description of the Brauer group Br (K) of K, using the theory of Brauer groups of two-dimensional local fields due to Kato [8], which we first recall briefly: Let P be ...
I~lventlones
Invent. math. 85, 379-414 (1986)
mathematicae (~) Springer-Verlag 1986
Arithmetic on two dimensional local rings Shuji S a i t o * Department of Mathematics, Faculty of Science, University of Tokyo, Hongo, Tokyo 113, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w1. Some notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . w2. A review on the local theory and the statements of main results . . . . . . . . . . . w3. The injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w Some results for special cases . . . . . . . . . . . . . . . . . . . . . . . . . . w5. Cohomological dimensions (by K. Kato) . . . . . . . . . . . . . . . . . . . . . w6. Some results for special cases (continued) . . . . . . . . . . . . . . . . . . . . . w Some investigations of the structures of Pic(X) and C s . . . . . . . . . . . . . . . w The completion of the proof of (2.10) . . . . . . . . . . . . . . . . . . . . . . . w9. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
379 382 382 384 386 388 394 400 402 406 408 413
Introduction Let A be a n excellent n o r m a l t w o - d i m e n s i o n a l h e n s e l i a n local ring w i t h finite r e s i d u e field F a n d q u o t i e n t field K. A p u r p o s e of this p a p e r is to give a d e s c r i p t i o n of the B r a u e r g r o u p Br (K) of K, u s i n g the t h e o r y of B r a u e r g r o u p s of t w o - d i m e n s i o n a l local fields d u e to K a t o [8], w h i c h we first recall briefly: Let P be the set of all p r i m e ideals of h e i g h t o n e i n A. F o r each p E P , let A~ b e the h e n s e l i z a t i o n of A at p a n d let Kp (resp. ~c(p)) b e its q u o t i e n t (resp. residue) field. T h e n , K~ is a h e n s e l i a n t w o - d i m e n s i o n a l local field in the sense of (2.2). F o r s u c h a field, K a t o c o n s t r u c t e d a c a n o n i c a l p a i r i n g (cf. [8] a n d w in this paper) (0.1)
( , ) K p " Br(Kp) x
K*~II)/TI,
* A part of this work was done while the author was enjoying the hospitality of the Department of Mathematics at Harvard University with the financial support from The Educational Project for Japanese Mathematical Scientists
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S. Saito
which induces a canonical homomobphism ~t% : Br (Kr) ~ D(K*),
(0.2)
where K* is endowed with a suitable topology (cf. (2.3)), and for a topological group G, D(G) denotes the group of all continuous homomorphisms x:G~O./7/ of finite order. Moreover, we have the following commutative diagram (0.3)
Sr(K)
*K~D(K*)
Br(Ar)_~Br(~(p))
, ~/TZ =D(Z),
where the right vertical arrow is the dual of ordKr" K* ~ 7l, and ~ is the wellknown isomorphism for the henselian discrete valuation field ~(p) with finite residue field (cf. [16]). Then, a result of Kato states that tbrr is almost an isomorphism (for the precise statements, see (2.7)). Starting with this local result, we put together all maps &Kp for p e P to obtain a description of Br(K) (the method is analogous to a standard argument in the classical class field theory): Let I K be the restricted product of K* for p e P with respect to subgroups A*, namely, I K consists of all elements a ---(ap)r~ P (ar~K*) such that areA* for almost all peP. Using (0.1), we define a pairing (0.4)
( , )r:Br(K)xItc-*Q/Z;
(o), a ) K = ~, (cot, at) %, reP
where for coeBr(K), cop denotes its image in Br(Kr). Since cow is contained in Br(Ar) for almost all peP, we see by (0.3) that the sum in the right hand is a finite sum, and ( , ) K is well-defined. Furthermore, the "reciprocity law" for A (cf. (2.9)) gives us the equality (e), a ) K = 0 for any coeBr(K) and aE1K in the diagonal image of K* (for the details, see w2). Hence, putting C K= Coker (K* ---,IK), we obtain a canonical pairing
(0.5)
( , )K: Br(K) x CK~II~/Z.
For a finite subset S of P, we define Cs=C~
F[ A*-*Cr)~-C~ ~eP--S
@ 7/)| ~P-S
K*)). p~S
Note that when S is empty, Cs is nothing other than the divisor class group of A. For a non-empty S, let As be the subring of K consisting of all elements which are integral at any t3eP-S. Then, for any p e P - S , the image of Br(As) in Br(Kp) lies in the subgroup Br(Ap). Hence, (0.3) implies that ( , )K induces a pairing (0.6)
< , >s: Br (As)x Cs--*tl~/TZ,
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381
and this pairing induces a canonical homomorphism (0.7)
(bs: Br (As) -o D(Cs).
Here, C s is endowed with the finest topology which is compatible with its group structure and for which the natural maps K*-o C s
for p~S
are continuous, namely, X: Cs-~ ~/~. is continuous if and only if its restriction to K* is continuous for any peS. Then, under a certain condition, which is known to be satisfied in a wide variety of contexts (cf. (1.3)), our main results assert that 4~s is injective, and give a description of its image (cf. w2). Noting
Br(K)=limmBr(As)
and
CK=~imC s,
S
S
we will obtain also a canonical homomorphism (0.8)
(btr Br(K) ~ D(Ct()
and similar results to the case of q~s. As one corollary, we have the following result which is an analogue of a well-known statement in the classical class field theory. (0.9) Theorem. Let S be as above. Then, co~(~Br(K~) lies in the diagonal image of Br(As) if and only if the character ~s peS
corresponding by (0.2) to co annihilates the diagonal image of A*. Lastly, in case that S is empty, the same argument gives us a canonical pairing (0. t0)
Br (X) • Pic (X) ~ ~/7/.
Here, X = S p e c ( A ) - { m } (m is the maximal ideal of A), Br(X) = Het 2 (X, Gin) and Pic(X)=H~,(X, ~3,,) (= the divisor class group of A). We have the following result which is analogous to the classical unramified class field theory (0.11) Theorem. Both groups Br(X) and Pic(X) are finite, and (0.10) is a perfect pairing of finite abelian groups. In w we give an analogue of (0.11) for the case that X is a proper smooth geometrically connected curve over k which is a complete discrete valuation field with finite residue field. Let K be the function field of X. Noting that for a closed point p of X, the henselization Kp of K at p is again a henselian twodimensional local field, we define a pairing in the same way as before (0.12)
Br(X) x Pie(X) ~ (I•/Z,
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S. Saito
This p a i r i n g was first defined b y L i c h t e n b a u m [11] by a different m e t h o d , a n d he p r o v e d that it induces an i s o m o r p h i s m (0.13)
Br(X)~
D(Pic(X)),
ignoring the p - p r i m a r y torsion p a r t when k is of characteristic p > 0 . Here, Pic(X) is e n d o w e d with a locally c o m p a c t t o p o l o g y d e t e r m i n e d by the usual adic t o p o l o g y o n k (cf. (9.4)). The r e a s o n why he neglected the p - p r i m a r y torsion p a r t is t h a t he r e d u c e d (0.13) to the following result due to T a t e [19], where the p - p r i m a r y t o r s i o n p a r t is not treated. Theorem. Let A be an abelian variety over k and B its dual abelian variety. Then, the derived cup product deduced from the relation B = Ext x (A, ~ , , ) gives an isomorphism (0.14)
H L(k, A (kSep))~- D (B (k)). In w we p r o v e directly (0.13) and deduce (0.14) from (0.13), giving a new p r o o f of (0.14) including the p - p r i m a r y t o r s i o n part. In a f o r t h c o m i n g p a p e r [13], using the results in this paper, we d e v e l o p the class field t h e o r y of the q u o t i e n t field K of A.
