N. Because a and N are homogeneous, we have an ∈ a and mn ∈ N for all homogeneous components an of a and mn of m. Consequentially almn ∈ aN for all l, n ∈ N. Ad (e): By the above points, all that is left to check is the case of radicals. For this let a A √ √ be homogeneous and a = P a ∈ a with a ∈ A for n ∈ . We prove a ∈ aPfor n∈N n n n N n k all n ∈ N by induction. For n = 0, choose k ∈ N such that a ∈ a and observe that the k k k homogeneous component of a in degree 0 is a0. By homogenity of a thus a0 ∈ a implying √ √ 0 √ a0 ∈ a. Now if a0, . . . , an ∈ a then a := a − a0 − a1 − · · · − an ∈ a. Thus we find 0k 0k k k ∈ N with a ∈ a and the homogeneous component of a in degree k(n + 1) is an+1. By k √ homogenity of a we derive an+1 ∈ a and thus an+1 ∈ a. Ad (f): Obvious by a = L (a ∩ A ) and b = L (b ∩ A ). √ n∈N n n∈N n Ad (g): a is homogeneous and the claim follows from the last point. Ad (h): Obviously if a is prime it has the stated property. Conversely, let a, b ∈ A with homogeneous components an, bn and ab ∈ a. If a∈ / a then there is k ∈ N minimal such that 0 0 ak ∈/ a. If follows that a := (a − a0 − a1 − · · · − ak−1) ∈ a and thus a b ∈ a. We now prove bn ∈ a for all n ∈ N by induction. The case n = 0 is trivial, for the homogeneous component 0 of a b in degree k is akb0, which lies in a by homogenity but ak ∈/ a by definition of k. If n > 0 0 0 0 and b0, . . . , bn−1 ∈ a then b := b − b0 − b1 − · · · − bn−1 ∈ a and thus a b ∈ a. Now again, the homogeneous component of a0b0 in degree k + n is a b ∈ a (by homogenity) but a ∈/ a. k n k

(3.8) Proposition. Let A be a graded ring and S ⊆ A a multiplicative system consisting of homogeneous elements of A.

−1 −1 (a) Then the localisation S A is a Z-graded ring in a canonical way by deﬁning (S A)n as the set of all a/s with a ∈ A, s ∈ S both homogeneous with deg a − deg s = n. In 2 −1 particular, if s ∈ A is homogeneous and Ss := {1, s, s ,... } then As = Ss A has a canonical Z-grading.

−1 −1 (b) Similarly, if M is a graded A-module then S M is a Z-graded S A-module in a −1 canonical way, again by deﬁning (S M)n as the set of all m/s with m ∈ M, s ∈ S both homogeneous with deg m−deg s = n. Moreover, if ϕ: M → N is a morphism of graded A-modules, the induced morphism S−1ϕ: S−1A → S−1B between the localisations is again a morphism of graded modules.

−1 Proof. Ad (a): Let us check that for n ∈ Z the set (S A)n is well-deﬁned. If a/s = b/t 6= 0 with a, b ∈ A, s, t ∈ S all homogeneous then (at − bs)u = atu − bsu = 0 for some u ∈ S. Because a/s = b/t 6= 0 it follows that atu, bsu 6= 0 and by atu = bsu, we have

deg(atu) = deg a + deg t + deg u = deg(bsu) = deg b + deg s + deg u,

−1 implying that the sets (S A)n are indeed well-defined. That this defines a Z-grading of S−1A is now obvious. Ad (b): The proof that S−1M is graded is analogous to (a) and S−1ϕ is indeed graded by its very definition and the definition of the gradings on S−1M and S−1N.

