Invent. math. 113,389-417 (1993)
Inventiones
mathematicae
9 Springer-Vertag 1993
e-factor of a tamely ramified sheaf on a variety
Takeshi Saito
Department of Mathematical Sciences, University of Tokyo, Tokyo 113, Japan
Oblatum 18-XII-1991 & 1-III-1993
For a smooth (-adic sheaf on a variety over a finite field, a formula for the e-factor
(the constant term of the functional equation of the L-function) is proved by
S. Saito [SS]. According to his formula, it equals the value of the determinant
character of the corresponding representation evaluated at the canonical cycle of
the variety. In this paper, we generalize this to a tamely ramified sheaf and also
prove an analogous formula for a variety over a local field. In our formula, the
canonical cycle is replaced by a refined object, the relative canonical cycle defined
in the cohomology with compact support, and a certain product of Gauss sums
appears. That for a variety over a local field is an arithmetic version of the
conductor formula of Bloch [B] and that of Kato [K1].
Let X be a proper smooth variety of dimension n over a perfect field F of
characteristic p and let U be an open of X such that the complement D is a divisor
of X with simple normal crossings. Then the relative canonical cycle Cx. v is defined
in the cohomology with compact support Hc2, (U, 77'(n)) for 7f' = / / q . p Zq. It is
( - 1)"-times of the relative top chern class of the partially trivialized locally free
sheaf O~/F(logD) of rank n defined in Sect. 1. A locally free sheaf ~ is said to be
partially trivialized if a family of surjections 8 lD, -} (gD, on closed subschemes Di
satisfying a certain property is given. As for O~/v(logD), the family of the residue
maps at the irreducible components Di of D provides the partial trivialization. The
observation that there should exist a good definition of the relative chern class of
a partially trivialized locally free sheaf is due to G. Anderson.
Assume F is finite and let : be a prime number ~ p and ~ be a smooth (-adic
sheaf on U. Then by Grothendieck, the L-function satisfies the formula
2n
L(U/F,~
t) = I-I det(1 - tpv t;H~ (U-i,f" ))~-~)'+'
i=1
where ~0r denotes the geometric Frobenius (the inverse of the Frobenius substitution) and F is an algebraic closure of F. Therefore if we put
2n
eo(U/F, o~ ) = [ I det( - ~ov;H~(U-f, ~))~-1),+,,
i=1
390
T. Saito
we have the functional equation
L(U/F, ~ , t) = eo(U/F, o~)'t - z~(ur'~)" L(X /F, R j , ~ *, (q"t) -1 )
by Poincar6 duality. Here ~ * is the dual of ~ , j: U ~ X and zc(U-~, , ~ ) =
~ ( - 1)i dim H~(U~, Y). In this paper, we consider an f-adic representation p of
rtl(U, ~)t~m~and the corresponding smooth f-adic sheaf ~-p on U tamely ramified
along D~ By virtue of the Poincar6 duality and the reciprocity map
H2" (U, 2U(n)) • (CHo (X) | Tip) ~ rcl (u)~b' tame, we can evaluate det p at - cx, u.
Then Theorem 1 asserts that
eo(U/F, ~p)'eo(U/F, ~e) -d~gp = detp( - cx, v)
9(a certain product of Gauss sums).
We also study varieties over a local field. Let K be a local field with finite
residue field F. The theory of e-factor [DI] assigns eo(K, V, ~, d x ) ~ , ; to an f-adic
representation Vofthe absolute Galois group of K, a non-trivial additive character
~: K ~ ~ e~ and an (-adic Haar measure dx. In this paper, we fix the Haar measure
dx such that ~,~ dx = 1 and we drop dx in the notation. Let X be a proper regular
and fiat scheme over the integer ring CK with smooth generic fiber and U be an
open of XK such that D = X - U is a divisor with simple normal crossings. Let
p be an f-adic representation of z~ (U,)~)tame and ~-o be the corresponding smooth
(-adic sheaf on U. If we fix ~, of order - 1, the local field analogue of e0 (U/F, ~p)
above is defined by
eo(U/K, Yp, if) = e0(K, RF~(U~, ~p), ~ ).
We define the relative canonical cycle Cx, v/o~ ~ n l (U) ab' tam~ in Sect. 2 for "tame"
(X, U) and generally in Sect. 3. For the tame case, we prove
eo(U/K, ~p, if) = det p( - Cx, u/r
"(a certain product of Gauss sums)
in Theorem 2. We also prove an analogous formula for the tame part of the
cohomology generally, in the same way assuming the purity for etale cohomology.
In Sect. 3, a general conjecture is formulated.
Here is a rough idea of the proof of theorems. By taking a Lefschetz pencil,
Theorem 1 is proved by induction using Theorem 2 by virtue of Laumon's product
formula [L]. An explicit computation of the vanishing cycles as in [DG] proves
Theorem 2.
1 Relative top chern class
In this section, we define the relative top chern class of a partially trivialized locally
free module in the cohomology with compact support and study its basic properties.
Let X be a scheme and D = (D~)i~1 be a finite family of closed subschemes Di of
X. For a subset J c I, we put Ds = ~ s D i . For a prime q invertible on X, we
define a complex Zq(n)x,o by
Z'~(n)x'~176
"
i
~
,it,
( ~' 7lq(n)~
IJl=m
""]
e-factor of a tamely ramified sheaf on a variety
391
Here the sheaves on closed subschemes are identified with their direct image on
X and the boundary maps are the alternating sum of the natural ones with respect
to some total order structure on I. Here a choice of order structure does not matter
since the complexes obtained are canonically isomorphic to each other. The
complex is canonically quasi-isomorphic to j! 7Zq(n)u for j: U = X - U i e l D i ---r S.
We put
H* (X mod D, Zq(n)) = H* (X, ;gq(n)x,D)
= H * (X, j! •q (n)v).
If D is a divisor of X with simple normal crossings, we use the same letter D for the
family (D~)~I of the irreducible components of D by abuse of notation.
Let D be a family of closed subschemes of X as above and do be a locally free
(gx-module of rank n. We call a family p = (p~)~1 of CD-morphisms p~: dolo, ~ ~O,
a partial trivialization of d~ on D when the following condition is satisfied. For
all J c l , p j = (~sP~:dolo~ ~(gJo~ are surjective. For a partially trivialized
(gx-module (do,p), we define the relative top chern class c,(do, p)eHE"(X mod D,
7Zq(n)) as follows. Let V = l,r(do) be the covariant vector bundle SpecS'(dualdo).
We define a family A = (A~)~I of closed subschemes of V by As = p * ( 1 ) ~ Vo,
where p~: Vo, ~ ~ , is induced by Pi: do]o, --* Co, and 1 denotes the 1-section of ~k 1.
Since As c~ {0} = 0 for J :# 0 and {0} = ( t h e 0-section), the Gysin map
H~
7s ~ H~g~(Vmod A, 7/q(n)) is an isomorphism by the relative purity. We
define the class of the 0-section [O]EHz"(VmodA, Tlq(n)) to be the image of
1~H~
7Zq) by
H~
7lq) ~ H ~ ( V m o d A, 7Zq(n))~ H2"( Vmod d, 7/q(n)).
Further since As is an affine space bundle over Ds for J c 1, the canonical
morphism H 2, (X mod D, 7/q(n)) ~ H 2, ( V mod A, 7/~(n)) is an isomorphism by the
homotopy property. Now we define the relative top chern class
c,(6, p)eH2"(X mod D, 2~q(n))
to be the inverse image of [-0]~H2n( Vmod A, Zq(r/)) by the canonical isomorphism.
There is a variant for K-cohomology. In this paragraph, we assume that every
scheme is regular, noetherian and satisfies the Gersten conjecture. This assumption
guarantees that the relative purity and the homotopy properties used in the
definition of the relative top chern classes hold. For a scheme X and an integer
n > 0, let ~ . , x denote the Zariski sheaf on X associated to the Quillen's
K-theory. Let X be a regular noetherian scheme and D = (D~)i~1 be a finite family
of regular closed subschemes of X such that every intersection Dj = ("]i~sD~ is
regular for all J c I. We define a complex ~,,X,D o n XZa r by
IJI = m
and the relative cohomology
H*(X mod D, .)C.) = H*(X, YY.,x,o).
392
T. Saito
Then for a partially trivialized (gx-module g, the relative top chern class
c,(g)~H"(X mod D, oug,) is defined in the same way as above.
Example. Let X be a regular noetherian integral scheme of dimension 1 and D be
a reduced divisor. Then
~- (
Hl(Xm~
Q
zeOK~/u~
x :closed pt
r
)/ K ~
xeD
is the so-called divisor class group with modulus D. Here K is the fraction field of
X and U~ = 1 + m~. Let (L~', p) be an invertible (9x-module partially trivialized on
D and f be a non-zero rational section. For xeD, let a~eK • be such that
ord~(a~) = ord~(:) and that px(af 1 . : ) = 1. Then we have
Cx(L~~
= E ~
x] + E c~.
x~D
xeD
This follows from Proposition 5 proved later in this section. In fact, replacing : by
a - l : with a~K • =- a~mod U~ for x~D, we may assume a~ = 1. For the etale
cohomology, the cycle map H 1(X mod D, 113,.)~ H 2 (X rood D, 77q(1)) preserves
the relative top chern class.
We establish some basic properties of the relative top chern class, used in the
following section to prove the main results. We state and prove the results only for
the etale cohomology, though they also hold for the K-cohomology under suitable
modifications. First of all, the relative top chern class is mapped to the usual one by
the canonical map H2"(XmodD, 2~q(n)) ~ H2n(X, 7Zq(n)). In the following of this
section, we omit to write the coefficient 71q(n) in the notation when the degree of the
cohomology is the twice of that of the Tate twist in the coefficient
H 2. (X rood O) = H 2. (X mod D, ~q (n)).
