Topic: Abstract varieties
Mitchell Faulk
March 7, 2016
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Spaces with functions
Definition 1.1. Let X be a topological space. We say that X is a space with functions if there is an
assignment OX which assigns, to each open subset U , a C-algebra OX (U ) of C-valued functions on U , called
regular functions on U , satisfying the following properties
(i) If Uα is an open cover of U , then a C-valued function f : U → C is regular on U if and only if each
restriction f |Uα is regular on Uα .
(ii) If f is regular on U , then the subset Uf = {u ∈ U : f (u) 6= 0} is open in U and
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f
is regular on Uf .
We sometimes call the assignment OX a sheaf.
Example 1.2. For example, if X is a topological space, then taking OX (U ) to be the set of continuous
functions U → C endows X with the structure of a space with functions.
Example 1.3. As another example, let V be a subspace of a space with functions (X, OX ). Then define
an assignment OV by the rule OV (U ) = OX (U ). This is well-defined because for the following reason: by
definition of the subspace topology if U is an open subset of V , then there is an open subset U 0 of X satisfying
U = U 0 ∩ V , and so U is open in X as well. We sometimes write OX |V for the assignment OV .
Definition 1.4. If (X, OX ) and (Y, OY ) are two spaces with functions, a morphism from X to Y consists
of a continuous map ϕ : X → Y such that whenever V is an open subset of Y and f is a regular function
on V , the composition f ◦ ϕ is a regular function on ϕ−1 (V ), that is—more succinctly—if f ∈ OY (V ), then
f ◦ ϕ ∈ OX (ϕ−1 (V )). Informally, one says that ϕ “pulls back regular functions.”
Example 1.5. Let X, Y be topological spaces, and let OX and OY be the sheaves of continuous C-valued
functions. We claim that any continuous map ϕ : Y → X is a morphism from (X, OX ) to (Y, OY ). Indeed,
whenever V is an open subset of Y and f : V → C is a continuous function, note that the composition f ◦ ϕ
is a continuous function on the open set ϕ−1 (V ). So ϕ indeed pulls back regular functions.
1.1
Affine varieties
Other examples of spaces with functions are affine algebraic subsets of Cn , which we describe now.
Recall 1.6. The Zariski topology on Cn is the one where the closed subsets are the affine algebraic
subsets. For an affine algebraic subset X ⊂ Cn , the Zariski topology on X is just the subspace topology
induced by the Zariski topology on Cn .
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Note 1.7. If X ⊂ Cn is an affine algebraic subset, any f ∈ R(X) determines an open set Uf defined by
Uf = {x ∈ X : f (x) 6= 0} = X \ V (f ).
Note that
Uf g = Uf ∩ Ug .
One can show that the collection of open sets {Uf } forms a basis for the Zariski topology on X.
Definition 1.8. Let X ⊂ Cn be an affine algebraic subset. We define a sheaf of regular functions OX on
X in the following way. If U is an open subset of X, we say that a C-valued function f : U → C is a
regular function if there are polynomials g, h ∈ C[x1 , . . . , xn ] such that h never vanishes on U and such
that f = g/h on U .
Example 1.9. Let X = C, and let a be a point of C. Consider the open subset Uf determined by the
polynomial f = x − a ∈ C[x]. This means that
Uf = {x ∈ C : x 6= a}.
We claim that all regular functions on Uf take the form
g(x)
(x − a)n
for g(x) ∈ C[x] and n > 0.
Indeed, if g/h is a regular function, then the requirement is that h never vanishes on Uf , meaning that h
has no possible roots other than a. We can write the ring of regular functions in the form
OX (Uf ) = C[x][f −1 ]
where f = x − a. Another way of writing this
OX (Uf ) = Sf−1 C[x]
where Sf is the set Sf = {f n : n > 0} of powers of f . This is the so-called “localization of the ring C[x] at
Sf .”
Fact 1.10. More generally, let X ⊂ Cn be an affine algebraic subset and let f be an element of the coordinate
ring R(X). Then f determines an open set
Uf = {x ∈ X : f (x) 6= 0}.
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The ring of regular functions on Uf is then given by
OX (Uf ) = R(X)[f −1 ] = Sf−1 R(X)
where Sf is the set Sf = {f n : n > 0} of powers of f . In particular, we find that
OX (X) = R(X)
when f = 1.
Fact 1.11. If (X, OX ) and (Y, OY ) are affine varieties such that R(X) ' R(Y ), then (X, OX ) and (Y, OY )
are isomorphic. This is saying, in some sense, that for an affine variety X, the sheaf OX is completely
determined by the “ring of global sections OX (X) ' R(X).”
1.2
Pre-varieties
Definition 1.12. A space with functions (X, OX ) is called a pre-variety if there is an open cover Uα of
X such that each (Uα , OX |Uα ) is isomorphic (as a space with functions) to an affine variety. A morphism of
pre-varieties is just a morphism of the spaces with functions.
