Products, Dimension

8 Products, Dimension

8.1 Product of affine spaces For any variety X, the set of all regular morphisms f:X m+n →C is in bijection with the set of all ordered pairs (g, h) where g : X →

and h : X m →C are regular morphisms. The projections p

n m+n

1 :C →C and

2 :C →C are regular, and the bijection is given in terms of the projections

m+n

Hom(X, C n ) → Hom(X, C ) × Hom(X, C ):f 7→ (p

m+n

1 ◦ f, p 2 ◦ f)

8.2 Categorical universal property The above shows that the ordered triple

(C n ,p

V of all quasi-affine and quasi-projective varieties.

m+n

1 ,p 2 ) is a product of C and C in the category

8.3 Tensor product: categorical characterisation In the category of com- mutative C-algebras, the tensor product has the following meaning. Let A and B

be commutative C-algebras, let A ⊗ C B their tensor product over C, which is again be commutative C-algebras, let A ⊗ C B their tensor product over C, which is again

C -algebra homomorphisms defined respectively by sending a 7→ a ⊗ 1 and b 7→ 1 ⊗ b. Then the triple (A ⊗ C 1 ,θ B, θ 2 ) is a co-product of A and B. That is, given any

triple (R, η 1 ,η 2 ) consisting of a commutative C-algebra R together with C-algebra homomorphisms η 1 :A → R and η 2 :B → R, there exists a unique C-algebra

homomorphism α : A ⊗ C B → R such that η i =α P

◦θ i for i = 1, 2. Note that α is

defined by the formula α( a k ⊗b k )=

η 1 (a k )η 2 (b k ).

We denote α by the symbol (η 1 ,η 2 ).

8.4 Tensor product of polynomial algebras Given m + n algebraically in- dependent variables x 1 ,...,x m ,y 1 ,...,y n over C, show that the pair of C-algebra homomorphisms η 1 : C[x 1 ,...,x m ] → C[x 1 ,...,x m ,y 1 ,...,y n ]:x i 7→ x i and η 2 :

C [y 1 ,...,y n ] → C[x 1 ,...,x m ,y 1 ,...,y n ]:y j 7→ y j induces an isomorphism of C- algebras (η ∼

1 ,η 2 ) : C[x 1 ,...,x m ] ⊗ C C [y 1 ,...,y n ] → C[x 1 ,...,x m ,y 1 ,...,y n ].

8.5 n Product of affine varieties Let X ⊂C and Y ⊂C be closed subvarieties, defined by radical ideals I ⊂ C[x 1 ,...,x m ] and J ⊂ C[y 1 ,...,y n ] respectively. Let

K ⊂ C[x 1 ,...,x m ,y 1 ,...,y n ] be the ideal generated by I ∪ J, where we regard I and J as subsets of C[x 1 ,...,x m ,y 1 ,...,y n ] under the inclusions C[x 1 ,...,x m ]֒ →

C [x 1 ,...,x m ,y 1 ,...,y n ]:x i 7→ x i and C[y 1 ,...,y n ]֒ → C[x 1 ,...,x m ,y 1 ,...,y n ]: y m+n

j 7→ y j . Then K is a radical ideal, and Z(K) ⊂C is the subset X ×Y⊂

×C n =C (where we have made the standard identification of C ×C with

n m+n m

C m+n ). This shows that X × Y is again an affine variety.

8.6 Universal property in

V Let p 1 :X × Y → X and p 2 :X × Y → Y again

denote the projections. These are regular, being restrictions of p m

2 :C →C . Show that the triple (X × Y, p 1 ,p 2 ) is a product of X and Y in the category

m+n

V of all quasi-affine and quasi-projective varieties.

1 # : O(X) → O(X × Y ) and p 2 : O(Y ) → O(X × Y ) be the pull-back homomorphisms. Then show that the induced homomorphism

8.7 # Correspondence with tensor product With notation as above, let p

(p #

1 ,p 2 ): O(X) ⊗ C O(Y ) → O(X × Y )

is an isomorphism. In terms of the defining ideals, this is the isomorphism

C [x 1 ,...,x m ]

C [y 1 ,...,y n ]

C [x 1 ,...,x m ,y 1 ,...,y n ]

(I ∪ J)

8.8 Note By the anti-equivalence of categories between affine varieties and finite- type reduced C-algebras, it follows that a product (X × Y, p 1 ,p 2 ) of X and Y in the category of affine varieties will exist and will correspond to O(X)⊗ C O(Y ). However, 8.8 Note By the anti-equivalence of categories between affine varieties and finite- type reduced C-algebras, it follows that a product (X × Y, p 1 ,p 2 ) of X and Y in the category of affine varieties will exist and will correspond to O(X)⊗ C O(Y ). However,

V of all quasi-affine and quasi-projective varieties.

