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INVERTIBLE SHEAVES
DAVID WHITE
- Defining Sheaves
Definition 1. Given a variety V , the ring of regular functions (or coordinate ring) is R(V ) = k[x 1 ,... , xn]/I(V ). The functions in R(V ) are called regular functions. Note that R(V ) is a finitely generated k-algebra.
Definition 2. Given a commutative ring R define the Spectrum as Spec(R) = {P ⊂ R | P is a proper, prime ideal}.
This can be augmented with the Zariski Topology in the following way: The closed sets in | Spec(R)| are those V such that there is some I ⊂ R s.t. V consists of all prime ideals in R containing I. Formally, V (I) = {x ∈ | Spec R| : f (x) = 0 ∀ f ∈ I} = {[P ] ∈ | Spec R| : P ⊃ I}. A point in | Spec(R)| is a prime ideal P strictly contained in R. We will denote this [P ] ∈ Spec(R). The only closed points are maximal ideals in R.
Define Γ(Df , OX ) = Rf to be the localization of R (this is also a sheaf). If P ∈ Spec(R) then the stalk at P is a locally ringed space which is a localization of R at P. Note that with this topology, | Spec(R)| is compact and has as basis {Df }f ∈R where Df = {prime ideals P such that f ∈/ P }. Also, Spec is a contravariant functor from the category of commutative rings to the category of topological spaces and all ring homomorphisms f : S → R induce continuous maps: Spec(f ) : Spec(S) → Spec(R). As a side note, | Spec(R)| is a structure sheaf, has a sheaf defined on it, and is a locally ringed space. We’ll talk more about this in a moment.
Examples:
(1) If R = Z then | Spec(R)| consists of primes union (0) (this is a special point because it’s dense).
(2) If R = C then | Spec(R)| consists of the complex line union (0). All points on the complex line are closed (because (x − a) is a maximal ideal) but (0) is dense in R.
(3) Spec(Cx) has only two points. It has the ideal (x) which is closed and it has the point (0) which is dense.
(4) If R = Zp then | Spec(R)| is just the point { 0 }
(5) Spec(Z[x]) has four types of points: (0), (p), irreducible polynomials f , and (f, p) where the polynomials are irreducible mod p.
For every f in R there is some related map on | Spec(R)|. Also, if [P ] ∈ | Spec(R)| then we have R → R/P , an integral domain. This maps to the Residue Field at x, K(x) which is just the quotient field of R/P. We define the value of f at [P ] as f (x) ∈ K(x)
Examples:
Date: April 2009. 1
2 DAVID WHITE
(1) If R = C, a ∈ R, and P = (x−a) then we have C[x] → C[x]/(x−a) ∼= C so the value of f (x) is f (a) ∈ C. If P = (0) then we have C[x] → C[x]/(0) → C[x] and f 7 → f + (0) 7 → f ∈ C[x]
(2) If R = Z, f = 15, and x = [P ] ∈ | Spec(R)| then if P = (0) we have Z → Z/(0) → Q and 15 7 → 15 7 → 15. If P = (p) for some prime p then we have Z → Z/(p) → Zp and 15 7 → 15 + (p) 7 → 15 (mod p)
We now define a sheaf of functions on X = | Spec(R)|, but note that shaves can be defined MUCH more generally on the open sets of any topology:
Definition 3. A sheaf is an assignment which assigns to every open U ⊂ X a set OX (U ) of regular functions on U (more generally, a set of “sections” on U ) s.t.
(1) V ⊂ U ⇒ OX (U ) → OX (V ) via a restriction of U to V
(2) We obtain a category X with open sets as objects and inclusions as arrows such that OX : Xop^ → Ring.
(3) Given an open cover U =
α Uα, if^ f^ ∈ OX^ (U^ )^ and^ f^ |Uα^ = 0^ for all^ α^ then^ f^ = 0. Also, if fα ∈ OX (Uα) and fα|Uα∩Uβ = fβ |Uα∩Uβ then there is some f ∈ OX (U ) such that f |Uα = fα for all α. This is referred to as the gluing property. If this property is omitted we get a pre-sheaf.
One immediate fact is that OX (X) = R. Also, given f ∈ R we can define Rf to be the localization where we invert all powers of f. We can then define Xf = | Spec R| \ V (f ) = {[P ] | f ∈/ P } ∼= | Spec Rf | and note that OX (Xf ) = Rf. These two facts determine OX. Note that if (X, OX ) is a ringed space then the sheaf OX is called the structure sheaf of X.
This is not a talk about sheaves in general, but I feel I must mention that a pre-sheaf can be turned into a sheaf by the functor of “sheafification” which is adjoint to the forgetful functor going from sheaves to pre-sheaves. I also feel I must define a stalk since you will hear people talking about them.
Definition 4. The stalk Fx of a sheaf F captures the properties of a sheaf ”around” a point x ∈ X as we look at smaller and smaller neighborhood of x. Formally, Fx := i−^1 F({x}), where i is the inclusion map: {x} → X.
The natural morphism F (U ) → Fx takes a section s in F (U ) to what is called its germ. This generalizes the usual definition of a germ.
- Schemes and Coherent Sheaves
Definition 5. An Affine Scheme is a pair (|X|, OX ) such that there exists a cover |X| =
α Uα and (Uα, OX |Uα ) ∼= (| Spec Rα|, OSpec Rα ). Here |X| is a topological space and OX is a sheaf of rings in |X|.
Perhaps an easier way to think of this is as a locally ringed space isomorphic to Spec(A) for some commutative ring A. A Scheme is a pair (|X|, OX ) which is locally an affine scheme (i.e. a locally ringed space which is locally isomorphic to the spectrum of a ring). Schemes form a category if we take as morphisms the morphisms of locally ringed spaces.
