1.1. ALGEBRO-GEOMETRIC TERMINOLOGY

7

has an affine open neighborhood U = Spec B such that B is etale over an algebra of

polynomials A[x\, ...,xn] contained in B\ x\, ...,xn are then called etale coordinates

onU. If IT : X — Y is smooth then the sheaf of Kahler differentials ftx/Y

1S

locally

free and its dual, TX/y, identifies with the sheaf Per

7 r

-i

C ) y

(0x,Ox)- We denote

by

(1.11) Kx/Y = det(ilx/Y)

the canonical bundle of X/Y; if Y is clear from the context we write Kx instead

of KX/Y- We also denote by

(1.12) T(X/Y) =

V(QX/Y)

= Spec Symm(nX/Y)

the tangent scheme of X/Y; if Y is clear from the context we simply write T(X)

instead of T(X/Y). Assume in the situation above that Y = Spec R with R a local

ring. Also assume P G X(R) and let PQ G X be the image of the maximal ideal of

R. Then the R—module

(1-13) TPX:=DerR{Ox,P0,R)

will be referred to as the tangent space of X at P.

1.1.5. Correspondences on schemes.

DEFINITION

1.4. Let us fix a ring R and consider the category

(1.14) C — CR := {schemes over R}.

By a correspondence over R we will always understand a correspondence in CR.

An object in CR is called trivial if it is isomorphic to Spec R. A correspondence

X = (X, X, G\, oi) in CR is said to be non-empty if the fibers of X/R are non-empty.

We say X has (relative) dimension d if both X and X have pure relative dimension

d over R. We say X has an irreducible base (resp. a geometrically irreducible base) if

X/R has irreducible fibers (resp. geometrically irreducible fibers). We say that X is

irreducible (resp. geometrically irreducible) if both X/R and X/R have irreducible

fibers (resp. geometrically irreducible fibers). We make similar definitions with the

word "irreducible" replaced by the word "reduced". We say X is of finite type over

R (resp. smooth over R) if both X and X are of finite type (resp. smooth) over R.

We say that X is left etale if o\ is etale. We say X is etale if both o\ and cr2 are

etale. A morphism

(TT,TT)

: X' = (X',X',a[,a'2)^X= (X,X,aua2)

is called an open immersion if both n and TT are open immersions. We refer to X7

as an open subcorrespondence of X; by abuse we usually view X' C X and X' C X

as open subsets and we write X ' c X .

Note that if X =

{X,X,G\,J2)

is a correspondence over R and i : Y — X is

an open immersion then the pull-back correspondence (cf. Equation 1.4) is given

by

i*x =

(y,(7r1(y)ri(72-1(y),(7i,t72)

and i*X is an open subcorrespondence of X.

We will later need to consider the following construction. Assume ( X ^

e

/ is a

family of correspondences over R with the same base, i.e. X* = (X, X^ o\i, v^i)-