Introduction

We start, in Section 1, by explaining our main motivation which comes from

the fact that categorical quotients of correspondences on curves tend to be triv-

ial in algebraic geometry. In Section 2 we explain how one can fix this problem

for a remarkable class of correspondences; this is done by developing a geometry,

called 5—geometry, which is obtained from usual algebraic geometry by adjoining

a "Fermat quotient operator1' S. If one views 6 as an analogue of a derivation with

respect to a prime number then £—geometry can be viewed as obtained from usual

algebraic geometry by replacing algebraic equations with "arithmetic differential

equations". In Section 3 we discuss relations between our theory and some other

theories.

The present Introduction is written in an informal style; a formal presentation

of this material will be made in the body of the book.

0.1. Motivatio n and strateg y

0.1.1. Correspondence s and their categorical quotients. It is conve-

nient to start in complete generality by considering an arbitrary category C. (Morally

C should be viewed as a category of "spaces" in some geometry.) A correspon-

dence in C is a tuple X = (X, X , 7i, a2) where X and X are objects of C and

7i,(72 : X — X are morphisms in C. We sometimes write X = (X,a), where

a := (X, 0-1,0-2)' Following the standard terminology of geometric invariant the-

ory [113] we define a categorical quotient for X to be a pair (Y, TT) where Y is an

object of C and TT : X — » Y is a morphism in C satisfying the following proper-

ties: 1) TT o o\ = TT o (J2; 2) For any pair (Y',TT') where Y' is an object of C and

TT' : X — » Y' is a morphism such that TT' O a\ = TT' O G2 there exists a unique

morphism 7 : Y — Y' such that 7 o TT — TT' . Categorical quotients are sometimes

referred to as co-equalizers. Of course, if a categorical quotient (Y, TT) exists then it

is unique up to isomorphism and we shall write Y = X/a. Correspondences form,

in a natural way, a category: a morphism X —• X ' between two correspondences

X = (X, X, (7i, 0-2) and X ' = (X7, X', G'X, G'2) is, by definition, a pair of morphisms

(71-, TT), TT : X — » X 7 , TT : X — X'', such that TT O ai = a[ o TT, i = 1, 2. We will assume,

for each category C we shall be considering, that a class of objects in C is given

which we refer to as trivial objects. (Morally trivial objects should be viewed as

spaces that reduce to a point.)

An important example of correspondences is provided by (discrete) dynamical

systems (i.e. self maps). Indeed if X* is an object in our category C and s :

X* —• X* is a morphism then one can attach to these data a correspondence

X* = (X*,X*,idx*,s). More generally if we assume C possesses fiber products

then one can consider, in a natural way, the pull-back X := J * x * of X* via any

xi