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Generic automorphisms of fields
ANNALS OF PURE AND APPLIED LOGIC ELSEVIER
Annals of Pure and Applied Logic 88 (1997)
Generic
165-180
automorphisms
of fields
Angus Macintyre* Mathematical Received
29 January
Institute,
1996; received
24-2Y St. Giles. oxford
0x1
in revised form 28 January
3LB.
UK
1997; accepted
12 March
1997
Abstract
It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected (via fixed fields) to Ax’s theory of pseudofinite fields, and analogues are obtained for most of Ax’s classical results. Some indication is given of the connection to nonstandard Frobenius maps. Automorphism;
Keywords: AMS’
Existentially
closed; Pseudofinite
03ClO; 03C60
C/usszjication:
0. Introduction This
paper
some
aspects
Theorem 1. (a) Any two algebraically closedfields of the same characteristic the same jirst-order sentences of the language of field theory. (b) lf @ is a sentence qf the language qf,field theory then
satisjj
of which
originated
in some
reflections
on the Lefschetz
The principle
has two parts:
: @ holds in (Z/P)“‘g}
{P E Spec(Z)
is open or closed in Spec(Z), and so contains it contains all hut jinitely many P. Moreover,
there
(respectively, Theorem
Principle,
I now recall.
are effective
of fields 1 is useful
aspects
of fixed
0 13 it contains
to this. The theory
characteristic)
infinitely
of algebraically
is decidable.
a result
over
(2) transferring
a result
over all @” to all other
Cc to all characteristic
zero cases
cases
[ 121;
[l].
* E-mail: [email protected] 0168-0072/97/$17.00 @ 1997 Published PIZ SOl68-0072(97)00020-1
closed
For all this see [7].
in two ways:
(1) transferring
many P $T
by Elsevier Science B.V. All rights reserved
fields
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
166
A functorial
way of expressing
I be a countable algebraically
Here
1 is via the ultraproduct
index set, and D a nonprincipal
ultrafilter
construction.
Let
on I. Let Ki (i E D) be
closed fields such that for each prime p
{i : Ki has characteristic and only, pose such
Theorem
165-180
p} $ D,
is of course due to cardinality IKil<2N”. Then fl, Ki N @. The isomorphism and is noncanonical. But now there is a feature missing in Theorem 1. Supgi E Aut(Ki) for all i. Then there is a well-defined element n, (Ti of Aut(&, Ki), that for any sentence 0 in the language of fields with automorphism
n,
ci
is a D-average
of the
ci,
and
(noncanonically)
is an element
of
Aut( a=). Consider the special case when I is the set of primes, KP is of characteristic p, and aP is Frobenius, i.e. x H xp. Then the automorphism no oP of @ deserves to be written x H xp, p a nonstandard prime, and be called a nonstandard Frobenius. What is special about these maps? There are two aspects, one tedious, and one fascinating: (1) (C, no crP) is Nl-saturated, i.e. has a kind of compactness property [lo]; (2) The elementary properties of &, aP relate to Weil’s Riemann Hypothesis for curves, and beyond to the Weil Conjectures. Already from Ax’s work [l] one knows that Fix(&, oP) is severely constrained, as &, Fix(ap), i.e. n, EPp,and so must be pseudofinite, i.e. have Galois group f and be such that every absolutely irreducible curve over it has a rational point. Moreover, that is essentially all one can say about n,Fix(crP) [l]. Let 0 = n, op. One wants to know about polynomial equations in xi,. . . ,x,, a(x, ), . ..) (T(x,), a2(x*),.. .,02(x,) )... solvable in @. By the usual device of adding extra variables this comes down to understanding polynomial systems C(xi, . . . ,x,, a(x, ), . . . , a(~,)). By the methodology of Robinson’s Test [8], if one understands enough about the solvability of such systems, then one understands the elementary theory of (C, a). This in turn means that one understands the theory (or the almost-all variant) of the class (F;lg , opp).This was my original question: P , op) decidable? Does it admit a reasonable elimination Is the theory of all ([Falg theory? What can one say about the nonstandard Frobenius maps on C? In [5] I will show that the answers to the first two questions are yes, and that the third has a beautiful answer in terms of an independent notion of generic automorphism of C. In fact, I succeed in generalizing naturally every result from Ax’s classic paper PI. This paper develops the theory of generic (or existentially closed) automorphisms of fields. Its sequel [5] will deal with the much deeper problem of showing that the nonstandard Frobenius maps are existentially closed.
A. Macintyrel
I formulated
Annals of Pure and Applied Logic 88 (1997)
the main problems
in the mid-1980s
165-180
167
and made the first progress
MSRI in 1990, when I showed that the theory of fields with an automorphism model companion.
At that time I had very helpful conversations
at
has a
with Zoe Chatzidakis,
Lou van den Dries and Carol Wood, whom I heartily thank. A result of Lou, that the fixed field of an existentially forth the conjecture
closed automorphism
that the nonstandard
Frobenius
is pseudofinite,
naturally
maps are existentially
brought
closed (equiv-
alently, Robinson generic). Later, Udi Hrushovski gave a neater formulation of the axioms for generic automorphisms (Axiom H in this paper) and this has turned out to be an inspired
move. I thank Udi for showing
me this insight.
1. Difference fields, Axiom H 1.1. Basic systems For reasons unknown to me, a pair (K, o), where K is a field and rr E Aut(K), is called a d@erence jield. The class of difference fields is axiomatizable, by an V’3 sentence, in the language of ring theory extended by a symbol for the automorphism. In both the algebraic and logical study of such structures, one must study (difference) systems, the natural generalization of polynomial equations (and inequations) to take account
of rr. Bear in mind the two simple observations
(1) and (2) below:
K/=‘X#OOK~((3/?)(a~-l=O),
(1)
K t= @(xi,. . . ,xn, 4x1), . . . , ah),
a2h 1,. . , a2M>
@ K I= ~y1...~y,[~(~l,...,~,,Yl,...,y,,~(Yl),...,~(Y,)) Ay1=o(x1)A
...
Ay,=o(x,)].
C(xi,. . . , xn) is equivalent
Then
solvability
C’(Yl,.
. ., y,,,), where C’ consists of polynomial . . ., a(~,). Such C* I call basic systems.
4Yl),
1.2. Existential
of systems
(2)
to solvability
equations
of systems
over K in yi,. . . ,y,,,,
closedness
Definition 1. (K, O) is existentially closed (e.c.) if whenever C(y) is a basic system of K, and z is solvable in an extension (K’, a’) of (K, a) then C is solvable in K. General
logical theory [ 1 l] tells us
Lemma 1. Any (K, CT)extends
to an e.c. (K’,o’),
with JK’( = max(lKI,Ns).
In the category of fields, the e.c. structures are the algebraically closed fields, which have the special property that they are axiomatized by the infinite set of conditions C, (1 dn
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In the category of fields with derivation, studied, under the name of differentially order axiomatizations,
165-180
the e.c. structures have also been extensively closed fields [9]. These too admit nice first-
see [lo].
Theorem 2. The class of e.c. d#erence jields is first-order axiomatizable. The proof will be given a little later. First, a simple lemma.
Lemma 3. If (K, cr) is e. c., K is algebraically closed. Proof. Any automorphism
of a field extends to its algebraic
closure.
0
Definition 2. Fix(o) = {X E K : a(x) =x}. Lemma 4. Let (K, o) be e. c. Then (i) Fix(o) is perfect; (ii) the absolute Galois group of Fix(a)
is f.
Proof. (i) Clear. (ii) Note that by the absolute Galois group I mean Aut(Fix(o)a’s]Fix(o)). (K,a) is e.c. I claim that for each II 32, there exists x,, in K with @(xn) =x,,
Suppose d(~,) #
x,, j < IZ. To see this, let z be a permutation of { 1,. . . , n} of order n, and let yl, be algebraically independent over K. Extend 0 (uniquely) to an automorphism
.
. , y, 6 of
via 6(yi)=yr(i). Then one of the y’s satisfies c?(y)=y, c?j(y) # y, K(YI ,...,y,) j
is exactly n. Finally,
since Aut(Fix(a)a’s/Fix(a))
Galois Theory, we get it to be f.
is procyclic
by elementary
0
Just as simple is
Lemma 5. Let (K, CJ) be e. c., and let f (x, y) = 0 be an absolutely irreducible plane curve over Fix(o) (by which we imply in particular (a,P)~Fix(o)~ with f(a,B)=O.
f E Fix(o)[x,
y]). Then there exists
over K. Let K’ = K[x, y]/( f), and extend cr to CT’on K’ by a’(x) =x, o’(y) = y. So there exists (cc’,,!?‘)E Fix(o’)*, f(c~‘,/3’) =O, and this drops to the e.c. (K,a). 0
Proof. f is irreducible
Van den Dries showed me these in June 1990 (and Lascar noted it independently). Together they give something extremely suggestive.