(3.9) Deﬁnition. Let A be a graded ring, p /A a prime ideal (not necessarily homogeneous) and S := {a ∈ A \ p | a homogeneous}. Then S is a multiplicative system and we 60 Chapter 6. Algebraic Preliminaries

gr −1 write Ap := S A, which we call the graded localisation of A at p. Moreover, we introduce the notation gr a gr A := (A )0 = ∈ A a ∈ A, s∈ / p both homogeneous and deg a = deg s . (p) p s p

gr Observe, that in the deﬁnition of Ap we did not require p to be homogeneous but eventually we will only be interested in the homogeneous case. In the same spirit, if s ∈ A is homogeneous, we deﬁne a A := (As)0 = ∈ As a ∈ A homogeneous, n ∈ and deg a = n · deg s (s) sn N

gr It is natural to ask how the graded localisation Ap , which is somewhat harder to grasp, compares with the usual (ungraded) localisation Ap.

gr (3.10) Proposition. Let A be a graded ring, p/A a prime ideal and ϕ: Ap → Ap, a/s 7→ a/s the morphism induced by the univeral morphism A → Ap. Then ϕ is injective and we gr may view Ap as a subring of the localisation Ap. In particular A(p) is a subring of Ap. P Proof. Let a = n∈ an ∈ A (with an ∈ An) such that ϕ(a/1) = 0 ∈ Ap. We shall show that gr N an/1 = 0 ∈ Ap for all n ∈ N by induction. For the case n = 0, observe that since ϕ(a/1) = 0 P there is some t = n∈ tn ∈ A \ p (with tn ∈ An) such that at = 0 and the homogeneous N gr component of at in degree 0 is a0t0. It follows that a0t0 = 0 and thus a0/1 = 0 ∈ Ap . For gr 0 gr the inductive step assume a0/1 = a1/1 = ··· = an/1 = 0 ∈ Ap , so that a/1 = a /1 ∈ Ap 0 0 with a := a − a0 − a1 − · · · − an. It follows that ϕ(a/1) = ϕ(a /1) = 0 and there is some t0 = P t0 ∈ A \ p (with t0 ∈ A ) such that a0t0 = 0. Again, the homogeneous component n∈N n n n of a0t0 in degree n + 1 is a t0 , so that a t = 0 and thus a /1 = 0 ∈ Agr. n+1 0 n+1 0 n+1 p

(3.11) Deﬁnition. (Proj Construction) Let A = L A be a graded ring. We deﬁne n∈N n

X := Proj A := {p ∈ Spec A | p is homogeneous and A+ 6⊆ p} and equip this set with a topology (again called the Zariski topology) by declaring the closed sets to be

VX (a) := V (a) := {p ∈ Proj A | a ⊆ p} , where a A is a homogeneous ideal. This does indeed define a topology because (as in the P affine case for Spec) we have V (A) = V (A+) = ∅, V (0) = Proj A, V (a) ∪ V (b) = V (a ∩ b) T P and i∈I V (ai) = V i∈I ai , which makes sense by (3.7). Again as in the affine case, the special (or basic or distinct) open sets

DX (a) := D+(a) := Proj A \ V (Aa) = {p ∈ Proj A | a∈ / p} for a ∈ A+ homogeneous form a basis (closed under finite intersections because D+(a) ∩ ∼ D+(b) = D+(ab)) for the Zariski topology and D+(a) = Spec A(a), p 7→ Aap ∩ A(a) = −1 Sa p ∩ A(a) is a homeomorphism (where D+(a) carries the subspace topology and Sa = 2 {1, a, a ,... }), which is just a restriction of the familiar homeomorphism DSpec A(a) → −1 Spec Aa, p 7→ Sa p from the affine case. Because we want Proj A to be a scheme, the only reasonable choice for a structure sheaf is OX (D+(a)) := A(a) with the obvious restriction maps induced by the universal property of the localisation, analogous to the affine case. Paragraph 4. Chow’s Lemma 61

(3.12) Example. For A any ring and n ∈ N>0, we deﬁne the projective n-space over A r PA := Proj A[X1,...,Xn], where the polynomial ring A[X1,...,Xn] is given the canonical grading, where homogeneous clements of degree i ∈ N are the homogeneous polynomes of degree i.