We study the changing of the partial trivialization. Let J be a subset of I, put
IJt = r and J ' = I - J, We let denote D]D, the family of ctosed subschemes D~ c~ Ds
of Ds indexed by ieJ'. Then a canonical morphism 7zq(n)o~.Dto,[ - r] --* 7Zq(n)x.o is
defined by
0
~
""
~
7/q(n)Dj
->
'"
--*
(~
Zq(n)o= ~ ' " ]
JcKcl
IKl=m
Z,(n)x -+ "'"-+
@ Z~(n)o~
Kr
JKI = r
"'"-+
0
Kcl
IKl=m
ll~(n)~=
~" " I
and it induces H=-'(DsmodDlo,,7Zq(n))--*H=(XmodD, 71q(n)). Note that the
canonical morphisms above may depend on the choice of an order structure but
r
is, :A 7~s | H=-r (Ds mod D [D,, Zq(n)) ~ H=(X mod D, 7lq(n)).
is independent of the choice. For a partially trivialized locally free sheaf (g, p) of
rank n, let gs = K e r ( p s : S l o j ~ d?s,) and Plm be the partial trivialization
(Pi:~JIo, ~ 13~~ (9o, ~ o,)i~s, of gs.
e-factor of a tamely ramified sheaf on a variety
393
Proposition 1 Let (g, p = (p~: o:[D, -~ (90,)i~t) be a partially trivialized locally free
sheaf of rank n and fieF(Di, (9{~,)for iEI. Let t7 = (a i = f i -x "pi:o~lD. ~ (9D,)i~l be
another partial trivialization of ~. Then
Cn(~'O')= JcIE(--1)r(i?s{fi}) u is*C'-r(ES, P]os)"
Here U denotes the cup-product with respect to an order structure of I,
i~J
{ } : F (Di, (9x ) ~ H 1(Di, 7Zq(1)) is induced by the Kummer sequence, ~ denotes the
cup-product and the dependence on the choice of the order structure cancels.
Corollary. If X is an IFp-scheme and every fl comes from IF~ , we have
Cn(0r162 =Cn(g'P)-- E {fl} L;Cn-I(~i, PID,).
iel
Proof of Corollary. Since H 2 of a finite field is 0, it follows immediately from
Proposition 1.
Proof We m a y assume f~ = 1 except for i = i0 by induction on the number of i's
such thatf~ 4= 1. We write 0 = io, E = Do a n d f = f o
and D' = (Oi)iel,. We show
c,(g,a)--c,(oC, p)=
for short and let I ' = I - {0}
--{f}wie.c,_l(g0,PlE).
In the proof, we write Zx = Zq(n)x,o,Z'x = 7Zq(n)x,o, and Z~ = 7Zq(n)~,olE for
short. Clearly we have Zx = c o n e [ Z ~ ~ Z e ] . We define a complex
Yx = c o n e [ Z ~ ~ Z ~ 2 ] and a morphism 6: Yx --' Zx by
Yx = [ Z ~
~l
~L0
Zx
= [Z'x
~
Z ~ 2]
--,
Zr].
.~ p r : - p r ,
Let A' = (p* (1))~r be the family of closed subschemes of V = V ( g ) as before and
let F = p * ( l ) ,
G = a * ( 1 ) = V E . We put Z'v=Zq(n)v,~,, Zr=7/q(n)F,~,l~,
ZG = Zq(n)G,3'lo and
let Yv = c o n e [ Z ~ - - ~ Z r @ Z ~ ] .
Since A) c~ {0} =
F c~ {0} = G c~ {0} -- 0 for J = I ' , 4= O, the class [O]~Hz"(V, Y v ) i s defined similarly as above. It is easy to see that the difference c,(g, a) - c,(g, p) is equal to the
image of [0]eHE"(V, Yv) by
6,
H2n( V, Yv)
~- H2n(X,
Yx)
~ H2n(X, Zx)=
H2n(XmodD).
The m a p 6 is factorized as
rx
YE
,~
z~[-1]
$
Zx
=
[z~, --, z t =]
$
A
:=[ZE ~
~,
=[o
~
il
Z~ 2]
~,pr2
--, z ~ ]
II
= [Z'x -~ Z ~ ] .
pr,
394
T. Saito
Hence we have a commutative diagram
H~"(V, r~)
L H~.(X, r~) -~
H~"(V~, r~) L H:"(~, r~) _Z,
H:"IX, Z~)
H~.-~(e,Z~),
where Yv~ = cone [Zv~ ~ Ze@Z~] and Zv~ = 7lq(n)z~ ~'lz~' Therefore it is sufficient to show that the image of [O]~H2n(V~,Yv~) in H2"-~(E,Z~)=
H 2n-1 (EmodDle, 7Zq(n))is equal to the cup product - {f} ~ c,-~(~o, P[~). Let
W = V(r
and consider the diagram
v ~
T
~ 0-section
W ~
E.
Let t be the coordinate of A a, il and iI be the sections of A~ defined by t = 1
and
t =f
respectively and
Y~ = c o n e [ Z k , ~ ix,Zr.~if,Z~]
where
Z~, = 7Zo(n)~, ~1 . Since Wc~ F = W n G = 0 and W meets A) transversely at
Aw, we have a commutative diagram
E
~
E'
DIE
.
H2"(VE, Yv~)
.
g-
H2"(A~,
Y~)
g-- H2"(E, YE)
T
T
H2,-2(WmodAw) ~- H2"-2(EmodDl~).
Here the vertical arrows are the Gysin maps of the 0-section and hence are equal
to the cup product with [ 0 ] ~ H : ( Z ~ , T ~ ) where Tz~=cone[2gq(1)~,-~
il, TZq(1)E(~if, TIq(1)E]. Since [O]eHZ'- Z(Wmod Aw) goes to [0]eHz"(VE, Yv,)
and to c._l(8o)eH2"-Z(EmodD[E), it is sufficient to show that [-0]~HE(A~x, T ~ )
is sent to
{f-1}eHl(E, 7lq(1))by Ra,Tt~ (0, pr:-pr,) 71q(1)e[ -- 1]. Consider the
commutative diagram
, H ~({1}LI {f}, Zq(1))
t
H~(A~, r.,)
H' (~
-- {0}, ~Eq(1)) ~ H~o}(A~, Zq(t))
deduced from the distinguished triangle
(il,Zq(1)@i:.7Zq(1))[ -- 1] ~ TA ~ Zq(1) --*.
Then it follows from the fact that [0]~H2(~_~,TA) is the image of
{ t } e H l ( A 1 - { 0 } , 71q(1)) and that the boundary map d is the opposite of the
canonical map. Thus Proposition 1 is proved.
For the relative top chern class, we have the following functoriality. Let
f : X ~ Y be a morphism of schemes and D and E be a family of closed subschemes
of X and of Y with the same index set I. Assume that f(D~) ~ E~ for all i~I. Then
~-factor of a tamely ramified sheaf on a variety
395
f * : H * ( Y mod E, Zq (n)) ~ H* (X mod D, 7Zq(n)) is defined and preserves relative
top chern classes.
Proposition 2
Let (g, p) be a partially trivialized locally free module of rank n on
X and D, 11 be a subset of the index set I of D and E = Ui~hDi. Let ~ be a locally
free (9x-module of rank m with a partial trivialization a: ~ ]E -* CE and go: ~ -~ ~ be
a locally splitting injection. Assume griD' =piogo for i~I1 and piogo = 0 for
i~I2 = I - 11. Let N = Coker(go: o~--, N) be a locally free Cx-module of rank
r = n - m and ~ be the partial trivialization induced by pifor i~I2. Then we have
c.(& p) = cm(g, a) ~ cr(~, ~).
Proof. Let V = V(N), L = V(o~), W = sr(fr and Av, AL, Aw be the family used in
the definition of the relative top chern class. Consider the diagram
L
~
V
0 - section
X
, W.
Since L meets Av, J transversally at AL.j for J c l l
and L g"~Av,j = O for
J c~ I2 4= 0, by the relative purity and the homotopy property, we have a commutative diagram
H2m(LmodAL)~
H2m(XmodDo)--}
H2"(VmodAv)
H 2 " ( W m o d ( A w H Do)) ~- H 2 " ( X m o d D ) .
Since the composite of the lower horizontal arrows is the cup product with c,(f#, ~),
we have the proposition.
Corollary.
Let the notation be as in Proposition 2 except that we assume
OlD, = m? 1. P i ~ gofor mi ~ IF[, for i ~ l o . Assume further that rk Y = 1. Then we have
c.(g,p) = c,(~,a).c,_~(fr
+ 2 {m~}'c,-l(g~).
i~lo
Proof. Follows immediately from Proposition 2 and corollary of Proposition 1.
In the rest of this section, we assume the absolute purity of etale cohomology.
Namely for a closed immersion i : X ~ Yof regular noetherian schemes of codimension d and a prime n u m b e r invertible on Y, the Gysin m a p Ri!Zq, r
7Zq( - d)x[ - 2d] is an isomorphism. In the application in the later sections, it is
satisfied since the schemes there are assumed to be smooth over a perfect field.
Proposition
3 Let E be a regular divisor of a regular noetherian scheme X and
~ ~ ~ be a morphism of locally free (gx-modules of the same rank n. Assume
= Coker go is a locally free (gE-module of rank m and let ) F = Kerq0lE. Let
j: K = V ( J ( ) ~ V = V(o~) and W = ~g(ff). Then we have
r
go*([0w]) = [0v] +j.(c~-l(OUF))
in H~gI ~1r
- H~gI(V)@ H~"(V).
~
Here j . : H 2 m - 2 ( K ) -~ H2"(V) and ~ = Coker(fl:o:*~A/'~/x ~ c~*ff) is a locally
free Or-module of rank m - 1 where ~: K ~ E is the projection, JVE/x = (gx(E)]r
396
T. Saito
is the normal sheaf and fl is induced by 1/a~ JVe/x ~ (x ~ x/a) ~ H o m ~ (off, c~)
s'(~*)|
Proof. By the absolute purity, we have
2n}~ x(V) ~- H ~ ) ( V ) ~ H2"(V) "~ H ~ (X) ~ H 2m-2 (K).
H/o
Since H ~
+ H~
- E) and H 2 m - Z ( K ) ~ H 2 m - z ( K - - {0}) are injective by the
relative purity, it is sufficient to show that [Ow]~H~gl(W ) is pulled-back to
[Ov]eH~}~(Vx-E) and to c m _ l ( ~ ) e H z m - z ( K - {0})---H~"_~o/(V - {0}). The first
assertion is clear and the second follows from the excess intersection formula below
(cf. IF, Theorem 6.3]).
Lemma. Let X be a regular closed subscheme of a regular noetherian scheme W of
codimension n and cp: V ~ W be a morphism of regular noetherian schemes.
Assume Z = rp*(X) is a regular subscheme of V of codimension m < n. Let
= Coker(JVz/v ~ go*JV'x/w) be the excess normal sheaf which is locally free
Cz-module of rank n - m. Then
q0*([X]) = Cn-m(~)
in H z " ( v ) ~ H z n - z m ( z ) .
Proof Here we given only an outline of the proof. By deformation to the normal
bundle (cf. [F, Chap. 5]), we may assume that W = Nx/w (resp. V = Nz/v) is
a vector bundle over X (resp. Z), that X --* W and Z ~ V are the 0-sections, that
qo:V~W
is linear and further that X = Z .