Notations A ring means a commutative ring with unit. For a ring R, ~R; the absolute differential module of R over 7z. For a field k, k~eP; a separable closure of k. For an integer i>0, K,(k); the i-th Minor K-group of a field k. For a discrete valuation field k, Ok; the ring of integers, ink; its maximal ideal, ordk: k*---,Tl; the normalized additive valuation of k, and for each integer n>0,
U~'):={x~k*lordk(X--1)>n}(n>l),
U~~ = k*,
~"); the subgroup of Kz(k) generated by all elements of the form {x, y} such that xeU~"1 and
yek*. For an abelian group G, G* (resp. G*or); the set of all homomorphisms Z: G~,/71 (resp. of finite order). When G is endowed with a topology compatible with its group structure, D(G); the set of all continuous homomorphisms Z: G ~ Q/Z of finite order. For an abelian group M and integer n > 0, M,:= Ker (M ~ For a prime number l, M(I): = ~
M) and M/n,= Coker (M-L~ M). MI~.
v
w 1. Some notations and preliminaries In this paper, A always d e n o t e s a t w o - d i m e n s i o n a l excellent n o r m a l henselian local ring, and use the following c o n v e n t i o n s : m; the m a x i m a l ideal of A, F = A / m ; the residue field of A, K ; the q u o t i e n t field of A0 X : = S p e c ( A ) - {m}, P ;
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383
the set of all prime ideals of height one in A. For p e P , Ap; the henselization of A at p, Kp (resp. ~:(p)); its quotient (resp. residue) field. A resolution of A is a proper morphism f : X ~ S p e c ( A ) satisfying the conditions: 3~ is a two-dimensional regular scheme, the reduced part Y of ~ @AA/1T[ is a geometrically connected one dimensional scheme over F such that any irreducible component of Y is regular and it has only ordinary double points as singularities, and f induces an isomorphism f: 3E- Y~-X. By [1] and [20], we can always find a resolution of A. For a fixed resolution of A, we use the following conventions: Yo (resp. I11); the set of all closed (resp. generic) points of Y. For r/GYI, K,; the quotient field of the henselization of the local ring of 3~ at q, ~/o; the set of all x c Y o lying on the closure {q} of r/ in 3s For x ~ Y o, Ax; the henselization of the local ring of 3~ at x. A x is an excellent regular twodimensional henselian local ring, and we define Fx, K x and Px for Ax in the same way as A. For xcYo, Y~; the subset of Px consisting of all elements lying over some element of YI. For t/6Y~ and x~q0, there is a unique qx~Y~ lying over q, and K0/x) is the henselization of K(q) at x. We denote by K,x the henselization of Kx at t/~. For each x~Yo, the cardinality of Y~ is one or two, according as x is a regular point of Y or not. For p~P, its closure in X contains a unique xcYo when we denote " p ~ x ". For x c Yo, W; the subset of P consisting of all p such that p ~ x . For pcpx, Px: =PAx is an element of P~ and A p "-~ _ ( A ~)p. Thus, px is identified with a subset of P~, and Px is the disjoint union of W and Yff. Here, we include some long exact sequences and isomorphisms, which will be used frequently in this paper: For a sheaf (or a complex of sheaves) ~ on X~t, we have the localization sequences (1.1)
...-~
~) H~IY, ' i* ~)-~ H~(Y, i* ~)-~ @ W(q, i* ~ ) - , . . . . xcYo
tlcYI
' i* ~ ) ~ H'(x, t~ * ~ ) - - , ,x~r'( @ U'(,7~, i* ~ ) - - , . . . , ...--*H~(Y, where i: Y~3E and i ~ : x ~ X for x c Y o are the inclusion maps. Moreover, we have isomorphisms (1.2)
H'(r/, i* o~ ) ~ H'(CK,,, ~ ) ,
H i(X, t9x, ~ - ) ~ _ H (A~, ~ ) , H i (rl~, i*~)~-H~((Y~,, .~). Lastly, we consider the following condition on A. (1.3) Artin's approximation property is satisfied for A (cf. [2]), namely, for any functor F: (A-algebras) ~ (Sets) which is locally of finite presentation, and given any ~r and integer c > 0 , there is a ~cF(A) such that ~ = ~ modulo m ~. Here, /1 denotes the completion of A. (1.3) is trivially true when A is complete. In [2], (1.3) was conjectured to be true in general (under the assumption that A is excellent) and was proved to
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S. Saito
hold when A is the henselization at a prime ideal of an algebra of finite type over a field or an excellent dedekind domain. A couple of more results to this direction are found in [22-24]. Recently, the author heard that D. Popescu had succeeded to give a positive answer to this conjecture.
w2. A review on the local theory and the statements of main results To state our main results, we first recall some local results in [8] I and II in the following items (2.1)~(2.8). (2.1) For a field A and a prime number l, we define an 1-primary torsion group Hi(A)(l) by;
Hi(A)(l)=lim r4i~/t ,,|
if /4:ch(A),
n
Hi(A)(1)=lim _____). H~(A, W, ~ A'-~ , log)
if
l=p:=ch(A)>O,
n
where W, QA, i-1Io, is the logarithmic part of the De Rham-Witt complex on Spec(A), t (cf. [7]) and the transition maps are induced by the multiplication by p. We put Hi(A) = @ HI(A)(1). l
We can see that
HI(A)~-D(Gal(Aab/A))
and
H2(A)~-Br(A).
On the other hand, in case 14: ch (A), we have the Tate's Galois symbol
Kj(A) -~ HJ(A, /i~), and in case l=ch(A), we have the differential symbol
Kj(A)~ H~ A, W, f2ja,lo~). Hence, using cup products on cohomologies, we have a pairing (2.1.1)
H'(A) • Kj(A)-~ H'+ J(A).
(2.2) Definition. A henselian two-dimensional local field A is an excellent henselian discrete valuation field whose residue field is a henselian discrete valuation field with finite residue field. We denote by (~A the ring of integers in A. Let A be as w1, and suppose the residue field F is finite. Then, for any p~P, K~ is a henselian two-dimensional local field. Conversely, for any A as (2.2) we can find A and ~0 such that A-~ K~. (2.3) Definition (cf. [8] I w Let A be as (2.2), and choose A, p~P, and an identification z: A~-Kp as above. For an integer n > 0 and a non-zero ideal I c A which is not contained in p, let U~(I) be the subgroup of K* generated by
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385
U~") and l + I ( c A * ) . We can consider A* a topological group by taking {~"(I)},,i as a basis of the neighbourhoods of the unit in A*. This topology depends only on A*, not on the choice of A, p e P and i. By [8] ! and II, we have a canonical isomorphism (2.4)
invA: H3(A)-Y~ Q/;~.
Hence, by (2.1.1) we have a canonical pairing (2.5)
( , )a: H2(A) x A* ~H3(A)~-Q/71.
Moreover, each o~Br(A) annihilates some open subgroup of A* in the pairing ( , )A (Cfi [8] I w SO we obtain a canonical homomorphism (2.6)
q~A: Br (A) -o D(A*).