2 2 (3.13) Example. Let C[ε] := C[X]/hX i with ε := [X], so that ε = 0. By defining C[ε]0 := C and C[ε]1 := Cε this becomes a graded ring and one easily calculates Proj C[ε] = ∅. More generally, if√A is a graded ring, we have Proj A = ∅ iff A+ ⊆ p ∀p ∈ Spec A iff T A+ ⊆ Spec A = 0.

(3.14) Deﬁnition. Let A be a graded ring, X := Proj A and M a Z-graded A-module. We deﬁne the sheaf of OX -modules Mf associated to M on the basic open sets by

MDf +(a) := M(a), with the obvious OX D+(a) = A(a)-module structure. One readily checks that OX = Ae.

(3.15) Proposition. Mf as in the deﬁnition is quasi-coherent.

(3.16) Definition. (Twisting Sheaf) Let A be a graded ring and M a Z-graded module over A. For n ∈ Z we write M(n) for the shifted module, defined by M(n)i := Mn+i, which is obviously again a Z-graded module on A. Now, if X := Proj A, we write Mf(n) := M(n)e and in particular OX (n) := A(n)e . The OX -module OX (1) is called the twisting sheaf and ∼ one has obvious isomorphisms Mf(n) = Mf ⊗OX OX (n), which suggests the following, more general definition: If F is an OX -module, we define the twisted sheaf F (n) := F ⊗OX OX (n).

(3.17) Proposition. If A is a graded ring with A = A0[A1] (i.e. A is generated by A1 as an A0-algebra), then for n ∈ Z the sheaf OX (n) is locally free of rank 1 and there is a ∼ ⊗n canonical isomorphism OX (n) = OX (1) .

r (3.18) Proposition. Let A be a ring, X → PA a closed immersion for some r ∈ N>0 and r OX (1) the pullback of the twisting sheaf on PA. If F is a ﬁnitely generated, quasi-coherent ⊕m OX -module, then there are m ∈ N, n ∈ Z and a surjection OX (n) → F .

4. Chow’s Lemma

(4.1) Definition. A morphism ϕ: X → Y of schemes is said to be projective (or globally projective) iff there is some n ∈ N such that ϕ factors as ϕ n X −→ Y = X → PY → Y, n n where X → PY is a closed immersion and PY → Y the standard projection. A morphism ϕ: X → Y of schemes is said to be quasi-projective (or globally quasi-projective) iff ϕ factors as an open immersion X → X0 followed by a projective morphism X0 → Y .

(4.2) Remark. There exist different definitions of a projective morphism, which are not equivalent but are so under mild assumptions. Our definition here is the same as in Hartshorne and Eisenbud-Harris but be aware the Grothendieck uses a different definition in EGA II.

(4.3) Theorem. (Chow’s Lemma) Let X be a scheme of ﬁnite type over another scheme S. If 62 Chapter 6. Algebraic Preliminaries

(a) S is Noetherian or

(b) S is quasi-compact and X has a ﬁnite number of irreducible components then there exists a quasi-projective scheme P over S and a surjective and projective morphism f : P → X of schemes over S such that there is an open set V ⊆ X for which −1 U ∼ U := f V lies dense in P and f restricts to an isomorphism f|V : U = V . Moreover, if X is reduced/irreducible/integral then so is P .

Proof. EGA 2, Théorème 5.6.1. (4.4) Corollary. Under the same hypotheses as in the theorem. If in addition X is proper over S (e.g. if S = Spec k for some ﬁeld k) then P is projective over S.

