Course Notes for Math 738 (Algebraic Geometry)

13. Varieties. Abstract Varieties. The sheaf of regular functions on an affine variety is defined, and abstract varieties are defined as “things” which are locally affine. We investigate some basic properties of varieties (which are analogues in algebraic geometry of differentiable or complex manfolds), define normality, and expand our definitions of closed immersions, dominant maps and finite maps to encompass all varieties. We finish with a discussion of normality and ramification.

Let V be an affine variety over a field k.

Definition: For each nonempty open subset U ⊆ V, the ring of regular functions on U is:

O_{V} (U) := {ϕ ∈ k(V) | U ⊆ dom(ϕ)}.

In other words, O_{V} (U) ⊆ k(V) is the subring consisting of all the rational functions which are “regular” (i.e. defined!) at all the points of U.

Proposition 13.1: Given V as above, the assignments:

(i) U ↦ O_{V} (U) ⊆ k(V) and

(ii) U ⊆ W ↦ ρ_{W,U} = the inclusion O_{V} (W) ⊆ O_{V} (U)

yield a sheaf (denoted O_{V} and called the sheaf of regular functions on V) of commutative rings with 1 on V. Moreover,

(a) If p ∈ V and m_{p} ⊂ k[V] is the corresponding maximal ideal, then the stalk of O_{V} at p (denoted O_{V,p}) is the local ring k[V]_{m_{p}} ⊆ k(V).

(b) If k is algebraically closed, then O_{V} (U_{f}) = k[V]_{f} for each f ∈ k[V].

Proof: Fix O_{V} (∅) = 0. It is immediate that O_{V} is a presheaf (see §0.1) and moreover that the additional sheaf requirements (see §0.2) hold.

Notice that the restriction maps ρ_{W,U} are all inclusions and each O_{V} (U) is contained in k(V), so it follows that the stalks (see §0.2 for the general definition) can also be realized as subrings of k(V) defined by:

O_{V,p} = {ϕ ∈ k(V) | p ∈ dom(ϕ)}

Thus ϕ ∈ O_{V,p} if and only if ϕ = \frac{g}{f} for some f ∉ m_{p}, so O_{V,p} = k[V]_{m_{p}}.

Part (b) is a generalization of Corollary 12.2. Clearly k[V]_{f} ⊆ O_{V} (U_{f}) since any rational function of the form \frac{g}{f^{n}} is regular at all points where f fails to vanish(!) On the other hand, if ϕ ∈ O_{V} (U_{f}), consider the ideal of denominators I of ϕ (as in the proof of 12.2). Then V (I) ⊆ V (f) by definition, so by the strong form of the Nulstellensatz, we have f^{n} ∈ I for some n, hence ϕ = \frac{g}{f^{n}} for some g.

Remark: Given the affine variety V, then each ϕ ∈ O_{V} (U) is in particular a well-defined function ϕ : U → k. If we endow k = A_{k}^{1} with the Zariski topology (i.e. the cofinite topology) and U with the induced topology, then it follows from Proposition 11.8 that each ϕ : U → k is continuous. That is, O_{V} ⊂ C_{V,k} is a subsheaf of the sheaf of continuous functions from V to k.

Definition: Suppose X is a topological space and O_{X} is a subsheaf of C_{X,k}. Then (X, O_{X}) is a prevariety over k if there exist:

(i) A covering of X by finitely many open subsets U_{1}, ..., U_{n} and

(ii) Homeomorphisms h_{i} : U_{i} → V_{i} to affine varieties over k satisfying h_{i}^{∗}(O_{V_{i}}(U)) = O_{X}(h_{i}^{−1}(U)) for all V_{i} and all open subsets U ⊂ V_{i}.

Given a prevariety X, an affine atlas of X will refer to any system {(U_{i}, h_{i})}_{i=1}^{n} of open subsets and homeomorphisms satisfying (i) and (ii).

Obviously affine varieties are prevarieties. Here’s another example:

Proposition 13.2: Suppose V is an affine variety and W ⊂ V is a nonempty open subset. Define the sheaf O_{W} by:

O_{W} (U) := O_{V} (U) for all open subsets U ⊆ W

with the same restriction maps. Then (W, O_{W}) is a prevariety.

Proof: We choose a particular V = V (f_{1}, ..., f_{l}) ⊂ A_{k}^{n}. Then W is a finite union: W = ∪_{i=1}^{m}U_{g_{i}} for g_{i} ∈ k[V] (we will be sloppy and also let g_{i} refer to the corresponding polynomials in k[x_{1}, ..., x_{n}]). Consider the affine “hyperbolas” V_{i} := V (f_{1}, ..., f_{l}, g_{i} · x_{n+1} − 1) ⊂ A_{k}^{n+1} and the rational maps

h_{i} : V − − > V_{i} given by h_{i}(p_{1}, ..., p_{n}) = (p_{1}, ..., p_{n}, \frac{1}{g_{i}(p_{1}, ..., p_{n})}).