Then the equality reads
~P*~P,([0v]) = C , - m ( ~ ) W [0] in H ~ ( V ) . By considering the Cartesian square
V ~
X
W
-~ V ( ~ ) ,
it follows immediately from the definition of the chern class.
Corollary. Let the notation be as in Proposition 3 and let X and D indexed by I be as
before. Assume Dj is regular and meets E transversally for all d r
and
m = rkE~ = 1. Let p and a be partial trivializations of 8 and ~ respectively on
D satisfying p = q~*a namely Pi = ai~
all i~I. Then ( J = Image q~lE, a [ y ) is
a partially trivialized locally free module of rank n - 1 on (E, Die) and we have
Cn(~,~ , a) = Cn(O~, p) + i, Cn-l ( J , a j )
where i,: H2n-2(EmodDIE) ~ H ~ n ( X m o d D ) ~ H2n(XmodD).
Proof By Proposition 3, we have c , ( ~
j: K = W(Ker~plr)-o V. By the cartesian square
k
K ~
Ve
0
where
--, v ( J ) ,
we see k , ( [ K ] ) = e n _ l ( ~ ,~, o'j~) in H2n-2(VrmodA]v~) -~ Hzn-2(EmodDIE).
Since j , = i, o k , , we have corollary.
e-factor of a tamely ramified sheaf on a variety
397
Let D be a family of closed subscheme of X as before. Fix an element 0 of the
index set I and put E = Do.~Let K denote the set of i e I ' = I - {0}such that
D / ~ E # 0. Define a family D = (O~)~r by/~i = Di w E for i ~ K and Di = Oi for
i r K. Then since U i~i Di = U i~r D~, the relative cohomology H* (X mod D) and
H* (X mod/)) are naturally identified.
Proposition 4 Assume that X and every Ds for J ~ I are regular and that E = Do is
a regular divisor meeting transversally D s f o r all J ~ I' = j - {0}. Let (~, p) (resp.
( ~ , a) ) be a partially trivialized locally free sheaf on (X, D) (resp. on (X, D ) ) of the
same rank n. Let (p: ~ ~ o~ be a morphism such that Coker (p is a locally free
(?)E-module of rank m. Assume that ~ri o qo = Pi on g tD, for ie I' and a o o qo = ~ i~i( Pl on
6ole. Then we have
c,(~,a)=c,(o~,p)
in
Hz"(XmodD)"~Hz"(XmodD).
Proof. We use the notation in Proposition 3. First we prove it under the assumption that r = Card(K) = 1 on E. We put A = q)*Aw. Then As is an affine space
bundle over Ds for all J c I and we have a commutative diagram
Hz"(WmodAw)
Hz"(XmodO)
q)*
~
Hz"(VmodA)
=
Hz"(XmodD).
We have ( P * [ 0 w ] = [ 0 v ] + j . c , , - l ( o ~ )
by Proposition 3. We prove
j.c,,_l(W)=0
in H 2 " ( V m o d A ) . For this, we show j . : HZm-z(K)--~
2n V mod A) is the 0-map. By deforming to the normal bundle, we may assume
Hv~(
that X is the normal bundle N = NE/x, E ~ N is the 0-section, V is the direct
product Ve x e N and Ai is the pull-back (A~ c~ Ve) x E N for all iel'. Let A' be the
family A with Ao replaced by the pull-back Ab = Ao xEN. Since A; c~ K = 0,
j . factors H ~ ( V m o d A ' ) .
It is easy to check that H ~ ( V m o d A ' ) ~H 2n 2 ( V E m o d A ' [ v E ) " ~ H Z n - 2 ( E m o d D I E ) -- 0 and j . = 0 is proved.
Thus it is sufficient to show the existence of an isomorphism
H2" ( Vmod A ) ~ H2" ( Vmod Av) preserving [0] and making the diagram
H2,(VmodA)
--. H 2 " ( V m o d A v )
Hz"(XmodD)
~_ H 2 " ( X m o d / ) )
commutative. Let A" = (A}')i~r be the family defined by A~' = As w (Ao c~ Av,~) =
(p/-l(1) c~ V v , ) w ( p 7 1 ( 1 ) c ~ ( a o O t p ) - a ( 1 ) c a VE). Then since A } ' c A v , i
for
~),
a
canonical
morphism
i~l'(resp.
Ui~lAi = U~x'
I|
A,,~
H * ( V m o d A v ) --* H * ( V m o d A " )
(resp. H * ( V m o d A ) --* H * ( V m o d A " )
is defined. By the homotopy property, we check that the canonical morphism
H* (X mod/)) ~ H* ( Vmod A") is an isomorphism. They form a commutative
diagram
H2"(VmodA)
~T
H2"(XmodD)
~
H 2 " ( V m o d A '') ~
~T
~_ H2"(Xmod/~)
H2"(VmodAv)
~T
=
H2"(Xmod/9).
398
T. Saito
It is easily checked that the upper arrows preserves [0] and the assertion follows.
Thus when r - 1 on E, Proposition 4 is proved.
In general case, we may assume that there aref~eF(X, t13~) for iEK satisfying
~ i ~ r f / = 1 since
H * ( X m o d D , Zq(*))
H*(X[t~' . . . . .
t,']l(~t~-l)modD[t~'
.....
t,']/(~t,-
1), Zq(,))
are injective. Consider another partial trivialization defined by replacing p~ by
f~-~ .p~ for i~K. Then by using Proposition 1, the same argument as above proves
Proposition 4 by induction on r.
Lastly we give the relation with the localized top chern class defined in
IF, Sect. 14.1]. In the rest of this section, we work with the K-cohomology and
write H" (X mod D) = H" (X mod D, ~ ) . We assume the Gersten conjecture so
that we have H'~(X)"~ CHalmx_m(Z) for closed subscheme Z of a regular noetherian X. The following proposition and its corollary are useful for an explicit
computation of the relative top chern class.
Proposition 5 Let (~, p) be a partially trivialized locally free module of rank n on
(X, D) and s be a section s~F(X, ~) such that pi(s) = 1 for all index i of D. Let
Z = Z(s) be the zero scheme ofs and 7Z(s)~CHdimX-n(Z ) be the localized top chern
class (loc.cit). We assume Z c~ D = O. Then we have
c,(o~, p) = i.(Z(s))
where i,: CHdimX_n(Z ) --% H ) ( X m o d D ) ~ H " ( X m o d D ) .
The following is clear from Propositions 5 and 1.
Corollary. Under the same assumption except that p~(s) = f ~ F ( D i , Gm)for all index
i of D, we have
( - 1 ) r (/l~Ij {fl} ) w is.c,-,(~s, PlD~)=i*(7l(s)) 9
Jcl
Proof Since D c~ Z = 0, i, is well-defined. By the assumption, we have a commutative diagram
s*
n~o~(VmodA) ~
H"(VmodA)
s*
~
n)(XmodD)
H"(XmodD).
Since the lower s* is the inverse of the canonical morphism, it is sufficient to show
that s* ([0]) = Z (s) in H~"(X mod D) ~ CHdim x - . (Z). We interchange 0 and s by
using the automorphism v ~ s - v of Vand put S = s(X). Then A c~ S = 0 and we
need to show that 0* IS] = 0 ! IS] in H~ (X mod D, 7/q(n)) ~ CHalmx-.(Z), where
the right hand side is the K-theory and the left is the intersection theory. We
consider the deformation to the normal bundle as in IF, Chap. 5]. Let t: ~,,,x --* V
be the map defined by r
(2) ~ , x x x V ~ V where z is the inverse and # is the
multiplication. Let Sbe the closure of the graph of t in ~kxx • x V. Then 5~c~ {1} = S
e-factor of a tamely ramified sheaf on a variety
399
and S c~ {0} is the normal cone N = Nz/s of Z in S. We have a commutative
diagram
(TZ.= )H~(VmodA)
H"z(XmodD)
~i*
Hs(Vz~modAz~)
i* H~(VmodA)
~,1 H. ,z , , I"X ~ , modDz~,) ~'*, H~(XmodD)
where the composition of the lower arrows is the identity. We need to show that the
K-theoretical and the intersection theoretical maps are the same at the left vertical.
It is sufficient to show that so they are at upper i~ and at the right vertrical. For io*,
we check that i~([S]) = [Nz/s] in both cases by factorizing it as
U s ( Vz~,mod A . , ) ~ [{o!] H~+ ~o~(Vz~,mod A ~ )
~- H~(VmodA).
F o r the right vertical, they are the composition of the map enlarging the support
and the inverse of the canonical isomorphism
U ~ ( V m o d A ) ~ H~.(VmodA) ~- H [ ( X m o d D )
and are the same. Thus Proposition 5 is proved.
2 Main results
Before stating the main results, we review some general facts on tamely ramified
fundamental groups, Gauss sums and e-factors. Let X be a regular noetherian
connected scheme and U be an open subscheme such that the complement
D = X - U is a divisor with simple normal crossings. F o r a geometric point ~ of U,
the tamely ramified fundamental group nl (U, .~)tame is defined. It is the quotient
group of n l ( U , ~ ) classifying the finite etale coverings of U tamely ramified
along D.
Example. Let L be a complete discrete valuation field with the residue field E of
characteristic p. We put X = Spec (-9L, U = Spec L and D = Spec E. Let GL, IL and
PL be the absolute Galois group of L, the inertia group of GL and the pro-pSylow subgroup of I r respectively. Then we have 7rl (U, ~)tame = GL/PL. If we put
Z'(1) = lim #, and take a prime element n of L, a canonical isomorphism
IL/PL ~ Z'(1)g is defined by a ~ (a(nl/n)/ztl/n),. H e r e / ~ is an algebraic closure
of E.
F o r each irreducible component Di of D, let pi be the characteristic of the
residue field x(th) of the generic point th of D~ and let G~, li and Pi be the
decomposition group, the inertia group and its pro-pi-Sylow group of X at D~
respectively, namely those for the completion of the fraction field of X with respect
to the valuation corresponding to D~. Then a canonical morphism
400
T. Saito
ai:Gi/Pi ~ n l (U, ~)tame is defined uniquely up to conjugacy. By [R, Proposition
5.21, the restriction ctili,/p' factors the quotient lim #., where n is invertible on Di,
~p~n
of l i / P i. The kernel of nl (U,)~)tame __~ 7Zl(X, .~) is topologically generated by the
conjugates of the images ~i(Ii/Pi). In this paper we consider the case where X is
a 7/<p)-scheme and that p is not invertible on any irreducible component Di. We
keep the notation 7Z' = I - [ p , , p7tp,. Then the canonical morphism above defines
~ i : Z ' ( 1 ) ~ 7 5 , ~ - - - ~ T I ' 1 ( U , ) ~ ) tame for each i up to conjugacy. The abelianization
nl(u)ab't"m[ is equal to the product nl(u)ab'Xnl(X)av b where n l ( u ) a b ' =
7~1 ( u ) a b ( ~ ) 7 Z ' and 7 ~ l ( X ) ~ b = 7 ~ l ( x ) a b ( ~ Z p .