(4.5) Corollary. If X is a normal scheme of ﬁnite type over a Noetherian scheme S, then there is some scheme P projective over S together with a dominant open immersion X → P over S. Chapter 7

ANALYTIC PRELIMINARIES

1. Analytic Spaces

n Recall that for U ⊆ C open a function f : U → C is called holomorphic iﬀ for each z = (z1, . . . , zn) ∈ U there is an open neighborhood V ⊆ U of z and a formal power series

X α cα(X − z) ∈ C[[X1,...,Xn]] n α∈N which converges in all of V to f|V . One can show that in this case, such a power series is n always locally uniformly and absolutely convergent. The n-dimensional complex space C n becomes a locally ringed space by equipping it with the structure sheaf HC given by

n : HC U = {f : U → C homolorphic} n with the obvious restriction maps. More generally, if D ⊆ C is open, we write HD for n the restriction of HC to D and call this the sheaf of holomorphic functions on D. As one can show, its stalk at a point z ∈ D is the ring of power series (in n indeterminates X1,...,Xn) convergent around z (i.e. in a neighborhood of z); in particular two holomorphic functions agree in some neighborhood of z iﬀ their power series expansions at z have the same coeﬃcients. It follows that Hz is a local ring having as maximal ideal mz the convergent power series with constant term 0.

n (1.1) Proposition. (Identitätssatz) Let D ⊆ C a connected open subset and f, g : D → C holomorphic. If there is some U ⊆ D open such that f|U = g|U then f = g on the whole of D.

n (1.2) Corollary. For any D ⊆ C open the sheaf HD is Hausdorﬀ (i.e. ΛHD is Haus- dorﬀ).

n (1.3) Deﬁnition. Let n ∈ N, D ⊆ C open, H := HD and I H an ideal sheaf of ﬁnite type (i.e. I U H U for all U ⊆ D open and for every x ∈PD there is some open P neighborhood U ⊆ D of x and sections f1, . . . , fm ∈ H U such that I U = f1H U + ··· + fmH U). Consider the quotient sheaf H /I and write

Y := Supp(H /I ) = {z ∈ D | (H /I )z = Hz/Iz 6= 0} = {z ∈ D | Hz 6= Iz}

Plainly, if z ∈ D and U ⊆ D is an open neighborhood of z, together with f1, . . . , fm ∈ H U such that I U = f1H U + ··· + fmH U then obviously

Y ∩ U = NU (f1, . . . , fm) = {w ∈ U | f1w = f2w = ··· = fmw = 0} , 64 Chapter 7. Analytic Preliminaries

for if we have w ∈ U such that there is some i ∈ {1, . . . , m} with fiw 6= 0 then germw fi ∈/ mw and thus germw fi ∈ Hw is invertible. But Iw = germw f1Hw + ··· + germw fmHw ⊇ germw fiHw = Hw. By defining HY := (H /I )|Y , we get a new locally ringed space (Y, HY ), which is called an analytic model space. To be more precise, it is the analytic model space defined by the ideal sheaf I H . P (1.4) Example. For any D ⊆ X the whole of (D, HD) is an analytic model space, it is defined by the zero-ideal 0 HD. The other extreme is ∅, which is the model space defined P by HD itself.

(1.5) Example. With the same notation as in the definition: For f1, . . . , fm ∈ H D, we can define I := f1H + ··· + fmH , which is obviously an ideal sheaf of finite type. The corresponding model space is now

ND(f1, . . . , fm) = {z ∈ D | f1z = f2z = ··· = fmz = 0} , whose structure sheaf is H /(f1H + ··· + fM H ).

(1.6) Definition. (Analytic Space) A locally ringed space (X, HX ) over C is called an analytic space iff X is Hausdorff and for every x ∈ X there is some U 3 x open such that (U, HU ) is isomorphic to some analytic model space. By definition of an analytic model space every stalk HX,x is a quotient of an algebra of convergent power series. We write AnSpc for the full subcategory of LSpc having as objects all analytic spaces. A morphism between C analytic spaces is called a holomorphic map an isomorphism of analytic spaces is called a biholomorphic map. Finally an HX -module is usually called an analytic sheaf .