The h_{i} are well-defined and continuous on U_{g_{i}} by Proposition 11.8. The projection maps π_{i} : V_{i} → U_{g_{i}} given by π_{i}(p_{1}, ..., p_{n+1}) = (p_{1}, ..., p_{n}) are inverses to the h_{i}, which are also continuous. So the h_{i} give homeomorphisms from U_{g_{i}} to V_{i}. Via the isomorphisms h_{i}^{∗} : k(V_{i}) ≅ k(V), we have:

h_{i}^{∗}(O_{V_{i}}(U)) = {h_{i}^{∗}(ϕ) ∈ k(V) | ϕ ∈ k(V_{i}) and U ⊂ dom(ϕ)}
= {ψ ∈ k(V) | h_{i}^{−1}(U) ⊂ dom(ψ)} = O_{V} (h_{i}^{−1}(U))

for all V_{i} and open subsets U ⊂ V_{i}. Thus W is a prevariety and {(U_{g_{i}}, h_{i})} is an affine atlas.

More generally than the proposition, we have:

Open subsets of prevarieties are prevarieties:

If X is a prevariety and W ⊂ X is an open subset with sheaf O_{W} defined as in the Proposition, then (W, O_{W}) is a prevariety.

Proof: Let {(U_{i}, h_{i} : U_{i} → V_{i})}_{i=1}^{n} be an affine atlas of X. Given the open subset W ⊂ X, consider the images W_{i} := h_{i}(W ∩ U_{i}) ⊂ V_{i}. Via the proposition, we know that each W_{i} is a prevariety, say with affine atlas {(U_{i,j}, h_{i,j} : U_{i,j} → V_{i,j})}_{j=1}^{n_{i}}. It is now an exercise to show that W is a prevariety with affine atlas {(h_{i}^{−1}(U_{i,j}), h_{i,j} ∘ h_{i})}.

The above was also true of premanifolds. However, here is a new property:

Irreducible closed subsets of prevarieties are prevarieties:

Given a closed subset Z ⊂ X of a prevariety, we endow Z with the induced topology. The sheaf O_{Z} is a little tricky to define. Here is the definition:

For each open set W ⊂ Z, there are, by definition, open sets U ⊂ X so that W = Z ∩ U. A continuous function f : W → k is defined to be in O_{Z}(W) if and only if there is such a U ⊂ X together with an open (finite) cover U = ∪_{i=1}^{n}U_{i} and regular functions f_{i} ∈ O_{X}(U_{i}) such that for each i, the f_{i} and f restrict to the same function from U_{i} ∩ Z to k.

We claim that if Z is irreducible, then (Z, O_{Z}) is a prevariety.

Proof: Let’s first see how this works when X = V is an affine variety. Then Z ⊂ V is an affine variety, with k[Z] ≅ k[V]/I(Z) since it irreducible and closed in V. As an affine variety, Z has the induced topology (§11). We want to see that its sheaf of regular functions (which I also denoted by O_{Z} but will forget for now) coincides with the sheaf O_{Z} defined above.

If ϕ ∈ k(Z) and W ⊂ dom(ϕ) (i.e. ϕ is regular on W), then because the ideal of denominators of ϕ is finitely generated, there exist a finite set of expressions ϕ ∼ \frac{f_{i}}{g_{i}} such that W = ∪_{i=1}^{m}(U_{g_{i}} ∩ W). Each f_{i}, g_{i} ∈ k[Z] lift to elements F_{i}, G_{i} ∈ k[V], which lets us lift \frac{f_{i}}{g_{i}} to ψ_{i} := \frac{F_{i}}{G_{i}} ∈ k(V) which automatically satisfy U_{G_{i}} ⊂ dom(ψ_{i}) and U_{G_{i}} ∩ Z = U_{g_{i}}. Choose any open U ⊂ V such that U ∩ Z = W, and let U_{i} = U_{G_{i}} ∩ U. It follows that the ψ_{i} are elements of O_{V} (U_{i}) with the desired properties, so that ϕ ∈ O_{Z}(W).

Conversely, given a collection of ψ_{i} ∈ O_{V} (U_{i}) determining an element ϕ ∈ O_{Z}(W), then by definition ψ_{i} ∈ k(V) with U_{i} ⊂ dom(ϕ_{i}). We may assume U_{i} ∩Z ≠ ∅, and it follows that the ψ_{i} determine well-defined elements ϕ_{i} ∈ k(Z) with U_{i} ∩ Z ⊂ dom(ϕ_{i}) and which coincide with ϕ as functions from U_{i} ∩ Z to k. But this means each ϕ_{i} = ϕ_{j} as a function from U_{i} ∩ U_{j} ∩ Z to k. Since open subsets are dense in an affine variety, and since rational functions are continuous on their domains, it follows that ϕ_{i} = ϕ_{j} for all i and j. In other words, ϕ comes from a single rational function ϕ = ϕ_{i} with W ⊂ dom(ϕ). Thus ϕ is a regular function on W.

Thus O_{Z} really is the sheaf of regular functions on the affine variety Z.