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Theorem 6. Zf (K, 0) is e.c., Fix(o)
is pseudojinite
165-180
169
(i.e. an infinite model of the theory
of jinite jields). Proof. See [2] for a proof that a field L is pseudofinite iff Aut(L”‘s/L) CYf, and every q absolutely irreducible plane curve over L has an L-rational point. Note.
It is quite easy to see (using Ax’s [l] characterization
of pseudofinite
fields) that every pseudofinite
of the absolute
L is elementarily
equivalent
for some e.c. (K, G). Just take K as an extension of r on the algebraic prime field of L, where Fix(r) is the field of absolute numbers of L. 1.3. o-action
in algebraic
numbers to Fix(o)
closure of the
geometry
Though it is not strictly necessary for the formulation of the crucial Axiom H, I find it valuable to give a systematic account of how rr E Aut(K), for K algebraically closed, induces
natural
actions in algebraic
1.3.1. CJacts on K” coordinatewise,
geometry
over K.
and on the power set of Kn by a(E) = {o(_?) :
TEE}. 1.3.2. ~7 acts on K[xl,. . .,x,1 by o on K, and the identity 1.3.3. If Zer( fi,.
on xi,. . . ,x,,.
..,fk)C K” is the set of zeros in K” of the fi E K[xI,. . . ,x,J, then
dZer(fi,...,fk)) = {dY):fi(Y))=
... =fk(Y)=O)
= Zer(fl( fl >,. . . , o( fk 1). 1.3.4. Thus, irreducibility.
r~ acts on A,(K) and P,(K), and TT acts on algebraic sets, respecting So rr is a homeomorphism for the Zariski topology, in both cases.
1.3.5. The a-action to radicals.
on K[xl , . . . ,x,1 extends to ideals, sending primes to primes, radicals
1.3.6. Evidently, 0 sends quasi-affine affine (respectively, quasi-projective)
(respectively, varieties.
quasi-projective)
varieties to quasi-
1.3.7. So g sends varieties to varieties [3, p. 151. Moreover, the action on polynomials extends to one on regular functions and morphisms, making cr functorial. Precisely: Let V be a variety, wlog affine, and U an open subset of V, containing the point P. Suppose f on U is given by g/h, g, h polynomials, h not vanishing on U. Then G(U) is an open subset of a(V) containing a(P), o(h) does not vanish on a(U), so O( f) =der o(g)/o(h) is (gives) a well-defined regular function at P.
and
This gives the action of o on regular functions. Notice that cr defines a continuous map (not a morphism) from V to o(V), and that for a regular function f : V -+ K, CJ(f) : (T(V) --) K is given by O( f) = afo-'. Now taking the definition of morphism of varieties given in [3, pp. 15, 161, f : VI ---f V2 is a morphism so is af g-’ =,+f c( f ).
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A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
Now the functoriality 1.3.8.
0 has obvious
natural
of IJ is evident. action
ring isomorphisms
structure).
on the rings O(Y) ? U(a(Y))
This in turn induces
1.3.9. Similarly,
165-180
O(Y),
Y ranging
(not of course
over K-varieties, respecting
giving
the K-algebra
action on the local rings @~,r (E Q.(p),.(r)).
there is an obvious
1.3.10. (T respects rational, dominant nonsingularity).
action on the fields K(Y), and birational
Y a variety.
points, and respects singularity
(and
1.3.11. Now let V be a variety (construed as topological space), and 0~ its sheaf of regular functions, so cOv(U) is the ring of regular functions on U. Now cr : V ---)a(V) is continuous (as is o-l), and by [3, p. 651 the direct image sheaf o*(V) is o,(Ov)(W) = 0,(a-‘(IV)). One easily checks that QT(V)(W) = {ogc-’
: 9 E Q*(@)(W)}
and that hH o-‘ha defines a map Q,(v) -+ LT*(@v)of sheaves of rings on cr( V). So, in fact, (cr, h H o-‘ho) defines a morphism (actually an isomorphism) (K(%)-,(@U,4r(V,) of locally ringed spaces. Write 0(~,(%)=(4V,G,,,) and this is clearly functorial
on morphisms
of varieties.
1.3.12. In [3, p. 781 one deals with the fully faithful functor t from varieties to schemes over K. o is entirely compatible with this, as I now show. Firstly,
one associates
to any topological
space X a set t(X)
consisting
over K of all
nonempty irreducible closed subsets of X, and topologizes this by taking as closed subsets all t(Y), Y closed in X. t is functorial on continuous maps f, by defining t(f)(E), E nonempty irreducible closed, as the closure of f(E). One also defines a continuous cx:X + t(X) by setting a(P) equal to the closure of {P}. The scheme attached to V is (t(V),cr,(0~)). Now clearly, t(a)(E) = a(E) all E E t(V). Also, when U is open in t( I’), cr*(Sv)( U) = &(a-l(V)) and ~*(~0(V))(~(~)(~))
= QT(V,(~-‘t(@(U)) = {oga-’
: g E U)y(Cr-l(CL-l(t(0)(U))}
= {ago-’
: g E coV(a-‘(u))}.
A. MacintyrelAnnals
Since
the scheme
attached
171
of Pure and Applied Logic 88 (1997) 165-180
to o(V)
is t(o( V)), a*(Qv,),
of 1.3.11 of r~ on (V, 0~) extends to a sheaf isomorphism
we see that the action (also denoted
0)
(V)>~*(~v))-+(~(4~))?x*(Lo,(V~)) which is t(o)
on t(V),
and gives a map
~*(QT(V,) + a*(@*(&)) of sheaves of rings on t(V) by h I-+ t(a)-‘/n(o) as before. Again this is functorial. Moreover, t(V) =&f (t(V), a,(Uv)) has a natural structure of scheme over K, with the morphism to Spec(K) being (f, f’), where f sends every point of t(V)
to 0, the unique point of Spec(K),
and
f # : QSpec(K) + f*&(V) is given by f #(X) = the constant
function
on t(V) with value X.
On the other hand, r~ induces an automorphism of Spec(K) by (id,@). So one easily sees that 0 is even functorial on t at the level of schemes over K. 1.3.13. Finally, how does cr act on schemes over K? Let (X,0X)-+(,-f,+) Spec(K) be a scheme S over K. Composed with (id, g) this gives another scheme a(S) over K. Of course, 00-l IS the identity. We retrieve all the earlier nonsense via t! 1.4. The axiom Now we return to earth (and logic). Let V be a closed subvariety by equations fi(xl,..., xn)= ... = fk(x, ,..., x,)-O. I assume K algebraically defined by (~fl)(Yl,...,
(of1
)(Yl,
. ..)
closed. Then g(V)
Yn)=
‘..
=(~fk)(yl,...,yn)=0,
y,)=
...
=(cJfk)(y
,,...,
of A”(K) defined
is also a closed subvariety
of A”(K)
yn)=O.
a(V) shares with V all natural properties like nonsingularity, dimension Now form V x a(V), the closed subvariety of A2”(K) given by
d, etc.
=(~fk)(Yl,...,Yn)=O. Suppose,
W is a closed subvariety
w-Vxcr(V) 711J V
\
712
o(V)
of V x r~(V), and suppose the two morphisms
172
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
are generically
surjective
(the xi are the natural projections).
point of W projects to generic points of V and a(V). intersects
165-180
This means that a generic
In this case, I say W potentially
the graph of cr.
Now I come to the axiomatic Axiom H. If W potentially
heart of the matter.
intersects
such that (x1 ,...,Xn,O(Xl),...,d(X,))E This extremely
the graph of (T, then there exists xl,. . . ,x,, E K w.
suggestive
principle
(in 1990) a more complicated version, order to axiomatize the e.c. (K,o).
was isolated involving
by Hrushovski,
various
V x o(V)
after I had used x . - . x am(V), in
It is crucial for what follows that one sees Axiom H as the conjunction of axioms Hk, for k= 1,2 ,..., where the Hk are (under the assumption that K is algebraically closed) first-order axioms. It is worth explaining this in some detail. Though I am only interested (now) in the case when K is algebraically closed, and the difference between variety and absolutely irreducible variety disappears, let me consider the variant of H where V and W are assumed absolutely irreducible. Fix an integer k and restrict H to the case in which there exists and n
1.5. Elimination
of quanti$ers
Lemma 5. Suppose
(K,o)
is e.c. Then Axiom
H holds in (K,o).
Proof. Suppose (K, a) e.c. Then K is algebraically closed. Go to a (K]+-saturated elementary extension (K, o) -t (KI, 01) [lo]. Then K1 contains a generic point (at,. . . , a,, PI,. . . , Pn) of W over K. We are using “generic” in the sense of [ 121. Now by generic surjectivity of ~1,712, (~(1, . . . ,a,) is V-generic over K and (fit,. . . , fin) is a~( V)-generic over K, i.e. a( V)-generic. Now define
d : K(Q ,...,a,)-~K(Bl,...,p,),
A. Macintyrel
Annals
of Pure and AppliedLogic 88 (1997) 165-180
by C’ = CJon K, o’(ai) = /3i. 0’ extends to an automorphism o”(4 ), . . . , a”(a,)) E W. Since (K, a) is e.c., we get some ..,4Y,))E
(Yl, ...,Yn,4YI)1. where yI,...,y,,EK.
