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.)