8.6 Vector Bundles and regular schemes

Most of the examples of vector space schemes that we have seen so far are vector bundles; these are vector space schemes that are ``locally'' isomorphic to $\mathbb {V}$ⁿ for some fixed n. In other words, E$\to$X is a vector bundle if there is a collection of open subschemes U_i $\subset$ X such that $\cup$ U_i(A) = X(A) for every finite local ring and E×_XU_i is isomorphic to $\mathbb {V}$ⁿ×U_i as a vector space scheme over U_i for every i. A collection of open sets satisfying the first property is referred to as an open cover of X. The vector bundle $\mathbb {V}$^q×X is called the trivial vector bundles on X. The number n is called the rank of the vector bundle.

Recall, that L was defined as the subscheme of $\mathbb {V}$^{p + 1}×$\mathbb {P}$^p consisting of pairs of tuples (b₀,..., b_p;a₀ : ^... : a_p) such that a_ib_j = a_jb_i for all i and j between 0 and p. An open cover of $\mathbb {P}$^p is given by the open subschemes U_i = V(0;X_i). We see easily that L×_{$\mathbb {P}$^p}U_i is given by the equations b_j = (a_j/a_i)b_i since a_i is a unit. Thus the map from $\mathbb {G}$_a×U_i to L×_{$\mathbb {P}$^p}U_i given by

(c;a₀ : ^... : a_p) $$\mapsto$$ $$\left(\vphantom{ (a_0/a_i)c,\dots,(a_p/a_0)c ; a_0:\cdots:a_p }\right.$$(a₀/a_i)c,...,(a_p/a₀)c;a₀ : ^... : a_p$$\left.\vphantom{ (a_0/a_i)c,\dots,(a_p/a_0)c ; a_0:\cdots:a_p }\right)$$

gives an isomorphism. Thus L is a vector bundle of rank 1 or a line bundle. Recall also that H was defined as the subscheme of $\mathbb {P}$^{p + 1} which is the complement of the point (0 : ^... : 0 : 1). The morphism H$\to$$\mathbb {P}$^p is the projection away from this point and the zero-section is V(X_{p + 1}). For each i between 0 and p we have a natural homomorphism s_i : $\mathbb {G}$_a×$\mathbb {P}$^p$\to$H given by

(c;a₀ : ^... : a_p) $$\mapsto$$ (a : 0 : ^... : a_p : c ^. a_i)

Note that this is an isomorphism outside the hyperplane V(X_i); in other words this is an isomorphism on U_i = V(0;X_i). Thus H is also a line bundle.

The automorphisms of the vector space $\mathbb {V}$ⁿ are given as the closed subscheme GL_n of $\mathbb {A}$^{n2 + 1} consisting of ((X_ij)_{i, j = 1}ⁿ, T) such that det((X_ij))T = 1. For any scheme X, any automorphism of the vector space scheme $\mathbb {V}$ⁿ×X corresponds naturally to a morphism g : X$\to$GL_n. Moreover, it is clear that GL_n is a group scheme.

Now let E be a vector bundle over a scheme X, {U_i} be an open cover of X and $\phi_{i}^{}$ be the isomorphism of vector space schemes $\phi_{i}^{}$ : E×_XU_i$\to$$\mathbb {V}$ⁿ×U_i. For any i and j it is clear that we get a morphism $\phi_{ij}^{}$ : U_i $\cap$ U_j$\to$GL_n by comparing the two isomorphisms of E×_X(U_i $\cap$ U_j) with $\mathbb {V}$ⁿ×(U_i $\cap$ U_j). These morphisms satisfy $\phi_{ij}^{}$ ^. $\phi_{jk}^{}$ = $\phi_{ik}^{}$ on U_i $\cap$ U_j $\cap$ U_k. Conversely, it is clear that we can use such a collection of morphisms $\phi_{ij}^{}$ : U_i $\cap$ U_j$\to$GL_n to construct a vector bundle on X by patching together the vector bundles $\mathbb {V}$ⁿ×U_i. More generally, we can easily show that for any vector space scheme V on X, the group scheme GL_n operates on V^{$\oplus$ n}. Thus we can use the $\phi_{ij}^{}$ to patch together V^{$\oplus$ n}×_XU_i to obtain a vector space scheme. This vector space scheme is denoted E $\otimes$ V and is called the tensor product of E with V. It is clear that $\mathbb {V}$¹ $\otimes$ V = V. One can show that H_n = H^{$\otimes$ n} and H $\otimes$ L = $\mathbb {V}$¹×$\mathbb {P}$^p.

As before we define the K-group of vector bundles of a scheme S as the quotient K₀(S) of the free abelian group on isomorphism classes of vector bundles by the subgroup generated by relations of the form [V] + [U] - [W] where 0$\to$V$\to$W$\to$U$\to$ 0 is an exact sequence of vector bundles. Note that any vector bundle is a vector space scheme and an exact sequence of vector bundles is also an exact sequence of vector space schemes. Thus we have a natural homomorphism K₀(S)$\to$G₀(S). When S is a regular scheme this is an isomorphism; usually one gives a definition of regular schemes in terms of ring theory and proves the equivalence, but we could equally well use this as a definition. As a particular case we have the ``Jacobian criterion'' which says that a scheme is regular if the Zariski tangent vector space scheme is a vector bundle; note however that this is not in general necessary. For example the subscheme of $\mathbb {A}$² defined by XY = p for some prime p is regular but its Zariski tangent space is not a vector bundle.

In fact the tensor product construction makes K₀(S) into a ring and G₀(S) a module over this ring.