173
(T” of Kl, and (~(1,. . . , a,,
w,
0
Now the crucial Nullstellensatz: Theorem 7. Suppose (K, C) satisJies Axiom H, and K is algebraically closed
Then
(K, o) is e. c. CT(D,)) be a basic system over K, with a solution Proof. Let C(vl, . . ..V.,~(Ul),..., a,) in an extension (L, a’) of (K, g) over K. Let V be the variety of (al,. . . , cc,,) (al,..., over K, so o(V) is the variety of (a’(~1 ), . . . ,~‘(a,)) over K, and let W be the variety of (c!,,..., a,,o’(al) ,...) cr’(a,)) over K. Then W 2 V x o(V). It is obvious that W potentially (P1,...,j3n)
meets the graph of O. So W has a K-point solves C.
(PI,. . .,/$,,o(/I~),
. . ., o(fin)), so
0
Note. 1. We use the assumption that K is algebraically closed to get V, W absolutely irreducible. 2. Inspection of the proof shows that in the above we can weaken H to H,,, where V, W are assumed nonsingular. The reason is that we can add to the equations for V (respectively (respectively, as usual.
W) inequations expressing that V (respectively, W) is nonsingular at cl (&o’(i)), and replace these inequations by equations in more variables
This shows that the class of e.c. (K, a) is elementary,
and so model-complete
[8]. So,
modulo Axiom H and the axioms that K is algebraically closed, every formula Q(V) is equivalent to an existential formula. We can do a little better, for the following reason. Lemma 8. Suppose (Kl,q), (K~,cz) are extensions of (Kj, ~3), where K3 is algebraically closed. Then (Kl, q), (K~,Q) can be jointly embedded in some W4,04
).
Proof. Trivial manipulation
of transcendence
bases over K3.
Cl
Corollary. If (Kl, 02) and (Kz, az) are e.c., and 01, CQagree on the algebraic closure of the prime jield then (Kl, rs1 ) 3 (K2,02). Proof. By Lemma
8 and model-completeness.
0
More generally, Lemma 8 shows that the (elementary) type of a tuple (al,. . . , a,) in an e.c. (K, o) is determined by 0 on the algebraic closure (in K) of the field generated
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by the sequence. . . , &z, o(w),
al,.
o-‘(cq),...
. . .,4&z), ) 6’(an),
(i.e. the smallest
o2(a1 ), . . ., 02(%), . . .
a-2(q),
. . .) C2(cl,),
...
~1, . . . , ~1, and closed under
field containing
of course, leads one to a quantifier-elimination. obvious counterexample to quantifier-elimination
~,a-‘).
This remark,
What is required is suggested by an in the original language. Consider
Q(V) below. @p(u)Z o(u) = u A (3)(?
= lJA o(t) # t).
That one cannot eliminate the existential quantifier is seen by taking independent transcendentals ui and ~2, taking o as the identity on the algebraic closure of Ka(ui) (Ka the prime field), while taking o to fix K:“(u2) but move the square roots of u2, and then extend to an e.c. (K,cr’). The predicates I use for quantifier-elimination are natural generalizations of those used in the theory of pseudofinite fields [2]. Let it, k be nonnegative integers, and let f,gi,hi
(i,
be members
of
9.. . ,XOn,Xll ,...,~ln,yll,...,Yln,~2l,...,~Zn,Y2l,...,Y2n,...,
Xkl,...,Xkn,Ykl,...,Ykn,tl.
Let ShQ,i(u~,. . .,u,)
3t
be
d(Un),
f(Ul,...,
t) = 0
[
1
A A ((T(SI.(Ul,...,~-k(un),l))=hi(Vl,...,a-k(U,),t)) idm
where ui,,..,~-~
(u,) abbreviates
the sequence
9
of arguments
ul,...,Un,~(~l),...,~(~,),~-1(~l),...,~-’(~,),~2(Ul),...,~2(~,), 8(v1)
)...,
l+(u,),..
To formulate the extension notation is useful:
.,&I*),.
lemma
. .,(Tk(U,),a-k(u~) )...) dyu,). that gives quantifier-elimination,
the following
Definition 3. Let (K, o) be a difference field and X G K. Then (X), denotes the smallest subfield of K containing X and closed under e and 0-l. Lemma 9. Let (K, CJ) be a difference jield, Ko be the prime field of K, and X 2 K. Then (X), = K&Y, a(X), o-‘(X),
02(X), a-*(X),
. . .).
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
Proof. Trivial. Lemma
175
165-180
0
10. Let Y be the relative algebraic closure of (X),
in K. Then (Y), = Y.
Proof. Suppose c( E Y. Then Bk,cr(B,)
WA,..., &I
), .*. 3 fJQ%f),
so CT(u) E Y.
(z(pk),~-l(B*),...,o-l(Bk),..., 0%
1,. . ., &AC),
and F a polynomial
for /h,...,fikEX,
F(o(B,),
,...,
. . . , ‘@k),
. . . ,a-‘+‘(jh
a) = 0
over Z. Then ), . . . , @-‘+‘(pk),
a(@))
= 0,
0
Lemma 11. Let (Ki,oi), i= 1,2 be dtjterence fields, Ki algebraically closed, and (Mi,ll/) dtjerence subjields of the respective (Ki,oi). Let 0: (Ml, $1)2:(A42,&) be an isomorphism of dtrerence ftelds, and suppose 0 respects all Sf,sk and TS~,~,~, (computed in the respective ambient Ki). Let h4; be the algebraic closure of Mr in Ki, and $: = o 1Mi. Then 0 extends to 0’: (Mi, &‘) N (AI,‘, *ii’>, 0’ also respecting the SrG,h and -SL8,i. Proof. By considering f = tP - x, we see that 0 respects being a pth power, so without loss of generality one can assume M/IMi is separable, 0’ will be constructed by a Tychonov argument, as a limit of extensions ON, where 0~ is defined on a finite Galois extension N of Mt. Fix such an N =A41 [a], say. Let F(t) = f (Yl, . ..> Yn,~l(Yl)r...,~l(Yn),~~l(Yl),...,~l-l(Y~),...,
)>. .
&Yl
2 &Yd,
qk(yl
>, . . ., qk(Yn>,
where F is over E, be the minimum normal closure Nr of Ml(a, q(a)). Then, by increasing a=goh,. &Yl
t),
polynomial
over A41 of some generator
(The latter is obviously
n, k if necessary
finite-dimensional
we can write
..,Yn,~l(Yl),...,~l(Yn),~~l(Yl),...,~~l(Y~),..., h f.. , &rn>>
qkbJ1
>, . . . >qk(Y?J?
6)
and 01 (a) = ho(Y1 ~~~.7Yn,~l1(Yl),~*~, &Yl),.
‘. 7 ~~:(Yn>, al-kh
with ga,hs over h. So M
~Sf,go,ha(Yl,...,Yn),
dYd>q’(n) ), . . . , al-k(Yn),
T..., 6)
a,-‘(Y,)
,...,
6 of the over Ml).
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
176
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whence
Obviously, f(@(Yl),
. . . > @(Yn),OZ(@(Yl)),.
&@h
is irreducible
>>>.
. .Y &@(Y,)),
..,02(~(yn)>,o~1(o(yI))
~$w(Yd),
,...,
. . ., q?o(y,)),
(T;l(o(y,))
,..‘)
t)
over A42. By the above, it has a root 6’ such that
(There may of course be several such roots 8.) If we choose one such 6’ then there is a unique field isomorphism 0’: Ml(S) N A42(8) extending 0, with O’(6) = 6’. Moreover, cQ(@‘(cc)) = O’(Oi(cc)). It may, however, happen that there is /3 in M,(6) and that oi(/?) is also in A41(6) and a2(@‘(/3)) # O’(oi(/?)). But, I claim there is a choice of 6’ such that this does not happen,
i.e. so that both
az(@‘(cc)) = @‘(o,(a)) and a2(@‘@>>
=
For, increasing P = (n(B)
@‘(w(B)).
n, k again if need be, we have
Sl(Yl,...,Yn,...,q =
h(Yl,.
-?Yl>,. . . ,Yn,.
. . , q-k(Yl),
. . >q-k(Yn),
6)
. . . , qk(Yn),
with 91, hi over 27. Let 8= (go,gl), i; = (ho,hi). Ml
8)
Then
i==f,g,~(YIMYn),
so
thereby clearly giving us 6’ that works for cc,p. There are only finitely many possible 8, so proceeding in this way we eventually get 8, and so O’, which works for all /3 such that p and o,(p) are in A&(d). Now let CIvary, and consider the projective system of all such 0’. Let 6 be a limit of this system. Then 6 is clearly a difference field isomorphism, evidently respecting all SJgh q 1 , and ~5”~i. I ,
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
From this, Lemma
8, and the standard
criterion
165-180
177
[lo] for quantifier-elimination,
one
concludes: Theorem 12. The theory of e.c. (K, c) has quantzjier-elimination extra predicates
in the language with
for all Sf,g,r,.