In general, let {(U_{i}, h_{i} : U_{i} → V_{i})} be an affine atlas for a prevariety X. If Z ⊂ X is closed and irreducible, let Z_{i} = h_{i}(Z ∩ U_{i}) ⊂ V_{i}. It is an easy exercise to see that each Z_{i} is closed and irreducible in V_{i}, hence an affine variety, and it follows as another exercise that Z is a prevariety with affine atlas: {U_{i} ∩ Z, h_{i}|_{U_{i}∩Z} : U_{i} ∩ Z → Z_{i}}.

Remark: It would be a lot simpler if we could define:

O_{Z}(W) := {f : W → k | ∃ g ∈ O_{X}(U) such that U ∩Z = W and f|_{W} = g|_{W} }

to avoid taking open covers of U. Unfortunately, the presheaf defined in this way may not satisfy property (ii) of sheaves, even when X is affine!

Next, we’d like to define a variety over k as a “Hausdorff” prevariety. Similarly, we’d like a notion of compactness. Unfortunately, because of the peculiar nature of the Zariski topology, we cannot use the topological definitions, as in the case of manifolds. Instead, we will use the category-theoretic replacements of “separatedness” and “properness”. To do this, we need to construct the category of prevarieties and to show that products exist in this category (see §0.3).

Definition: If (X, O_{Y}) and (Y, O_{Z}) are prevarieties over a field k, then a mapping Φ : X → Y is called a regular map if:

(i) Φ is continuous, and

(ii) Φ^{∗}(O_{Y} (U)) ⊆ O_{X}(Φ^{−1}(U)) for all open U ⊆ Y.

Example: (a) Let V, W be affine varieties and Φ : V − − > W a rational map such that V = dom(Φ). If ϕ ∈ k(W), if U ⊂ dom(ϕ), and if Φ^{−1}(U) ≠ ∅, then Φ^{∗}(ϕ) ∈ k(V) is well-defined and Φ^{−1}(U) ⊂ dom(Φ^{∗}(ϕ)), so Φ is a regular map of prevarieties. Conversely, if Φ : V → W is regular as a map of prevarieties, let x_{1}, ..., x_{n} ∈ k[W] ⊂ O_{W} (W) be the coordinate functions for W ⊂ A_{k}^{n}. Then Φ = (Φ^{∗}(x_{1}), ..., Φ^{∗}(x_{n})) and Φ^{∗}(x_{i}) ∈ O_{V} (V) ⊂ k(V). Thus Φ is a rational map which is everywhere defined. In case k is algebraically closed, this condition on a rational map is equivalent to being regular (exercise) so in that case, the two notions of a regular map coincide.

(b) If W ⊂ X is an open subset or Z ⊂ X is an irreducible closed subset, then the inclusion maps are regular maps of prevarieties, when W and Z are thought of as prevarieties as above. Indeed, the inclusions are continuous because the topology is induced. And the definitions of O_{W} and O_{Z} are set up so that the inclusions satisfy (ii) of the definition.

Definition: (a) The isomorphism classes of prevarieties over k with regular maps are a category, called the regular category of prevarieties over k.

(b) A regular map Φ : X → Y of prevarieties is an open (resp. closed) immersion if Φ(X) ⊂ Y is open (resp. closed and irreducible), and Φ is an isomorphism to its image. It is dominant if the image is dense in Y.

(c) A prevariety is called affine if it is isomorphic to an affine variety and quasi-affine if it admits an open immersion to an affine variety.

Examples: (a) The open subsets U_{g} ⊆ V of an affine variety are affine since they are isomorphic (by Proposition 13.2) to affine “hyperbolas.”

(b) Conversely if k is algebraically closed, W ⊆ V is a nonempty open subset, and if k[V] is a UFD, then the quasi-affine W is affine if and only if W = U_{g} for some g ∈ k[V]. (Exercise!) For example, A_{k}^{n} − {(0, ..., 0)} is not affine if n ≥ 2 and k is algebraically closed.

(c) If X is a prevariety and {U_{i}, h_{i} : U_{i} → V_{i}} is an affine atlas, then from the definition of an atlas, each h_{i}^{−1} : V_{i} → X is an open immersion.

Here is a useful lemma.

Lemma 13.3: Suppose X is a prevariety and X = ∪X_{i} is an open cover.

(a) If Y is a prevariety and Φ : X → Y is a mapping with the property that each Φ_{i} := Φ|_{X_{i}} : X_{i} → Y is regular as a map of prevarieties, then Φ is also regular as a map of prevarieties.

(b) If {U_{i}, h_{i} : U_{i} → V_{i}} is an affine atlas for X, then each composition h_{j} ∘ h_{i}^{−1} is the restriction of a birational isomorphism Φ_{i,j} : V_{i} − − > V_{j}.