Example 1.13. Let X = C∗ be the complex plane without the origin. We claim that X is a pre-variety.
Indeed, let Y be the affine variety Y = {xy = 1} ⊂ C2 . Note that projection onto the first coordinate
π : Y → X gives a homeomorphism between X and Y (when both are equipped with the Zariski topology).
An inverse of π is the map ϕ : X → Y given by ϕ(t) = (t, t−1 ).
Under a suitable choice of sheaf OX , we claim that π is an isomorphism of varieties. Indeed, for open
V ⊂ X, we just take
OX (V ) = OY (π −1 (V )).
In particular, then, we have
OX (X) = OY (Y ) = R(Y ) = C[x, y]/(xy = 1) ' C[x, x−1 ].
Example 1.14. Let X = C ∪ {0∗ } be the “complex line with the origin doubled.” Then X is a pre-variety.
Indeed we can cover X by X1 = C and X2 = C \ {0} ∪ {0∗ }, which are both affine varieties.
From another viewpoint, let us take X1 = X2 = C. We can view the pre-variety C∗ as a subset of both
X1 and X2 in a natural way, and the identity map idC∗ : C∗ → C∗ is an isomorphism of pre-varieties. We
can then form the topological space
X = X1 ∪idC∗ X2 ≡ (X1 ∪ X2 )/ ∼
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where ∼ is determined by x ∼ x for x ∈ C∗ . So we have a diagram
C∗
id
i1
id
X2
/ X1
i2
/X
−1
and a subset U ⊂ X is open if and only if i−1
1 (U ) ⊂ X1 and i2 (U ) ⊂ X2 are both open. For any open set
U ⊂ X, we can define
OX (U ) = {(f1 , f2 ) : fj ∈ OXj (i−1
j (U )), f1 |i−1 (U )∩C∗ = f2 |i−1 (U )∩C∗ }.
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2
In this way, X is a pre-variety.
Example 1.15 (Complex projective space). Again let X1 = X2 = C. But now consider the isomorphism of
pre-varieties J : C∗ → C∗ given by J(z) = z −1 , and glue X1 to X2 along J:
X = X1 ∪J X2 .
This means that X is the quotient X1 ∪ X2 / ∼ where x1 ∼ J(x1 ) for each x1 ∈ C∗ ⊂ X1 . We have natural
maps ij : Xj → X as above. A sheaf is determined by
OX (U ) = {(f1 , f2 ) : fj ∈ OXj (i−1
j (U )), f1 |i−1 (U )∩C∗ = f2 ◦ J|i−1 (U )∩C∗ }.
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2
One can check that X is homeomorphic to CP1 , and thus, this construction gives a pre-variety structure on
CP1 .
Let us consider the coordinate ring of CP1 , given by OX (X). A regular function in OX (X) consists of
P
a pair (f1 , f2 ) where fj ∈ OXj (Xj ) ' R(Xj ) ' C[x] such that f1 |C∗ = f2 ◦ J|C∗ . But if f1 = n an xn and
P
f2 = m bm xm , then
X
f2 ◦ J =
bm x−m
m
and so f1 |C∗ = f2 ◦ J|C∗ if and only if f1 and f2 are both constant polynomials, equal to the same constant.
It follows that we can identify OX (X) with C. This says that the only global regular functions on CP1 are
constants.
Example 1.16 (Gluing pre-varieties). More generally (X1 , OX1 ) and (X2 , OX2 ) be pre-varieties and let
U1 ⊂ X1 and U2 ⊂ X2 be sub pre-varieties, with ϕ : U1 → U2 an isomorphism of pre-varieties. We can form
the topological space X = X1 ∪ϕ X2 = (X1 ∪ X2 )/ ∼ where x1 ∼ ϕ(x1 ) for each x1 ∈ U1 . We have natural
maps ij : Xj → X so that a subset U ⊂ X is open if and only if each i−1
j (U ) ⊂ Xj is open. A sheaf OX on
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X is described by
OX (U ) = {(f1 , f2 ) : fj ∈ OXj (i−1
j (U )), f1 |i−1 (U )∩U1 = f2 ◦ ϕ|i−1 (U )∩U2 }.
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2
Then X is a pre-variety “obtained by gluing X1 and X2 along U1 and U2 via the isomorphism ϕ.”
Example 1.17 (Product of pre-varieties). The product of pre-varieties is a pre-variety. Indeed, suppose
that (X1 , OX1 ) and (X2 , OX2 ) are pre-varieties. Then there are open covers X1 = ∪α X1α and X2 = ∪β X2β
with each X1α and X2β isomorphic to an affine variety. The collection {X1α × X2β } forms an open cover of
the product X, with each element of the collection being isomorphic to an affine variety. A topology on X
is defined by saying that a subset U ⊂ X is open if and only if each intersection U ∩ (X1α × X2β ) is open in
X1α × X2β . Moreover a function f : U → C is regular if and only if each restriction f |X α ×X β is.
