4
1. PRELIMINARIES FROM ALGEBRAIC GEOMETR Y
If a categorical quotient (Y, n) exists then it is unique up to isomorphism and
we shall write
(1.3) Y = X/a.
Correspondences form, in a natural way, a category: a morphism X X '
between two correspondences X = (X, X , crl5 a 2) and X7 = (X ; , X', o~[, af2) is, by
definition, a pair of morphisms (7r,7r), TT : X X' , 7r : X X'', such that
TV o at = cr^oTr, i = 1,2. Note that if TT is an isomorphism and rr is an epimorphism
then X/cr exists if and only X'/a' exists and, in this case, the two are isomorphic.
If fiber products exist in C and one is given a correspondence X = (X, X , cri, a2)
and a morphism / : Y X then one can define the pull back correspondence
(1.4) / * X := (Y,(Y xf,x,ai X) xpr^pri (X xa3tXJ Y),pn opri,pr2 opr2).
1.1.2. R i n g s . We denote by Z , Q , R , C the rings of integers, rational num-
bers, real numbers, and complex numbers respectively; Z
+
will denote the set of
non-negative integers. We denote by Zp the ring of p—adic integers, by Qp the field
of p—adic numbers, and by Fp = Z/pZ the prime field with p elements. We denote
by
Zu/ = [jZp[Q
c
the maximum unramified extension of Z
p
; here £ runs through the set of roots of
unity of order prime to p. Denote by Z'pr the p—adic completion of Z^ r .
Unless otherwise stated all rings will be assumed commutative with identity
and all ring homomorphisms are assumed to preserve the identity. (Some non-
commutative rings or commutative rings without identity will also occur in our
discussion but, when this is the case, sufficient warning will be given.)
If A is a ring we denote by
(1.5) A[e]=A®eA, e2 = 0,
the ring of dual numbers on A.
If L is a field we denote by La an algebraic closure of L.
Number fields are subfields of C and are always assumed finite over Q. A place
in a number field F will always mean a finite place identified with a maximal ideal
in the ring of integers OF of F.
Let A be a ring, p a prime integer, and M a maximal ideal in A; we usually
denote by
(1.6) i , Afor
the p—adic completion and the M—adic completion of A respectively. We say A is
p—adically complete if A A.
D E F I N I T I ON 1.2. Let p 0 be a prime number. A local p—ring is a discrete
valuation ring of characteristic zero with maximal ideal generated by p. A global
p—ring is a ring A such that:
1) p is an non-zero divisor in A,
2) the principal ideal (p) is prime in A,
3) A is p—adically separated.
(Note that A is then necessarily an integral domain.)
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