Proof: (a) If U ⊂ Y is open, then each Φ_{i}^{−1}(U) ⊂ X_{i} is open, because the Φ_{i} are continuous, hence Φ^{−1}(U) = ∪Φ_{i}^{−1}(U) is open, so Φ is continuous. If ϕ ∈ O_{Y} (U), then each Φ_{i}^{∗}(ϕ) ∈ O_{X_{i}}(Φ_{i}^{−1}(U)) = O_{X}(Φ^{−1}(U) ∩ X_{i}) because the Φ_{i} are regular. But by the second sheaf property for O_{X}, this implies Φ^{∗}(ϕ) ∈ O_{X}(Φ^{−1}(U)). Thus Φ is regular.

(b) Consider the open subset h_{i}(U_{i} ∩U_{j}) ⊂ V_{i}. By the the definition of an affine atlas, the map h_{j} ∘ h_{i}^{−1} : h_{i}(U_{i} ∩ U_{j}) → V_{j} pulls back rational functions defined on V_{j} to rational functions on V_{i} defined on h_{i}(U_{i} ∩ U_{j}). If we choose a particular V_{j} ⊂ A_{k}^{n}, and let x_{1}, ..., x_{n} ∈ k[V_{i}] ⊂ O_{V_{i}}(V_{i}) be the coordinate functions, then the rational map Φ_{i,j} := ((h_{j} ∘h_{i}^{−1})^{∗}(x_{1}), ..., (h_{j} ∘h_{i}^{−1})^{∗}(x_{n})) is defined and agrees with h_{j} ∘h_{i}^{−1} on h_{i}(U_{i}∩U_{j}). It is a birational isomorphism, because it has a rational inverse, namely Φ_{j,i} (!)

Proposition 13.4: If V and W are affine varieties over k, then their product V × W in the category of affine varieties is also a product of prevarieties.

Proof: Suppose X is a prevariety and p : X → V and q : X → W are regular maps of prevarieties. We need to prove that (p, q) : X → V × W is also regular as a map of prevarieties.

Suppose {U_{i}, h_{i} : U_{i} → Y_{i}} is an affine atlas for X. It follows from the definition of an atlas that each p ∘ h_{i}^{−1} : Y_{i} → V and q ∘ h_{i}^{−1} : Y_{i} → W is a regular map of prevarieties, hence rational as a map of affine varieties (see the earlier example). But since products in the regular category of affine varieties agree with products in the rational category (HW §11) it follows that the maps (p∘h_{i}^{−1}, q ∘h_{i}^{−1}) : Y_{i} → V ×W are also rational maps, hence regular as maps of prevarieties. It follows that the restrictions (p, q)|_{U_{i}} : U_{i} → V ×W are regular maps of prevarieties. But now by Lemma 13.3(a), we have that (p, q) is itself regular, as desired.

Proposition 13.5: Products of prevarieties over k always exist. Moreover, the product agrees with the Cartesian product.

Proof: Suppose we are given prevarieties X and Y over k with affine atlases {S_{i}, g_{i} : S_{i} → V_{i}} and {T_{j}, h_{j} : T_{j} → W_{j}} respectively. I want to prove that there exists a topology and sheaf on the Cartesian product so that {S_{i}×T_{j}, (g_{i}, h_{j}) : S_{i}×T_{j} → V_{i}×W_{j}} is an affine atlas for X×Y. (Here V_{i}×W_{j} is the ordinary product of affine varieties and (g_{i}, h_{j})(x, y) := (g_{i}(x), h_{j}(y)).) By definition, then, (X × Y, O_{X×Y}) would be a prevariety, and moreover,

(i) The projection map π_{X} : X × Y → X restricts to the projections π_{i,j} : S_{i} × T_{j} → S_{i} ⊂ X. Each of these would be regular maps of prevarieties (they are really the projections V_{i} × W_{j} → V_{i} in disguise), and by Lemma 13.3(a), it would follow that π_{X} (and similarly π_{Y}) is regular.

(ii) If Z is a third prevariety with regular maps p : Z → X and q : Z → Y, let U_{i,j} := p^{−1}(S_{i}) ∩ q^{−1}(T_{j}) ⊂ Z. The “Cartesian” lift (p, q) : Z → X × Y restricts to (p, q) : U_{i,j} → S_{i} × T_{j} ⊂ X × Y which would all be regular maps by Proposition 13.4 (again, think of S_{i} × T_{j} as the affine V_{i} × W_{j} in disguise). But then by Lemma 13.3(a) it would follow that (p, q) is regular.

In other words, it would follow that (X × Y, O_{X×Y}) is the product!

Construction of the topology and sheaf: Let Φ_{i,k} : V_{i} − − > V_{k} and Ψ_{j,l} : W_{j} − − > W_{l} be the birational isomorphisms from Lemma 13.3(b). Since products agree in the regular and rational categories of affine varieties, there is a birational isomorphism (Φ_{i,k}, Ψ_{j,l}) : V_{i} × W_{j} − − > V_{k} × W_{l} whose domain includes the open subset dom(Φ_{i,k}) × dom(Ψ_{j,l}) and which is the map (Φ_{i,k}, Ψ_{j,l})(x, y) = (Φ_{i,k}(x), Ψ_{j,l}(y)) on that open set. We now construct the desired topology and sheaf by “gluing”. Namely, consider the data:

• The affine varieties V_{i} × W_{j}

• Open subsets U_{i,j,k,l} := (g_{i}, h_{j})(S_{i} ∩ S_{k}, T_{j} ∩ T_{l})

• Birational isomorphisms Φ_{i,j,k,l} := (Φ_{i,j}, Ψ_{k,l}) : V_{i} × W_{j} − − > V_{k} × W_{l}.