For a prime number f, a finite dimensional ~:-representation (p, V) of a profinite group G is called an f-adic representation if there is a E-structure VE of V for
some finite extension E of Qc such that p is defined on VE and that p : G ~ GL(VE)
is continuous. Assume f is invertible on X. Then an f-adic representation p of
n l ( U , 2)tame is called quasi-unipotent if the restriction Pi = ~*(P) of p to Ii/Pi is
quasi-unipotent for all Di, namely there exists an open subgroup Ji c Ii/Pi such
that PiIJ, is unipotent. By [ D G Proposition 1.1], if every finite extension of the
residue field K(~i) for every Dg contains finitely many roots of unity, every ~-adic
representation of nl (U, 2)tam~ is quasi-unipotent. This condition is satisfied if X is
of finite type over 71 or 7Zp.
Next we define the Gauss sums. Let F be a finite field with q elements and f be
a prime invertib]e in F. Let ~ f be the category of quasi-unipotent f-adic representation V of 7E'(1)F satisfying q* V ~ V: Here q * V is the pull-back of V by the
automorphism, multiplication by q, on Z ' ( l ) t . For a non-trivial additive character
~/01 F ~ Qr
- x , we define a character of the Grothendieck group Ze(, ~bo): K0(C~e)
~ ; as follows. The simple objects o f ~ v are of the form P(Z) = ~ - o 1 qi*z. Here
Z is a character of Z'(1) of order m and f i s the order of q in (TZ/m) • Then the
character is determined by
ZF(P(Z), ~0) = Zl~s(X, 111oo Tr/~s/F)"
On the right hand side, E s is the extension of F of degree f in if, X is also used
to denote the character Z: E ] ~ (~;
induced by the isomorphism
(7s162 _ 1))(1)~ ~- E ; and
zE(qLIP)= ~
~P-'(a)0(a)
acE •
is a usual Gauss sum. It is easy to check that the character r e ( , ~b0) is well-defined.
Let K be a local field, namely a complete discrete valuation field with finite
residue field F. F o r a general theory of the ~-factor of f-adic representation of local
field, we refer to [D1]. In this paper, we always take the Haar measure dx of the
additive group K such t h a t ~mx dx = 1. For a quasi-unipotent E-adic representation
V of the absolute Galois group GK = Gat(K~eP/K) and a non-trivial additive
character ~b: K ~ Q-r•
e(K, V,~b) and ~o(K, V , ~ ) ~
are defined. F o r the
product formula of the e-factor, we refer to [L]. The ~-factor of tamely ramified
representations and the Gauss sums are related as follows.
Lemma 1 (1) Let K be a local field with finite residue field F. Let p be the
characteristic o f f and f be a prime 4: p. Let cgr be the category of tamely ramified
d-adic representation of Gr. Then the restriction to the inertia group defines an exact
e-factor of a tamely ramified sheaf on a variety
401
functor cb: ~K ~ ~v and a surjection K o ( ~ K ) ~ K o ( ~ F ) . Let ~ be an additive
character of K of order - 1, namely ~(CK) 4= 1 and ~(mK) = 1, and ~o be the
induced additive character of F = CK/mK. Then for V~ ob(~K),
eo(K, V, ~) = ( -- 1)dimV'c,~(~(V), ~o).
(2) Let E be a finite extension o f F and r: ~'(1)2 --, ~'(1)~ be the multiplication by an
integer r > O. Here we identify E = F. Let r = m" s be the decomposition into the
prime-to-p part m and the p-part s. For Vcob(CgE), let Ind Vcob(C#r) denotes the
induced virtual representation of V b y r : 77' (1)~ ~ 77' (1)p and det V be the determinant
character of the coinvariant E • = (77'(1)~)GE. Then we have
~ [~
~[E:F].dimV
ze(Ind V, [/.t0)[E:F] = (det V)(m)'zE(V, q/d) oTr E/FJ'\~F,m,qlo]
On the right hand side, t~): a ~ Oo(a ~) is a character o f F and
f
Ze'm'~'~ =
(F)((~)zv((ff),
2
//m"~'
~)q
\/
=~p)
Oo))
m-~
P 4=2
m~ 1
2
p=2.
, i f p * 2and (ff):(7Z/8)
i/m\
,, . . . .
1 or d
p = 2. In particular when r = 1, we recover theJormula of Hasse-Davenport
ze(V, ~k0)IE:FI = ze(V, ~ko o Tre/r).
Proof. (1) The functor is defined since every object of ~K is quasi-unipotent. The
surjectivity on Ko is checked easily. The formula is [D1 (5.16.1)].
(2) It is sufficient to treat the cases
A. E = F a n d r i s a p o w e r o f p .
B. r is prime to p
separately. In case A, it is clear by the transport of the structure since the m a p r is
an isomorphism. We consider the case B. Take a tamely ramified extension L of
K with residue extension E and ramification index m and take a character ~ of K of
order - 1 and inducing ~ko on F. Then $ o TrLir is of order - 1 and induces
OoO(m.Tr~/F). Take a virtual object W of ~L such that q~L(W)= V. Then
q~r(Ind~ W) = [ E : F ] I n d V and by (1) above, we have
zv(Ind V, ~o) re:r] = (
-
1)[L:KldimVg,o(K , Ind~
W, ~O).
F o r V of dimension 0, by I-D1, (5.6.1), (5.12), (5.4)] and (1) above, we have
eo(K, Ind~ W, ~,) = eo(L, W, ~ o TrL/s) = (det V)(m).zE(V, ~o ~ TrE/r)
and the assertion is proved. Therefore it is sufficient to prove the equality for the
trivial representation V -- Qe. Namely we are reduced to show
co(K, M, ~) = ( -
I~[L:K].c[
E:F]
,
r.,,,*o
for M = Ind~ll~e. W e may assume E = F and L is a totally ramified of degree m.
We prove the equality by the induction on ord2m. First we assume m is odd. Then
402
T. Saito
by Proposition 2 of [H] and by the fact that 6 = det M is unramified, we have
e'( M ) = g ( 6 )m under the notation there. Since the Artin conductor a( M ) = m - 1,
we have
m--I,
e(K, M, ~b). e(K, 6, ~)-"* = q 2
by [D1, (5.5.2)]. Substituting det ( - FrF, M t) = -- 1 and e(K, 6, t~) = 6 (Fr~ 1 ) =
_+ 1, we obtain
m--I
e(K, M, ~b) = - 6 ( F r r ) . q
Since zv
, ~Oo
=
(/)
2
q, it is sufficient to prove
m--1
6(Ere) =
,
By the quadratic reciprocity law and its second complementary law for p = 2, the
right hand side is equal to the Jacobi symbol (~) and it is easy to check
~(FrF) ----(~).
Finally we assume m is even and p 4: 2. Let L' be the subextension of L of
degree ~ over K and Z be the quadratic character of L' corresponding to L. Then
since M = Ind~'(Z - 1) + 2Ind/~'l, we have
e(K, M, O) = e(L', Z -- 1, q/o o TrL/r)e(K, Ind,' 1, ~b)2.
From
the
(~)ZF((~),
assumption
of
the
induction
and
e(L',z-
1,~bo ~
0 0 ) , the required equality follows.
Let X be a proper and smooth scheme of dimension n over a perfect field F of
characteristic p. Let U be an open of X such that the complement D = X - U is
a divisor with simple normal crossing. Let (Di)~x denote the family of irreducible
components of D. Let O~/F(lOgD) be the locally free (gx-module of rank n of
differential 1-forms on X with logarithmic poles along D. We consider the partial
trivialization of 8 defined by the residue morphisms rest: gID, ~ (gD, for i~I. We
define the relative canonical cycle
Cx.v = ( - 1)" c,( f2~/F(logD), res)~H2"(X m o d D, ~'(n))
by using the relative top chern class defned in Sect. 1. Note that this definition
differs from that in [SS] by ( - 1)-times. It is easy to check that deg Cx. v = ]~(U-~)
where fi is an algebraic closure of F and z~(U-~) = E( - 1)qdimH~(UT, Q:) for
: 4: p is the Euler characteristic with compact support.
Now assume the base field F is finite. We identify ~ ~-Gal(/7/F) by 1
the geometric Frobenius tpr (the inverse of the Frobenius substitution). We define
the relative canonical cycle Cx, v in h i ( U ) ab'tame as follows. We have
n l ( U ) ~ b ' t ~ = nl(u)~b'• nl(X)~ b. By Poincar6 duality, we have n l ( U ) ~b'~H~2n (U, Z'(n)). Hence Cx, v/F~H~2n (U, 71'(n)) above defines the prime-to-p component. F o r the p-component, we define it to be the image of the usual chern class
( - 1)"c.(O~c/v(log D)) by the reciprocity map C H o ( X ) ~ h i ( X ) ~b. F o r a prime
e-factor of a tamely ramified sheaf on a variety
403
n u m b e r E 4: p and an (-adic representation p of nl (U, ~)tame thus
d e t p ( - Cx, v ) s O j is defined. F o r iel, let F~ be the constant field of D~. We put
D* = Di - Ui' * iDi and ci = degv, Co,,D; where dege : H"- l (Dimod DID,, • , - 1 )
^t
--.H~
7l. The Euler n u m b e r zc(D*-L)=c i. Also for i~I, 7I
(1)~5,) is
canonically identified with ~ ' ( l ~ , and hence a canonical_ morphism
cti 7/'(1) L ~ nl(U, if)tame is defined upto conjugacy. Let Oo:F ~ Q 2 be a nontrivial additive character. Then since an f-adic representation p of n~ (U, y)tamo is
quasi-unipotent in this case, the above definition applies to the restriction
Pi = ~*(P) o f p to 2~'(1)~, and zF,(Pi, ~bo~
is defined for all i~1. We define
~DIF(P, ~o) = 1~ ~F,(Pl, ~o ~Tre,/e) ~'.
i~l
We can easily check that this is independent of ~bo.
Theorem 1 Assume that X is projective. Let ~ p be the smooth d-adic sheaf on
U corresponding to a tamely ramified (-adic representation p of nl (U, ~)tame. Then
the e-factor
2n
eo(U/F,~p) = I-[ det( - q~r,H~(Ue,
. i _ ~0))(- 1),+,
i=0
satisfies
eo(U/F, ~ p ) = d e t p ( - Cx, v)'zo/v(p, ~9o)'eo(U/F, ~ e ) d~gp.