1.6. Decidability We can prove decidability
of the theory
of all e.c. (K,a),
Lemma 7 or the preceding quantifier-elimination. Instead of the formulae Sr,g,i I now use sentence
using the corollary
to
S‘,#,a, where n = k = 0 (cf. intro-
duction of the predicates). So now f, S,i are polynomials over Z in a single variable. We know of course that two e.c. (Ki, ai) (i = 1,2) are elementarily equivalent if and only if they agree on such sentences Sf,g,h. But a little more is obvious on (short) reflection. We want to restrict somewhat the f we need consider. We restrict to f E Z[x] which satisfy: f is irreducible over Q, of degree m say, and with splitting field generated over Q by a single root c1 of f. Then the roots of f are ho(a), hl (a), . . . , h,_l(a) where each hi E Q[t], ho = t. Any 0 on Q(a) is determined by a choice of c( and one hi. If we do not specify E, but only hi, g is determined exactly up to conjugacy in Gal(Q(cc)). Now the algebraic closure UY’s of Q is a limit of such Q(a), compactness argument the conjugacy class of c in Gal(ClY’s) is Sf,s,h, where f ranges as above, g(t)=rt some r E Z, r# 0 and above are of form h/r, r E Z, h E Z[t].) But now it is clear that if (K, 0) is e.c., so is (K, z-‘oz) for any any p E Gal(Qa’s) extends to an automorphism of any algebraically
and by a simple determined by all h E Z[t]. (The hi r E Aut(K). Since closed K > Q, the
above yields, in characteristic 0, that the theory of an e.c. (K,o) is determined by the Sf,rt,h in it, as above. So, effectively any sentence @ is equivalent (in characteristic 0, and so in all characteristics Sf.,rl,h as above. So the decision
problem
p 3 PO(@), po recursive)
to a Boolean
combination
of
for @ reduces to
(i) Is Q, satisfiable in some characteristic p < PO(@)? (ii) Is the Boolean combination satisfiable in any other characteristic, including O? We may clearly assume, by loading a bit more into (i), that if Sf,rr,h or its negation occurs in the Boolean combination then r # 0 mod p for p 3 PO(@). For future reference (in [5]) we pay attention to the set of characteristics p in which @ is satisfiable, and if p#O we examine the possible restrictions to @” of cr satisfying
@.
Case (i): We examine each prime p< po separately. Fix one such. Now the discussion over Q can be replaced by one over EPp,considering f irreducible over EP (the assumption on generation by a single root is redundant), and we can clearly take
178
A. Macintyrel
f manic.
Annals of Pure and Applied Logic 88 (1997)
So now Q, is effectively
equivalent
need for r’s). But now by constructing and running
through
all (TE Gal(N)
to a Boolean
a finite extension
(all powers
165-180
combination
of Sf,t,h (no
N of lFP splitting
of Frobenius)
each
f,
we just test if Q, is
satisfiable. Case (ii): Without loss of generality, with the fi satisfying fi,
the earlier
Qi is a Boolean combination
conditions.
of S’,r,t,h, 1 6 i
Let L be the splitting
field over Q of
..,fN,presented effectively as Q[/3], with G(b) = 0, G manic irreducible
over Q.
We relegate to Case (i) any primes p for which the p-adic value of any coefficient of G is nonzero. Fix a /I, and effectively list the conjugates of p as F&3), . . . ,I$(/?), the L$ E Q[x], Fe(x) =x. Write all the roots of all the fi as polynomials Hil over Q in /I. Discard all primes p dividing the denominator of some coefficient of some such polynomial. The truth value of S’,r,t,h depends only on the restriction of o to L. This restriction is given by P H Fj(P)
some j,
and its action on all roots of the fi is explicitly
calculable.
In this way we can effec-
tively find the automorphisms
which satisfy @. This gives the decision procedure in characteristic 0, and in passing effectively describes the cr that satisfy Cp. These form a union of conjugacy classes. In finite characteristic we use Cebotarev’s Theorem. Note that if @ is not satisfiable in characteristic 0 then Qi is not satisfiable in characteristic p 3 pl(@), where pi is effectively calculable. Relegate the primes
surviving
prime).
We have the familiar
L -L,p-Fp
(*)
T
T
T
0 -
op-
Fp
EP the residue field of Lg. Suppose (TE Gal(L), o(P) = P. Then g extends uniquely to 5 E Gal(LpJQ&), which is naturally of order [Lq: Q,] (=ord(a)). Gal(LpIQ,) is naturally isomorphic to Gal( lF9)Fp), which is generated by x H x P, the Frobenius automorphism F’. This lifts back to a unique (~9 with op(Y)=P. As 9 varies above p, ~9 varies through a conjugacy class 9$ in Gal(L). Cebotarev’s Theorem [2] says that if %? is a conjugacy class in Gal(L), then the p with $, = 59 have density card(VQ/card(Gal(L)). In
A. MacintyrelAnnals
of Pure and Applied Logic 88 (1997) 165-180
179
particular, every u E Gal(L) is CJ~ for infinitely many 9’. I refer to the sets {p : %$= 9) as cebotarev sets. In particular, each of the conjugacy classes of automorphisms
has associated Consider
Cebotarev
sets.
first the j such that p -Fj(/?)
satisfies
@. Fix one such, and consider
(*), where 9 and p are chosen so that rsg is /? -4(/Q. It is quite evident that hold in (Ep,&). To deal with the finitely many my &,t,h, holding in (LB -g(p)) relevant SJ,,,, false in (L,/3 ++Z$(P)) we have to prepare a little, and discard some primes. Here is the procedure for dealing with one such S’,rt,h. Write the roots of f as polynomials Ht in a over Q, with coefficients integral for all surviving p. We have by assumption rN(4(P))
# h(N(B)),
so
G t (rHd4(t)) - Wfdt)). Since G is irreducible
over Q, we can find at(t), bt(t) E Q[t] so that
ar(tMt) + b/(t) . [rHI(f$(t)) - Wdt))l=
1.
Discard all p dividing any denominator of a coefficient of any at or bt. Then for the p surviving, ~&“,~~,h holds in (Eq,&). So we have shown (effectively) that, for all but finitely many p of the Cebotarev class attached to b-5(/3), @ is satisfiable in characteristic p by x H xp. Consider on the other hand, a LP’such that @Jis not satisfiable in characteristic p by x H xp on I&P,9 unramified. Then @ is not satisfied on L by 08, for the same reasons as above. Again,
if @ is satisfiable
in characteristic
p, it is by x ++xpm, some m, on lF9. As
above, it follows that o,$ satisfies @ in L. We have proved: Theorem 13. (a) The theory of e.c. (K,o)
of characteristic 0 is decidable.
(b) Given a sentence 8 the set of characteristics p in which 0 is satisjiable in an e. c. (K, CJ) consists of either (bl) a jinite set of finite characteristics, or (b2) the union of: {0}, a finite set of jinite characteristics, and a jinite number of effectively computable cebotarev sets for an eflectively computable L, possibly with finitely many effectively computable elements deleted. (c) The theory of all e. c. (K, a) is decidable. (d) Given 0 one can calculate an N and a prime po such that in all jinite characteristics p for which @ is satisfiable it is satisfiable by an extension of x H xP on IF ilg, tf p 2 pi, and by an extension of x H xPn on @, tf P-C pi, for some n
A.
180
2. Concluding
Macintyrel Annals of Pure and Applied Logic 88 (1997) 165-180
remarks
I expect to present, in [5], a proof that nonprincipal ultraproducts of fields (Kp, x H xp”‘), where K is algebraically closed of characteristic p, are e.c. provided the resulting
automorphism
Weil Conjectures,
is not a finite power
intersection
of Frobenius.
theory and desingularization.
The proof will use the By the preceding
it will
yield decidability of Frobenius, and, in characteristic 0, identify, up to elementary equivalence, nonstandard Frobenius with generic. Hrushovski [4] has independently obtained the same result, which is about to be submitted as this paper is being revised (February 1997).
Acknowledgements The research reported here was supported by a NSF grant to MSRI, by a Senior Research Fellowship of EPSRC, and by Merton College. This assistance is gratefully acknowledged. I am grateful to Gabriel Carlyle for helpful discussions, reading of earlier versions.
and for careful
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [l l] [12] [ 131
J. Ax, The elementary theory of finite fields, Ann. Math. 88 (1968) 239-271. M. Fried, M. Jarden, Field Arithmetic, Springer, Berlin, 1986. R. Hartshome, Algebraic Geometry, Springer New York, 1977. E. Hrushovski, The elementary theory of Frobenius, Preprints, Jerusalem (1996-1997). A. Macintyre, Nonstandard Frobenius and generic automorphisms, in preparation. B. Poizat, Groupes stables, Nur al-Mantiq wal-Ma’rifah, 2, Bruno Poizat, Lyon, 1987. M.O. Rabin, Decidable Theories, in: J. Barwise (Ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 595-629. A. Robinson, On the Metamathematics of Algebra, North-Holland, Amsterdam, 1951. A. Robinson, On the concept of a differentially closed field, Bull. Res. Council of Israel Sect. F SF (1959) 113-128. G. Sacks, Saturated Model Theory, Benjamin, Reading, MA, 1972. H. Simmons, Existentially closed structures, J. Symbolic Logic 37 (1972) 293-310. A. Weil, Foundations of Algebraic Geometry, Amer. Math Sot. Colloq. Publ., vol. 29, American Mathematical Society, New York, 1946. A. Weil, Sur les courbes algkbriques et les vari&s qui s’en deduisent, Hermann, Paris, 1948.