Let Z := (∐ V_{i} × W_{j})/ ∼, where p ∼ q if p ∈ U_{i,j,k,l} and Φ_{i,j,k,l}(p) = q. There is a natural bijection Z ↔ X × Y, and the desired properties are obtained from the Gluing Construction (HW # 8).

Definition: A variety over k is a separated prevariety over k. Regular maps of varieties are the same as for prevarieties.

Examples: (a) Affine varieties are varieties by 13.4 and HW §11.

(b) Quasi-affine varieties are varieties. If V is affine and W ⊂ V is open, then W × W ⊂ V × V is open (why?), and by the universal property, this must be the product. But the diagonal Δ_{W} ⊂ W × W is the intersection Δ_{V} ∩ (W × W), which is closed since V is separated.

(c) More generally, if Φ : X → Y is an open (resp. closed) immersion and Y is a variety, then the natural map X × X → Y × Y is an open (resp. closed) immersion and X is also a variety.

(d) Consider the “affine line with doubled origin” formed by applying the gluing construction (see HW # 8) to the following data:

• V_{1} = V_{2} = A_{k}^{1}

• U_{1,2} = U_{2,1} = A_{k}^{1} − {0},

• Φ_{1,2} = Φ_{2,1} = the identity map.

The resulting prevariety X is not a variety. If {U_{i}, h_{i} : U_{i} → A_{k}^{1}} is the associated affine atlas, then the two open immersions Φ_{i} := h_{i}^{−1} : A_{k}^{1} → X have the property that {Φ_{1} = Φ_{2}} = A_{k}^{1} − {0} is not closed in A_{k}^{1}.

Here is an important result from commutative algebra:

Finiteness of Integral Closure: Suppose A is an integral domain which is finitely generated as an algebra over a field k. If K ⊂ L is a finite field extension of the field of fractions of A, then the integral closure of A in L is a finitely generated A-module. In particular, it is finitely generated as an algebra over k.

(See Zariski-Samuel, Commutative Algebra I for a proof.)

Definition: A prevariety X is normal at p ∈ X if O_{X,p} is integrally closed. X is normal if it is normal at all points p ∈ X.

Proposition 13.6: Every prevariety X is normal at some point, and the set N(X) := {p ∈ X | X is normal at p} is open in X.

Proof: Let {U_{i}, h_{i} : U_{i} → V_{i}} be an affine atlas for X. Since N(X) = ∪N(U_{i}), it suffices to show that each N(U_{i}) ⊂ U_{i} is open and nonempty, i.e. that each N(V_{i}) ⊂ V_{i} is open and nonempty. So we can assume X = V is an affine variety. Let \overline{k[V]} be the integral closure of k[V] in k(V).

By the finiteness theorem, \overline{k[V]} is finitely generated as a k[V] module, say by ϕ_{1}, ..., ϕ_{m} ∈ k(V). If we write each ϕ_{i} = \frac{a_{i}}{b_{i}} and let f = ∏_{i=1}^{m} b_{i}, then \overline{k[V]} ⊂ k[V]_{f}, and it follows that k[V]_{f} is integrally closed, and each k[V]_{m_{q}} is integrally closed whenever f ∉ m_{q}. (See Exercise # 5.) By 13.1(a), U_{f} ⊂ N(V), which is therefore not empty. If p ∈ N(V), we can choose the b_{i} above so that b_{i} ∉ m_{p} since k[V]_{m_{p}} ⊂ k(V) is integrally closed, so if f is chosen as above, then f ∉ m_{p} and p ∈ U_{f} ⊂ N(V) which is then open.

Definition: Suppose X is a prevariety over k. Let

k(X) := {(U, ϕ) |U ⊂ X is open and nonempty, and ϕ ∈ O_{X}(U)}/ ∼

where (U, ϕ) ∼ (V, ψ) if ϕ = ψ as functions from U ∩ V to k. We call k(X) the field of rational functions on X.

Observations: (a) k(X) is a field. If X is affine, then this agrees with the earlier definition of the field of rational functions.

(b) If Φ : X → Y is a dominant regular map of prevarieties, then Φ^{∗} determines a well-defined injection of fields Φ^{∗} : k(Y) ↪ k(X). If Φ is an open immersion, then Φ^{∗} is an isomorphism of fields.

(c) There is a natural inclusion O_{X,p} ⊂ k(X) whenever p ∈ X. I claim that k(X) is the field of fractions of O_{X,p}. To see this, observe that there is an open immersion Φ : V → X of an affine variety with p ∈ Φ(V). Then Φ^{∗} : O_{X,p} ≅ k[V]_{m_{p}} and Φ^{∗} : k(X) ≅ k(V). The same may not be true of O_{X}(U) ⊂ k(X) for an open U ⊂ X. This integral domain can be quite small.