Remark. O n the right hand side, eo(U/F,~<) is the alternating product
1-[j=Ie(Dj/F) (-1)lJI. For proper smooth Y over F of dimension m,
e(Y/F) = ( - 1)P~qmz~/2. Here q is the order of F, Zr = degecr = z(Y~), which is
even if m is odd, and Pr is the multiplicity of eigenvalue qm/2 on Urn( Y~, ff)~()if m is
even and is 0 if m is odd. An exact formula for the sign will be given in a paper in
preparation.
Proof will be given later in this section.
We proceed to varieties over a local field. Let K be a complete discrete
valuation field with a perfect residue field F. Let X be a proper fiat and regular
scheme of dimension n over the integer ring (9K such that the generic fiber XK is
smooth over K. Let U be an open of X contained in the generic fiber X~. Assume
that the complement D = X - U is a divisor with simple normal crossings of X.
This implies that the reduced special fiber (Xv)~ed is a divisor with simple n o r m a l
crossing. Let (D~)~t be the family of irreducible components of D as before. We
assume that Ds, r = ~i~sDi |
K is smooth over K for all J ~ I. Let Io be the
subset ofieI such that D~ ~ XF. For i~Io, let r~ be the multiplicity of Xv at D~, m~ be
the prime-to-p part and s~ be the p-part of r~. We define the tame part of the relative
canonical cycle by
C ~ X ' U / ~ = i * ( - -~~H
m2~"C~D( X" ~m ~
'~
u 2 " - ~^ (D~modD, Z'(n)) --~n2"~(XmodD, ~'(n)) is that induced by
Here i . : ,.Tdielo~t~
the canonical m a p 71'(n)D,,DIo,[ -- 1] ~ 71'(n)x,o used in Proposition 1 in Sect. 1.
Note that i. is the opposite of the b o u n d a r y map. The class c~, u/e~ lies in the kernel
of H~'~ (X m o d D, 71'(n)) ~ H~:"~(X, Z'(n)). In fact the image of a prime-to-p integer
m in n 2 ( ( g r m o d F , Z'(1)) = Z ' is ordpm = 0.
404
T. Saito
The pair (X, U) is said to be tame over r
1-forms with logarithmic poles defined by
ff~l
O ~,e~ (log D /log F ) =
(da - a |
if the sheaf of relative differential
,
•
X/e~ ~ ( ( g K |
(gU )
ca j,(9~ ), 1 | b;(b~K • ))
(cf. [K2, (2.2)]) is locally free (of rank r = n - 1). We let dlog a denote 1 | a. On the
generic
9 fiber, fJx/~(logD/logF)]x~
~
as
" f2x~/K(logDr)
~
and is locally free of rank r.
O n the reduced closed fiber E = (Xv),od, there is an exact sequence
CE ~ f2~x(1OgD)[~ ~ f2~x/~(logD/logF)]~ ~ O.
Here the m a p on the left is 1 ~ d log ~ where ~zis a prime of (gK. Since ~2~x(log D) I~
is locally free of rank n by the exact sequence (cf. Theorem 2.4 loc.cit.)
0 ~ O~o,/v(logDIo,) ~ O~(logO)lo, r~,, (9~, ~ 0
for i~Io, the pair (X, U) is tame if and only if dlogTz is nowhere vanishing in
f2~ (log D)IE.
Example. If every multiplicity r~ of the closed fiber is prime to p, then (X, U) is tame
over CK. In the case of curves, the following holds by [ST Theorem 3]. Let U = XK
be a proper smooth curve of genus > 2 and X be the minimal n.c.d, model in the
sense of loc.cit. Then (X, U) is tame over (9r if and only if the action of PK on
H ~(X~, Q t ) is trivial for f =1=p.
If (X, U) is tame over CK, we define
CX, U / ~ K = C X ,
t
u/~ K9
In this case we have
i6lto
where 15 = {ielo;pXri}. In fact for i~I~ = Io - Ito, d log ~ is nowhere vanishing in
f2~,/F(logOID,) and we have CD.D* = O.
Assume K is a local field namely F is finite. Let (X, U) over the integer ring C r
be as before. By Poincar6 duality, we have ~l(u)ab'~--H~(X,j!77'(n)). Hence
ctx.v/~Ke~l(U) ab'c~zl(U) ab'tam~ is defined. It lies in the kernel of
~ I ( U ) ab' -~ 7h(X) ab. For an f-adic representation p of ~zl(U,:~)t,m~,
d e t p ( - C xt, v/oK)sff~l
- • is defined. F o r ielo, let F~ be the constant field of D~,
ei: Z'(1)f, ~ zx(U, ~)t~m~ and ci = degv, CD,,D~'be as before. Let ~ be a non-trivial
additive character O : K --* Q ~ of ord(O) = - 1 and let ~o be the induced character Oo:F = CgK/mK ~ ( ~ . Then we define
Zb/F(P, ~Po) = I] ~f,(Pi, O~')~
c'.
ielo
Here for ieIo, Pl = ct*p and, for a power s of p, ~]): a ~ ~o(a'). If(X, U) is tame
over (-gr, we define ZD/F(P, ~o) = zto/e(P, ~o). In this case, we have
ZO/F(P, ~o) = 1~ zv,(pl, qJo~
ielto
since ct = 0 for i e I ~ as before.
~'
e-factor of a tamely ramified sheaf on a variety
405
Theorem 2 Assume (X, U) is tame over (PK and let t) be an additive character
~: K ~ (~{ oford(~b) = - 1. Let o~p be the smooth f-adic sheaf on U corresponding to a tamely ramified #-adic representation p of nl (U, 2)tamL Then the e-factor
defined by
eo(U/K, .~p, ~b) = ~o(K, RF~(U2, ~p), ~)
satisfies
eo(U/K, Yp, ~b) = d e t p ( - CX,U/C,.~)'ZD/F(p,~o)
\F/
J
P=
Here
zc(U2) = ~ mici[Fi:F]
and
z~(Us ~r ~ ( _ 1)idimHi(U2, Qt),s~.~"
i~Io
= ~ ci[Vi:F]
ielo
are the Euler number and the alternating sum of the dimension of the fixed part by the
inertia group Ir of the semi-simplification respectively and M = I~i~om~i ' [F~:F].
Proof of Theorem 2 Changing slightly the notation, we assume that the residue
field F is algebraically closed and do not assume that X is proper in the following
computation of the vanishing cycles. Let p be a quasi-unipotent {-adic representation of n ~(U, ff)tam~and ~ p be the corresponding smooth (-adic sheaf on U. We fix
a subset J = I and a connected component E of Dj and study the restriction
R~,~p[E, of the complex of the nearby cycles R~o~o on E* = E - Wi~sDi. For
ieJ, let I~ and Pi be similarly as before. Taking a geometric point ff on the generic
point of the strict henselization of X at the generic point of E as the base point, we
may assume that the image of cq: Ii/P~ ~ n l (U, ff)t"m~ is commutative to each other
by the lemma of Abhyankhar. Let Jo = J c~ Io and M be the complex concentrated
on degree 0 and - 1,
#
ZJo -.
7Z:e i ~
r i.
Let the inertia group IK and its pro-p-Sylow subgroup PK be as before. Then the
natural morphism
f~: 1-] Ii/Pi ~ I r / P r
i~Jo
is canonically identified with #|
monodromy action is finite.
First we consider the case where the
Proposition 6 Assume that the restriction Ps of p to I]i~sIi/Pi by l~i~scq factors
a finite quotient and that (X, U) is tame over Cr. Then there is a canonical
isomorphism
q
Rq~b ~p[~, ,~ RO~O~-ol~,| A H o m ( H t (M)(1), Qr).
406
T. Saito
The sheaf R ~
is smooth on E* and tamely ramified along the boundary
E - E*. As a representation oflK/PK,
( I~Alr/Pr
^
*llUImage~ ( p K e r # ) , / f j = J o
R~ f f o It* ~- ( O,
otherwise.
Here p~r~ is the fixed part of the representation space of PJ by Ker fi regarded as
a representation of Image ft.
Proof. First we reduce it to the case where ~-p = Q~ and every m / = 1. The question
being local, we m a y assume that D~ is defined by 7r, = 0 for i~J. For a c o m m o n
multiple d of mi for i6Jo, let Kd = K(nl/a),~Pa:Xa = X [ ( n m ' / d ) i E j ]
--~ X and
Ud = r
Here zr is a prime element of K and we put m / = 1 for i(~Jo by
convention. The Galois group Gd of Xd over X is the quotient II~s(7Z/(d/mi))(1) of
~[~jI~/P~. Take a sufficiently large d so that q~* ffp extends to a smooth sheaf ~-d
on Xd. Then since f ' o = (~Od,~o*Yp) G~, we have
R~ ~-p "~ ~l)d,(~ d IXa, F (~ R~ ~,, u.) ~ .
Since the question is etale local, we may assume Xa is a scheme over (9/~. Let Rtpa
denote the nearby cycle functor with respect to (fird. By the definition of the nearby
cycle functor, we have R~bQtv, = Ind~[ R~'d~<U~. We assume Proposition 6 for
(Xa, Ua) over (fir~ and for ~ = @~. Note that every m~ for Xa over (gK~ is 1. Then
for J ' = J such that [J - J'[ = 1, we see R~b~plv, is smooth and tamely ramified
*
* "S'
along E* since the covering (Xd • X Ds' ) ~ -~ Dj' is etale and tamely ramified along
E*. On E* we have
q
R~~ ~-, I~* -~ ( ~ I~*| ~nd~ ~ ) ~ |
HomeH~~M)~), ~ ) .
Here we identify the sheaf on (p* (E*) with that on E* by a h o m e o m o r p h i s m ~gdIE*.
On the right hand side, the action of Ga on Ind~ot~ f is the restriction by
12d: Ga -~ IK/IKd induced by ft. Therefore we have
( ~ d 1~* |
Ind~,~ t ~ ) ~d ~ ((~a ]E*)Keru~ | I n d ~ ~ ) lmue
~_Ind~g[Pf(~a IE,) Ker€ .
Thus Proposition is proved by assuming the case where every m~ is 1 and ~ 0 = ~ e .
We compute the vanishing cycle assuming that every m~ is 1 and ~ o = ~e. It is
easily checked that we may assume U = XK by induction on dimension. Since
question is etale local and (X, U) is tame over OK, by an elementary computation,
Proposition 6 is rephrased as follows.
Proposition 6' Let
X = Spec(fir[tl . . . . .
Then there is a canonical isomorphism
Rq07Ze_~ACoker
tk]/(n - tl Iqk=2t f``) and
7Z~(-- 1)x~-~
Di =
V(ti).
