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)-
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