8.9 n Product of quasi-affine varieties Let X ⊂C and Y ⊂C

be locally

closed subvarieties. Then show that the subset X m+n ×Y⊂C ×C =C is a locally closed subvariety of C m+n . Let p

1 :X × Y → X and p 2 :X × Y → Y again

denote the projections (these are regular, being restrictions of p m

1 :C →C and

m+n

2 :C →C ). Show that the triple (X × Y, p 1 ,p 2 ) is a product of X and Y in the category

m+n

V of all quasi-affine and quasi-projective varieties.

Product of projective spaces

Let V = C n with linear coordinates (x

0 ,...,x m ), and let W = C with linear coordinates (y 0 ,...,y n ). Let T = V ⊗ C W be the tensor product, with induced

linear coordinates (z i,j ) where 0 ≤ i ≤ m and 0 ≤ j ≤ n. If v = (x 0 ,...,x m ) ∈V and w = (y 0 ,...,y n ) ∈ W are any two vectors, then the vector v ⊗ w ∈ T has linear coordinates z i,j =x i y j in terms of the basis (v i ⊗w j ) of T where (v i ) and (w j ) are the standard bases of V and W . We thus get a morphism of varieties

t:V × W → T : (v, w) 7→ v ⊗ w

8.10 Exercise: Image of t Show that the image of the above map is the closed subvariety b X ⊂ T defined by the ideal I ⊂ C[z i,j ] generated by all the monomials of the form z i,j z k,ℓ −z i,ℓ z k,j where 0 ≤ i, k ≤ m and 0 ≤ j, ℓ ≤ n. In terms of tensor product, this in particular shows that the set of so called ‘decomposable tensors’ in

V ⊗ W (means those which can be expressed as v ⊗ w) is Zariski closed.

8.11 mn+m+n The Segre map s : P ×P →P If one of v and w is multiplied by a scalar, v ⊗ w gets multiplied by the same scalar. This shows that we get a

well-defined set-map

s:P mn+m+n ×P →P

given in terms of homogeneous coordinates by ( hx 0 ,...,x m i, hy 0 ,...,y n i) 7→ hx 0 y 0 ,...,x 0 y n ,x 1 y 0 ,...,x 1 y n ,...,x m y n i

Note that in the above, P n ×P just denote the product set (we have not yet given it a topology or the structure of a variety). Let Z mn+m+n ⊂P denote the closed subvariety defined by the homogeneous ideal

I ⊂ C[z i,j ] defined above.

8.12 Proposition With notation as above, the map s is injective with image Z,

− 1 so it induces a bijection ψ : P m ×P → Z. The maps π

1 =p 1 ◦ψ :Z →P and

2 =p 2 ◦ψ :Z →P are regular morphisms, where p 1 and p 2 are the projections

(regarded as set-maps). The resulting triple (Z, π n

1 ,π 2 ) is a product of P and P in the category V.

In view of the above proposition, we will define the variety P n ×P to be the projective variety Z mn+m+n ⊂P constructed above, together with projections π

and π 2 . The bijection ψ and the equations p i =π i ◦ ψ show that if we forget the variety structure (that is, if we apply the forgetful functor from varieties to sets) the

product of P n and P as varieties reduces to the product as sets.

8.13 n Exercise Show that if m, n ≥ 1 then the Zariski topology on P ×P =Z is not the product of the Zariski topologies on the factors.

8.14 n Product of quasi-projective varieties Let X ⊂P and Y ⊂P be

locally closed subvarieties. Then show that the subset X n ×Y⊂P ×P is locally closed, and the triple (X × Y, p 1 ,p 2 ), where the π i are restrictions to X × Y of the

projections P n ×P →P and P ×P →P , has the universal property of product of X and Y in the category V.