7Z~(-- 1)o, 9
i=1
Proof. We show it by induction on k and e = m a x i ( e , 0). I f k = 1 (and e = 0), it is
clear since X is smooth. First assume one of e~ say e2 is 0. By the assumption of
induction, the assertion holds for Y = Spec (fir [s~ . . . . . s,]/(zr - s l l-Ik=asP:). Let
C = V ( t l , t 2 ) ~ X. Since X - C is locally isomorphic to Y, it is sufficient to show
e-factor of a tamely ramified sheaf on a variety
407
the assertion at C. Let C' = V(sx, S2) C Y~ (~9:X' --.4- Y b e the blowing-up of Y a t C'
and E = q~*(C'). Then t : X ~ x ' ; S l w - ~ t l , s i ~ - - , t l
s2
(i=~ 1) and
Y~X';
si~--*s~
(i + 2), s2 ~ s2 defines an o p e n ' c o v e r i n g of X'. We have a distinguished triangle
S1
~j!(R~TZt, x,lE_,(c) ) ~ (R~O7Z<x,)IE ~ (RffTZr
--.
where j: E - t(C) ~ E. Applying Rip., we obtain a distinguished triangle
--, (R ~ 7zt, t i c , ) ( - 1) E - 2] -~ (RffZt, r)Ic' --, (r~zt,
x)lc
-,
F r o m this, we deduce an exact sequence
0 --* (Rq~, Tie, r)Ic, ~
(r~'7Zt, x)lc
-* (R q- 1 [~/~f, v)lc, ( - 1) ~ 0
and the assertion for X.
N o w we assume every e~ > 0. Let M be an arbitrary separable extension of K of
degree p and of ramification index p. Let n' be a prime of M and put n = u. n 'p.
Then the integral closure Y of X |
m is SpecCM[Sl, t2, 9 9 . , 6]/
(n' -- &l--It p"'-') by tl ~-* u'SPl. By the assumption of the induction on e, the
assertion holds for Y over Ore. The assertion for X follows from this since the
formation of nearby cycle commutes with base change [D2, Proposition 3.7] and
the restriction of the map Y ~ X |
on the closed fiber is a homeomorphism.
Thus the proof of Propositions 6 and 6' is completed.
In general quasi-unipotent case, we have the following.
Corollary 1 Assume p is quasi-unipotent. Then Rq~ ~olE, is smooth on each stratum
E* and is tamely ramified along the boundary E - E*. The action of I t on it is tamely
ramified, namely the restriction to Pr is trivial, and is quasi-unipotent. If J = {i} for
some ielo, we have R q ~ - p l E . = Ofor q 4= 0 and
t
~IK/P K
-- lno~,/v, Pi
R~
as a representation of lK/Pr, l f J is otherwise, we have
[R~bffolE,] = 0,
where [ R ~ , ~ I ~ , ] = ~ q ( - 1)q [R~ff fro [~* ] is the alternating sum in the Grothendieek group of the smooth sheaves on E* tamely ramified along the boundary E - E*
with a quasi-unipotent action of l~/Pr.
Corollary 2 Assume further that X is proper. Then RFc(U2, ~o) is tamely ramified
and
=
CF, i ' l n ( l l d p ,
Pi"
i~lo
Here the left hand side denotes the alternating sum in the Grothendieck group of
tamely ramified quasi-unipotent f-adic representation of I t and cv, i = degr co,, D~. In
particular, we have zc(U~) = ~i~loCF, i" ml and )~(U~) = ~i~oCe, i.
Proof of Corollary 2 By the spectral sequence Hv(xF, Rq~ ~ , ) ~ H~+q(u~, ~p),
it follows immediately from Corollary 1.
Proof of Corollary 1 We use the n o t a t i o n in I-D3, (1.8.10)]. Let o ~ p [ E * ] be the
smooth ~-adic sheaf defined loc.cit, on the ~Sm-bundle T -
w ~ j Ti over E* where
408
T. Saito
T is the normal bundle of E* in X. Then by using the logarithmic monodromy
operators Ni, we see that there exists a filtration F on ~ p [ E * ] by smooth
subsheaves such that on each subquotient Gr'F~p[E*] the action of the monodromy group Hi~jIUP~ factors finite quotient. Locally, F induces a filtration on
=~, itself and we have a spectral sequence RPO(Gr~r~p) ~ RP+ql/l~o. For each
Grvq ~'a, Proposition 6 applies and the induced filtration by F on the limit patches
itself globally. Thus Corollary 1 follows from Proposition 6.
Now we prove Theorem 2. Let the notation be as in the statement of
Theorem 2. By Corollary 2 of Proposition 6 and Lemma 1, we have
eo(U/K,~o,~) = ( - 1 ) x ~ ( u ~ ' ~ . ) . z r ( i ~ o C V , i. Ind,r/~Kpi,~o )
= ( -- 1)zctv~)'dego x H (detpi(mi)'ZF.(Pi,
isIo
[F~:F].degp
.... ~o )c,
O~')~
= H detpi(mi) x ZDIF(p, ~PO)
ielo
(
x
-
/ m .\c'
//
\
\E,~lo(m,--1)c,[F,:F]\degp
((-1)x<Iu,:)Hi~lot~) . z r t t ~ ) , O o )
_
/m \to,
,
,
)
)
,
\degp
p4:2
,,: 2.
Therefore it is sufficient to show that
det pi(mi • CD.,DT)= detpi(mi) c'.
It follows from the commutativity of the diagram
H2._,(D,,~,(n)
HZx~(X,j! Z'(n))
)
x ....
>
Fi •
$
reciprocity> 7I"1(u)ab, tame
+-
2~'(1)~,
~,
4 - - 71' 1
(V,)~)tame.
Here the left vertical map is that used in the definition of etx, Vle~, and, in the right
square, ei: :~'(1)~
n l ( U ) "b't"me factors the coinvariant Fi = ( (1)~,)c~. We
prove the commutativity. By restricting to a regular one-dimensional closed
subscheme of X flat over (gK which meets D~ transversally, it is reduced to the case
of finite extension. Then it means the following fact on the local class field theory
deduced from IS, Chap. XIV Proposition 8, corollary]. F o r a local field L with
finite residue field E, the canonical map E • -.* Gal(Lab'tame/L) induced by the
isomorphism ~'(1)2-~ IL/PL of ramification theory coincides with that
E" ~ L •
--* Gal(L~b'tam*/L) induced by the reciprocity map sending a prime
to a geometric Frobenius. Thus the proof of Theorem 2 is completed.
If we use [DG, Theorem 3.3], we obtain an analogous formula for the e-factor
of the tame part without assuming (X, U) is tame over ~K. Let
do(U/K, ,~p, t#) = co(K, Rr,(vrr ,~o) e', O).
Then if we replace Proposition 6' by Theorem 3.3 (loc.cit), the same argument as in
the proof above shows
e-factor of a tamely ramified sheaf on a variety
409
Theorem 2' (using purity for etale topology (cf.loc.cit)) Let the notation be as above.
Then we have
S~o(U/K, o ~ , 6 ) = d e t p ( -- dx.v/e~).r~Olv(p, 60)
p+2
x
""
p=2.
def
Here Z~(U~) = ~ ( - 1)idimHi(U~, If)z) P~ = ~ o m ~ c i [ f i :
f].
Corollary. I f X is a curve over (9K namely if n = 2, the conclusion of Theorem 2' is
true.
Proof Since purity is k n o w n for H 1, Theorem 2' implies corollary.
Next we prove Theorem 1.
Proof of Theorem 1 We prove the theorem by induction on n = d i m X . It is clear
when n = 0. T h o u g h it is not logically necessary, we check the case n = 1 as an
example. F o r a closed point x e X , let Fx be the residue field of x and Kx be the
completion of the function field of X at x. Take a non-trivial rational section m of
Q~w(logD) such that, for all xeD, ordx(co) = - 1 and resx(co) = 1. Since we fix
an additive character 6o, it defines an additive character 6x: Kx ~ ~ / by
a ~ 6o(TrFxw(resx(aog))) for x e X . We let e(0)(Kx, V, co) denote S~o)(K:,, V, q&).
Then by L a u m o n ' s product formula ILl, we have
so(U/F, ~ p )
eo(U/F, ~,)dcgp
s(K~, ~p, co)
: ~ s(Kx, ~,~, co)degp
r~
eo(K~, ~p, co)
: X D s o ( g x ' (I~,, O))degp
We have 8(K~,~p, co)=detp(~p~)~162162
acgp for x e U by I-D1,
(5.5.3)] and eo(K~, ~ p , co) = zF~(p~, 6o oTrF~/F).eo(K~, ~ t , co)dcgp for x e D by
Lemma 1. Since Cx, v = - ~ v o r d ~ ( c o ) .
Ix]eCHO(X,D) by example in Sect. 1,
we have
so(U/F, ffp) = d e t p ( - CX,U)'ZDIF(P, 6O)"So(U/F, ~e)dr
and Theorem 1 is proved for n = 1.
We assume that Theorem 1 holds for lower dimensions.
Step 1 We prove the theorem under the following assumption.
I. There is a proper flat morphism f : X - - * Y to a proper smooth curve over
F satisfying the following property. Let V = f ( U ) and T = Y - V .
Then
fly: X v --* V is smooth, Dv is a relative divisor with simple normal crossings and
f:(X, D) ~ ( Y, T) is log smooth i.e. O~lr(logD/log T) is locally free of rank r = n - 1.
Let Ky for a closed point y e Y, a non-zero rational differential ~o, etc. be as
above. Then by L a u m o n ' s product formula ILl,
s o ( U / F , ~ o)
so(V/F, R f ~ o)
~o(U/F, ~ t ) ~~ = ~o(V/F, Rf,~t) de*"
= [I ~(I,:. Rr~(u~.,~), co) x l-I ~o(U,,JG, ~ , co)
~ve(K~, RG(U~, Q~), ~)deg. y~Tso(UK~/K~, ~ , co)do~"
410
T. Saito
By [D1, (5.5.3)], for y~ V, we have
e(Kr, Rr~(U~,, ~ ) , o3)/~(K,, Rr~(u~,, ~:), o,)~ .
= (eo (UFJFy, ~,~p)/eo (UvJFy, ~:)deg p )ora~(~,).
Further by the assumption of induction, we have
eo (Ur~/Fy, ,~o)/eo (UF~./Fy, ~:)d~
= det p (
-- Cx~,, U~/F,.)"~O~/r~(P, ~ o ~TrF/F).
