0.2.ROUGH OUTLINE OF THE THEORY xv

F = Q, p = (p), we have Z(p) = Zp, the ring of p—adic numbers, so we simply

write i?p =

Zpr

in place of (Z^)~. By the way, for a general F and p the natural

morphism Rp — i?p is an isomorphism so the ring Rp depends only on p and not

on F and p.

0.2.2. £—geometry. With the notation above we let S = Sp : Rp — » i?p be

the "Fermat quotient operator" defined by fa = (4(x) —

xp)/p.

We consider a

"geometric category", C«$, which morally underlies S—geometry at p. The objects of

C 5 will be called 5—ringed sets; a £—ringed set Xs consists of an underlying set Xset

equipped with a family (Xs) of subsets and a family of "structure rings" (Os) of

.Rp—valued functions on the Xs's such that / G Os implies So f G Os. Here s runs

through a monoid S and one assumes that Xst = Xs D Xt for s, £ G 5. One also

assumes compatibility with restrictions i.e. / G Os implies f\xst £ ^st- A (5—ringed

set is called trivial if Os = Rp for all s. There is a natural concept of morphism

of 5—ringed sets; with these morphisms the £—ringed sets form a category C$. Any

correspondence in C$ has a categorical quotient. So the existence of categorical

quotients is not here the issue; the issue will always be their non-triviality.

0.2.3. Passing from algebraic geometry to ^—geometry. Let Xp = X

be a smooth scheme of finite type over Rp with irreducible geometric fibers. We

will attach to the scheme X a S—ringed set Xs] the underlying set Xset of Xs will

be the set X{Rp) of Rp—points of X and the structure rings Os will consist of

certain functions P i— ip(P) that locally, in coordinates x G Rp1 look like

F(x,fa,

...,5rx)

^ = G{x,6x,...,6rx)

where F, G are restricted power series with Rp—coefficients. Let us make this

precise. First, a function / : X(Rp) — • Rp is called a S—function of order r if

for any P G X(Rp) there is a Zariski open set U C X, P G U(RP), and a closed

embedding £/ C A

d

such that f\u(Rp) is given in coordinates x G i?p by

/(x) = F(x,fa,...,^

r

x)

where F is a restricted power series with coefficients in Rp. Recall that a power

series is called restricted if its coefficients converge p—adically to 0. Denote by

Or{V)

the rings of £—functions of order r on a Zariski open set V C X. Then

V i— •

Or(V)

defines a sheaf

Or

of rings on X for the Zariski topology. Define a

S—line bundle to be a locally free sheaf of

Or—

modules of rank one. Consider the

ring W := Z[£] C End{Rp) and let W+ be the set of all w = J2ai^ € w w i t h

ai 0. The multiplicative monoid W acts on Rx by Aw = Hift(\))ai, X E Rx.

For any S—line bundle L one can define a S—line bundle Lw by acting with w upon

the defining cocycle of L. Let L = K~l be now the anticanonical bundle on X,

viewed as a 5—line bundle. We consider the graded ring

0

H°(X,LW).

wEW+

Its homogeneous elements will be called 5—sections. We let 5 be the monoid of

S—sections of weight ^ 0, not divisible by p, in the above ring. For any s e S of

degree w$ we let

Xs := {P G X(Rp) | s(P) £ 0 mod p},