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
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