Annals of Pure and Applied Logic 88 (1997)
Generic
165-180
automorphisms
of fields
Angus Macintyre* Mathematical Received
29 January
Institute,
1996; received
24-2Y St. Giles. oxford
0x1
in revised form 28 January
3LB.
UK
1997; accepted
12 March
1997
Abstract
It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected (via fixed fields) to Ax’s theory of pseudofinite fields, and analogues are obtained for most of Ax’s classical results. Some indication is given of the connection to nonstandard Frobenius maps. Automorphism;
Keywords: AMS’
Existentially
closed; Pseudofinite
03ClO; 03C60
C/usszjication:
0. Introduction This
paper
some
aspects
Theorem 1. (a) Any two algebraically closedfields of the same characteristic the same jirst-order sentences of the language of field theory. (b) lf @ is a sentence qf the language qf,field theory then
satisjj
of which
originated
in some
reflections
on the Lefschetz
The principle
has two parts:
: @ holds in (Z/P)“‘g}
{P E Spec(Z)
is open or closed in Spec(Z), and so contains it contains all hut jinitely many P. Moreover,
there
(respectively, Theorem
Principle,
I now recall.
are effective
of fields 1 is useful
aspects
of fixed
0 13 it contains
to this. The theory
characteristic)
infinitely
of algebraically
is decidable.
a result
over
(2) transferring
a result
over all @” to all other
Cc to all characteristic
zero cases
cases
[ 121;
[l].
* E-mail: [email protected] 0168-0072/97/$17.00 @ 1997 Published PIZ SOl68-0072(97)00020-1
closed
For all this see [7].
in two ways:
(1) transferring
many P $T
by Elsevier Science B.V. All rights reserved
fields
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
166
A functorial
way of expressing
I be a countable algebraically
Here
1 is via the ultraproduct
index set, and D a nonprincipal
ultrafilter
construction.
Let
on I. Let Ki (i E D) be
closed fields such that for each prime p
{i : Ki has characteristic and only, pose such
Theorem
165-180
p} $ D,
is of course due to cardinality IKil<2N”. Then fl, Ki N @. The isomorphism and is noncanonical. But now there is a feature missing in Theorem 1. Supgi E Aut(Ki) for all i. Then there is a well-defined element n, (Ti of Aut(&, Ki), that for any sentence 0 in the language of fields with automorphism
n,
ci
is a D-average
of the
ci,
and
(noncanonically)
is an element
of
Aut( a=). Consider the special case when I is the set of primes, KP is of characteristic p, and aP is Frobenius, i.e. x H xp. Then the automorphism no oP of @ deserves to be written x H xp, p a nonstandard prime, and be called a nonstandard Frobenius. What is special about these maps? There are two aspects, one tedious, and one fascinating: (1) (C, no crP) is Nl-saturated, i.e. has a kind of compactness property [lo]; (2) The elementary properties of &, aP relate to Weil’s Riemann Hypothesis for curves, and beyond to the Weil Conjectures. Already from Ax’s work [l] one knows that Fix(&, oP) is severely constrained, as &, Fix(ap), i.e. n, EPp,and so must be pseudofinite, i.e. have Galois group f and be such that every absolutely irreducible curve over it has a rational point. Moreover, that is essentially all one can say about n,Fix(crP) [l]. Let 0 = n, op. One wants to know about polynomial equations in xi,. . . ,x,, a(x, ), . ..) (T(x,), a2(x*),.. .,02(x,) )... solvable in @. By the usual device of adding extra variables this comes down to understanding polynomial systems C(xi, . . . ,x,, a(x, ), . . . , a(~,)). By the methodology of Robinson’s Test [8], if one understands enough about the solvability of such systems, then one understands the elementary theory of (C, a). This in turn means that one understands the theory (or the almost-all variant) of the class (F;lg , opp).This was my original question: P , op) decidable? Does it admit a reasonable elimination Is the theory of all ([Falg theory? What can one say about the nonstandard Frobenius maps on C? In [5] I will show that the answers to the first two questions are yes, and that the third has a beautiful answer in terms of an independent notion of generic automorphism of C. In fact, I succeed in generalizing naturally every result from Ax’s classic paper PI. This paper develops the theory of generic (or existentially closed) automorphisms of fields. Its sequel [5] will deal with the much deeper problem of showing that the nonstandard Frobenius maps are existentially closed.
A. Macintyrel
I formulated
Annals of Pure and Applied Logic 88 (1997)
the main problems
in the mid-1980s
165-180
167
and made the first progress
MSRI in 1990, when I showed that the theory of fields with an automorphism model companion.
At that time I had very helpful conversations
at
has a
with Zoe Chatzidakis,
Lou van den Dries and Carol Wood, whom I heartily thank. A result of Lou, that the fixed field of an existentially forth the conjecture
closed automorphism
that the nonstandard
Frobenius
is pseudofinite,
naturally
maps are existentially
brought
closed (equiv-
alently, Robinson generic). Later, Udi Hrushovski gave a neater formulation of the axioms for generic automorphisms (Axiom H in this paper) and this has turned out to be an inspired
move. I thank Udi for showing
me this insight.
1. Difference fields, Axiom H 1.1. Basic systems For reasons unknown to me, a pair (K, o), where K is a field and rr E Aut(K), is called a d@erence jield. The class of difference fields is axiomatizable, by an V’3 sentence, in the language of ring theory extended by a symbol for the automorphism. In both the algebraic and logical study of such structures, one must study (difference) systems, the natural generalization of polynomial equations (and inequations) to take account
of rr. Bear in mind the two simple observations
(1) and (2) below:
K/=‘X#OOK~((3/?)(a~-l=O),
(1)
K t= @(xi,. . . ,xn, 4x1), . . . , ah),
a2h 1,. . , a2M>
@ K I= ~y1...~y,[~(~l,...,~,,Yl,...,y,,~(Yl),...,~(Y,)) Ay1=o(x1)A
...
Ay,=o(x,)].
C(xi,. . . , xn) is equivalent
Then
solvability
C’(Yl,.
. ., y,,,), where C’ consists of polynomial . . ., a(~,). Such C* I call basic systems.
4Yl),
1.2. Existential
of systems
(2)
to solvability
equations
of systems
over K in yi,. . . ,y,,,,
closedness
Definition 1. (K, O) is existentially closed (e.c.) if whenever C(y) is a basic system of K, and z is solvable in an extension (K’, a’) of (K, a) then C is solvable in K. General
logical theory [ 1 l] tells us
Lemma 1. Any (K, CT)extends
to an e.c. (K’,o’),
with JK’( = max(lKI,Ns).
In the category of fields, the e.c. structures are the algebraically closed fields, which have the special property that they are axiomatized by the infinite set of conditions C, (1 dn
168
A. MacintyrelAnnals
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In the category of fields with derivation, studied, under the name of differentially order axiomatizations,
165-180
the e.c. structures have also been extensively closed fields [9]. These too admit nice first-
see [lo].
Theorem 2. The class of e.c. d#erence jields is first-order axiomatizable. The proof will be given a little later. First, a simple lemma.
Lemma 3. If (K, cr) is e. c., K is algebraically closed. Proof. Any automorphism
of a field extends to its algebraic
closure.
0
Definition 2. Fix(o) = {X E K : a(x) =x}. Lemma 4. Let (K, o) be e. c. Then (i) Fix(o) is perfect; (ii) the absolute Galois group of Fix(a)
is f.
Proof. (i) Clear. (ii) Note that by the absolute Galois group I mean Aut(Fix(o)a’s]Fix(o)). (K,a) is e.c. I claim that for each II 32, there exists x,, in K with @(xn) =x,,
Suppose d(~,) #
x,, j < IZ. To see this, let z be a permutation of { 1,. . . , n} of order n, and let yl, be algebraically independent over K. Extend 0 (uniquely) to an automorphism
.
. , y, 6 of
via 6(yi)=yr(i). Then one of the y’s satisfies c?(y)=y, c?j(y) # y, K(YI ,...,y,) j
is exactly n. Finally,
since Aut(Fix(a)a’s/Fix(a))
Galois Theory, we get it to be f.
is procyclic
by elementary
0
Just as simple is
Lemma 5. Let (K, CJ) be e. c., and let f (x, y) = 0 be an absolutely irreducible plane curve over Fix(o) (by which we imply in particular (a,P)~Fix(o)~ with f(a,B)=O.
f E Fix(o)[x,
y]). Then there exists
over K. Let K’ = K[x, y]/( f), and extend cr to CT’on K’ by a’(x) =x, o’(y) = y. So there exists (cc’,,!?‘)E Fix(o’)*, f(c~‘,/3’) =O, and this drops to the e.c. (K,a). 0
Proof. f is irreducible
Van den Dries showed me these in June 1990 (and Lascar noted it independently). Together they give something extremely suggestive.