For each prime number l, let q~a(l) be the map induced on the /-primary torsion part, and let k be the residue field of A. (2.7) Theorem (Kato [8] I and II). (1) ~A is injective. (2) @a(l) is an isomorphism for 14:ch(k). (3) For l=ch(k), CbA(l) is an isomorphism if k is complete. (4) We have the following commutative diagram Br(A)
I
Br ((gA)~_Br (k)
~ D(A*)
~--, II~/Z = D(7'Z)
(41 the explanation below the diagram (0.3)). (2.8) Remark. If )~e(A*)tot * has a finite order prime to ch(k), it is automatically continuous, namely, xeD(A*). Now, fix A as w1 with finite residue field F. Recall the argument in the introduction. We have a well-defined pairing (0.4)
( , )r:Br(K)xlr~Q/Z;
(co, a ) r = ~, (c%, a~)~ . oeP
The reciprocity law for A is stated as follows (cf. [10, 14]). (2.9) Theorem. Let c~H3(K), and for each peP, let co be the image of c in H3(K~). Then, we have cp =O for almost all p e p and ~, invK~(cv)--0. r~EP
By definition, (2.9) implies that (co, a ) x = 0 for any o2~Br(K) and a~I K in the diagonal image of K*. As was explained in the introduction, this fact gives us the fundamental tools (0.7), (0.8) and (0.10) with which we play our game. Now, under the assumption that (1.3) holds for A, our main results are the following
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S. Saito
(2.10) Theorem. Let S be a non-empty finite subset of P. (1) The maps ~s and ~K are injective. (2) The image of ~s (resp. ~K) consists of all elements whose restrictions to O(K*p) come from Br(Kp)for every p~S (resp. P). The proof of (2.10) will be completed in w8. For a prime number l, let ~s(l) (resp. 4~K(/)) be the map induced by q~s (resp. ~ ) on the/-primary torsion parts. (2.11) Corollary. (1) qbs(1) and ~K(I) are isomorphisms if 14:ch(F). (2) For l=ch(F), ~s(l) (resp. (b~(l)) is an isomorphism if .['or each p~S (resp. P), either ch(K(p))--0 or 1r is complete. (2.12)
Corollary. I f A is complete, ~s and @K are isomorphisms.
(2.11) and (2.12) follow at once from (2.7) and (2.10). (2.13) Remark. By the definition of the topology of Cs, any z~(Cs)*~r of the order prime to ch(K) is continuous. As more corollaries, we obtain (0.9) and (0.11) stated in the introduction. In fact, (0.9) follows from (2.10), noting that Coker (A* ~ @ K*) p~S
can be viewed naturally as an open subgroup of C s. (0.11) follows from (2.10) and the finiteness of Pic(X) proved in (7.1), together with the following (2.14) Remark. (2.10) for cbK implies (2.10) for q~s for any S, and if we assume the finiteness of Pic(X), it also implies (0.11). In fact, for each peP, the localization theory on Spec(Ap) gives us an isomorphism (2.15)
H 3 ( a , , Gm)--~Coker (Br (A,) ~ Br (K,)).
Therefore, by (2.7) we have an injective homomorphism (2.16)
H3(A,, G,,)--+ D(A*),
which is an isomorphism if either ch(K(p))=0 or to(p) is complete. Hence, our assertion follows from the following commutative diagram 0-
_,
Br (As)
0
~ D(Cs) -
,
Br (K)
, D(C~)--
~
,
,
@) D(A*), peP--S
where the upper horizontal sequence comes from the localization theory on X, and S may be empty.
w 3. The injectivity Let A be as w1 with finite residue field F, and assume (1.3) for A. The purpose of this section is to prove (2.10.1), that is,
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387
Proposition. The maps (0.7) and (0.8) are injective.
(3.1)
Proof By definition and (2.7), it suffices to show the injectivity of the map B r ( K ) ~ H Br(Kp). peP
Thus, let co~Br(K) have the trivial image in Br(Kp) for any peP. (3.1.1) Claim. w is contained in Br(X). In fact, this follows from the following localization sequence 0 ~ H 2 (X, ~3m) ~ n 2 (K, Gin) ~ @ H i (X, ~m), peP
noting an isomorphism (2.15). Now, we fix a resolution of A as w1. (3.1.2)
Claim. The image of co in Br(K,) is trivial for any ~/eY1.
Proof In view of the injectivity of the map Br(K~)~ tq Br(K,x)
(cf. (8.1) and [8] III),
it suffices to show that the image of co in Br(K,~) is trivial for any ~/~Y, and xe~/o. Let goxeD(C~ ) be the image of co under the homomorphism (0.8) for A x. Recall that I ~ consists of local components K* for p e P x and K*Ilx for ~/~eYff (cf. w1). The assumption on co implies that &x is trivial on the image of
K*-+I~--+CK~
for any p e p ~.
On the other hand, by (2.3) and a standard approximation theorem for a finite family of discrete valuations on K*, we can see the density of the diagonal image of K* in the product of any finite number of local components of I~:. So we can conclude that &x=0. Hence, our assertion follows from the injectivity of the map (2.6) for d = K , . Now we can complete the proof of (3.1). We consider the following commutative diagram ~r(,@H~(~.]t 113,.) @ HZ(Spec(A~) - x , G,.)
XEYO
,
H~( i, ~ )
--+
H:(X, Gr.)
--+
]
H3(i,,~y, @ G,.)
@ HZ(K,, ~m)
ring1
He(y, G,,)= Br(~) where the upper horizontal sequence and the middle vertical sequence are exact. By the regularity of A~, we have (cf. [6] (6.1)) H2 (Spec (A~) - x, ~ ) = Br (Spec (Ax) - x) = Br (A~) "~ Br (x) = 0.
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S. Saito
Hence, (3.1.1) and (3.1.2) implies that co comes from Br(JE). So it suffices to show that Br(~)=0. This is reduced to the fact that Br(Y)=0 in the same argument as [6] (3.1), except that we use (1.3) instead of the result in [5]. Consider the localization sequence @ H~(Y, tim) ~ Hz(K (13m)~ @ HZ(q, (13m)~ @ H2(Y,, l~Im). x s Yo
~l~ Yl
x~ u
We can easily see that 2 Hx(K(~,.)=0
and
3 Hx(Y, l~m)~ x@egBr(tlx ).
Hence, we obtain the following exact sequence O--*Br(Y)--* @ Br(r/)~-~ @ rl~ Yl
@ Br(qx).
x ~ Yo rlx~ Y~c
Consequently, our assertion is reduced to the injectivity of cr which is a wellknown fact in the classical number theory.
w 4. Some results for special cases Let A be as w1 with finite residue field F. In this section, we treat the following special case of (2.10.2). (4.1) Proposition. Let I be a prime number and z~D(CK)(I), and assume that the image of Z in D(K*) comes from Br(Kp) for any peP. Suppose further that A is regular, and that if c h ( K ) = 0 and l=p:=ch(F), the following condition (4.2) is satisfied. Then, Z belongs to the image of el)it. (4.2) K contains a primitive p-th root of unity, and one of the following conditions is satisfied; (4.2.1) there exists a unique element p e p which divides the ideal (p), and A/p is a henselian discrete valuation ring, (4.2.2) there exist exactly two elements p~, p~ of P which divide the ideal (p). For v=c~ and fl, A/O, is a henselian discrete valuation ring and t 0 c c + p f l = n l A, (4.3) Corollary. Let l be a prime number and coe@Br(Kp)(l), where S is a nonp~S
empty finite subset of P. Assume that A is regular and that if ch(K) = 0 and l =p: =ch(F), A satisfies (4.2). Then, the same conclusion as (0.9) holds for co. (4.3) follows from (4.1), noting (2.14). The proof of (4.1) will be completed in w In this section, we first prove (4.1) assuming l Z =0. First, suppose l+ch(F). Then, we may suppose that F contains a primitive l-th root of unity. Then, using the isomorphism
Kz(K)/I~-Br(K)~|
z
(cf. [123),
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389
and the Gersten sequence in K-theory (4.4)
K 2 (K) ~ @ K(p)* ~ Z ~ 0, peP
we obtain the following exact sequence (4.5)
0 ~ Br (K)t ~, @ H 1(to(p), Z / I Z ) ~ #~ -
1 ~ O.
p~P
Here, the injectivity of l follows from the localization sequence H2 (X, th) --' H2 (K,/h) ~ @ H3 (X, #,) -~ @ p~P
H 1 (K(p),
Z/l~),
peP
together with the fact HE(x, ~) =0, which follows from the regularity of A. On the other hand, since Pic(X)=O, we have
Coker (A*
l-[ A*) c,,. pEP
Hence, noting the canonical isomorphisms A*/l~c(p)*/I
and
A*/l~-F*/l~-p;,
we have a canonical isomorphism C~r ~- Coker (#~ ~ H K(p)*//). pep
Consequently, we have a commutative diagram 0 - - - - - - , Br(K) t
9 @H'(K(p), Z / I Z ) - - -
, ~?-1
O-
* @ D(H 1(tc(p),/~t))--
' D(/~t),
, D(CK) t
p~P
where the middle vertical arrow is an isomorphism by the Tate duality for ~c(p) (cf. [15]). This completes the proof in this case. Next, we suppose c h ( K ) = 0 and l = p : =ch(F). In the following, we will give the proof assuming (4.2.2). The proof for the case (4.2.1) is similar and easier, and we omit it. Put
= /5/and P ' = P - { p = ,
p~}, and let K,. be the henseliza-
tion of K at Pv for v=c~ and ft. Using the localization theory on Spec(R), we have an exact sequence 0
, B r ( R ) p - - - - - * Br(K)p
,
| Hi(X, pEP' /
(4.6)
|
~p'
- - , "
Z/pZ)
0.
