Nearly perfect complexes and Galois module structureT. Chinburg�; M. Kolster��; G. Pappasy; V. Snaith��January 3, 1998; revised June 27, 1998Abstract: We de�ne a generalization of the Euler characteristic of a perfect complex ofmodules for the group ring of a �nite group. This is combined with work of Lichtenbaumand Saito to de�ne an equivariant Euler characteristic for Gm on regular projectivesurfaces over Z having a free action of a �nite group. In positive characteristic we relatethe Euler characteristic of Gm to the leading terms of the expansions of L-functions ats = 1.Mathematics Subject Classi�cation (1991): 19F27, 11G45, 11G40, 14G15, 14J20.Key Words: Galois structure, Euler characteristics, surface, L-function.� Partially supported by NSF grant DMS-9701411.�� Partially supported by NSERC grantsy Partially supported by NSF grant DMS-9623269.1

1. IntroductionA perfect complex of modules for a ring R is a bounded complex of �nitely gen-erated projective R-modules. The Euler characteristic of a perfect complex lies in theGrothendieck groupK0(R) of all �nitely generated projective R-modules. Perfect com-plexes and their Euler characteristics have been a basic tool in homological algebra,topology, algebraic geometry, and more recently the theory of Galois module structure.Suppose G is a �nite group and that R = ZG is the integral group ring of G.The main object of this paper is to de�ne invariants in K0(ZG) associated to certaincomplexes of ZG-modules which are of obvious arithmetic interest, but which are notperfect. Such complexes arise naturally from the cohomology of class �eld theory. Inx2 we de�ne the concept of a `nearly perfect complex' of ZG-modules, and we show inTheorem 2.3 that each such complex has a generalized Euler characteristic in K0(ZG).The motivation for this paper is the work of Lichtenbaum [L1] and Saito [Sa] onthe cohomology of Gm on surfaces X which are proper over Spec(Z). In particular,Lichtenbaum discovered that for surfaces over a �nite �eld, one could use duality theo-rems to assign a numerical Euler characteristic to Gm, despite the fact that some of thecohomology groups of Gm may not be �nitely generated. The cohomology groups of aperfect complex of ZG-modules must be �nitely generated, so that the cohomology ofGm on X cannot in general be computed by a perfect complex. The duality theoremsused by Lichtenbaum identify the divisible subgroups of the cohomology groups of Gmon X with the Pontryagin duals of �nitely generated groups. Theorem 2.3 arose fromthe idea that in de�ning Euler characteristics, one can compensate for the existenceof divisible subgroups in the cohomology of a complex provided one has a �xed iso-morphism between these divisible groups and the Pontryagin duals of other �nitelygenerated groups.In x3 we describe an arithmetic application to surfaces X having a free action of a�nite group G. We assume that X is regular, geometrically connected and proper overSpec(Z). We will also assume that Brauer group of X is �nite, which is conjecturedto always be the case. Let K(X) be the function �eld of X. We show how theabove work of Lichtenbaum and Saito leads via Theorem 2.3 to an Euler characteristic�G(X;Gm) for Gm on X which lies in K0(ZG) if K(X) has characteristic p > 0, andin K0(Z[1=2][G]) if K(X) has characteristic 0. When K(X) has characteristic p, weshow how Lichtenbaum's work on the leading terms of zeta functions at s = 1 leads toa formula of the form f(�G(X;Gm)) = LX;1: (1:1)2

Here f : K0(ZG)! G0(ZG) is the forgetful homomorphism to the Grothendieck groupof all �nitely generated ZG-modules. The class LX;1 in G0(ZG) has order 1 or 2, andis de�ned by the signs of all the conjugates of the totally real algebraic numbers whichare the leading terms at s = 1 of the L-functions of symplectic representations of thegroup G of the cover X ! X=G. The class LX;1 actually pertains to the di�erencebetween an Euler characteristic of Ga and Gm (c.f. Proposition 3.10). One arrives at(1.1) from the fact that LX;1 = �LX;1 and by using work of Nakajima [N] to show thatthe class associated to Ga in G0(ZG) is trivial because G acts freely on X. In a laterpaper we will prove a generalization of (1.1) to the case in which G acts tamely on Xin which both multiplicative and additive Euler characteristics occur.The results in the paper were described in a talk at the Durham meeting onarithmetic algebraic geometry in July of 1996 (c.f. [E]). The authors would like tothank Steve Lichtenbaum for useful conversations related to this work.