A. MacintyrelAnnals
of Pure and Applied Logic 88 (1997)
Theorem 6. Zf (K, 0) is e.c., Fix(o)
is pseudojinite
165-180
169
(i.e. an infinite model of the theory
of jinite jields). Proof. See [2] for a proof that a field L is pseudofinite iff Aut(L”‘s/L) CYf, and every q absolutely irreducible plane curve over L has an L-rational point. Note.
It is quite easy to see (using Ax’s [l] characterization
of pseudofinite
fields) that every pseudofinite
of the absolute
L is elementarily
equivalent
for some e.c. (K, G). Just take K as an extension of r on the algebraic prime field of L, where Fix(r) is the field of absolute numbers of L. 1.3. o-action
in algebraic
numbers to Fix(o)
closure of the
geometry
Though it is not strictly necessary for the formulation of the crucial Axiom H, I find it valuable to give a systematic account of how rr E Aut(K), for K algebraically closed, induces
natural
actions in algebraic
1.3.1. CJacts on K” coordinatewise,
geometry
over K.
and on the power set of Kn by a(E) = {o(_?) :
TEE}. 1.3.2. ~7 acts on K[xl,. . .,x,1 by o on K, and the identity 1.3.3. If Zer( fi,.
on xi,. . . ,x,,.
..,fk)C K” is the set of zeros in K” of the fi E K[xI,. . . ,x,J, then
dZer(fi,...,fk)) = {dY):fi(Y))=
... =fk(Y)=O)
= Zer(fl( fl >,. . . , o( fk 1). 1.3.4. Thus, irreducibility.
r~ acts on A,(K) and P,(K), and TT acts on algebraic sets, respecting So rr is a homeomorphism for the Zariski topology, in both cases.
1.3.5. The a-action to radicals.
on K[xl , . . . ,x,1 extends to ideals, sending primes to primes, radicals
1.3.6. Evidently, 0 sends quasi-affine affine (respectively, quasi-projective)
(respectively, varieties.
quasi-projective)
varieties to quasi-
1.3.7. So g sends varieties to varieties [3, p. 151. Moreover, the action on polynomials extends to one on regular functions and morphisms, making cr functorial. Precisely: Let V be a variety, wlog affine, and U an open subset of V, containing the point P. Suppose f on U is given by g/h, g, h polynomials, h not vanishing on U. Then G(U) is an open subset of a(V) containing a(P), o(h) does not vanish on a(U), so O( f) =der o(g)/o(h) is (gives) a well-defined regular function at P.
and
This gives the action of o on regular functions. Notice that cr defines a continuous map (not a morphism) from V to o(V), and that for a regular function f : V -+ K, CJ(f) : (T(V) --) K is given by O( f) = afo-'. Now taking the definition of morphism of varieties given in [3, pp. 15, 161, f : VI ---f V2 is a morphism so is af g-’ =,+f c( f ).
170
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
Now the functoriality 1.3.8.
0 has obvious
natural
of IJ is evident. action
ring isomorphisms
structure).
on the rings O(Y) ? U(a(Y))
This in turn induces
1.3.9. Similarly,
165-180
O(Y),
Y ranging
(not of course
over K-varieties, respecting
giving
the K-algebra
action on the local rings @~,r (E Q.(p),.(r)).
there is an obvious
1.3.10. (T respects rational, dominant nonsingularity).
action on the fields K(Y), and birational
Y a variety.
points, and respects singularity
(and
1.3.11. Now let V be a variety (construed as topological space), and 0~ its sheaf of regular functions, so cOv(U) is the ring of regular functions on U. Now cr : V ---)a(V) is continuous (as is o-l), and by [3, p. 651 the direct image sheaf o*(V) is o,(Ov)(W) = 0,(a-‘(IV)). One easily checks that QT(V)(W) = {ogc-’
: 9 E Q*(@)(W)}
and that hH o-‘ha defines a map Q,(v) -+ LT*(@v)of sheaves of rings on cr( V). So, in fact, (cr, h H o-‘ho) defines a morphism (actually an isomorphism) (K(%)-,(@U,4r(V,) of locally ringed spaces. Write 0(~,(%)=(4V,G,,,) and this is clearly functorial
on morphisms
of varieties.
1.3.12. In [3, p. 781 one deals with the fully faithful functor t from varieties to schemes over K. o is entirely compatible with this, as I now show. Firstly,
one associates
to any topological
space X a set t(X)
consisting
over K of all
nonempty irreducible closed subsets of X, and topologizes this by taking as closed subsets all t(Y), Y closed in X. t is functorial on continuous maps f, by defining t(f)(E), E nonempty irreducible closed, as the closure of f(E). One also defines a continuous cx:X + t(X) by setting a(P) equal to the closure of {P}. The scheme attached to V is (t(V),cr,(0~)). Now clearly, t(a)(E) = a(E) all E E t(V). Also, when U is open in t( I’), cr*(Sv)( U) = &(a-l(V)) and ~*(~0(V))(~(~)(~))
= QT(V,(~-‘t(@(U)) = {oga-’
: g E U)y(Cr-l(CL-l(t(0)(U))}
= {ago-’
: g E coV(a-‘(u))}.
A. MacintyrelAnnals
Since
the scheme
attached
171
of Pure and Applied Logic 88 (1997) 165-180
to o(V)
is t(o( V)), a*(Qv,),
of 1.3.11 of r~ on (V, 0~) extends to a sheaf isomorphism
we see that the action (also denoted
0)
(V)>~*(~v))-+(~(4~))?x*(Lo,(V~)) which is t(o)
on t(V),
and gives a map
~*(QT(V,) + a*(@*(&)) of sheaves of rings on t(V) by h I-+ t(a)-‘/n(o) as before. Again this is functorial. Moreover, t(V) =&f (t(V), a,(Uv)) has a natural structure of scheme over K, with the morphism to Spec(K) being (f, f’), where f sends every point of t(V)
to 0, the unique point of Spec(K),
and
f # : QSpec(K) + f*&(V) is given by f #(X) = the constant
function
on t(V) with value X.
On the other hand, r~ induces an automorphism of Spec(K) by (id,@). So one easily sees that 0 is even functorial on t at the level of schemes over K. 1.3.13. Finally, how does cr act on schemes over K? Let (X,0X)-+(,-f,+) Spec(K) be a scheme S over K. Composed with (id, g) this gives another scheme a(S) over K. Of course, 00-l IS the identity. We retrieve all the earlier nonsense via t! 1.4. The axiom Now we return to earth (and logic). Let V be a closed subvariety by equations fi(xl,..., xn)= ... = fk(x, ,..., x,)-O. I assume K algebraically defined by (~fl)(Yl,...,
(of1
)(Yl,
. ..)
closed. Then g(V)
Yn)=
‘..
=(~fk)(yl,...,yn)=0,
y,)=
...
=(cJfk)(y
,,...,
of A”(K) defined
is also a closed subvariety
of A”(K)
yn)=O.
a(V) shares with V all natural properties like nonsingularity, dimension Now form V x a(V), the closed subvariety of A2”(K) given by
d, etc.
=(~fk)(Yl,...,Yn)=O. Suppose,
W is a closed subvariety
w-Vxcr(V) 711J V
\
712
o(V)
of V x r~(V), and suppose the two morphisms
172
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
are generically
surjective
(the xi are the natural projections).
point of W projects to generic points of V and a(V). intersects
165-180
This means that a generic
In this case, I say W potentially
the graph of cr.
Now I come to the axiomatic Axiom H. If W potentially
heart of the matter.
intersects
such that (x1 ,...,Xn,O(Xl),...,d(X,))E This extremely
the graph of (T, then there exists xl,. . . ,x,, E K w.
suggestive
principle
(in 1990) a more complicated version, order to axiomatize the e.c. (K,o).
was isolated involving
by Hrushovski,
various
V x o(V)
after I had used x . - . x am(V), in
It is crucial for what follows that one sees Axiom H as the conjunction of axioms Hk, for k= 1,2 ,..., where the Hk are (under the assumption that K is algebraically closed) first-order axioms. It is worth explaining this in some detail. Though I am only interested (now) in the case when K is algebraically closed, and the difference between variety and absolutely irreducible variety disappears, let me consider the variant of H where V and W are assumed absolutely irreducible. Fix an integer k and restrict H to the case in which there exists and n
1.5. Elimination
of quanti$ers
Lemma 5. Suppose
(K,o)
is e.c. Then Axiom
H holds in (K,o).