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S. Saito
Here, the surjectivity of c~ follows as before from (4.4). On the other hand, defining I , = C o k e r ( R * ~ ~ K*) and Io = [-[ A*, v=~,fl
peP'
we have an exact sequence and canonical isomorphisms
0-~ Io ~ CK ~ I 1 ~ 0 , D(lo)p~ @ D(A*/p)~- @ D(~c(p)*/p)~_ @ HX(~c(p), Z/pZ). p~ 1~"
P~P'
pEP"
Combining these with (4.6), we are reduced to prove (4.7)
Claim. The induced map Coker(Br(R)p~ @ Br(K,.)p)~(10* v = ~, fl
is injective. From now on, fix a primitive p-th root of unity ~eA* and put
Ki(L) = Ki(L)/p
for a field L and i = 1, 2.
Then, by [12], we have isomorphisms
Br(Kv)p~F,z(K~)(v=~,fl)
and
Br(K)r-~/s
).
Combining this with (2.5), we have a pairing < , >~: I~2(K~)@K,(K~)-*Z/pZ
(v=~, fl).
Since the image of the map
K2(R)~ ~
K2(K~) ~- H Br(K,,)v
v=~,t~
~-~,~
lies in the image of Br (R)p, (4.7) is reduced to the following (4.8)
Claim. For i = 1 and 2, define Ai=Im(Ki(R) -" I~ Ki(Kv)) 9 v = ~, ,/]
Then, A~ is the annihilator (A ~)• of A ~ in the pairing
(,)~+(, >~: ]7 g~(K~)| I-[ g~(K~)~Z/pe. v=a.t~
(4.9)
v=~,fl
Remark. (4.8) implies a canonical isomorphism Br (R)p | gp_~ A2,
for we have already known that Br(R) injects into (A~)• by (3.t).
A r i t h m e t i c on two d i m e n s i o n a l local rings
391
Now, we define A~ =the image of Ker(K~(R)-~t~(Kp)) i n / ( I ( K , ) , A~ =the image of Ker(Kz(R)-+K2(K=) ) in/(2(K~),
A~ =Im(KI(R)-~t~1(Kfl) ),
A 2 =Im(K2(R)-.',K,.2(Ko)).
We have exact sequences
O--+A~--+A'--+A~--+O and O--+A~-+A2-+A2-+O. Then, (4.8) is reduced to the following (4.10) Claim. In the pairing ( , ),., A2 coincides with the annihilator of A~ for v = c~ and ft. We define, for each integer n > 0 (resp. i>0), a subgroup V~"~ of /(2(K,,) (resp. U~~ of KI(K,.)) to be the image of V,Kv (") (resp. UKv] ~)~ (cf. Notations). Then, {_V~"~},>=o (resp. {U,!i~}i>=o) defines a decreasing filtration on /(2(K0 (resp. K~(K,.)) such that V,}"~=0 for n>f,. (resp. U~i)=O for i>fv ), where we put s
~
We define
Gr(") V,.= V,}"'/V~"+1) and Gr ~i) U,.=U~(i)/U~i+ ') Since if n + i > f v (x, y ) v = 0 for any xeV,}"1 and yeU,} i) (cf. [8] I w ( , ) v induces, for each integer ne[0,f~], a pairing ( , ) ~ : Gr ~")V,,|
the pairing
~sv "~U,.--+Z/pZ.
Let Gr (") A2 (resp. Grt~ be the image of AZm V~") in Gr(")V~ (resp. A~c~ U~') in Gr (~ U0. Then, (4.10) follows from (4.11) Claim. In the pairing ( , ) ~ (O- - j } (resp. ~k~ rr(J)~J in k j ( 1 - f ) k~ (resp. k*/p). (B) For v=fi, we take n=ur b=(ua)n/P(u~)f~/p in (ii), and c=(ua)"(u,)I" in (iii). -
Case(I) Case(II)
Gr(~
(~+~) and Gr(Y~)A~=,4a(f~). Gr(")A~=Aa/p and Gr(f"-")A~=Aa/p. ~k13
Case (llI) Gr (")A~ = f 2 ~ and Gr (f~-")A~ =Aa. Case (IV) Grtf"'A~=0 and Gr(~ {~ U~. Now, (4.11) follows at once from these descriptions. Finally, supposing IZ=0 with l=p:=ch(K), we prove (4.1). Fix a regular parameter u of A, and put R--A].~/ and P ' = P - { ( u ) } . Then, we have the [ltJ following commutative diagram
o
,
0
, D(I1) p
-,
(K),
o ,H 3(x, G.),
BriR, BrI
~K
,0
tO
, D(CK),~
@ D(A*)p paP"
with the exact horizontal sequences. Here, K~ is the henselization of K at (u) and 11 =K*/R*. The upper sequence is obtained from the localization theory on Spec(R) with SGA4X (5.1), and 0 is the map (2.16). The assumption of (4.2) implies c~(Z)eIm(0). Hence, we are reduced to show the following (4.12) Claim. The image of Br(R) in Br(K,) coincides with the annihilator of the image of R* i n / ( I ( K , ) in the pairing (4.1 3)
Br (K~)p|
(K,) --,"Z/p 71.
394
s. Saito
This can be proved by the similar argument to the previous case. But, instead of the isomorphism /(2(K~)__Br(K~)p and a decreasing filtration {V2")},_>0, we here use the isomorphism 0: M~: = f2rj(1 - 7 )
~C~K~, d =
0 -~ Br (K~)p,
and an increasing filtration ~M~")~ t ~ )n>= -- I of M~ which is defined as follows: M~~> (n>O) is the image of the subgroup of ~r~ generated by all elements of the
dy
form; x - - with y~K* and x~K~ such that ord r ( x ) > - n , and M~ 1~=0. Y Putting G r ~")M ~ = M ,~")/ M ,~"- ~) for n > 0 , (4.13) induces a pairing, for each n > 0 , (,)~": G r I") M | Gr ~") V~~ Z/pZ. The descriptions of the structures of G r (") M~ and the above pairings are given in [8] I w4 and II w3.5. The rest of the proof is similar to the previous case and we omit it.
w5. Cohomologieal dimensions This section is due to K. Kato. Let A be as w1. The purpose of this section is to prove the following Theorem. Suppose that F is algebraically closed. Then, cdz(K)=2 jbr any prime number l which is different from ch (K). (5.1)
In case 14=ch(F), this result was already proved by O. Gabber with a different method (cf. [4]). Here, we can include the case c h ( K ) = 0 and 1 = c h ( F ) > 0 . First, we note that for any p e P such that ch(~c(p))4=/, we have
c d I(K) >__c d~(K ~) = c dt(lc (p)) + 1 = 2. So, we have to prove that cdl(K)
if q=3, otherwise.