On the other hand, by Theorem 2 applied to X ~ , we have
~o(UK/l';,, ~ , ~)/~o(U,,,/K~, Q:, ,o) ~e~"
= detp( - Cxe~: u~#%)"
"~DoKjF,(P'~0 ~
-
Therefore we have
eo ( U/F, ~p ) = A. B. C. eo ( U/F, (~:)deg v
where
B = ~I (ZD~jF,(p, ~o~
~176
yeV
C = 1-I ~D~K/F,(P,~ o ~
9
yeT
It is sufficient to show that
A = detp(-
Cx, v), B = I~ zv,(p, OooTrv,/v)
and
i~ll
C = [I zv,(p, ~ho~
ir
where 11 = {izl;D~ flat over Y} and Io = 1 - 11. By corollary of Proposition 2
applied to the exact sequence
0 --+f* Ql(log T) ~ Q l ( l o g D ) ~ Q~/r(log D/log T) ~ 0
and by - Cy, v = ~y~v ordy(~o)" [y], we have
Cx, v = y~v
~ ordy(o)).Cx~,/u~, + y~w Cxe~ t:~,/e~,"
Thus the first equality is shown. The same argument applied to D~ for
i~I1 shows
ci=degF, CD,,D,=
~v(Ordy(~)" j ~~,degF,,(CDj,,O*)'[FJ,,:F,]
)
'
jy
'
,
where (Dj, y)g~z~is the irreducible components of D~,F~and F~,y is the constant field
of Dj.y. Thus the second follows from Lemma 1. The last is clear and Step 1 is
completed.
Step 2 We prove the theorem assuming the existence o f a Lefschetz pencil. Namely
we prove it under the following assumption.
II. There is a pencil (Ht)~p satisfying the following properties.
(1) The axis of the pencil meets transversally D~ for all J = I.
e-factor of a tamely ramified sheaf on a variety
411
(2) There exists an open V c P such that for all te V, the hyperplane H, meets
transversally Ds for all J c I.
(3) F o r all t~P and all J c I, the hyperplane Ht meets transversally Ds except at
isolated ordinary quadratic singularities.
For J ~ I, let Sj = {xeDs; x is a singular point of H, ~ Dj for some t~P}. Then
(4) Sj and Ss, are disjoint for all J + J ' ~ I
We show if the condition II is satisfied, a certain modification of the pair (X, U)
satisfies the condition I. Let S be the disjoint u n i o n S = [ I j c t S s and let
~01:X~ --* X be the blowing-up of X at the intersection of X with the axis of the
pencil. Let f l : X1 ~ P be the m a p defined by the pencil, T = f ( q ) * ( S ) ) and
V = P - T. Let X2 be the blowing-up of X1 at all xe~o*(S). For x e S s , J + O, let
Bx be the intersection of the exceptional divisor Ex and the proper transform of Ds
in X2. Let X3 be the blowing-up of X2 at all Bx for xeS, r SO and put ~03:X3 ~ X
and f 3 : X 3 -* P. We show that the condition I is satisfied for f 3 : X 3 ~ P and
U3 = f * ( V ) c~ q)~(U). We only need to check that the complement D3 = X3 - U3
is a divisor with simple normal crossings and that f3: (X3, D3) ---, (P, T) is log
smooth. It is clear outside q~*(S). At ~o*(S), it is checked by an elementary
calculation of the blowing-ups at each xeS. Let xESs, J = {m + 1, -.. , n} and ~ be
a prime element at fl (x). Then etale locally at x, we can take a system (z~i)~=1.....,
of local coordinates at x such that ~ is a local equation of D~ at x for i~J and that
f * ~ = ~ YCi~I'i+I(-1- U/[2) Jr- ~ gj "~ higher terms
i<m
:odd
j>m
where the term u~ 2 appears when m is odd and u is a unit. F r o m this, it immediately
follows that D3 is a divisor with simple normal crossings and that the multiplicities
of bad fibers at the components are 1 or 2. So there only remains to check that
f3: (X3, D3) ~ (P, T) is log smooth when p = 2. Even in this case, by straightforward calculation we check that f * d l o g n is non-vanishing at x, using the equality
above. We leave the detail to the reader.
Now by Step 1, Theorem 1 holds for (X3, U3). Therefore to complete Step 2, it
is sufficient to show the following.
Lemma 2 Let X, U and p be as in Theorem 1 and assume Theorem 1 for lower
dimension. Assume that (p: X' ~ X and U'= ~o*(U) are defined as in one of the
following 4 ways. Then Theorem 1 for (X, U,p) is equivalent to that for
(X', U', p' = q)* p).
(1) Let E be a regular divisor of X meeting Dj transversally for all J c I so that
D' = D u E has simple normal crossings. Then X' = X, U' = X - D' and (p is the
inclusion.
(2) Let Z be a regular subscheme of X of codimension 2 meeting D s transversally for
all J c l. Then (p: X ' ~ X is the blowing-up of X at Z and U' = q)*(U).
(3) Let x be a closed point of D. Then (p: X ' -* X is the blowing-up of X at x and
U' =
q,*(U).
(4) Let J be a subset of I. Then q): X ' --* X is the blowing-up of X at Ds and
U' = ~ * ( U ) .
412
T. Saito
P r o o f o f L e m m a 2 (1) In this case, we have an exact sequence
(p*
0 ~ q~*g2}(logD) ~ (2},(logO') ~ (9E ~ 0
and Image ~o*[E = f2k(logD]~). Therefore by corollary to Proposition 3, we have
Cx,v = Cx, v, + c~. v~ and
detp( - Cx, v) = detp( - Cx, v , ) . d e t p ( - cE, c~).
F o r the Gauss sums, we have
9o/~ (p, ~o) = ~,/~ (p, 4'o)" ~ ~ o/~(p, r
by a similarly obtained formula CO,,D, = Co,, D~* + CE ~ D,, E ~ D',* for i~1 and by the
formula of Hasse-Davenport (Lemma 1). O n the other hand, we have a distinguished triangle
Rrr
~o*~ ) -~ Rrr
~ ) --~ RF~((E c~ U)~, o~olE ) ~ .
Hence we have
co(U/F, ~ p )
=
~o(U'/F, ~'~o)" eo(E n U / F , ~p).
Therefore by using Theorem 1 for E, we have the assertion.
(2) Let E be the exceptional divisor. Then we have exact sequences
0 ~ (p*f2~(logD) e* , s
---rO1/z -~ 0
and
0 ~ CE(1)--* Image~o* I~ ~ (p* Qzl (logDIz) -~ 0.
Therefore by corollary to Proposition 3, we have Cx,,v, = ~p* Cx, v + ~o*(Cz, vz) w
cl ((9e(1)) and ~0. Cx,, v, = Cx, v + Cz, vz. O n the other hand, we have a distinguished
triangle
RFc(U-~, ~ o ) -* RFc(U~, q2*~o) -* g r ~ ( ( Z ~ U)~, ~olz)( - 1)[ - 2] - * .
Similarly as in the case 1, by using Theorem 1 for Z, we have the assertion.
(3) Let E be the exceptional divisor. Then by an elementary calculation, we check
that (p*: ~p*s
~2},(log~p*D) satisfies the assumption of Proposition
4 with J = {i~l; x ~ D i } and m = ]J]. Therefore we have Cx,.v, = ~P*Cx, v. Further
deg~(x)cE,~, = 1 i f m = 1 and = 0 ifm > 1. O n the other hand, r is the identity on
U. The assertion is proved similarly as above.
(4) Let E be the exceptional divisor. Then (p*: cp*f2}(logD) ~ O},(log~p*D) is an
isomorphism and the assumption of Proposition 4 is satisfied with J and m = 0.
Therefore we have the assertion in the same way as in (3).
Step 3 The following lemma is an easy consequence of the theory of Lefschetz
pencil [KZ, th6or~me 2.5].
L e m m a 3 Let X c IP be a projective smooth scheme over a field k and Z c X be
a smooth closed subscheme. Let U be the open of the Grassmann Gr(1, IP) of the lines
L of the dual projective space ~ satisfying the following conditions.
e-factor of a tamely ramified sheaf on a variety
413
(1) L is a Lefschetz pencil for X and Z.
(2) The singularity of the hyperplane sections Ht n X and Ht c~ Z are disjoint for all
teL.
If we replace the original embedding X ~ 9 by the d-uple one for d >=2, then U is
non-empty.
Proof The condition (2) means that the pencil L ~ IP does not meet the image of
P ( N x l z ) which is of codimension > 2.
By the lemma, we can take a pencil satisfying the condition II in Step 2 if we
extend the base field F to the rational function field F (T). Hence there is an open
V of IP~ such that the condition II is satisfied for each fiber of X • r V ~ V.
Therefore by Step 2, the assertion of the theorem holds for every fiber of X v ~ V.
Now by Cebotarev density, the proof of Theorem 1 is completed.
3 Wild case
First we generalize the definition of the relative top chern class in Sect. 1 to
coherent modules satisfying a certain property. Let X be a scheme, D = (Di)~t be
a family of closed subschemes of X and (g, p) be a partially trivialized locally free
sheaf on (X, D) as in Sect. 1. Let ~ be a locally free 6x-module such that
rkg-rk~=n-1
and let H = { r k C o k e r f < n} be an open subscheme of
Homx ( ~ , g ) where f is the universal homomorphism. Let Z = {rk Coker f = n}
be a closed subscheme of H of codimension n, A be a family of closed subschemes
Ai = {piof = 0} c HD~ for i~I of H and T be a Gm-torser I s o m z ( 5~ Cz) over
Z where ~ is the invertible (gz-module K e r f l z . Let z: Z ~ H denote the immersion
and t: T ~ Z be the projection. Then we define the universal relative top chern class
Cn([ ~" ~ d~])~HZ"(T, t* Rz ! A7Zt (n)n.a)
as follows. If we forget p and D, HZn(T, t*RT:!~'(n)H) is naturally identified with
H~
~') and c , ( [ ~ ~ ~ ] ) goes to 1, which is the usual localized chern class
C~z([~- --, ~]).
We put ~ = Rz~'(n)u.a[2n]. Since the immersion Z c~ As ~ As is locally
isomorphic to a section of an affine space bundle of relative dimension n - IJI
9
for J c l , we have y?' - - j (Se)---@lJl=j z t (J)Aj~z
by relative purity. We need to
define a canonical class in H ~
If we put : ~ = R z ~ ( [ Z ' ( n ) n ~
# ' ( n ) a , ] ) [ 2 n ] for ieI, we see there is a canonical isomorphism t~)~lSel ~ ~ .
Hence we may assume l I[ = 1. Then our ~ ( ~ e) = 2~: and ~ - 1 (se) = Z'(1)an z. We
show that the extension class is c1(~*)~HZ(A n Z,~'(1)). We may assume
D = X. Further, by working locally on H, we may assume g = (9x and rk ~ = 1.