Proof. Suppose (K, a) e.c. Then K is algebraically closed. Go to a (K]+-saturated elementary extension (K, o) -t (KI, 01) [lo]. Then K1 contains a generic point (at,. . . , a,, PI,. . . , Pn) of W over K. We are using “generic” in the sense of [ 121. Now by generic surjectivity of ~1,712, (~(1, . . . ,a,) is V-generic over K and (fit,. . . , fin) is a~( V)-generic over K, i.e. a( V)-generic. Now define
d : K(Q ,...,a,)-~K(Bl,...,p,),
A. Macintyrel
Annals
of Pure and AppliedLogic 88 (1997) 165-180
by C’ = CJon K, o’(ai) = /3i. 0’ extends to an automorphism o”(4 ), . . . , a”(a,)) E W. Since (K, a) is e.c., we get some ..,4Y,))E
(Yl, ...,Yn,4YI)1. where yI,...,y,,EK.
173
(T” of Kl, and (~(1,. . . , a,,
w,
0
Now the crucial Nullstellensatz: Theorem 7. Suppose (K, C) satisJies Axiom H, and K is algebraically closed
Then
(K, o) is e. c. CT(D,)) be a basic system over K, with a solution Proof. Let C(vl, . . ..V.,~(Ul),..., a,) in an extension (L, a’) of (K, g) over K. Let V be the variety of (al,. . . , cc,,) (al,..., over K, so o(V) is the variety of (a’(~1 ), . . . ,~‘(a,)) over K, and let W be the variety of (c!,,..., a,,o’(al) ,...) cr’(a,)) over K. Then W 2 V x o(V). It is obvious that W potentially (P1,...,j3n)
meets the graph of O. So W has a K-point solves C.
(PI,. . .,/$,,o(/I~),
. . ., o(fin)), so
0
Note. 1. We use the assumption that K is algebraically closed to get V, W absolutely irreducible. 2. Inspection of the proof shows that in the above we can weaken H to H,,, where V, W are assumed nonsingular. The reason is that we can add to the equations for V (respectively (respectively, as usual.
W) inequations expressing that V (respectively, W) is nonsingular at cl (&o’(i)), and replace these inequations by equations in more variables
This shows that the class of e.c. (K, a) is elementary,
and so model-complete
[8]. So,
modulo Axiom H and the axioms that K is algebraically closed, every formula Q(V) is equivalent to an existential formula. We can do a little better, for the following reason. Lemma 8. Suppose (Kl,q), (K~,cz) are extensions of (Kj, ~3), where K3 is algebraically closed. Then (Kl, q), (K~,Q) can be jointly embedded in some W4,04
).
Proof. Trivial manipulation
of transcendence
bases over K3.
Cl
Corollary. If (Kl, 02) and (Kz, az) are e.c., and 01, CQagree on the algebraic closure of the prime jield then (Kl, rs1 ) 3 (K2,02). Proof. By Lemma
8 and model-completeness.
0
More generally, Lemma 8 shows that the (elementary) type of a tuple (al,. . . , a,) in an e.c. (K, o) is determined by 0 on the algebraic closure (in K) of the field generated
A. MacintyreIAnnals
174
of Pure and Applied Logic 88 (1997)
165-180
by the sequence. . . , &z, o(w),
al,.
o-‘(cq),...
. . .,4&z), ) 6’(an),
(i.e. the smallest
o2(a1 ), . . ., 02(%), . . .
a-2(q),
. . .) C2(cl,),
...
~1, . . . , ~1, and closed under
field containing
of course, leads one to a quantifier-elimination. obvious counterexample to quantifier-elimination
~,a-‘).
This remark,
What is required is suggested by an in the original language. Consider
Q(V) below. @p(u)Z o(u) = u A (3)(?
= lJA o(t) # t).
That one cannot eliminate the existential quantifier is seen by taking independent transcendentals ui and ~2, taking o as the identity on the algebraic closure of Ka(ui) (Ka the prime field), while taking o to fix K:“(u2) but move the square roots of u2, and then extend to an e.c. (K,cr’). The predicates I use for quantifier-elimination are natural generalizations of those used in the theory of pseudofinite fields [2]. Let it, k be nonnegative integers, and let f,gi,hi
(i,
be members
of
9.. . ,XOn,Xll ,...,~ln,yll,...,Yln,~2l,...,~Zn,Y2l,...,Y2n,...,
Xkl,...,Xkn,Ykl,...,Ykn,tl.
Let ShQ,i(u~,. . .,u,)
3t
be
d(Un),
f(Ul,...,
t) = 0
[
1
A A ((T(SI.(Ul,...,~-k(un),l))=hi(Vl,...,a-k(U,),t)) idm
where ui,,..,~-~
(u,) abbreviates
the sequence
9
of arguments
ul,...,Un,~(~l),...,~(~,),~-1(~l),...,~-’(~,),~2(Ul),...,~2(~,), 8(v1)
)...,
l+(u,),..
To formulate the extension notation is useful:
.,&I*),.
lemma
. .,(Tk(U,),a-k(u~) )...) dyu,). that gives quantifier-elimination,
the following
Definition 3. Let (K, o) be a difference field and X G K. Then (X), denotes the smallest subfield of K containing X and closed under e and 0-l. Lemma 9. Let (K, CJ) be a difference jield, Ko be the prime field of K, and X 2 K. Then (X), = K&Y, a(X), o-‘(X),
02(X), a-*(X),
. . .).
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
Proof. Trivial. Lemma
175
165-180
0
10. Let Y be the relative algebraic closure of (X),
in K. Then (Y), = Y.
Proof. Suppose c( E Y. Then Bk,cr(B,)
WA,..., &I
), .*. 3 fJQ%f),
so CT(u) E Y.
(z(pk),~-l(B*),...,o-l(Bk),..., 0%
1,. . ., &AC),
and F a polynomial
for /h,...,fikEX,
F(o(B,),
,...,
. . . , ‘@k),
. . . ,a-‘+‘(jh
a) = 0
over Z. Then ), . . . , @-‘+‘(pk),
a(@))
= 0,
0
Lemma 11. Let (Ki,oi), i= 1,2 be dtjterence fields, Ki algebraically closed, and (Mi,ll/) dtjerence subjields of the respective (Ki,oi). Let 0: (Ml, $1)2:(A42,&) be an isomorphism of dtrerence ftelds, and suppose 0 respects all Sf,sk and TS~,~,~, (computed in the respective ambient Ki). Let h4; be the algebraic closure of Mr in Ki, and $: = o 1Mi. Then 0 extends to 0’: (Mi, &‘) N (AI,‘, *ii’>, 0’ also respecting the SrG,h and -SL8,i. Proof. By considering f = tP - x, we see that 0 respects being a pth power, so without loss of generality one can assume M/IMi is separable, 0’ will be constructed by a Tychonov argument, as a limit of extensions ON, where 0~ is defined on a finite Galois extension N of Mt. Fix such an N =A41 [a], say. Let F(t) = f (Yl, . ..> Yn,~l(Yl)r...,~l(Yn),~~l(Yl),...,~l-l(Y~),...,
)>. .
&Yl
2 &Yd,
qk(yl
>, . . ., qk(Yn>,
where F is over E, be the minimum normal closure Nr of Ml(a, q(a)). Then, by increasing a=goh,. &Yl
t),
polynomial
over A41 of some generator
(The latter is obviously
n, k if necessary
finite-dimensional
we can write
..,Yn,~l(Yl),...,~l(Yn),~~l(Yl),...,~~l(Y~),..., h f.. , &rn>>
qkbJ1
>, . . . >qk(Y?J?
6)
and 01 (a) = ho(Y1 ~~~.7Yn,~l1(Yl),~*~, &Yl),.
‘. 7 ~~:(Yn>, al-kh
with ga,hs over h. So M
~Sf,go,ha(Yl,...,Yn),
dYd>q’(n) ), . . . , al-k(Yn),
T..., 6)
a,-‘(Y,)
,...,
6 of the over Ml).
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
176
165-180
whence
Obviously, f(@(Yl),
. . . > @(Yn),OZ(@(Yl)),.
&@h
is irreducible
>>>.
. .Y &@(Y,)),
..,02(~(yn)>,o~1(o(yI))
~$w(Yd),
,...,
. . ., q?o(y,)),
(T;l(o(y,))
,..‘)
t)
over A42. By the above, it has a root 6’ such that
(There may of course be several such roots 8.) If we choose one such 6’ then there is a unique field isomorphism 0’: Ml(S) N A42(8) extending 0, with O’(6) = 6’. Moreover, cQ(@‘(cc)) = O’(Oi(cc)). It may, however, happen that there is /3 in M,(6) and that oi(/?) is also in A41(6) and a2(@‘(/3)) # O’(oi(/?)). But, I claim there is a choice of 6’ such that this does not happen,
i.e. so that both
az(@‘(cc)) = @‘(o,(a)) and a2(@‘@>>
=
For, increasing P = (n(B)
@‘(w(B)).
n, k again if need be, we have
Sl(Yl,...,Yn,...,q =
h(Yl,.
-?Yl>,. . . ,Yn,.