2. Weierstraß’ Preparation Theorem

3. Riemann Extension Theorem

(3.1) Theorem. (Riemann Extension Theorem) Let X be a normal complex space, A ⊆ X thin and f : X \ A → C holomorphic. If f is bounded near A or A is thin of order 2, then f has a unique holomorphic extension f˜: X → C to the whole space X. Proof. Grauert & Remmert – Coherent Analytic Sheaves, p.144. (3.2) Corollary. If X is a connected normal complex space and A ⊆ X thin, then X \ A is connected, too.

Proof. Grauert & Remmert – Coherent Analytic Sheaves, p. 145. <++>

4. Dimension Theory

(4.1) Theorem. (Ritt’s Lemma) Let X be an analytic space, A ⊆ X an analytic set. Then dimx A < dimx X for all x ∈ A if and only if A is nowhere dense in X.

Proof. Grauert & Remmert – Coherent Analytic Sheaves, p.102. Chapter 8

GAGA

1. Analytiﬁcation of Schemes

As the analytiﬁcation of a scheme boils down to the fact that a certain functor is representable recall that by (1.4), if C is a category, R ∈ C and F : Cop → Sets, then a natural transformations ϕ: C(−,R) ⇒ F is completely determined by r := ϕR1R because by naturality ϕC a = (F a)r for all C ∈ Ob(C), a ∈ C(C,R). In particular, a representation for F is completely determined by a representing object R ∈ Ob(C) together with an arrow r ∈ FR.

(1.1) Main Theorem. Any scheme X locally of ﬁnite type over C (viewed as a locally ringed space over ) deﬁnes a functor LSpc (−,X): LSpcop → Sets, represented by X. C C C When restricted to AnSpc the resulting functor LSpc (−,X): AnSpcop → Sets C an an is again representable by an analytic space X , together with λX ∈ LSpc(X ,X), whose an ∼ an underlying map restricts to a bĳection X = X(C) and which for x ∈ X induces an isomorphism

λX,x OX,λx / OXan,x

∼ ObX,λx / ObXan,x bλX,x between the completed local rings. In particular, λX is faithfully flat. (1.2) Remark. If we unfold the definition of a representable functor in this particular case, the theorem says that for any scheme X locally of finite type over C, there is an analytic an an space X (unique up to isomorphism), together with a morphism λX : X → X such that for every analytic space X, the map ∼ an LSpcC (Y,X) = AnSpc(Y,X ), ϕ 7→ λX ◦ ϕ is a bĳection. One can interpret this as a sort of “partical adjoint” for the inclusion functor AnSpc ,→ LSpc (it’s partial because X must be a scheme, locally of finite type over and C C in general, we do not have such a bĳection for an arbitrary locally ringed space over C). Proof. SGA 1, Exposé XII, Théorème et Définition 1.1 As for the flatness of λX : By the Weierstraßpreparation theorem, all the stalks of Xan are Noetherian. Furthermore, since X is locally of finite type, the Hilbert Basissatz implies that all the stalks of X are Noetherian, too. Now, we use (9.11). 66 Chapter 8. GAGA

2. GAGA for Projective Spaces

n an n In the following section let us write λ := λ n for the canonical morphism P → P from PC C C the analytiﬁcation.

(2.1) Lemma. If F is a coherent sheaf on n then λ∗F is a coherent sheaf on n an, so PC PC that λ∗ restricts to a functor

λ∗ : QCoh( n ) → QCoh( n an), PC PC which is exact, by ﬂatness of λ. INDEX OF NOTATION