(d) O_{X}(U) = ∩_{p∈U}O_{X,p} ⊂ k(X). This is simply the following statement: “a function is defined on U iff it is defined at all the points of U”(!)

Proposition 13.1(b) will play a prominent role in the rest of this section. For this reason, we will now (and quite frequently in the future) assume k is algebraically closed. Varieties over general fields were studied in the early part of this century by similar techniques, but the theory of schemes, due to Grothendieck (where the analogue of 13.1(b) does hold) provides a much better framework for studying such varieties. For this reason, we will not worry about the loss of generality inherent in assuming k is algebraically closed, referring the interested reader to Hartshorne’s Algebraic Geometry for an exhaustive introduction to the scheme-theoretic approach.

Assume k is algebraically closed.

Proposition 13.7: If Φ : V ↪ X is an open immersion from an affine variety to a variety over k, then Φ(V) ⊂ N(X) if and only if k[V] is integrally closed. In particular, an affine variety V is normal iff k[V] is integrally closed.

Proof: By Proposition 13.1(a), the Nullstellensatz and HW §12 # 2(d),

k[V] = ∩_{p∈V} k[V]_{m_{p}} and each k[V]_{m_{p}} = O_{V,p} ≅ O_{X,Φ(p)}

so k[V] = \overline{k[V]} iff O_{X,q} is integrally closed for all q ∈ Φ(V) (HW §12 # 3(d)).

Definition: A regular map Φ : X → Y of varieties over any field is finite if there is an affine atlas {U_{i}, h_{i} : U_{i} → V_{i}} of Y with the following properties:

(a) Each Φ^{−1}(U_{i}) is affine, say by an isomorphism g_{i} : Φ^{−1}(U_{i}) → W_{i}.

(b) Each regular map Φ_{i} : h_{i} ∘ Φ ∘ g_{i}^{−1} : W_{i} → V_{i} is a finite regular map of affine varieties.

Proposition 13.8: If Φ : X → Y is a finite map of varieties over k and V ⊂ Y is open and affine, with the property that W := Φ^{−1}(V) ⊂ X is also affine, then O_{X}(W) is integrally dependent upon Φ^{∗}(O_{Y} (V)). Thus the new definition agrees with the old one if X and Y are affine varieties.

Proof: I’m going to be sloppy and write k[W] = O_{X}(W), k[V] = O_{Y} (V), assuming W and V are literally affine varieties. Similarly, instead of the affine atlas of the definition, let’s write V_{i} ⊂ Y and W_{i} ⊂ X instead of the open immersions. The only effect of this sloppiness is to reduce notation(!)

First, we assume V ⊂ V_{i} for some i. In that case, cover V = ∪U_{f_{j}} with basic open sets U_{f_{j}} ⊂ V_{i}, and further cover U_{f_{j}} = ∪U_{h_{j,l}} with basic open sets U_{h_{j,l}} ⊂ V. Since by assumption k[W_{i}] = O_{X}(W_{i}) is integrally dependent upon k[V_{i}] = O_{Y} (V_{i}), it follows that each k[W_{i}]_{Φ^{∗}(f_{j})} = O_{X}(Φ^{−1}(U_{f_{j}})) (13.1(b)) is also integrally dependent upon k[V_{i}]_{f_{j}} = O_{Y} (U_{f_{j}}). Let ϕ ∈ k[W] = O_{X}(W). Then each image of ϕ in O_{X}(Φ^{−1}(U_{f_{j}})) is integrally dependent upon O_{Y} (U_{f_{j}}), hence each further image of ϕ in k[W]_{Φ^{∗}h_{j,l}} is integrally dependent upon k[V]_{h_{j,l}}. Recall that the U_{h_{j,l}} cover V so that the h_{j,l} generate k[V] as an ideal (Nullstellensatz). It is an exercise now to see that ϕ is integral over k[V] = O_{Y} (V).

In general, cover V by affines U_{g_{m}} each of whose image lies in some V_{i}. Then by the above argument, each k[W]_{Φ^{∗}g_{m}} is integrally dependent upon k[V]_{g_{m}} and as before, the fact that the g_{m} generate k[V] as an ideal implies that k[W] is integrally dependent upon k[V].

Observations: (a) A finite map of varieties over k is closed and the preimage of every point is a finite set. This follows from Proposition 12.4 and the fact that Z ⊂ X is closed iff Z ∩ U_{i} is closed in U_{i} for each U_{i} in an open cover.

(b) The degree of a finite dominant map Φ : X → Y is well-defined. It is the degree of the field extension Φ^{∗}k(Y) ⊂ k(X), which is finite by Proposition 12.6 because it is the degree of Φ^{∗}k(V) ⊂ k(W) for any affine V with affine preimage W := Φ^{−1}(V) by Proposition 13.8.

Definition: A dominant, finite map Φ : X → Y of prevarieties is:

(a) ramified at p ∈ Φ^{−1}(q) ⊂ X if Φ^{∗}(m_{q}) does not generate m_{p}, where m_{q} and m_{p} are the maximal ideals in O_{Y,q} and O_{X,p}, respectively.