Example: The smallest non-affine scheme is |X| = {p, q 1 , q 2 } with open sets ∅, {p}, X 1 = {p, q 1 }, X 2 = {p, q 2 }, X and sheaf OX (∅) = 0, OX ({p}) = C[x], OX (X 1 ) = OX (X 2 ) = OX (X) = Cx
4 DAVID WHITE
Definition 9. A Weil divisor is a finite formal linear combination of codimension 1 subvarieties
A Cartier divisor is a collection of {Ui, fi} such that {Ui} is an open cover of X and fi ∈ K(Ui) the function field of rational functions.
Given a (Weil) divisor we may define a relation D < D′^ if D′^ − D has non-negative coefficients. Then we get a vector space of functions L(D) = {f /g | (f /g) > −D} = {f /g | (f /g) + D > 0 }. Finally, we get an equivalence relation D ∼ D′^ if there is some f /g with D + (f /g) = D′^ under the above notion of equality. It is an easy proposition to see that D ∼ D′^ ⇒ L(D) ∼ L(D′) and the converse also holds.
To patch with divisors we need fi/fj 6 = 0 anywhere on Ui ∩ Uj. With this condition, we can get a line bundle from a divisor by simply taking the cross product of our open sets with C. All we need is the patching function gα,β which is non-zero on Ui ∩ Uj. We can simply define gα,β = fα/fβ and we have this property.
To get an invertible sheaf from D = {Ui, fi} define L(D)(Ui) = OX (Ui)-module generated by 1/fi. This means it’s {f /fi | f ∈ OX (Ui)}. Because fi ∈ OX (Ui) we know that 1 is in our module.
Recall that D ∼ D′^ iff L(D) ∼= L(D′). This implies L(D 1 +D 2 ) ' L(D 1 )⊗L(D 2 ). We can therefore define the Picard group as the group of line bundles mod this isomorphism, as the group of divisors mod this isomorphism, or as the group of invertible sheaves mod this isomorphism.
On a smooth curve C we get a canonical divisor KC which captures tangency information and is associated to the cotangent bundle.
Now come some extra sections that I won’t have time to talk about but which I find interest- ing.
- Grothendieck Topology
Definition 10. A Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.
With this notion we can define sheaves on a category and get closer to derived categories and stacks.
The motivation for this concept is the Weil conjectures. Andr Weil proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they defined. His conjectures postulated that there should be a cohomology theory of algebraic varieties which gave number-theoretic information about their defining equations. This cohomology theory was known as the “Weil cohomology”, but using the tools he had available, Weil was unable to construct it.
A Grothendieck topology on C is a collection of sets (called coverings) for every object x which act as the categorical morphisms. This collection will be denoted Cov(x) and satisfies:
(1) For all objects x, {x
id → x} ∈ Cov(x) (2) If {Xα → X}α ∈ Cov(X) and Y → X is any arrow then the fiber product Xα ×X Y exists and is in Cov(Y ) for all α
(3) If {Xα → X} ∈ Cov(X) and for all α, {Xαβ → Xα}β ∈ Cov(Xα) then {Xαβ → Xα → X} ∈ Cov(X)
INVERTIBLE SHEAVES 5
Note that the fiber product is the limit of the following diagram:?^ //
?
Y
Xα //X
Example: If P and Q are properties of morphisms of schemes and Y is a fixed scheme then the P −Q site on Y is a category called the full subcategory of schemes over Y. It’s objects are P morphisms and its arrows are commutative diagrams (where f 1 , f 2 ∈ P ) X 1 // f 1
X 2
~~^ f^2 Y
This example leads to
(1) Big/small site of a topological space (Q is the property of being a homeomorphism onto an open subset) (2) Big/small Zariski site
(3) Etale Site (Q is etale maps)
(4) Big faithfully flat finite presentation site (Q is flat and finitely presented) (5) Lisse-Etale site (P is smooth, Q is etale)
A topos is a category equivalent to the category of sheaves on a site.
- Etale
If F is a sheaf over X, then the tale space of F is a topological space E together with a local homeomorphism π : E → X; the sheaf of sections of π is F. E is usually a very strange space, and even if the sheaf F arises from a natural topological situation, E may not have any clear topological interpretation. For example, if F is the sheaf of sections of a continuous function f : Y → X, then E = Y if and only if f is a covering map.
The tale space E is constructed from the stalks of F over X. As a set, it is their disjoint union and π is the obvious map which takes the value x on the stalk of F over x ∈ X. The topology of E is defined as follows. For each element s of F (U ) and each x ∈ U , we get a germ of s at x (i.e. an equivalence class of functions). These germs determine points of E. For any U and s ∈ F (U ), the union of these points (for all x ∈ U ) is declared to be open in E. Notice that each stalk has the discrete topology. Two morphisms between sheaves determine a continuous map of the corresponding tale spaces which is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.
This gives an example of an tale space over X. An tale space is a topological space E together with a continuous map π : E → X which is a local homeomorphism such that each fiber of π has the discrete topology. The construction above determines an equivalence of categories between the category of sheaves of sets on X and the category of tal spaces over X. The construction of an tale space can also be applied to a presheaf, in which case the sheaf of sections of the tale space recovers the sheaf associated to the given presheaf.
The map π is an example of what is sometimes called an tale map. “tale” here means the same thing as “local homeomorphism”. However, the terminology “tale map” is more common in contexts where the right analogue of a local homeomorphism of manifolds is not characterized by the property of being a local homeomorphism. This is the case in algebraic geometry.
