0.1. MOTIVATION AND STRATEGY xiii

of X/a in this enlarged geometry. The present book is an attempt to develop such

an enlarged geometry and to apply it to the study of categorical quotients.

Before proceeding to explain our approach it might be useful to put things in

perspective by discussing some general aspects of quotient theory.

0.1.3. T w o viewpoint s in quotient theory. Assume we are given a cor-

respondence X = (X, a) in some category of "spaces" (e.g. smooth manifolds,

varieties, schemes, etc.) There are (at least) two possible paths towards the idea

of "quotient" of X by a which we shall call here, for convenience, invariant theory

and groupoid theory.

In invariant theory one seeks to construct a categorical quotient as a "genuine

space" X/a such that the (locally defined) functions on it identify with the (locally

defined) ua—invariant functions" on X. (Here a locally defined function ^ o n l

is called a—invariant if ip o G\ = p o o2 o n their common domain of definition).

This viewpoint is actually quite successful in algebraic geometry, in case (X, a)

comes from a reductive algebraic group action f i : G x I ^ I ; geometric invariant

theory codifies the situation in this case. On the other hand the invariant theoretic

approach in algebraic geometry utterly fails for correspondences with Zariski dense

equivalence relation (a); these will be, by the way, the correspondences that will

be at the heart of this book.

In groupoid theory one assumes that the correspondence X = (X, X, o\, a2)

has a groupoid structure (y, L, e) and one defines the "space X/a" to be the groupoid

(X, X,cri,(j2,^, i,e) itself regarded up to an appropriate equivalence on the class

of groupoids. (This equivalence should allow roughly speaking to transfer modules

from one groupoid to another in a sense that depends on the particular context

we are in.) Non-commutative geometry and stack theory adopt this viewpoint.

By the way, one of Connes' original motivations for developing non-commutative

geometry [36] was to address the "basic pathology" above as it manifests itself in

topology and differential geometry; this has been enormously successful and can

be adapted to algebro-geometric situations [127], [99]. On the other hand passing

from algebraic varieties to algebraic stacks, as one sometimes does in moduli space

theory, is not sufficiently drastic to make the "basic pathology" go away !

0.1.4. Strategy of th e present approach. Our approach towards the "ba-

sic pathology" above will use the viewpoint of invariant theory (as opposed to that

of groupoid theory); but in order that invariant theory be non-trivial we will have

to enlarge algebraic geometry by adjoining to it some new functions. Having "more

functions" will increase our chances to find, as we will, interesting a—invariant func-

tions. Now there is a general recipe to enlarge algebraic geometry by adjoining to

its functions an extra function, £, satisfying "polynomial compatibility conditions"

with respect to addition and multiplication. Cf. [20]; see also the last Section of

Chapter 2 in our book. Under a certain genericity condition it turns out, as we

shall see, that "locally" there are exactly 4 types of such 5's which can be referred

to as derivation operators, difference operators, p—derivation operators (= "Fermat

quotient operators"), and p—difference operators. If S is a derivation one is led to

the differential algebra (and corresponding geometry) of Ritt [117] and Kolchin

[84]; cf. also [32], [12], [14], [15], [16]. If S is a difference operator one is led to

the difference algebra (and corresponding geometry) of Ritt-Cohn [35]; cf. also the

work of Chatzidakis-Hrushovski [33] and Hrushovski [70]. If S are p—derivation