3

2. Euler characteristics of nearly perfect complexes.Let G be a �nite group. If A is an abelian group, let Adiv be the subgroup of alldivisible elements of A, and let Acodiv = A=Adiv .De�nition 2.1. A nearly perfect complex of G-modules is a triple (C�; fLigi; f�igi)of the following kind.a. C� is a bounded complexC� : � � � ! Cm ! Cm+1 ! � � � ! Cn ! � � � (2:1)of cohomologically trivial ZG-modules. If some of the Ci are non-zero, we let m( resp. n ) be the smallest ( resp. largest ) integer for which Ci 6= 0. If all of theCi = 0, let n = �1 and m = 0. De�ne length(C�) = n�m+ 1.b. For each integer i, Li is a torsion-free �nitely generated ZG-module. Let Hi(C�)be the i-th cohomology group of C�. We require �i to be a G-isomorphism �i :HomZ(Li;Q=Z)! Hi(C�)div.c. For all i, the group Hi(C�)codiv is �nitely generated over Z.De�nition 2.2. Two nearly perfect complexes (C�; fLigi; f�igi) and (C 0�; fL0igi; f� 0igi)are quasi-isomorphic if the following is true:a. There is an isomorphism between C� and C 0� in the derived category of the ho-motopy category of ZG-modules.b. There is a ZG-module isomorphism L0i ! Li for each i.c. There is a commutative diagram of isomorphismsHomZ(Li;Q=Z) �i�! Hi(C�)div??y ??yHomZ(L0i;Q=Z) � 0i�! Hi(C 0�)divin which the left vertical isomorphism is induced by (b) and the right one by (a).Theorem 2.3. Suppose (C�; fLigi; f�igi) is a nearly perfect complex.a. One can de�ne an Euler characteristic �(C�; fLigi; f�igi) 2 K0(ZG) which de-pends only on the quasi-isomorphism class of (C�; fLigi; f�igi).b. The image of �(C�; fLigi; f�igi) in G0(ZG) is equal toXi (�1)i � ([Hi(C�)codiv]� [HomZ(Li;Z)]):4

c. Suppose that the cohomology groupsHi(C�) are �nitely generated as ZG-modules,so that the Li and �i are trivial. In this case the usual construction ( c.f. [H,Lemma III.12.3], [M, p. 263] ) produces a perfect complex P � of ZG-moduleswhich is quasi-isomorphic to C�, and �(C�; fLigi; f�igi) is the Euler characteristicin K0(ZG) of P �.Remark: Suppose R is a Dedekind ring, and that the fraction �eld F of R is a number�eld. One can then replace Z by R and Q=Z by F=R in De�nitions 2.1 and 2.2 andin Theorem 2.3. This leads to Euler characteristics in K0(RG) for nearly perfectcomplexes over RG.We now outline the construction of �(C�; fLigi; f�ig).>From (C�; fLigi; f�ig) we will construct a nearly perfect complex (D�; fL0igi; f� 0ig)with the following properties (c.f. Corollaries 2.10 and 2.8). If length(C�) = n�m+1 =0, then both C� and D� are the zero complex. If length(C�) = 1, then Cn is the onlynon-zero term of C�, but Cn may not be projective. In this case, Dn�1 will be theonly non-zero term of D�, and Dn�1 will be a �nitely generated projective ZG-module.Finally, if length(C�) > 1, then length(D�) < length(C�).If length(C�) � 1 we will de�ne �(C�; fLigi; f�ig) via the class of the projectivemodule Dn�1 in K0(ZG), as in De�nition 2.11. If length(C�) > 1, then length(D�) <length(C�), and we de�ne �(C�; fLigi; f�ig) in terms of �(D�; fL0igi; f� 0ig) in De�-nition 2.11. This leads to an inductive de�nition of �(C�; fLigi; f�ig), which onemust show is independent of the choices involved in constructing (D�; fL0igi; f� 0ig) from(C�; fLigi; f�ig).De�ne �i : Ci ! Ci+1 to be the i-th boundary map in C�, and let Zi = ker(�i)be the i-th cycle group. Thus Zn = Cn, since C� has no terms above degree n. Theexact sequence 0! Zn�1 ! Cn�1 ! Cn ! Hn(C�)! 0 (2:2)de�nes an extension class � = �n 2 Ext2ZG(Hn(C�); Zn�1). The main idea in theconstruction of (D�; fL0igi; f� 0ig) is to replace the �nal terms� � � ! Cn�2 ! Cn�1 ! Cnof C� by � � � ! Cn�2 ! Dn�1where Dn�1 is a cohomologically trivial ZG-module constructed using the extensionclass � and the given isomorphism �n : HomZ(Ln;Q=Z)! Hn(C�)div. The �rst step5