(b) ramified over q ∈ Y if it is ramified at some p ∈ Φ^{−1}(q).

Exercise: This agrees with the definition of ramification in §12.

Finally we answer a question that was begged by the examples of §12.

Proposition 13.9: Let Φ : X → Y be a finite and dominant map of degree d of prevarieties over k and q ∈ Y. Let e be the number of points in Φ^{−1}(q).

(a) If Φ is not ramified over q, then e ≥ d.

(b) If q ∈ N(Y), then e ≤ d, and e = d iff Φ is not ramified over q.

We will need two more tools from commutative algebra for this proof:

Nakayama’s Lemma: If A is a local ring with maximal ideal m and F : M → N is a homomorphism of finitely generated A-modules with the property that the induced map \overline{F} : M/mM → N/mN of vector spaces over k = A/mA is surjective, then F is also surjective.

Coefficients of the Minimal Polynomial: Suppose A is an integral domain which is integrally closed and K ⊂ L is a finite extension of the field of fractions of A. If b ∈ L is integrally dependent upon A, then the coefficients of the minimal polynomial of b are in A (as opposed to K).

Proof: By definition of finiteness, there is an affine neighborhhod V of q ∈ Y with the property that Φ^{−1}(V) ⊂ X is also affine. Moreover, if q ∈ N(Y), we can find a basic open set U_{f} ⊂ V which is a neighborhood of q contained in N(X) such that k[U_{f}] is integrally closed, by 13.6 and 13.7. Note that Φ^{−1}(U_{f}) = U_{Φ^{∗}f} ⊂ Φ^{−1}(V) is also affine. Thus in (a) we can assume X and Y are affine, and in (b) we can assume that k[Y] = \overline{k[Y]}.

(a) It follows from from 12.3 and 12.4 that if Φ is not ramified over q, then k[X]/I ≅ k^{e}, where I = m_{q}k[X]. Choose a_{1}, ..., a_{e} ∈ k[X] whose images generate k[X]/I. I would like to say that the a_{i} generate k[X] as a k[Y]- module, because the proof of 12.6 would imply that the a_{i} generate k(X) as a field extension of k(Y). Unfortunately, this may not be true. However, if we let M = k[Y]−m_{q} and localize everything at M, then Nakayama’s lemma tells us that the homomorphism of k[Y]_{M}-modules:

⊕_{i=1}^{e}k[Y]_{M} → k[X]_{M}; (b_{1}, ..., b_{e}) ↦ ∑ \frac{a_{i}}{1} Φ^{∗}(b_{i})

is surjective, because k[X]_{M}/m_{q}k[X]_{M} is isomorphic to k[X]/m_{q}k[X] and because the images of the a_{i} generate this vector space. In other words, the a_{i} generate k[X]_{M} as a module over k[Y]_{M}, and the same argument as in 12.6 shows that they generate the field extension, so d ≤ e.

(b) Suppose first that Φ is not ramified over q. We can find a ∈ k[X] which takes different values at each p ∈ Φ^{−1}(q). Since k[Y] is integrally closed, it follows from the coefficient lemma that the minimal polynomial satisfied by a for the field extension k(Y) ⊂ k(X) has the form:

x^{n} + Φ^{∗}(b_{f−1})x^{n−1} + ... + Φ^{∗}(b_{0}) = 0

where each b_{i} ∈ k[Y] and, of course, n ≤ d. If we evaluate each b_{i} at the point q ∈ Y, then we get a polynomial equation with coefficients in k which is satisfied by each of the e distinct values a(p) for p ∈ Φ^{−1}(q). But this means that e ≤ n ≤ d, and using (a) we get equalities all around.

Finally, suppose Φ is ramified at p ∈ Φ^{−1}(q). By definition, this means that Φ^{∗}(m_{q}) does not generate m_{p}. We can therefore find an a ∈ k[X] with the property that a(p) = 0, the image of a in O_{X,p} is not in the ideal generated by Φ^{∗}(m_{q}), and the values a(p) are distinct and nonzero for the other p ∈ Φ^{−1}(q). It follows that the minimal polynomial for a as above satisfies b_{0}(q) = 0 and b_{1}(q) = 0, hence the root at p counts for two zeroes, and we get e < d.

Eighth Homework Assignment

1. The topology on a prevariety X is called the Zariski topology. Prove:

(a) X is irreducible, so open subsets are dense.

(b) Every open cover of an open subset of X has a finite subcover.

(c) Every closed subset of X is a union of finitely many irreducibles.

2. If k is algebraically closed, show that the regular category of affine varieties over k is a full subcategory of the regular category of prevarieties over k.

3. Suppose V is an affine variety over an algebraically closed field and k[V] is a UFD. If W ⊂ V is open and affine as a prevariety, show that W = U_{g} for some g ∈ k[V]. Conclude that A_{k}^{n} − {(0, ..., 0)} is not affine if n ≥ 2.