Arrows Sets category of small sets Cat 2-category of small categories → arrow (not necessarily a 1- CAT meta-2-category of all cate- arrow) gories ⇒ 2-arrow CD functor category monomorphism Cb category of presheaves on C epimorphism (a.k.a. cocompletion of C) ∼ op −→= canonical isomorphism C op-dual of C (1-arrows re- =∼ canonically isomorphic versed) co −→∼ , C co-dual of C (2-arrows re- ' versed) −→ isomorphism/equivalence coop ' isomorphic/equivalent C coop-dual of C (1-& 2- arrows reversed) y = yC Yoneda embedding y: C → Cb 0 0 y = yC alternative Yoneda embedding y: C → (SetsC)op Categorical Constructions L poset of open subsets functor Top → Catop 19 Q ×, product PShA(X) A-valued sheaves on X 19 +, q, ` coproduct PSh(X) Sets-valued presheaves on X Lanb a left Kan extension,7 19 Ranb a right Kan extension,7 ShA(X), − ⊗ P left Kan extension of P ∈ Cb A(X) A-valued sheaves on X 20 along y, 16 Sh(X) Sets-valued sheaves on X 20 R P category of elements of P ∈ a sheaﬁﬁcation functor C Cˆ , 16 SpcR ringed spaces over R 37 LSpcR locally ringed spaces over R 37 OX -Mod Sheaves of OX -modules 41 Geometric Constructions Sheaves TxX cotangent space of a locally ringed space X 45 F |U restriction of a sheaf F to U TxX tangent space of a locally 24 ringed space X 45 Hom(F,G) sheaf hom 43 n (X, O ) ringed space 37 PA projective n-space over A 61 X OX structure sheaf of a ringed space 37 (ϕ, ϕ#) morphism of (locally) ringed spaces 37 Set Theory

∩, T intersection ∪, S union ∪˙ , S˙ union of disjoint sets P power set

Categories and Functors BIBLIOGRAPHY

Atiyah, M. F.; Macdonald, I. G.: Introduction to Commutative Algebra, Addison-Wes- ley, London, 1969

Grauert, H.; Remmert, R.: Coherent Analytic Sheaves, Springer-Verlag, Berlin Heidel- berg, 1984

Grothendieck, A.: Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): I. Le langage des schémas (“EGA 1”), Publications Mathématiques de l’IHÉS 4, Paris, 1960

Grothendieck, A.: Revêtements Étales et Groupe Fondamental (“SGA 1”), Springer Lec- ture Notes in Mathematics 224, Springer-Verlag, Berlin, 1971

Tennison, B. R.: Sheaf Theory, Cambridge University Press, Cambridge, 1976 INDEX

Symbols functor, 35 map, 35 Śheaf Coherent sheaf, 47 analytic, 64 Colimit formula, 10 Connection A Galois, 34 Constant Adjunction presheaf, 32 Tensor-Hom, 16 sheaf, 32 Aﬃne scheme, 25, 38 Constructible subset, 53 Algebra Cotangent space, 45 graded, 56 Counit Analytic of a Kan extension,6 model space, 64 Covering sieve, 22 sheaf, 64 Cross-section space, 64 of a presheaf, 19 Associated sheaf, 32 D

B Degree, 56 of a homogeneous element, 56 B-sheaf, 25 Depth Base change, 35 of a prime ideal, 55 functor, 35 Dimension map, 35 of a ringed space, 45 Basic open set, 26, 60 of a ringed space at a point, 45 Biholomorphic Distinct open set, 26, 60 map, 64 Dual ring, 49 Bundle, 27 pullback, 35 E

C Element homogeneous, 56 Category Elements of elements, 16 of a presheaf, 16 of OX -modules, 41 Essential Čech complex, 48 Čech Étalé cohomology, 48 space, 30 complex, 48 Étale essential, 48 bundle, 30 Change of base, 35 associated to a presheaf, 28 70 Index

map, 30 Hyperconnected space, 30 Extension of scalars, 49 Extension problem,6 I

F Ideal, 42 homogeneous, 56 Faithfully ﬂat Ideal sheaf, 42 Module, 49 Integral Morphism, 52 ringed space, 41 Flat Irreducible module, 48 point, 41 Morphism, 52 space, 30 Function holomorphic, 63 J