Indeed, by using (5.1) and the fact
H~(X,Z/nZ)~-H q
2(N(p),fn~(-1)):O
for q > 4 ,
the localization sequence on X shows
Hq(X, TZ/n7Z)=O for q > 4 . Furthermore, we have the following d i a g r a m of exact sequences H2(K,f.|
K 2(K)/n
( ~ H p3 ( X , f .| )
, ( ~ ~:(p)*/n
H3(X,f.|
--~ Z/nZ
-
, H3(K,f.|
-~ O,
p~P
from which we obtain a canonical i s o m o r p h i s m H 3 (X, fin@2) ,~ 7Z/nZ. Lastly, the vanishings of Hq(X, 7l/nTl) for q = 1 and 2 follow from
Hi(X, 71/nZ)~-H 1(Spec (A), Z/nT]) = 0 Pic (X) = Br (X) = 0, which are consequences of the regularity of A.
(cf. SGA1X(3.1)),
400
S. Saito
w6. Some results for special cases (continued from w4) Let A be as w
For peP, let ip be the composite homomorphism H3(K,/~n@2) ~ H3(Kv,/2~2)"-Z/n7],
where n is an integer prime to ch(K), and the second isomorphism comes from (2.4). By (2.9), we have the following complex 0 ~ H 3 ( K , #o2)_L~ ( ~ Z / n Z - ~
(6.1),
71/n7Z--.O,
pEP
where t sends c6H3(K, #~2) to (tp(c))p~e and a is the addition map. (6.2) Lemma. Let l be a prime number. Suppose A is regular. (1) If l:t:ch(F), (6.1)1 is exact. (2) If ch(K) = 0 and l=p: =ch(F) and the condition (4.2) holds, (6.1)1 is exact.
Proof Let /1 be the strict henselization of A and define /s and /5 for /1 as w1. We have an isomorphism
H3(K, iz~, z)~_H'(r, H2(/~,/t,~ z)), which follows from (5.1) together with the fact Cdl(F)= 1. First, suppose /#ch(F). In this case, we have the following exact sequence (cf. (4.5));
o
, n2(~, ~ 2 ) _ _ _ ~
(6.3)
|
(~0), ~ , ) - - - ~
Z/lZ
-~ o,
~e
| from which we have an exact sequence
H~
(6.4)
M ) ~ Z / l Z ~ H 3 ( K , #~2)~HI(F, M)~7Z./IZ--.O,
where M denotes the middle term of (6.3). Notice that over each peP, there lie d~ distinct b~/5, where dp is the degree of the residue field F~ of ~c(p) over F. Hence, if we fix, for each peP, one ~06/5 lying over p, we have an isomorphism
H~(F, M)~- (~ U~(Fp, ~c(b)*/l). puP
Furthermore, noting isomorphisms
H'(Fp, to(b)*)=0 and
HZ(F~, ~c(~)*)~_nr(~(p))~Q/Z,
we can see
U'(Fp, tc(b)*/l)~-Br(~(p))z,,,Z/lZ
and
H~
~c(~J)*/l)~-tc(p)*/l.
Arithmetic on two dimensional local rings
401
Consequently, (6.4) is rewritten as follows;
@ g(p)*/l ~ Z/lgZ .-~H 3(K, I ~ 2)~_~@ ~./IZ ~
Z / I Z -->0,
p~P
p~P
where ~ is the composite of a ( = the addition map) and the map @ dp ord~lp): @ K(p)*/l --+ @ 7l/1Z peP
pEP
p~p
So it is surjective by the regularity of A. This proves (6.2) in this case. Next, we treat the second assertion. We suppose A satisfies (4.2.2). The proof for the case (4,2.1) is similar and we omit it. For v = ~ and fl, there exists a unique element iJ~ lying over p,.. We put P ' = P - { t % , P ~ } , / 5 , = / 5 _ { ~ , b p } We have the following exact sequence (cf. (4.6)) O-
-*Br(/~lp
' (~_ ~(b)*/P
' Br(/~)r - -
~EP'
-~0,
H2(K,/~v) from which we obtain an exact sequence
Ht(F, Br(R)p|
H3(K, flp@2)-~H i ( F , (~_ K(~)*/p)--~O. ~EP'
As before, we have an isomorphism H i ( F , ( ~ l,:(~)*/p)~ @ ~,/p7]...
fo~P'
p~P"
On the other hand, from the results in w we can describe the structure of Br (/~)v| g; (cf. (4.9)): There exists an exact sequence; 0 - ~ 3 ~ ~ Br(/~)p|
~ 3~2 ~ 0 ,
and z]~z ( v = ~ and fl) has a filtration whose successive subquotients Gr"z] 2 are described as in w (A) (I) ~ (IV) and (B)(I)~(IV), although we here replace ,4~ with its strict henselization ,4/'Pv and k,, with the quotient field ~c(~,,) of/]/-p,,. We can see that the only subquotient Gr , A-zv for which H1(F, Gr . A-2v) is nontrivial is G r 0 ~2 ~ tc(b~,),/p' so we have
H1 (F, Br (fl)p N ~v) ~- H1 (F, ~c(iJ,)* /p) ~-Z/p 7Z. Consequently, we have an exact sequence
g/p2g ~H3(K, ij~2)~ @) 7I/pTl ~O, p~p'
from which we obtain our assertion.
402
S. Saito
Now, we can complete the proof of (4.1). Thus, let zeD(CK)(t ) be as in (4.1). If I=p:=ch(K)>O, Br(K) is p-divisible, so (4.1) follows at once from the result in w4. Next, assume l + ch (F). We consider the following commutative diagram Br(K)(/)-
0
~ D(CK)(l)
~ @ H 3 ( X , ~3,,)(I) -
~ -. H3(X, ~m)(l)
~-~-~ @ O(A*)(l) p~P
Noting (2.16), the assumption implies ~0(z)elm(tp). Hence, we are reduced to prove the injectivity of the induced map p: Im (2) ~ Coker (~0). Consider the following localiation sequence @H3(X,
lAi)
--+ H 3 ( X , ] A , ) _
> H 3 ( K , IAI)-
a
p6P
>
@H~(X,#/),
p~P
where the bijectivity of fi follows from the fact that H2(X, ~3,,)=0 by the regularity of A. By (6.1), we see that e is injective, and this imples that the image of the map 2 in (6.5) contains H3(X, ~3,,)~, and by the result in w we can see Ker(p)cnH3(X, 113,.)t =0. Since H 3(X, G,,) is a torsion group, this completes the proof in this case. The proof for the case c h ( K ) = 0 and l=p:=ch(F) is similar to the previous case: Instead of ~J,,, x and g~, x used above, we use j~ ~,,, v and j~ g~, v respectively, where
is the inclusion map. The details are left to the readers.
w7. Some investigations on the structures of Pic (X) and
Cs
Let A be as w1 with finite residue field F. In this section, we give some investigations of the structures of Pic(X) and C s. First, we prove the following (7.1)
Proposition. Pic(X) is a finite group.