. . , q-k(Yl),
. . >q-k(Yn),
6)
. . . , qk(Yn),
with 91, hi over 27. Let 8= (go,gl), i; = (ho,hi). Ml
8)
Then
i==f,g,~(YIMYn),
so
thereby clearly giving us 6’ that works for cc,p. There are only finitely many possible 8, so proceeding in this way we eventually get 8, and so O’, which works for all /3 such that p and o,(p) are in A&(d). Now let CIvary, and consider the projective system of all such 0’. Let 6 be a limit of this system. Then 6 is clearly a difference field isomorphism, evidently respecting all SJgh q 1 , and ~5”~i. I ,
A. Macintyrel Annals of Pure and Applied Logic 88 (1997)
From this, Lemma
8, and the standard
criterion
165-180
177
[lo] for quantifier-elimination,
one
concludes: Theorem 12. The theory of e.c. (K, c) has quantzjier-elimination extra predicates
in the language with
for all Sf,g,r,.
1.6. Decidability We can prove decidability
of the theory
of all e.c. (K,a),
Lemma 7 or the preceding quantifier-elimination. Instead of the formulae Sr,g,i I now use sentence
using the corollary
to
S‘,#,a, where n = k = 0 (cf. intro-
duction of the predicates). So now f, S,i are polynomials over Z in a single variable. We know of course that two e.c. (Ki, ai) (i = 1,2) are elementarily equivalent if and only if they agree on such sentences Sf,g,h. But a little more is obvious on (short) reflection. We want to restrict somewhat the f we need consider. We restrict to f E Z[x] which satisfy: f is irreducible over Q, of degree m say, and with splitting field generated over Q by a single root c1 of f. Then the roots of f are ho(a), hl (a), . . . , h,_l(a) where each hi E Q[t], ho = t. Any 0 on Q(a) is determined by a choice of c( and one hi. If we do not specify E, but only hi, g is determined exactly up to conjugacy in Gal(Q(cc)). Now the algebraic closure UY’s of Q is a limit of such Q(a), compactness argument the conjugacy class of c in Gal(ClY’s) is Sf,s,h, where f ranges as above, g(t)=rt some r E Z, r# 0 and above are of form h/r, r E Z, h E Z[t].) But now it is clear that if (K, 0) is e.c., so is (K, z-‘oz) for any any p E Gal(Qa’s) extends to an automorphism of any algebraically
and by a simple determined by all h E Z[t]. (The hi r E Aut(K). Since closed K > Q, the
above yields, in characteristic 0, that the theory of an e.c. (K,o) is determined by the Sf,rt,h in it, as above. So, effectively any sentence @ is equivalent (in characteristic 0, and so in all characteristics Sf.,rl,h as above. So the decision
problem
p 3 PO(@), po recursive)
to a Boolean
combination
of
for @ reduces to
(i) Is Q, satisfiable in some characteristic p < PO(@)? (ii) Is the Boolean combination satisfiable in any other characteristic, including O? We may clearly assume, by loading a bit more into (i), that if Sf,rr,h or its negation occurs in the Boolean combination then r # 0 mod p for p 3 PO(@). For future reference (in [5]) we pay attention to the set of characteristics p in which @ is satisfiable, and if p#O we examine the possible restrictions to @” of cr satisfying
@.
Case (i): We examine each prime p< po separately. Fix one such. Now the discussion over Q can be replaced by one over EPp,considering f irreducible over EP (the assumption on generation by a single root is redundant), and we can clearly take
178
A. Macintyrel
f manic.
Annals of Pure and Applied Logic 88 (1997)
So now Q, is effectively
equivalent
need for r’s). But now by constructing and running
through
all (TE Gal(N)
to a Boolean
a finite extension
(all powers
165-180
combination
of Sf,t,h (no
N of lFP splitting
of Frobenius)
each
f,
we just test if Q, is
satisfiable. Case (ii): Without loss of generality, with the fi satisfying fi,
the earlier
Qi is a Boolean combination
conditions.
of S’,r,t,h, 1 6 i
Let L be the splitting
field over Q of
..,fN,presented effectively as Q[/3], with G(b) = 0, G manic irreducible
over Q.
We relegate to Case (i) any primes p for which the p-adic value of any coefficient of G is nonzero. Fix a /I, and effectively list the conjugates of p as F&3), . . . ,I$(/?), the L$ E Q[x], Fe(x) =x. Write all the roots of all the fi as polynomials Hil over Q in /I. Discard all primes p dividing the denominator of some coefficient of some such polynomial. The truth value of S’,r,t,h depends only on the restriction of o to L. This restriction is given by P H Fj(P)
some j,
and its action on all roots of the fi is explicitly
calculable.
In this way we can effec-
tively find the automorphisms
which satisfy @. This gives the decision procedure in characteristic 0, and in passing effectively describes the cr that satisfy Cp. These form a union of conjugacy classes. In finite characteristic we use Cebotarev’s Theorem. Note that if @ is not satisfiable in characteristic 0 then Qi is not satisfiable in characteristic p 3 pl(@), where pi is effectively calculable. Relegate the primes
surviving
prime).
We have the familiar
L -L,p-Fp
(*)
T
T
T
0 -
op-
Fp
EP the residue field of Lg. Suppose (TE Gal(L), o(P) = P. Then g extends uniquely to 5 E Gal(LpJQ&), which is naturally of order [Lq: Q,] (=ord(a)). Gal(LpIQ,) is naturally isomorphic to Gal( lF9)Fp), which is generated by x H x P, the Frobenius automorphism F’. This lifts back to a unique (~9 with op(Y)=P. As 9 varies above p, ~9 varies through a conjugacy class 9$ in Gal(L). Cebotarev’s Theorem [2] says that if %? is a conjugacy class in Gal(L), then the p with $, = 59 have density card(VQ/card(Gal(L)). In
A. MacintyrelAnnals
of Pure and Applied Logic 88 (1997) 165-180
179
particular, every u E Gal(L) is CJ~ for infinitely many 9’. I refer to the sets {p : %$= 9) as cebotarev sets. In particular, each of the conjugacy classes of automorphisms
has associated Consider
Cebotarev
sets.
first the j such that p -Fj(/?)
satisfies
@. Fix one such, and consider
(*), where 9 and p are chosen so that rsg is /? -4(/Q. It is quite evident that hold in (Ep,&). To deal with the finitely many my &,t,h, holding in (LB -g(p)) relevant SJ,,,, false in (L,/3 ++Z$(P)) we have to prepare a little, and discard some primes. Here is the procedure for dealing with one such S’,rt,h. Write the roots of f as polynomials Ht in a over Q, with coefficients integral for all surviving p. We have by assumption rN(4(P))
# h(N(B)),
so
G t (rHd4(t)) - Wfdt)). Since G is irreducible
over Q, we can find at(t), bt(t) E Q[t] so that
ar(tMt) + b/(t) . [rHI(f$(t)) - Wdt))l=
1.
Discard all p dividing any denominator of a coefficient of any at or bt. Then for the p surviving, ~&“,~~,h holds in (Eq,&). So we have shown (effectively) that, for all but finitely many p of the Cebotarev class attached to b-5(/3), @ is satisfiable in characteristic p by x H xp. Consider on the other hand, a LP’such that @Jis not satisfiable in characteristic p by x H xp on I&P,9 unramified. Then @ is not satisfied on L by 08, for the same reasons as above. Again,
if @ is satisfiable
in characteristic
p, it is by x ++xpm, some m, on lF9. As
above, it follows that o,$ satisfies @ in L. We have proved: Theorem 13. (a) The theory of e.c. (K,o)
of characteristic 0 is decidable.
(b) Given a sentence 8 the set of characteristics p in which 0 is satisjiable in an e. c. (K, CJ) consists of either (bl) a jinite set of finite characteristics, or (b2) the union of: {0}, a finite set of jinite characteristics, and a jinite number of effectively computable cebotarev sets for an eflectively computable L, possibly with finitely many effectively computable elements deleted. (c) The theory of all e. c. (K, a) is decidable. (d) Given 0 one can calculate an N and a prime po such that in all jinite characteristics p for which @ is satisfiable it is satisfiable by an extension of x H xP on IF ilg, tf p 2 pi, and by an extension of x H xPn on @, tf P-C pi, for some n
A.
180
2. Concluding
Macintyrel Annals of Pure and Applied Logic 88 (1997) 165-180
remarks
I expect to present, in [5], a proof that nonprincipal ultraproducts of fields (Kp, x H xp”‘), where K is algebraically closed of characteristic p, are e.c. provided the resulting
automorphism
Weil Conjectures,
is not a finite power
intersection
of Frobenius.
theory and desingularization.
The proof will use the By the preceding
it will
yield decidability of Frobenius, and, in characteristic 0, identify, up to elementary equivalence, nonstandard Frobenius with generic. Hrushovski [4] has independently obtained the same result, which is about to be submitted as this paper is being revised (February 1997).
Acknowledgements The research reported here was supported by a NSF grant to MSRI, by a Senior Research Fellowship of EPSRC, and by Merton College. This assistance is gratefully acknowledged. I am grateful to Gabriel Carlyle for helpful discussions, reading of earlier versions.
and for careful
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [l l] [12] [ 131
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