Then H = V ( . ~ - * ) = S p e c S ( ~ )
and z : Z = A - - + H
is the 0-section. Let
j : H - {0} ~ H
be
the
open
immersion.
Then
~q = Rztj!~'(1)[2] ~c o n e ( [ j ! ~ ' ( 1 ) ~ R j . Z ' ( 1 ) ] ) [ 1 ] . F r o m this and 5 a = ~ on Z, the claim follows
immediately. It implies that there is a canonical m a p F ( T , ( S ~ ( t * ~ z n a ) ) •
H~
t*~r). Therefore the universal trivialization of ~ on T defines the desired
class in H~
t*~).
Let (X, D) be as above and fr be a coherent (9x-module of tor-dimension < 1
with a partial trivialization p on D. Here similarly as in Sect. 1, a partial trivialization is a family p~: if[o, --' (90, such that ps: ~[Ds ~ (gJD,is surjective for all J ~ I.
414
T. S a i t o
Assume f# is locally generated by n sections and locally free of rank n - 1 on
a dense open of X. Let W be the closed subset of X such that f# is locally free of
rank n - 1
outside W. Further assume that there is an isomorphism or:
Tor~x(N, (gw) m Ow with a suitable subscheme structure on W. Then, by pullingback the universal relative top chern class above, we define the localized relative
top chern class
c,(f#, p, a)~H~v"(X mod D, ~'(n))
as follows. We make an additional assumption that f# admits a surjection g ~ f#
from a locally free Cx-module g of finite rank. The kernel Y = K e r ( g ~ ~ ) is
locally free by the assumption on tor-dimension and rk g - rk f f = n - 1. Since
the inclusion (p: f f ~ g satisfies rk Coker~o = rk f# < n, it defines a section s:
X ~ H. Since s*Z = W as a closed subset of X, cr defines a section W ~ T by
replacing the scheme structure of W i f necessary by that of W c~ s*Z. Therefore by
functoriality, the universal class c , ( [ ~ - ~ g ] , p )
is pulled-back to define
c,((#, p, a)~H~.(X m o d D , ~'(n)). It is easily checked to be independent of choice of
g~ff.
Example. Assume X is the spectrum of a discrete valuation ring CK and D is the
spectrum of the residue field F. Let ~ be a monogenic torsion CK-module with
a partial trivialization p: ~ |
~ F. F u r t h e r let n = 1, W = D and a be an
isomorphism
Torlr
F) ~ F.
Then
cl (~, p, a)~H~v(X mod D, ~'(1)) =
K•174
' is the class of a~K • defined as follows. Take a surjection (9~(~ f#
such that p ( 1 ) = 1 and let I be the kernel. Then Tor~'*(ff, F ) = I @ F and a is
a generator of I such that a(a) = 1. Another characterization of a is given as
follows. Let 5 ~ be the invertible CK-module det ft. Then there is canonical isomorphism L P Q K = K since f f |
= 0 and also a canonical isomorphism
@ F = Horn (Tory's(f#, F), ff @e~F) -~ F:f~--, p ofo a - 1 (l), N o w a - 1 is a generator of &,a going to 1 by s 1 7 4
considered as an element of K by
~|
N o w we apply the above construction to geometry of a variety over a local field.
Let K be a complete discrete valuation field with a perfect residue field F and
X = U be a scheme over the integer ring Cr of dim X = n as in Sect. 2. Let f# be the
sheaf of relative differential 1-forms with logarithmic poles fJ~/o~(log D / l o g F) and
W b e the locus where ff is not locally free. We define the localized relative top chern
class of (# as follows. By an elementary computation, we see the tot-dimension of
(# is < 1. Since f2~/e~(logD/logF)lx~ = f2~dK(1OgDK) is locally free of rank n -- 1
on the generic fiber, W is supported in the closed fiber, W ~ Xe. O n the reduced
closed fiber E = (XF)~d, by an elementary c o m p u t a t i o n we see there is a canonical
isomorphism
fr174L(9 E~--1-(gE~ f2~(logD)[E:I ~ d l o g z ]
for a prime element ~ of K. Here O~(logD)lE is locally free of rank n and is put on
degree 0. Therefore for i~Po={i~Io;p,,~h}, we see W c ~ D ~ = O and for
i e I ~ = {ielo;plr~} the residue m a p res~:(#[o.-*(9o, is defined. Further for
i~lx = {iel, D~ fiat over g0K} there is an exact sequence
0 ~ I2~,/~(logDIo,/logF) ~ c~lo,
res,
' (9o, --* 0.
e-factor of a tamely ramified sheaf on a variety
415
Let I ' = I~' w 11 and D' =(Di)iei,. Then the family of the residue maps
p = res = (resi)i~r and its modification p' = res w, where p~ = m~-1 .pi for i~1'~ and
Pl = P~ for i ~ l l , define partial trivializations of ~ and there is a canonical isomorphism H * ( X m o d D , ~'(n)) .z, H * ( X m o d D ' , 77'(n)). Further if we give W a closed
subscheme structure of E = ( X F ) r e d , the description of ~ | IL
e'x (fig above defines
a canonical isomorphism a = resv: Tor~x(ff, 6~w) -~ (~w. Therefore the definition
above applies to (~, p, or) and (if, p', o). We define the wild part of the relative
canonical class by
Cx,
w u/c~ = ( -- 1)n c , ( ~ x1/ ~ ( l o g D / l o g F ) , res TM,r e s e ) ~ H ~ ( X mod D, ~'(n)).
The relative canonical class is defined by
w
2n
Cx, v/~.~ = c~x,u / ~ + cx, v/e~ ~ H x~(X mod D, Z'(n)).
The latter has a simpler description
Cx, u/e~ = -
~ mi w CD,,D. + ( -- 1)"c,(~;/a~ (logD/logF), res, resr)
which follows from an analogue of Proposition 1 below.
Proposition 1'. Let X be a scheme with a family D indexed by I of closed subschemes,
(,~ be a coherent (fix-module of tor-dimension < 1 and of rank n - 1 and let W be the
closed subset where ~ is not locally free as above. Let p = (Pi: r
(fio.)i~1 be
a partial trivialization o f ~ and a: T o r ~ ( ~ , (fiw) =~ (fiw be a trivialization on W.
Further let fi~F(Di, (fi~,) for i~I and p' = (p~ = f f l " Pi: g]D. ~ (fiD,)i~t be another
partial trivialization of ~. Then
Here N j = K e r ( p j : . ~ I D ~ - - * (fi~,), PlD~ is the induced partial trivialization (Pi:
~s ID, ~ O~-~ Co, ~ o~)i~s, of Ns and o l w~, D~ is also the induced trivialization.
Proof goes similarly as that of Proposition 1.
If (X, U) is tame over (~K, we have c~,v/e, = 0 since W is empty.
Example. Let L be a finite separable extension of K with the residue field E and
X = Spec (ill ~ U = SpecL. Then Cx, v / ~ L • 1 7 4
77' is the class of a e L • such
that T r L m ( a . m L ) ~ m K and TrL/K(ax)=Tre/v(~) for all x~(fi~. In fact for
5 ~ -- det ~ ~,,/e~ (log E/log F ) = Homc~ (mz, mr), the trivialization 50 | L -~ L is
given by Tr~/~ and that 5 0 |
= Home(E, F ) ~ E is given by Tre/e.
In a similar way as above, the canonical cycle Cx, u/e~ is defined in H"x~(X,/;ft,).
Its image by the canonical morphism H"x~(X, ~ , ) -~ C H o ( X e ) coincides with that
defined by Kato [K2]. As for the degree dege: H"x~(X, ~ , ) -~ H~((fi/~, ~ , , ) = 7/of
Cx, v/e~, Kato formulates the following conjecture (loc.cit) analogous to that of
Bloch [B].
Conjecture. Under the notation above
degvcx, v/t~ = - S w r H * (X2, ~ ) .
Here the right hand side is the alternating sum of the Swan conductor.
416
T. Saito
N o w assume the residue field F is finite. By the Poincar6 duality, the pairing
H}~.(XmodD, ~'(n)) x H i ( u , (Q/7Z)')
Trx/e~r )n3((flK, ((I~/Z)'(1))
~_HZ(K, (~/7Z)'(t)) = (@/7])'
is perfect. Thus a canonical isomorphism H2"'XmodD,'~'(n))
xA
--* n l ( U ) ~b' is
defined. Also we have the reciprocity map of the class field theory
CHo(Xr) ~ nl (X) "b. Therefore the relative canonical cycle Cx, v/c~K is defined in
n l ( U ) ab'tame similarly as in Sect. 2. F o r iEI1, let Fi be the residue field of the
constant field of Di and for ieIo,let F~ be the constant field of Di as before. Then as
in Sect. 2, the canonical m a p ai: 7l'(1)~, ~ n l (U, y)tame is defined. F o r i e Io, let ci be
as before and for ieI~, let c~ = degr, Co,,O*,/eK,where Ki is the unramified extension
of K with the residue extension FdF. Let {, ~k, ~Po and an d-adic representation p of
n~ (U, y)t~m~ be as in Sect. 2. Let p~ = a * p for i~I be the restriction. Then we define
rDIF(P, O0) = H ZF'(Pi' 0 O~TrF,/F)c'"
iel
F o r (X, U) tame over Or, this definition coincides with that given in Sect. 2 as is
easily checked. We formulate the following conjecture.
Conjecture. Under the above notation.
eo(U/K, o~ o, ~) = d e t p ( - ex, u/o~)" ro/r(p, ~o)" eo(U/K, ~ t , ~)d~,p.
Example. If n = 1 i.e. for a finite extension of a local field, conjecture is shown as
follows. Let L be a finite separable extension of K with residue field E and p be
a tamely ramified representation of Gal(L~P/L). Then by example above, we have
eo(L/K, p - d e g p - (~t, ~)
= eo(L, p - d e g p "if)t, O~
(by [D1, (5.6.1)])
= detp(a-1).eo(L, p - d e g p . t ~ t , ~bOTrL/r(a.))
= d e t p ( - Cx, v/~)'zE/r(p, q/o)
(by loc.cit. (5.4))
(by L e m m a 1).
Acknowledgements. The author is indebted to G. Anderson for the idea of the relative chern class
and B. Gross who introduced him to the paper [KT] whose subject is closely related to that of this
paper. He also wishes to express his gratitude to K. Kato, who informed him of a work I-A] of
Anderson and S. Bloch for their encouragement and discussions. The referee informed him that
results related to this paper and to [A] are obtained in [Lo] and [Lo-Sa] respectively. The author
wishes to dedicate this paper to his late colleague Osamu Hyodo for the memory of the last
discussion with him, whose subject was that of this paper.
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