Sheaves, Local Ringed Spaces, Affine Schemes, Preschemes

Sheaves, Local Ringed Spaces, Aﬃne schemes, Preschemes

September 25, 2007

Related readings: [Hart]: pp. 60-82; [E-H]: pp. 7-35; [M1]: pp. 65-80.

1 Patching: naive discussion

Let a set1 X be covered by two subsets A, B ⊂ X so that X = A ∪ B; putting X˜ := A t B (the disjoint union) we have a surjection X˜ → X . To give a function (say real-valued) on X is the same as to give a pair consisting of a function on A and a function on B that coincide on the intersection A ∩ B. —–picture here—–

Form the ﬁber square of this mapping

X˜ ×X X˜ = A ×X A t A ×X B t B ×X A t B ×X B = = A t A ∩ B t B ∩ A t B.

We have a little diagram with two natural mappings

p1,p2 X˜ ×X X˜ −→ X˜ −→ X. If F denotes real-valued functions, we get a little diagram −→ F(X) −→ F(X˜) −→ F(X˜ ×X X˜) and another way of saying that we have a pair consisting of a function on A and a function on B that coincide on the intersection A ∩ B is just to say that we have a function h on X˜ that has the property that h · p1 = h · p2; in standard parlance: we have a function h that equalizes p1 and p2. Such functions h are in one-one correspondence with “global” functions on X. In simple terminology, the diagram

pi,p2 F(X) −→ F(X˜) −→ F(X˜ ×X X˜) is exact, which by deﬁnition means that F(X) is identiﬁed with the subset of F(X˜) equalizing p1 and p2. 1or you can take it to be a topological space, or anything else, appropriately modifying the language above

1 2 Presheaves and sheaves

Deﬁnitions, maps of presheaves, restriction of presheaves; note that to deﬁne sheaf the topology here plays a role, in terms of what is to count as a cover.

3 Examples

Topological spaces, diﬀerentiable manifolds, analytic manifolds—structure sheaves.

4 Notation for sections of sheaves

Let F be a sheaf (of anythings, say groups) on a topological space X. Alternate Notation: For U ⊂ X an open of X we sometimes denote Γ(U, F) := F(U) and refer to this as the group of sections of F over U; the group of global sections of F is Γ(X, F) := F(X).

—–discuss examples—–

5 Stalks

If X is a topological space, F a sheaf (of sets, groups, ...) on X and x ∈ X a point, the stalk of F at x, denoted Fx is the direct limit

Fx := lim F(U). U3x

—–discuss direct limits over index sets, over directed categories—–

Any global section determines an element in each stalk, as is (in turn) determined by these elements, i.e., we have an injection: Y F(U) ,→ Fx. x∈U

(Because if two elements f, g ∈ F(U) go to the same element in Fx there is an open Ux such that x ∈ Ux ⊂ U such that f, g go to the same element under the mapping F(U) → F(Ux); so they go to the same element in the cover {Ux}x∈U and therefore by the sheaf axiom they are the same.) Thus, maps of sheaves as determined by the induced mapping on stalks; i.e., Y Hom(F, G) ,→ Hom(Fx, Gx). x∈X The guts of the workings of sheaves is visible on the level of stalks.

2 6 The structure sheaf for aﬃne schemes

Pν Let A be a ring and X := Spec A. If 1 = i=1 fi ∈ A, setting ν ˜ Y A := Afν i=1 we have that pi,p2 A −→ A˜ −→ A˜ ⊗A A˜ is exact. Let us say that it is a sheaf on the standard base of opens. This is the essential ingredient to the assertion that the presheaf of rings U 7→ A(U) determines a well-deﬁned sheaf of rings. Namely:

Exercise 1 Show that a “sheaf on the standard base of opens” extends in a unique way to an honest sheaf.

The honest sheaf of rings on X = Spec A that is determined by the sheaf U 7→ A(U) on the standard base of opens is sometimes denoted OX . So, for any f ∈ A, if U = Xf = Spec Af we have

Af = Γ(Xf , OX ) = OX (Xf ) and as for global sections, we retrieve the ring A itself:

A = Γ(X, OX ) = OX (X).

IfP ⊂ A is a prime ideal, viewed as point [P ] ∈ Spec A, the stalk of OX at the point [P ] is just

(OX )[P ] = AP , the local ring at P .