Proof First, by [21] w1 Proposition 16, we may assume that A is complete. Fix a resolution of A as w1, and for each integer n > 0 , put ~n=~@aA/mn+1. There
Arithmetic on two dimensional local rings
403
is an exact sequence
@ ~E~=~ Pic(3s
Pic(X)--, 0,
~/~ YI
where c~ sends 1 ~ Z at r/to the class of the divisor {0} on 3s On the other hand, by E G A I l I (5.1.6), we have an isomorphism Pic (3s -~ lira Pic(3s rt
F r o m these facts, (7.1) is reduced to the following (7.2)
L e m m a . (1) For each n >0, the cokernel o f the composite @ 7~ - % Pic(3s ~ Pic(3s qEYl
is finite. (2) There exists an integer n o such that the natural map
Pic(3s
1) --* Pic(3s
is an isomorphism fi)r any n > n o. P r o o f o f (7.2.1). First, we consider the composite m a p
~o,: Pic(3s
Pic(Y)~
@ Z
(recall Y=(3s
~IEY1
where the m a p d is obtained by taking the degree on each irreducible component of E By a standard argument, we see that Ker(~o,) has a filtration whose successive subquotients are either a finite dimensional vector space over F or the g r o u p of F-rational points of a semi-abelian variety over F, so it is finite. Hence, it suffices to show the finiteness of the cokernel of the composite map @Z~ Pic(3s Pic(3s 'P"> @ Z , t/eY1
rteYl
which follows from the non-degeneracy of the intersection pairing (*)
@ Z x @ Z~Z r/~Ya
(cf. [173).
r/EY1
P r o o f o f (7.2.2). Let J be an ample invertible (5'.~-ideal such that Supp ((_9~/,,r = Y.. The existence of such o~r follows from the negative definiteness of the intersection pairing (,) and the numerical cbiterion of ampleness. F o r each integer v > 0 , let .~,, be the closed subscheme of 3s defined by J~. It suffices to show that there exists v o > 0 such that for any v > Vo, we have an i s o m o r p h i s m
Pic(~;~+ 1) ~-~ Pic(~,,). Indeed, we have an exact sequence of sheaves on 3s
404
S. Saito
where the map /3 sends a local section c of j , . / j v + , to the section l + c of ((9~/j~+ 1),. From this, we have an exact sequence H i (3E,J " / 3 ~"+ 1) ~ Pie ( ~ +x) ~ Pie (~ ,.) ~ 0. Hence, our assertion follows from EGAIII (2.2.2). Let S be a non-empty finite subset of P and U = X - S, and let 2: U -* X be the inclusion map. We have a canonical isomorphism
Ht(X, 2~G,.)-~ Cs=Coker(K*-~( @ 7/)| p~P-S
pES
which we can see by using the exact sequence H~ K, r
(~) H~ (X, 2, r
~ H x(X, 2, r
H ~(K, G,")=0,
peP
together with isomorphisms
H~(X, 2, ffk..)~_~Coker((9* ~K*)=7l 9
if p~S,
(K*
if p~S.
To control the structure of Cs, we fix a resolution of A and put
Ts=Ha(Y, i* R j,(2, G,.)). where i: Y~3E and j: X ~ X are the inclusion maps (cf. w1). We have a natural map t)s: Cs~ H 1(X, ,t.!ffj,,)--, Ts. The key points are that Os induces an isomorphism on the character groups (cf. (7.4)) and Ts has an expression mainly in terms of the groups K,* (xeY o and t/xeYX), which reduces our problem to the theory of the Brauer group of K, using a theorem of Kato (8.1) (cf. w8). Now, using (1.1) and (1.2) for J~=Rj,(2~ 113,.), we have an isomorphism (7.3)
Ts_~Coker(@ K, qeYt
@ [( @ Kp)@( @xKnx)]/Rx, s), xeYo
pES~
qxegl
where S ~ = S ~ W (cf. w and R~, s is the affine ring of Spec(Ax)• An explicit description of ~'s is given as follows: Noting that P~ is the disjoint union of W and Y~, we have a map, for each x~ Yo, ~ s . : ( @ Z ) G ( @ K*) pEU~
p~S ~
~ [ ( @ Z ) O ( @ Kp)O( @ K,..)J/K~ p~U ~
~eS])
n~EY~
I(
[ ( @ K~)@( @ K.x)]/nx. s ~3ESx
Ilx~ Y 1
where we wrote UX= W - S X Then, ffs is induced by @ ~Sx. x e Yo
Arithmetic on two dimensional local rings
405
(7.4) Lemma. Assume (1.3)for A. Then, ~s induces an isomorphism :tr
~
$
~J*: (Ts),or---~ (Cs),o r.
Proof It suffices to prove that 6s induces an isomorphism Cs/r"~Ts/r for any integer r>O. For a finite subset S' of P containing S, put U' = X - S ' . We have a commutative diagram @
A*
, C v,
, Cv-----*
@ A:
, r v,
, Tv
pES' - S
p~S'-S
0
--,0,
where the horizontal sequences are exact. Hence, it suffices to show (7.4), replacing S with S', so we may suppose the followings; (7.4.1)
if ch(K)=0, S contains any p e p such that chffc(p))=ch(F),
(7.4.2) the classes [p] of pcS in Pie(X) generates Pie(X), in other words, Pie(U) =0 (here, we used (7.1)). First, suppose that r is prime to ch(K). Then, we have the following commutative diagram
0
-* Cs/r
-+
0 - - - - - ~ Ts/r
H2(X, 2! Pr)
--~ H 2 (X, 2! I]~Jrn)
, Ha(Y, i* R j,(2! #r)),
where the horizontal sequences come from the Kummer sequence which exists on U by (7.4.1). The right vertical arrow is an isomorphism by the proper base change theorem for etale cohomology. Hence, it suffices to show that H Z ( x , 2! I ~ r , ) = 0 . W e have an isomorphism H2(X, 2~G,,)= Ker (Br (X) --* @ Br fie(p))), peS
which follows from the following exact sequence @ H I ( p , i * ~ , , ) ~ H 2 ( X , 2, Gin)~ H2(X, ~ m ) ~ (~)He(p, i* ~,.), peS
peS
(i~ : p -~ X is the inclusion map) together with isomorphisms Hi(p, i*~,,)~_HI(A,, ~ m ) = 0
and
HZ(p, i*~,.)~_H2(A,, G,,),-~Br(K(p)).
On the other hand, by (3.1) we can consider Br(X) a subgroup of Pie(X)*. Then, for weBr (X), the fact that its restriction to Br(~(p)) is trivial means that ~o annihilates the image of the class [p]ePic(X). Hence, by (7.4.2), we have co :0.
406
S. Saito
In case r is not prime to p=ch(K), we may suppose r = p C Then, our assertion is proved in the same way using the exact sequence 0-~
-~ ~.~ff~,,/r ~O
on
X~.
Lastly, we need a topological consideration on Ts. For an integer n >0, we put U" Ts = the image of @
~(~r;~l + m "+1 C~,
in T.s.
x~ro ~x
(7.5) Lemma. Let zE(Ts)*or, and suppose ~(z)~D(Cs). Then, Z annihilates U ~ Ts _ * * for some n >--O, and its image in (K~x)tO~ is in D ( K * ). For a subgroup H of @ K * (resp. J of K*), we denote by /4 (resp. a*) its peS
image in Ts. By definition, (7.5) is reduced to (7.6) Lemma. Let H be an open subgroup of @ K* and let r be an integer > I. Then, we have the Jollowings : ~s (1) For a sufficiently large n, U" Ts~ffI + r Ts. (2) For a sufficiently small open subgroup J of K *, J ~ 121+r Ts. (2) follows at once from (1). As for (1), if r is prime to ch(K), it follows from the fact U" T s ~ r T s ~ H for n large enough. In general, we see that the image of U"Ts under the map Ts/r (r
~ Cs/r_~ Pic (X)/r
coincides with the image of Ker(Pic(X)--*Pic(X,)) under the map Pic(~)~Pic(X)/r. So it is deduced essentially from (7.2.2), and we omit the details.
w8. The completion of the proof of (2.10) Let the notations be as in w In this section, we complete the proof of (2.10). First of all, we must recall the following result due to K. Kato (cf. [-8] III): Fix t/eY~. For co, eBr(K,) and for X~qo, let CO,x be the restriction of co to Br(K,x) and &,x be the element of D ( K * ) corresponding to ~O,x by (2.6). (8.1) Theorem. The correspondence B r ( K . ) ~ D . : = 1-I D(K* );
cg~(~b.~)~.o
xE~/0
gives an isomorphism between Br(K~) and the subgroup of D consisting of all elements (Z,x)x~,o satisfying the following conditions;
(8.1.1) there exists an integer n > 0 such that Z,x vanishes on the subgroup 1 + m " (9~,~ for any x~tlo.
