F = Q, p = (p), we have Z(p) = Zp, the ring of p—adic numbers, so we simply
write i?p =
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)
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
^ = 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
such that f\u(Rp) is given in coordinates x G i?p by
/(x) = F(x,fa,...,^
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
the rings of £—functions of order r on a Zariski open set V C X. Then
V i—
defines a sheaf
of rings on X for the Zariski topology. Define a
S—line bundle to be a locally free sheaf of
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
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},
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