Deﬁnition 1 A local ringed space is a pair (X, FX ) where X is a topological space and FX is a sheaf of rings on X whose stalks are local rings.

If (X, FX ) is a local ringed space, the sheaf FX is called the structure sheaf of the local ringed space. For every point x ∈ X we also have a residue ﬁeld at x (namely the residue ﬁeld kx of the local ring Fx). Any global section of the structure sheaf f ∈ Γ(X, FX ) can —sort of—be thought of as a function on X with the strange properties that

• its value at a point x lies in the ﬁeld kx (this ﬁeld may change from x to x), • it is not necessarily determined by its values al all points x ∈ X,

• its germ at x ∈ X lies in Fx, and it is determined by its germ at each point x ∈ X.

3 Discuss the category of local ringed spaces; deﬁnition of morphism, examples.

Proposition 1 The contravariant functor A 7→ (X, OX ) where X = Spec A is a fully faithful functor from the category of rings (commutative, with unit) to the category of local ringed spaces.

To see this, ﬁrst a short argument that it is indeed a functor; that is a homomorphism of rings h : B → A gives rise, in a functorial manner to a homomorphism h :(X, OX ) → (Y, OY ) ) where X = Spec A and Y = Spec B .

Then, let A, B be rings, (X, OX ) and (Y, OY ) their corresponding local ringed spaces, and η : (X, OX ) → (Y, OY ) a morphism of local ringed spaces. Deﬁne h : B → A by setting

h = ηX : B = Γ(Y, OY ) −→ Γ(X, OX ).

What needs to be proved is:

Lemma 1 η = h

Proof:

Step 1. First let us show that these morphisms of local ringed spaces are equal on underlying topological spaces. So, let x ∈ X correspond to P ⊂ A and put h(x) = y and η(x) = y0 corresponding to Q, Q0 ⊂ B where Q = h−1(P ). Now since η is a morphism of local ringed spaces, we get when the dust clears that we have a commutative diagram

B - A

? ? - BQ0 AP

0 which by the deﬁnition of morphism of local ringed space brings the maximal ideal Q BQ0 in BQ0 onto the maximal ideal PAP in AP . This, by considering the kernel of the diagonal homomorphism 0 B → AP , gives Q = Q.

Step 2. If we have an open V ⊂ Y given, say, by an element f ∈ B (V = Yf ) let U ⊂ X be the inverse image under η, so—by Step 1—it is given by g = h(f) ∈ A (U = Xg) we see that

ηV : Γ(V, OY ) = Bf −→ Γ(U, OX ) = Ag is the morphism pictured by the bottom horizontal arrow in the commutative diagram

4 B - A

? ? - Bf Ag

but by the universal property of the left vertical arrow B → Bf there is only one way to make the diagram commutative; i.e., the bottom horizontal arrow is unique, so must be equal to the evident ring homomorphism i.e., hV .

Deﬁnition 2 An aﬃne scheme is a local ringed space of the form (X, OX ) where X = Spec A.

The categories of rings and of aﬃne schemes are “anti”-equivalent.

Deﬁne OX -module.

Let M be any A-module and for U = Xf set M(U) = Mf := Af ⊗A M. The exact same proof that you will be using to check Exercise 24 of [A-H] for the homework on Thursday will give you that the presheaf of A-modules U 7→ M(U) is—or rather extends to— a sheaf (of OX -modules).

Deﬁnition 3 A quasi-coherent sheaf over an aﬃne scheme X = Spec A is a sheaf of OX - modules M that arises from an A-module M by the construction above.

Project: Explain why in the world we’d use such terminology.

7 Patching via local ringed spaces

Deﬁnition 4 A prescheme is a local ringed space (X, OX ) for which there is an open cover {Ui}i∈I of X such that the restriction of (X, OX ) to Ui, i.e., (Ui, OX |Ui) is an aﬃne scheme for all i ∈ I; that is, for every i ∈ I, there is a ring Ai and an isomorphism of local ringed spaces

(Spec Ai, OSpec Ai ) ' (Ui, OX |Ui).

8 Morphisms of preschemes

Give explicit description

5 Proposition 2 If X is a prescheme and A a ring we have a natural bijection

Hom(X, Spec A) ' Hom(R, Γ(X, OX )).

Give proof.

Give good examples of preschemes; funny examples