In this subsection we fix an integer n which is prime to the characteristic of the field k.
Our goal is to extend the construction of the trace mapping from § 5.4 and § 6.2 to any separated smooth morphism <p : Y -^ X of pure dimension d, i.e., to construct a canonical homomorphism of sheaves
Tr^R^^-^ZWx.
As in the Theorems 5.4.1 and 6.2.1, we will characterize the trace mapping by certain properties.
Let <p : Y -> X be a Hausdorff morphism of ^-analytic spaces, and let,/: X' -> X be a morphism of analytic spaces over k. They give rise to a cartesian diagram
Y -^ X t-Y' -^ X'
Suppose we are given two mappings a : R ^ y ^ ^ y ) - > (Z/nZ)x and a':R2 d<p;«^)-.(Z/^Z)^.
We say that a and a' are compatible with base change if the following diagram is commutative
f^^i^)) -^ R^K^y)
/"•(a) y.' V ^
/-((Z^Z)x) —^ (Z/TZZ)^
Here the upper arrow is the base change morphism, and the lower isomorphism is the canonical one.
Furthermore, let 9 : Y -^ X and ^ : Z —^ Y be Hausdorff morphisms whose dimen-sions are at most d and e, respectively. Suppose we are given three homomorphisms a : R^ 9,«y) -^ (Z/^, (3 : R26 +,(^^) -> (Z^Z)y and
Y : R2^e^ ) , « Y ) ^ ( Z ^ Z ) ^ .
Using the Leray spectral sequence 5.2.2 and Corollary 5.3.8, we get an isomorphism
R^WL (^) ^ R
2' ^.(R
26^.(^-z')).
Corollary 5.3.11 gives an isomorphism
R'^i^^^R'^.^z)®^^.
We get a mapping
R^W). (t^) ^ R
2' p^R
2' ^«z) ® <Y)
^R2d9I«Y)->(Z^Z)^
The composition mapping is denoted by a n P. We remark that if a and (3 are isomor-phisms (resp. epimorisomor-phisms), then a n (B is also an isomorphism (resp. epimorphism).
We say that the mappings a, (B and y are compatible with composition if y == a D (3.
Note that the operation D is transitive.
7.2.1. Theorem. — One can assign to every separated smooth morphism 9 : Y -> X of pure dimension d a trace mapping
Tr,:R2d9,(^y)->(Z/7^Z^.
These mappings have the following properties and are uniquely determined by them:
a) Tr<p are compatible with base change;
b) Try are compatible with composition^
c) if d ==0 (i.e., 9 is etale), then Tr<p is the trace mapping 9,(Z/nZ)y -> (Z/nZ)x from § 5.4;
d) if X ==^(A), ^ z^ algebraically closed and </===!, then T^v^ is the trace mapping Try : H^(Y, (JLJ ^ Z/nZ /r^ § 6 . 2 .
Furthermore^ if the fibres of 9 <zr^ nonempty, then Tr z'j flw epimorphism. If in addition r^/
the geometric fibres of 9 are nonempty and connected and n is prime to char(^), then Tr^ is an isomorphism.
Proof. — First of all, let 9 be the morphism r^ : Ad ->^{K). It is the ana-lytification of the morphism of schemes ^ : ^ ->Spec{k). One has a trace mapping Tr^ : R^^p^rf) -> (Z/7zZ)sp^) (which is an isomorphism). Its analytification (Corol-lary 7.1.4) gives rise to a trace mapping Tr^</: R^ Tcf(^Arf)-> (Z/wZ)^.^ (which is also an isomorphism). Furthermore, if 9 is the morphism TT^ : A^ == X x A? -> X, then we define Tr^rf as the base change of Tr^rf (Corollary 7.1.5).
7.2.2. Lemma. — Let 9 : Y —> X be a separated smooth morphism which can be repre-sented as a composition of an etale morphism f: Y -> A^ with the projection rc^: A^ -> X. Then the mapping
Tr, = Tr^ o Tr/ : R2" y,« ^ -> (Z/nZ)^
A^ no^ depend on the representation.
Proof. — We may increase the field k and assume that its valuation is nontrivial.
If d == 1, the statement is obtained from the case of curves (Theorem 6.2.1), using the Weak Base Change Theorem 5.3.1. Suppose that d^ 2. The Weak Base Change Theorem 5.3.1 reduces the situation to the case when we are given a separated connected smooth ^-analytic space X for algebraically closed k and elements /i, .. .,^ e ^(X) such that the morphism/: X -> A? that they define is ^tale. We have to verify that the mapping
Tr^' • - ^) = Tr^ o Tr/: H^(X, ^) ^ H^A^, ^)-^ Z/^Z does not depend on the choice of the elements/i, .. .,^.
First of all, this mapping is independent of the ordering of the elements f^ ...,/, since the group GL^) acts trivially on H2,^, ^) = H^A4, ^).
