8 1. PRELIMINARIES FROM ALGEBRAIC GEOMETR Y

Then we can consider the disjoint union correspondence

(1.15) X:=JJX

i :

=(X,]J Xi:;Q aH,]Ja2i).

i i i i

Here is another construction which will play a role later. Assume

X=(X,X,7i,72)

is a correspondence over R. Then, for each n 2, we may consider the scheme

Xon

:= X xx X xx • • • xx X, (n times)

where for each X xx X the left X is viewed as a scheme over X via a2 while the

right X is viewed as a scheme over X via o\. In other words

X xx X = X x(72:x,a1 X.

E.g., for n — 3 we have a diagram

i I io~i

X°2

- X ^ X

i lri

X ^ X

X

with Cartesian squares. We have natural morphisms

on ^ o n . y o n v

Gx , ( j

2

. A —v• A

defined, on 5—points (5 any R—algebra), by the formulae

T°1n(Pl,...,Pn)=T1(P1),

a°2n{Pl,...,Pn)=a2{Pn).

E.g., in the picture above,

a^3

is the composition of the left vertical arrows while

a23 is the composition of the top horizontal arrows. So we may consider the n—th

power correspondence in CR,

Xon . / v v o n ^ o n ^ o n

:= [X,X

,GX

,cr2

J.\

Of course, if X is smooth over R and left etale (resp. etale) then X o n is smooth

over R and left etale (resp. etale).

We will also need to consider line bundles on correspondences. As usual a line

bundle on a scheme X is a locally free Ox~module of rank one. The set Pic(X)

of isomorphism classes of line bundles is a group under ®.

DEFINITION

1.5. If X = (X, X, 0-1,0-2) is a correspondence over R and L is a

line bundle on X then a linearization of L is an isomorphism (3 : a^L ~ o\L. A

line bundle on X is a pair L = (L,(3) consisting of a line bundle L on X and a

linearization (3 of L. Two line bundles L = (L, (3) and L/ = (Z/, f3') on X are called

isomorphic if there exists an isomorphism u : L —

Lf

such that following diagram

is commutative:

G^L

— o\Li

o~2u

i i

°~iu

,-r* T ' ^ v ,-r* T

f