Arithmetic on two dimensionallocal rings
407
(8.1.2) for almost all xetlo, Z,x vanishes on the subgroup R*, where R~ is the aJfine ring of Spec (A~) x ~ X, (8.1.3) Jbr any acK*, we have Z,x(a) =0 X~r/O
(note that (8.1.1) and (8.1.2) imply that the sum is a finite sum). Fix a non-empty finite subset S of P, and put U = X - S investigate the structure of Br(As) (=Br(U)), we introduce
and Y s = Y i - U . To
~s: = H2(Ys, i~ Rjs , 113,,,v), where Js: U - , ~ map
and is: Ys--~Y, are the inclusion maps. We have the natural
Os: Br (As)~_ H 2 (~, R j s , 113~,v ) ~ Os" (8.2) Lemma. The map 0 s is surjeetive. This is reduced to the proper base change theorem for etale cohomology. The argument is similar to the proof of (7.4) and we omit it. Now, by the similar computation to (7.3), we have the following exact sequences ~s---* ( @ Br(K.))G((~Br(K~))~ (~ Hx3(Ys, i* Ujs , l13m,V), )l~ rl
p~S
x E Yo
Br(R~,s)~( @ Br(K,~))| ~E Y~
3 .. Rjs. , ffJm C)" Br(Kp)) ~ H~(Ys, 15 p~S~
This proves that the image of c~ consists of all elements ((o.).~r,, (oop)~Es)~F: =( @ Br(K.))|
Br (Kp))
which satisfy the following condition; (8.3) for x~Y0, the element ((6%x).~r ;, (eJ~).~s~)e( @ Br(K.~))G( @ Br(K~)) tl~e r i ~
peS ~
is contained in the image of Br(R~,s). Now, we define
K*,x) | ( |
5=(|174 xego
rlxeg~
PeS x
K *0)]) -* =H
I I ( K , x*)
* • H(K,*)*.
r/eYl Xe~o
pES
By (8.1), (2.7) and (7.3), both groups F and (Ts)t*r are viewed as subgroups of S. We define homomorphisms ~t
f: Os , F ~ , ir ~* (0~)-' g: , - s, to~
, (Ts)~*o~-~ Z
(cf. (7.4)).
408
S. Saito
In view of the explicit description of ~bs (cf. w anti-commutativity of the following diagram Br (As)
Br (As)
__
os
(2.9) for A x (xeYo) implies the
r
>Qs-----,
~
os , D(Cs)~(Cs)*o~. l
Now, let zeD(Cs)(I ) be as (2.10.2). First, we suppose that either /=l=ch(F) or l = c h ( K ) > 0 . By (8.1) and (7.5), we can see that g(•) comes from an element co of F, and (4.3) implies that co satisfies (8.3). This proves (2.10.2) in this case. Next, assume c h ( K ) = 0 and l = p : = c h ( F ) . First, we suppose K contains a primitive p-th root ~ of unity. We blow up in advance 3~ successively at some closed points so that A x satisfies (4.2) for any x e Yo- Then, the proof is just the same as above. In general, let L =K((), B be the integral closure of A in L and T be the set of all prime ideal of B lying over S. We have the norm maps
NL/K:Br(Br)~Br(As)
and
NL/K: C T ~ C s,
and
RK/L : Cs-* C T,
and the natural maps
RK/L : Br(As)-* Br(BT)
and we have the following commutative diagrams
Br(As)(p )
, D(Cs)(p )
Br (BT)(p)
---~ D(Cr)(p )
Br(As)(P)
, D(Cs)(p )
Br (Br)(P)--
, D(Cr)(p ).
The commutativities of the diagrams follow from [8] I w5. Since the composite maps NL/KRr/L are the multiplications by [L: K], we may replace A with B for the proof. This completes the proof.
w 9. A p p e n d i x
Let k be a complete discrete valuation field with finite residue field F and let (9k be the ring of integers of k. Let X be a projective smooth geometrically connected curve over k, K its function field. Then, for any closed point p of X, the henselization of K at p is a two-dimensional local field (cf. (2.2)), and in the same argument as w2, we obtain the pairing (9.1)
Br(X) x Pic(X) --*II~/Z.
Only a difference is that we use the results in [14] Ch. II, w1 instead of (2.9).
Arithmetic on two dimensional local rings (9.2)
409
Theorem. The pairing (9.1) induces an isomorphism
Br(X)~-D(Pic(X)), where Pic(X) is endowed with the topology defined below in (9.4). The pairing (9.1) was first defined by Lichtenbaum [11] by a different method (the coincidence of two definitions is easily checked), who reduced (9.2) to the following result due to Tate. (9.3) Theorem (cf. [19]). Let A be an abelian variety over k and B its dual
abelian variety. Then, the "derived cup product" deduced from the fact B = Ext 1 (A, G,,) gives an isomorphism Hi(k, A(k~r
D(B(k)).
In [19], (9.3) is proved ignoring the p-primary torsion part in case k is of characteristic p >0. In this section, we will prove (9.2) directly, and deduce (9.3) from (9.2), giving a new proof of (9.3) including the p-primary torsion part. (9.4) We define the topology of Pic(X). Let Pico(X ) be the subgroup of divisor classes of degree 0 in Pic(X), and let J be the Jacobian variety of X over k. By I l l ] Lemma 1, we can see that Pico(X ) is viewed as a subgroup of J(k) (=the group of k-rational points of J). Moreover, it is open and compact with respect to the usual adic topology on J(k). We endow Pico(X ) with the induced topology, and Pic(X) with the unique topology which is compatible with its group structure, and for which Pico(X ) is open in Pic(X), and which induces on Pic 0 (X) the above topology.
Proof of (9.2). Let 4: Br(X) ~ (Pic (X))*or be the map induced by (9.1). We fix a regular model X of X over (gk, that is, a two-dimensional regular proper flat scheme over Spec((gk) such that Y.| and Y:=(~| F)red is a geometrically connected proper onedimensional scheme over F whose irreducible components are all regular and which has only ordinary double points as singularities. The existence of such X follows from [1] and [20]. Starting with these notations, we define the same conventions as w1, and all facts stated there remain true. Thus, the injectivity of q~ is proved in the same argument as w It remains to prove that I m @ ) coincides with the subgroup D(Pic(X)). We first prove that we may assume that X(k) is non-empty. For a separable finite extension k'/k, put X ' = X | We have the following commutative diagram Br (X')
1o
- -
N ,
Br(X)
lo
( Pi c(X ))tor-----~(Pic(X))*o~, ,
,
R*
410
S. Saito
where N (resp. R*) is the norm map (resp. the dual of the natural map). From (9.4), we can see that R* induces a surjection D(Pic(X')) --~ D(Pic(X)). Hence, we may replace X by X' for the proof, and may assume that X ( k ) is non-empty. Now, our assertion is proved in the similar argument to w and w8. First, we consider the natural homomorphism : Pic ( X ) ~ T: = H ~(Y, i* R j , G,,), where i: Y ~ isomorphism
and j: X~3~ are the inclusion maps. We have a canonical
(9.5)
T~-Coker(@ K.* ~eYl
~
9 (~ ( @ K .*~ ) / R x), xeYo
~eg~-
where R x ( x e Y o ) is the affine ring of Spec(Ax)x IX. For each integer n>O, let U" T be the image in T of (~
(~ l+m~+lCK.x,
x~Yo t/x~ Y~
where m k is the maximal ideal of CkOn the other hand, we have the natural homomorphism 0: B r ( X ) ~ f2:--- H2(K i* R j . ~ 3 , . ) which is surjective as (8.2), and we have a homomorphism o~: f2 ~ F : = @ Br(K,)
whose image consists of all elements co =(co,)~rl satisfying that (9.6)
COx:=(09, ), ~r~e qxeY~C @ Br(K, ) x
x
x
is in the diagonal image of Br(Rx) for each x e Y o, where ~o,x is the restriction of c% to Br(K,~). By (0.9), (9.6) is equivalent to the condition (9.7)
the image of cox in ( ~
K,*)* annihilates the image of R*.