Jacobson radical, 51 G Jacobson space, 54 Galois connection, 34 Germ, 27 K Global sections functor, 39 Globally projective morphism, 61 Kan extension Globally quasi-projective morphism, 61 absolute,8 Gluing left,6 of sheaves, 25 pointwise, 12 Gluing axiom, 20 preservation of,8 Graded right,6 algebra, 56 Kan lift,6 concentrated in degree 0, 57 localisation, 60 L module, 56 ring, 56 Lemma concentrated in degree 0, 57 Nakayama, 46 Limit formula, 12 Local H homeomorphism, 30 Height Local dimension of a prime ideal, 55 of a ringed space, 45 Holomorphic Localisation function, 63 graded, 60 map, 64 Locally closed subset, 53 Hom, 43 Locally constructible subset, 53 Homeomorphism Locally irreducible local, 30 ringed space, 41 Homogeneous Locally ringed space, 37 component, 56 element, 56 M ideal, 56 submodule, 56 Map Index 71

biholomorphic, 64 separated, 28 holomorphic, 64 Presheaf criterion, 14 Model space, 64 Presheaf tensor product, 43 Module Projective faithfully flat, 49 morphism, 61 flat, 48 space over a ring, 61 graded, 56 Pullback shifted, 61 bundle, 35 Morphism faithfully flat, 52 Q flat, 52 globally projective, 61 Quasi-coherent sheaf, 47 globally quasi-projective, 61 Quasi-homeomorphism, 54 of graded algebras, 58 Quasi-projective of graded modules, 58 morphism, 61 of graded rings, 58 of locally ringed spaces, 37 of ringed spaces, 37 R projective, 61 quasi-projective, 61 Reduced point, 41 ringed space, 41 N Regular point, 45 Nakayama Lemma, 46 Replete subcategory, 21 Normal Residue field point, 41 at a point, 44 ringed space, 41 Restriction, 19 Restriction of scalars, 49 O Riemann Extension Theorem, 64 Ring Open set dual, 49 basic, 26, 60 graded, 56 distinct, 26, 60 Ringed space, 37 special, 26, 60 integral, 41 OX -module, 41 locally, 37 flat, 52 locally irreducible, 41 normal, 41 reduced, 41 P Ritt’s Lemma, 64 Point irreducible, 41 S normal, 41 reduced, 41 Scalars regular, 45 extension of, 49 smooth, 45 restriction of, 49 Presheaf Scheme, 40 constant, 32 affine, 25, 38 on a space, 19 locally finitely generated, 40 72 Index

locally of finite type, 40 generated by a family of subsets, 20 of finite type, 40 Subset Section constructible, 53 of a presheaf, 19 locally closed, 53 Separated locally constructible, 53 presheaf, 28 very dense, 53 Sheaf, 20 Subsheaf, 24 associated, 32 generated by a family of subsets, 35 B-, 25 basic, 25 T coherent, 47 constant, 32 Tangent space, 45 finitely generated at a point, 45 Tensor product ideal, 42 of presheaves, 43 locally finitely generated, 45 of sheaves, 43 locally finitely presentable, 47 Tensor-Hom adjunction, 16 of finite type, 45 thm:base change, 35 of finite type at a point, 45 Transpose of holomorphic functions, 63 of a functor, 13 of local morphisms, 43 Twisted of O-modules, 41 sheaf, 61 of relation finite type, 47 Twisting sheaf, 61 of rings, 37 quasi-coherent, 47 twisted, 61 U twisting, 61 Unit Sheaf Hom, 43 of a Kan extension,6 Sheaf tensor product, 43 Sheafification, 32 Shifted module, 61 V Sieve, 22 Very dense subset, 53 covering, 22 Skyscraper sheaf, 19 Smooth Z point, 45 Space Zariski topology, 25, 60 analytic, 64 Zero hyperconnected, 30 locus, 44 irreducible, 30 set, 45 Special open set, 26, 60 Stalk, 27 Stalk-space, 28 Structure sheaf, 37 Subcategory replete, 21 Submodule generated by a family of subsets, 42 homogeneous, 56 Subpresheaf, 20