4. (a) Prove that if Z ⊂ X is a closed irreducible subset of a prevariety, then the object O_{Z} constructed in the notes really is a sheaf.

Let Z = V (xy − zw) ⊂ A_{k}^{4}, and let ϕ ∼ \frac{x}{w} ∈ k(Z).

(b) Prove that k[Z] is not a UFD for any infinite field k.

(c) Find dom(ϕ), and let Y = Z − dom(ϕ). Suppose k is algebraically closed, and there is a polynomial f ∈ k[x, y, z, w] such that V (f) ∩ Z = Y. Conclude that y^{n} − fg ∈ ⟨xy − zw⟩ for some g ∈ k[x, y, z, w]. Conclude that there is a homogeneous F ∈ k[x, y, z, w]_{d} such that V (F) ∩ Z = Y. Derive a contradiction by considering the images of everything in V (xy −zw) ⊂ P_{k}^{3}. Conclude finally that the “simpler” construction of O_{Z} suggested in the notes does not produce a sheaf for this choice of Z ⊂ V = A_{k}^{4}.

5. If A is an integral domain and M ⊂ A is a multiplicative subset, prove that \overline{(A_{M})} = (\overline{A})_{M}, where the integral closures are taken in the field of fractions of A. Show how this completes the proof of Proposition 13.6.

6. If A ⊂ B is an inclusion of integral domains, if f_{1}, ..., f_{m} ∈ A generate A as an A-module, and if b ∈ B has the property that the image of b in each B_{f_{i}} is integral over A_{f_{i}}, conclude that b is integral over A. This is the key algebraic lemma we needed to complete Proposition 13.8.

7. Look up a proof of Nakayama’s Lemma and the result on coefficients of the minimal polynomial in a book on commutative algebra, then close the book and reproduce the proof in your own words. Or better yet, come up with proofs on your own.

8. The Gluing Construction: Suppose we are given the following data: • A finite collection of birational affine varieties V_{1}, ..., V_{m}.

• Nonempty open subsets U_{i,j} ⊆ V_{i} for each i, j = 1, ..., m (with U_{i,i} = V_{i}).

• Birational isomorphisms Φ_{i,j} : V_{i}−− > V_{j} with the following properties:

(o) U_{i,j} ⊆ dom(Φ_{i,j}) and Φ_{i,j}(U_{i,j}) = U_{j,i} for all i, j

(i) Φ_{i,i} : V_{i} → V_{i} is the identity map for all i. (ii) Φ_{i,j}^{−1} = Φ_{j,i} : U_{j,i} → U_{i,j} for all i, j and (iii) Φ_{i,j}(U_{i,j} ∩ U_{i,k}) = U_{j,i} ∩ U_{j,k} for all i, j, k and:

Φ_{j,k} ∘ Φ_{i,j} = Φ_{i,k} as functions from U_{i,j} ∩ U_{i,k} to V_{k}.

Let X = ∐_{i=1}^{m} V_{i}/ ∼ where p ∼ q ⇔ p ∈ U_{i,j}, q ∈ U_{j,i} and Φ_{i,j}(p) = q.

(a) Prove that ∼ is an equivalence relation, so the definition of X makes sense, and prove that each f_{i} : V_{i} → ∐ V_{i} → X is injective. Let U_{i} = f_{i}(V_{i}).

Put the topology on X generated by the f_{i}(U), where U range over all open subsets of V_{i}. We call this the Zariski topology on X.

(b) Prove that f_{i} : V_{i} → U_{i} are homeomorphisms, and let h_{i} = f_{i}^{−1}. Definition: The sheaf O_{X} is defined as follows:

(i) O_{X}(U) = {f : U → k | f|_{U∩U_{i}} ∘ f_{i} ∈ O_{V_{i}}(h_{i}(U ∩ U_{i})) for each i}. (ii) ρ is the restriction of functions.

(c) Prove that X is a prevariety and {U_{i}, h_{i}} is an affine atlas.

(d) Show that (o)-(iii) are satisfied for the data in Proposition 13.5, and finish the proof of that proposition.

(e) Given a prevariety X with affine atlas {U_{i}, h_{i} : U_{i} → V_{i}}, prove that the data V_{i}, U_{i,j} := h_{i}(U_{i} ∩ U_{j}), Φ_{i,j} := h_{j} ∘ h_{i}^{−1} (as in 13.3(b)) satisfy (o)-(iii), producing a prevariety isomorphic to Y.

(f) If X is a prevariety over an algebraically closed field, with affine atlas {U_{i}, h_{i} : U_{i} → V_{i}}, let W_{i} be the affine variety associated to the integral closure of k[V_{i}] in k(V_{i}), and let Φ_{i} : W_{i} → V_{i} be the resulting finite, dominant regular map. Show that the Φ_{i,j} induce birational isomorhisms from W_{i} to W_{j} and show that the data W_{i}, Φ_{i}^{−1}(h_{i}(U_{i} ∩ U_{j})), Φ_{i,j} produce a prevariety Y which is normal together with a finite, dominant regular map π : Y → X. This is called the normalization of X.