Let ^i, . . . , g^ be another system of elements in <P(X) for which the corresponding morphism g : X -> A^ is ftale. Take an arbitrary point A; e X(A) (such a point exists because the field k is algebraically closed, and its valuation is nontrivial). Replacing/
^fi —fiW and gi ^ gi — &(^)» we may assume that the elements/i, ...,/, g^ ..., ^ are contained in the maximal ideal m^ of the local ring ^x,a?* But then (/i, ..../,) and (^i, • • •?^d) are regular systems of parameters for ^x,a;? l•e•? ^^ f01'111 two bases
of the A-vector space m^/m^. Applying Steinitz Exchange Theorem for these bases, we can find a finite chain of systems of elements
( f f\ _ f f(l) f(m / f(2) f(2)\ i f(m} r(w)\
\Jl9 • • ' ) J d ) — \Jl ? • • •5J<i )f \Jl ? • • • 9 J d / ) • • • ? Ul 3 • • • ? J d ^
= (^n • • • . & )
such that
flj each/^ is one of the elements/i, .. .,,/p^i, . . . ? & ;
^ each system gives rise to a basis of mjrn^;
c ) each system (/i^^, .. .,/J1"^) arises from (/^), .. .,/J^) by the replacement of just one element.
By Proposition 3.3.10, each of the systems {f^\ .. .yf^) gives rise to an ^tale morphism y^ : X' -> A^, where X' is a nonempty Zariski open subset of X. We remark now that ifY = X\X', then dim(Y) ^ d — 1, and therefore the canonical homomorphism H^(X', pij ->H^(X, pij is bijective. Thus it suffices to show that if/i, .. .,/d-i,<?
andYi, .. ',fci-i9 h are two systems of elements in <P(X) which give rise to ^tale mor-phisms ( p r X - ^ A ^ and ^ : X - > Ad, respectively, then Tr^'-'-'^-r^ = Tr^-'-'^-r^.
Let TT be the projection A? -> Ad~l on the first d — 1 coordinates. Since TT o 9 == TT o ^, it follows, by the case rf == 1, that Tr^oTr^p == Tr^oTr^. We have
Tr^i- -^-i^) = (Tr^-i D TrJ o Tr, == Tr^-i D (Tr, o Tr,)
== TrArf-i a (Tr^ o Tr^) = (TrA</-i D TrJ o Tr^
^Tr^'-'^-r^
The lemma is proved. •
The construction of the trace mapping for arbitrary 9 is obtained from Lemma 7.2.2 in the same way as the corresponding construction in Theorem 6.2.1 is obtained from Corollary 6.2.8.
It remains to show that if the geometric fibres of <p are nonempty and connected
cv
and n is prime to char (A), then Tr^p is an isomorphism. By the Weak Base Change Theorem 5.3.1, it suffices to show that i f ^ i s algebraically closed and X is a separated connected smooth ^-analytic space of dimension d, then Tr^ : H^(X, ^) -^ Z/wZ. We remark that it suffices to find for such a space X an dtale covering (U, -> X),gi with separated U^ such that all Tr^j. are isomorphisms. This is verified by induction. To use the induction, it suffices to show that for any point x e X there exists a separated ^tale morphism/: U -> X and a smooth morphism 9 : U - > V of pure dimension 6ne to a separated smooth ^'analytic space V such that x e/(U) and the geometric fibres of <p are nonempty and connected. For this we shrink X and take an dtale morphism X -> A^.
Let ^ be the composition of the latter morphism with the projection Ad -^A1"1. The morphism ^ : X -^A1""1 is smooth of pure dimension one. Applying Theorem 3.7.2, we get the required morphisms f and 9. •
7.2.3. Corollary. —Let 9 : W ->SK be a separated, smooth morphism of pure dimension d between schemes of locally finite type over Spec(j^), where ^ is a k-affinoid algebra. Then the diagram
(i^vM^ -^ R^W^n)
(Trq,)*"^
(Z/%Z)^.an
^ commutative.
Let <p : Y -> X be a separated smooth morphism of pure dimension d. By Corol-lary 5.3.11, for any P eD(X,Z/%Z) there is a canonical isomorphism
Ry.^P^) [2^]) ^PI)R9,«Y) [2</].
Theorem 7 . 2 . 1 gives a morphism (in D(X, Z/yzZ)) R9.«y)[2^^(Z^Z^.
Therefore we get a morphism
Tr,:R9,(9-PW[2^])->F-which will also be called a trace mapping. For F eS(X,Z/yiZ) the latter morphism is induced by a homomorphism of sheaves R2^ y^y* F(rf)) ->F. It is an isomorphism if the geometric fibres of 9 are nonempty and connected and n is prime to char(^) because R^y-F^^F-SR^,^).