We say that a morphism of A-analytic spaces 9 : Y -^ X is almost smooth if, for any point x e X, there exist an open Hausdorff neighborhood ^ of x and affinoid domains Vi, .. ., V^C % such that Vi u . . . u V^ is a neighborhood of x and all the induced morphisms y'^V,) -^V, are smooth. Or course, smooth morphisms are almost smooth.
(And we believe that the converse implication is also true.)
7.8.1. Theorem. — Let f\ X' -> X be an almost smooth morphism of k-analytic spaces, and let 9 : Y -> X be a morphism of analytic spaces over k, which give rise to a cartesian diagram
Y -^ X /' / Y' -^ X'
Then for any (Z/^Z) ^-module F, where n is an integer prime to char(^), and for any q ^ 0 there is a canonical isomorphism
/-(R^F)^R^:CTF).
Proof. — First of all, the reasoning from the beginning of the proof of Theorem 7.3.1 reduces the situation to the case when/is a smooth morphism of good ^-analytic spaces.
Furthermore, since the statement is local with respect to X' and is evidently true when/
is Aale, it suffices to assume that/is of pure dimension one.
Let x ' be a point of X', and set x = <p(A:'). One has
(/*(R^ <p, F)),, = (R^ 9, F), = lin^ IP(Y x,, U, F),
where the limit is taken over all 6tale morphisms U -> X with a fixed point u e U over x and with a fixed embedding of fields ^(u) <^^{x)8 over ^f(;v). Similarly, one has
(R° 9:(/" F))^ = I™ H^(Y' Xx- W,/" F),
where the limit is taken over all ftale morphisms W -> X' with a fixed point w e W over x ' and with a fixed embedding of fields J^{w) ^^{x')8 over Jf(^'). Therefore, it suffices to show that for any ^tale morphism (W, w) -> (X', x ' ) there exist ^tale mor-phisms g : U -^ X and A : U" -^ U' = W X x U such that the point w is contained in the image of U" and for any q > 0 one has
Im^V',/" F) -> H^V",/" F)) C Im^V, F) -> H^V",/'- F)), where V = Y X ^ U , V = Y x ^ U ' and V" = Y X^U". We can shrink W and assume that W -> X is a separated smooth morphism of pure dimension one.
By Corollary 7.3.6, there exist separated ftale morphisms g : U -> X and A : U " ->U' = W X x U
W —> X t tI I'
U' -^> U
u" t^
such that the point w is contained in the image of U" and there is a factorization
Rxtd^n.u") —^ R+.(^,v)
\
^
<\ / (Z/»Z)^ [- 2]
where < is induced by the trace mapping. Consider the fibre product of the previous diagram with Y over X
Y X x W —> Y
f t ,
I I"
V —"-> V
t- y
V"
We claim that for any q > 0 there is a factorization IP(VV*F) —^ H^V'V^F)
\ /
H'(V,F)
where the homomorphism H^V, F) —>- W ^ M " , / ' * F) is the canonical one.
Indeed, by the Base Change Theorem for Cohomology with Compact Sup-port 7.7.1, there is a factorization
Rx^»,v") —^ R^»,v)
\ /
(Z/nZ)^ [- 2]
where t' is induced by the trace mapping. By Theorem 7.4.1, we have R«K(r(F|v)) ^^"(R+;(^,r), F|v[- 2]),
Rx:(r(F|y)) ^.^m(Rx:(^,y),F|^[- 2]).
Since ^foOT((Z/HZ)y [— 2], F|v[— 2]) ^ F[v, then there is a commutative diagram R4-:(r(F|v)) —^ Rx:(x*(F|v))
F|v
where the morphism F[y -^ Rx^X'^Fjv)) is the canonical one.
We now apply to the latter commutative diagram the derived functor RFy, where Fy : S(V, Z/yzZ) -> s/b is the functor of global sections on V. Since Fy o ^ == Fy and Fy o ^ = 1^5 we get a commutative diagram
RIV(r(F|^)) -^ Rr^,(x"(F|^))
RFv(F|v)
which gives, for any q > 0, a commutative diagram H»(V',/"F) —^ H^V'VF)
\ / H'(V, F)
in which the homomorphism H^V, F) -> H<(V",/'• F) is the canonical one. The theorem is proved. •
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INDEX OF NOTATIONS T | y : 1 . 1 (13).
0, OK (system of affinoid spaces) : 1.2 (15).
^, av/u : 1 . 2 . 3 (16).
9^ (9 morphism of analytic spaces) : 1.3 (27).
^Yo/Xo,^Y/x : 1 . 3 (28).
XQ, ^x, (X Hausdorff strictly analytic space) : 1.6 (35).
Xo', Fo (F coherent 6'x-module) : 1.6 (37).
Mod(Xo), Coh(Xo), Pic(Xo) : 1.6 (37).
A(X) (the skeleton of X) : 3.6 (71).
INDEX OF TERMINOLOGY
Boundary of an analytic space : 1.5.4 (34).
Closed immersion : 1.3 (28).
Coherent sheaf : 1.3 (25, 26).
Compact map : 1.1 (14).
Conormal sheaf : 1.3 (28).
Dense class of affinoid spaces : 1.2 (16).
Dense collection of subsets : 1.1 (13).
Dimension of an analytic space : 1.2 (23).
Dimension of a morphism : 1.4 (30).
Duality morphism : 7.3 (135).
Elementary curve : 3.6 (71).
Elementary fibration : 3.7.1 (76).
fitale topology on an analytic space : 4.1 (82).
fitale topology on a germ : 4.2 (86).
(p-Family of supports : 5.1.2 (97).
Fibre of a morphism at a point : 1.4 (30).
Flabby sheaf : 4.2 (88).
Geometric ramification index of an etale morphism : 6.3 (119).
Germ of ^-analytic space : 3.4 (64).
Germ of analytic space over k : 3.4 (65).
Gluing of analytic spaces : 1.3 (24).
Good analytic space : 1.2 (22).
Ground field extension functor : 1.4 (30).
Hausdorff map : 1.1 (14).
Interior of an analytic space : 1.5.4 (34).
Locally (G-locally) closed immersion : 1.3 (28).
Net : 1 . 1 (14).
Open immersion : 1.3 (24).
Paracompact germ : 4.3 (91).
Paracompactifying (p-family of supports : 5.1.2 (97).
Picard group : 1.3 (25, 26).
Quasicomplete field : 2.3.1 (43).
Quasiconstructible sheaf : 4.4.2 (94).
Quasi-immersion : 4.3.3 (90).
Quasi-isomorphism : 1 . 2 . 9 (19).
Quasinet : 1 . 1 (13).
Relative boundary of a morphism : 1.5.4 (34).
Relative interior of a morphism : 1.5.4 (34).
Sheaf of differentials : 1.4 (30).
Smooth pair : 7.4 (143).
T-Special covering : 1.2 (20).
0-Special domain : 1.2 (21).
T-Special set : 1.2 (20).
Stalk of a sheaf : 4.2 (87).
Strong morphism : 1.2.7 (18).
Structural sheaf of an analytic space : 1.3 (25).
Support of a coherent sheaf : 1.3 (25).
Support of a section of a sheaf : 4.2 (87).
G-Topology on an analytic space : 1.3 (25).
Trace mapping : 5.4 (106), 6.2 (112), 7.2 (130).
Zariski closed (open) subset : 1.3 (28).
Department of Theoretical Mathematics The Weizmann Institute of Science P.O.B. 26, 76100 Rehovot, Israel
Manuscrit refu Ie 28 janvier 1991, Revise Ie 13 mai 1992.
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