V Let X and Y be objects of V. As shown earlier, any locally closed subvariety X m ⊂C is isomorphic to the locally closed subvariety

8.15 Product of varieties in

0 (X) ⊂P . As a product is defined by its universal property, this allows us to replace any variety by an isomorphic variety and thereby assume that both X and Y are quasi-projective varieties. As we have constructed a product of quasi-projective varieties in

V, we get a product of any two objects of V.

V, show that the image of the morphism

8.16 Exercise: Separatedness If X is any object of

(id X , id X ):X → X × X : P 7→ (P, P ) is a closed subset ∆ X ⊂ X × X. This property is expressed by saying that varieties

in

V are separated. Note however that the topology on X × X is generally not the product topology, so ∆ X can be closed without X being Hausdorff. The property of separatedness is partly a substitute for the property of being Hausdorff in the

geometry of varieties.

Dimension of varieties

8.17 Definition Let X be a variety. The dimension of X is the supremum of all integers n such that there exists a chain of distinct irreducible closed subsets

X 0 ⊂X 1 ⊂...⊂X n = X. The dimension of the empty variety is −∞.

8.18 Caution The above definition only used the topology on X and not anything about regular functions, so the definition makes logical sense for arbitrary topological spaces. But it does not give the ‘correct’ notion of dimension for the usual topological spaces that are important in algebraic topology or differential geometry or analysis. For example, instead of giving the answer n for the dimension of the Euclidean space R n , it will give the answer 0.)

8.19 Dimension of an affine variety as Krull dimension of O(X) Recall that if X n ⊂C is an affine variety, then irreducible closed subsets of X correspond

to prime ideals in O(X). From this conclude that dim(X) is equal to the Krull dimension of the ring O(X).

8.20 n Exercise Determine the dimension of the affine variety C .

8.21 n Exercise Show that P is irreducible.

8.22 n Exercise Determine the dimension of the projective variety P . Is it equal to the Krull dimension of n O(P )?

8.23 n Exercise Show that for any quasi-affine variety X ⊂C , dim(X) = dim(X) where X n ⊂C is the closure of X. Show that for any quasi-projective variety

X n ⊂P , dim(X) = dim(X) where X ⊂P is the closure of X. Conclude in particular that varieties are finite dimensional.

8.24 Local nature of dimension We define the dimension of a variety X at

a point P to be the infimum of the dimensions of open neighbourhoods of P in

X. Show that this is the same as the maximum of the dimensions of irreducible components of X which contain P . Show that the dimension of X is the maximum of the local dimensions of X at all points.

8.25 Krull dimension of local ring Show that the local dimension of X at P equals the Krull dimension of the local ring of germs O X,P .

Function field of an irreducible variety

8.26 Rational functions Let X be an irreducible variety. Consider all ordered pairs (V, f ) where V ⊂ X is a non-empty open subset and f ∈ O(V ). We say that two such pairs (V, f ) and (W, g) are equivalent if f |

V ∩W (notice that V ∩W is again non-empty by irreducibility of X). Let C(X) denote the set of equivalence classes of such pairs. Each element of C(X) is called a rational function on X. We

V ∩W =g | V ∩W =g |

and their product

(V, f )(W, g) = (V ∩ W, (f|

V ∩W )(g |

V ∩W ))

which can be shown to be well-defined operations on C(X).

8.27 Exercise Prove that the above sum and product are well-defined and make

C (X) and extension field of C.

8.28 Exercise Prove that if X is an irreducible variety and V ⊂ X a non-empty open subset, then V is irreducible and restriction of functions gives an isomorphism

C ∼ (X) → C(V ) over C of the function fields.

8.29 Dimension of an irreducible variety as transcendence degree Show that the dimension of an irreducible variety X is the same as the dimension of any non-empty open affine subvariety of X, which in turn equals the transcendence degree over C of its function field. Hence dimension of an irreducible variety X is the same as the transcendence degree of C(X) over C.

8.30 Application of Krull’s principal ideal theorem Let X be an irreducible variety, and let f ∈ O(X) − {0}. Let Y ⊂ X denote the closed subset consisting of all P ∈ X such that f(P ) = 0. Let Z be any irreducible component of Y . Then show that dim(Z) = dim(X) − 1.

Is the above conclusion true without the hypothesis that X is irreducible?