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Fact 1.18. If X and Y are affine varieties, then
R(X × Y ) ' R(X) ⊗C R(Y ).
where ⊗C denotes the tensor product as C-algebras.
1.3
Projective varieties
Let π : Cn+1 \ {0} → CPn denote the natural projection map.
Definition 1.19. A polynomial f ∈ C[x0 , . . . , xn ] (note that there are n + 1 variables) is called homogeneous of degree k if f (λx) = λk x for each scalar λ ∈ C and each point x ∈ Cn+1 . A homogeneous
polynomial f ∈ C[x0 , . . . , xn ] determines a well-defined subset Z(f ) of CPn described by
Z(f ) = {π(x) ∈ CPn : f (x) = 0 for x ∈ Cn+1 }.
In a similar way, a collection T of homogeneous polynomials in C[x0 , . . . , xn ] determines a subset Z(T ) of
CPn given by their common zero locus. A projective variety is an irreducible subset of CPn of the form
Z(T ) for some collection of homogeneous polynomials T ⊂ C[x0 , . . . , xn ].
Definition 1.20. A projective variety is a space with functions in a natural way. Let X ⊂ CPn be a
projective variety and U ⊂ X an open subset. Say that f : U → C is a regular function if there are
homogeneous polynomials g, h ∈ C[x0 , . . . , xn ] of the same degree such that h never vanishes on U and
f = g/h on U .
Lemma 1.21. A projective variety X ⊂ CPn is a pre-variety.
Proof. Say that X = Z(f1 , . . . , fk ) for some homogeneous f1 , . . . , fk ∈ C[x0 , . . . , xn ]. (We may assume the
list of polynomials is finite because the ring C[x0 , . . . , xn ] is Noetherian.)
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Consider the open subset Uj of CPn given by Uj = {[x0 , . . . , xn ] : xj 6= 0}. We claim that the open
subset Uj ∩ X is affine.
Indeed, using the coordinates
ϕj : Uj → Cn
x0
xj−1 xj+1
xn
[x0 , . . . , xn ] 7→
,...,
,
,...,
xj
xj
xj
xj
−1
n
we see that X ∩ Uj corresponds bijectively to the affine variety V (f1 ◦ ϕ−1
j , . . . , fk ◦ ϕj ) ⊂ C . Note that
fi ◦ ϕ−1
is just the polynomial in C[x0 , . . . , xj−1 , xj+1 , . . . , xn ] obtained from fi by setting xj = 1. The map
j
ϕj is continuous with a continuous inverse and hence a homeomorphism.
Moreover the map ϕj is a morphism of spaces with functions for the following reason: if g is a polynomial
in C[x0 , . . . , xj−1 , xj+1 , . . . , xn ], then g ◦ ϕj is a quotient of polynomials each of which is homogeneous of the
−1
same degree (Why?), so if g/h is regular on some open subset of U ⊂ V (f1 ◦ϕ−1
j , . . . , fk ◦ϕj ), then g/h pulls
back to some quotient of polynomials each of which is homogeneous of the same degree and which does not
vanish on ϕj−1 (U ). Finally the inverse ϕ−1
is a morphism of spaces with functions for the following reason:
j
the pullback of f ◦ϕ−1
j of a polynomial in C[x0 , . . . , xn ] is just the polynomial in C[x0 , . . . , xj−1 , xj+1 , . . . , xn ]
obtained from f by setting xj = 1. We conclude that ϕj provides an isomorphism of X ∩ Uj onto the affine
−1
variety V (f1 ◦ ϕ−1
j , . . . , fk ◦ ϕj ).
1.4
Varieties
Definition 1.22. A pre-variety (X, OX ) is called separated if the diagonal ∆ is closed in X × X.
Remark 1.23. Remember that a topological space X is Hausdorff if and only if the diagonal ∆ is closed
in X × X with the product topology. However, for pre-varieties, we do not take the product topology on
X × X, so separated is not the same notion as Hausdorff. However, the definition of separated suggests that
it is some sort of analogous property to Hausdorff.
Example 1.24. The “complex line with the origin doubled” is not separated. Indeed, if X denotes this
pre-variety, then X × X is the complex plane with axes doubled and four origins. The diagonal ∆ has two
of these origins, but its closure must contain all 4, and hence the diagonal is not closed.
Fact 1.25.
(a) Any affine variety or projective variety is separated.
(b) A product of separated pre-varieties is separated.
Definition 1.26. A pre-variety (X, OX ) is called a variety if it satisfies the following extra conditions
(i) X is irreducible as a topological space
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(ii) X is separated.
Example 1.27. The disjoint union C ∪ C is a separated pre-variety. However, it is not irreducible as a
topological space, because it is disconnected, so it is not a variety.
Fact 1.28.
(a) Any affine variety or projective variety is a variety.
(b) Any product of varieties is a variety.
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