Now, we put =(|
- lq xe-Yo tlx~Yl
H (K.x) 9
qeYl x~rlo
By (8.1) and (9.5), both groups F and (T)*o, are viewed as subgroups of Z. Then, (9.7) and (8.1) imply that Im(~)=(T)*o~, and that Im(c 0 as a basis of neighbourhoods of O. Then, the composite homomorphism ~b^: Pic(X)-~
T-~ T A
is a homeomorphism. As is well-known, (9.8) follows from the following (9.9) Lemma. (1) 0 ~ is continuous. (2) 0 ^ is an isomorphism of abstract groups. Proof of (1). Fixing a e X ( k ) which was assumed not empty, we have a canonical map f: X ~ J defined over k. Let f": X'---,J be the sum o f / ' with itself r times, where X" is the r times product of X. If r is large enough, f f is smooth, so the induced m a p
~o: X'Ik)
=X(k)
x ... x
X(k)~J(k)
is continuous and has an open image. Moreover, by definition, ~0 sends (a 1. . . . . a,)~X'(k) to r
class ((a~) -(a))~d(k). r=l
Consequently, it suffices to prove the fact: For any n > 0 , if we take b e X ( k ) sufficiently near to a, the image of class ((b)-(a)) in T belongs to U"T. In fact, the same fact is proved in [9] p. 256. Proof of (2). For each integer n > 0 , put ~ , = X | be the image in Pic(X) of
(gk/m"+1 and let U" Pic(X)
Ker (Pic(30 --* Pic (X,,)). We have the following commutative diagram 0--
~Z-- '
, @Z--
~ , Pic ( t ) - - - - - - *
Pic(X)
--,0
q ~ YI
0
[
71
Z
Pic
+ Pic(X)/U" P i c ( X ) - -
, O.
tie YI
Here, ~ is defined as in the proof of (7.1) and t sends l e Z to (n,),~r,, where n, is the multiplicity of t/ in the divisor 3~| The exactness of the horizontal
412
S. S a i t o
sequences are easily seen except at @ Z in the sequences, where it follows tie Y1
from the intersection theory on @ 7Z (cf. [17]). From the diagram and the fact tl~ YI
Pic(X)~_limm Pic0i,)
(cf. EGA 1II (5.1.6)),
n
we can see Pic (X) _~ li,mm Pic (X)/U" Pic (X). n
Hence, we are reduced to prove (9.10)
Lemma. For each n >=0, O (U" Pic (X)) c U" T and we have Pic(X)/U" Pic(X)_~ T/U" T.
Proof. We consider the natural homomorphisms
(p: Pic(X)~ O : = Hi(Y, i* C~), (p,: Pic (X,) --+ 6),: = H* (17, i* (5)* ~,,, where i: Y-~X and i,: Y-*X, are the inclusion maps. The map q~ is an isomorphism by SGA4, VIII (1.1). On the other hand, using (1.1) and (1.2), we have isomorphisms O---Coker(@(9* ~ @ ( @ rleYt
6'*K.)/ A*x),
xeYo ttx~Y~
~
Pic(31,)~-O,~-O/U"O,
where U"O is the image in O of @
@ I +m;, +~
x~go rlxeY~
(QK'Tx"
By the explicit descriptions of T and O and by isomorphisms i ( ~ C* ]/A*~[ (~ K* ]/R* qxeY~
nx
rl~EY~
(xeYo) ,
fix
we have a natural homomorphism /~: O--+ T under which U " T is the image of U"O. Hence, (9.10) follows from the commutative diagram
@7l
, Pic(3~)
, Pic(X)--
,0
treY1
@Z r/~Y1
--*
O
-~
T
--*
0,
Arithmetic on two dimensional local rings
413
with the exact horizontal sequences. Here, ~' is defined as follows; for each r/eY1, we fix a prime element =, of K,. Then, ct' sends l e Z at ~/to the image of (q)xl(rr~))x~ro in O. The independence of ~' from the choices of ~z, (q~Y1) is easily seen. Lastly, we deduce (9.3) from (9.2). Lichtenbaum's argument in [11] implies that in case X(k) is not empty, (9.2) is equivalent to (9.3) for A = J . On the other hand, for any A as (9.3), we can find an exact sequence of fiat sheaves on Spec(k); C ~ B ~ A ~ O , where B and C are extensions of finite flat group schemes over k by products of Jacobians of projective smooth curves over k. Hence, a standard argument in homological algebra reduces (9.3) for A to (9.3) for Jacobians and Tate-Shatz duality for flat cohomologies of k (cf. [15] and [18]). This completes the proof of (9.3).
Acknowledgement. I would like to thank to Professor K. Kato who gave me the proof of (5.1) and other helpful suggestions, and encouraged me to make these researches.
References 1. Abhyankar, S.: Resolution of singularities for arithmetical surfaces. In: Arithmetical Algebraic Geometry. New York: Harper and Row, 1963, pp. 111 152 2. Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S. 36, 23-58 (1969) 3. Stein, M.R., Dennis, R.K.: K 2 of radical ideals and semilocal rings revisited. Lect. Notes Math. 342, 281 303 (1973) 4. Gabber, O.: A lecture at I.H.E.S. on March in 1981 5. Greenberg, M.: Rational points in henselian discrete valuation rings. Publ. Math. I.H.E.S. 23 (1964) 6. Grothendieck, A.: Le groupe de Brauer III. In: Dix expos~ sur la cohomologie des sch6mas. Amsterdam: North-Holland 1968 7. Illusie, L.: Complexe de De Rham-Witt et cohomologie cristalline. Ann. Sci. Ec. Norm. Sup. 12, 501-661 (1979) 8. Kato, K.: A generalization of local class field theory by using K-groups, 1. J. Fac. Sci., Univ. Tokyo, Sec. IA 26, 303 376 (1979); II, ibid. 27, 602 683 (1980); Ill, ibid. 29, 31-43 (1982) 9. Kato, K., Saito, S.: Unramified class field theory of arithmetical surfaces. Ann. Math. 118, 241275 (1983) 10. Kato, K., Saito, S.: Global class field theory of arithmetic schemes (Preprint) 11. Lichtenbaum, S.: Duality theorems for curves over l~-adic fields. Invent. Math. 7, 120-136 (1969) 12. Mercurjev, A.S., Suslin, A.A.: K-cohomology of Severi-Brauer varieties and norm residue homomorphism (Preprint) 13. Saito, S.: Class field theory for two dimensional local rings. (to appeal) 14. Saito, S.: Class field theory for curves over local fields. Journal Number Theory 21, 44-80 (1985) 15. Serre, J.-P.: Cohomologie galoisienne. Lect. Notes Math. 5, 1965 16. Serre, J.-P.: Corps locaux. Paris: Hermann 1962 17. Shafarevich, I.R.: Lectures on minimal models and birational transformations of two dimensional schemes. Tara Institute of Foundamental Research, Bombay, 1966 18. Shatz, S.S.: Cohomology of artinian group schemes over local fields. Ann. Math. 79, 411-449 (1964)
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19. Tate, J.: WC-groups over b!-adic fields. Sdminaire Bourbaki; 10e ann6e. 1957/1958 20. Hironaka, H.: Desingularization of excellent surfaces. Lectures at Advanced Science Seminar in Algebraic Geometry. Bowdoin College, Summer 1967, noted by Bruce Bennett 21. Bourbaki: Alg6bre commutative. Chapitre 7. Paris: Hermann 1965 22. Kurke, H., Pfister, G., Popescu, D., Roczen, M., Mostowski, T.: Die Approximationseigenschaft lokaler Ringe. Lect. Notes Math. 634, 1978 23. Brown, M.L.: A class of 2-dimensional local rings with Artin's approximation property. The Journal of the London Mathematical Society 27 Ser. 2, 29-34 (1983) 24. Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math. 30, 947-977 (1979)
Oblatum 6-XII-1984 & 18-VII-1985 & 16-XII-1985