in this is to use � to construct a class in Ext1ZG(Mn; Zn�1) for a suitable module Mnwe now de�ne.We have an exact diagram of the following kind:0 0 0??y ??y ??y0 �! HomZ(Ln;Z) �! Mn �! Nn �! 0??y ??y ??y0 �! HomZ(Ln;Q) �! HomZ(Ln;Q)L Fn �! Fn �! 0??y ??y ??y0 �! HomZ(Ln;Q=Z) �! Hn(C�) �! Hn(C�)codiv �! 0??y ??y ??y0 0 0 (2:3)To explain this diagram, note �rst that Ext1Z(Ln;Z) = 0 since in De�nition 2.1(b)Ln is a free �nitely generated Z-module. Hence the left column is exact. In the bot-tom row we have identi�ed HomZ(Ln;Q=Z) with Hn(C�)div using the isomorphism �nof De�nition 2.1(b). Because Hn(C�)codiv is �nitely generated, we can �nd a �nitelygenerated projective module Fn along with a homomorphism from Fn to Hn(C�)as in the middle column of diagram (2.3) which induces a surjection from Fn toHn(C�)codiv. We will further require that Fn = f0g if Hn(C�)codiv = 0. The ho-momorphism HomZ(Ln;Q) ! Hn(C�) in the middle column of diagram (2.3) is theone compatible with the left column and the bottom row. The modules Mn and Nnare de�ned to be the kernels of the vertical homomorphisms, and (2.3) is exact by thesnake Lemma.Applying the functor HomZG(�; Zn�1) to the middle column of (2.3) gives a longexact sequence� � � �!Ext1ZG(Hom(Ln;Q)MFn; Zn�1)�!Ext1ZG(Mn; Zn�1)�1�!Ext2ZG(Hn(C�); Zn�1)�!Ext2ZG(Hom(Ln;Q)MFn; Zn�1)! � � � : (2:4)Lemma 2.4. The module Mn in (2.3) is a �nitely generated torsion-free ZG-module.The boundary map Ext1ZG(Mn; Zn�1) �1�!Ext2ZG(Hn(C�); Zn�1) (2:5)6

in (2.4) is an isomorphism.Proof: Since Nn and HomZ(Ln;Z) are �nitely generated over Z, diagram (2.3) showsMn is as well. Because HomZ(Ln;Q)L Fn is torsion-free, so is Mn.To analyze (2.4) we use the spectral sequenceHp(G;ExtqZ(D1;D2)) => Extp+qZG (D1;D2) (2:6)for G-modules D1 and D2. The projective dimension of Z is 1 ( c.f. [K, p. 171 and191] ). Therefore ExtqZ(D1;D2) = 0 if q � 2: (2:7)If D1 is �nitely generated and torsion free, then Ext1Z(D1;D2) = 0. If D1 is a �nitelygenerated projective ZG-module, then HomZ(D1;D2) is a summand of an induced G-module, so HomZ(D1;D2) is G-cohomologically trivial. Any Q-vector space is alsoG-cohomologically trivial.One �nds from (2.6), (2.7) and the above remarks that if q > 1 thenExtqZG(Hom(Ln;Q)MFn; Zn�1) =ExtqZG(Hom(Ln;Q); Zn�1)MExtqZG(Fn; Zn�1) = 0 (2:8)and Ext1ZG(Mn; Zn�1) = H1(G;HomZ(Mn; Zn�1)): (2:9)The homomorphismExt1ZG(Hom(Ln;Q)M Fn; Zn�1)! Ext1ZG(Mn; Zn�1) (2:10)in the long exact sequence (2.4) is trivial, since its domain is a Q-vector space and, by(2.9), its range is a group annihilated by #G. Thus (2.4), (2.9) and (2.10) establishthe isomorphism stated in Lemma 2.4.Corollary 2.5. There is a unique class � 2 Ext1ZG(Mn; Zn�1) which maps to theextension class � 2 Ext2ZG(Hn(C�); Zn�1) of the sequence (2.2) under the boundaryisomorphism �1 of Lemma 2.4.De�nition 2.6. Let 0! Zn�1 ! Dn�1 !Mn ! 0 (2:11)7

be an exact sequence representing the extension class � of Corollary 2.5. Let Di = Ciif i < n� 1, and let Di = 0 for i > n� 1. LetD� : � � � �!Dm�!Dm+1�!� � ��!Dn�1�!0�!� � �be the complex such that the boundary �i : Di ! Di+1 is as follows. If i < n � 2then �i is the boundary map �i : Ci ! Ci+1 of C�. If i � n � 1, then �i is thezero homomorphism. Finally, if i = n � 2 then �n�2 : Dn�2 = Cn�2 ! Dn�1 is thecomposition of the boundary map �n�2 : Cn�2 ! Zn�1 with the inclusion Zn�1 !Dn�1 in sequence (2.11).Proposition 2.7. The module Dn�1 is cohomologically trivial for G.Proof. By Corollary 2.5, the cup product of � with the extension class of the middlecolumn 0!Mn ! HomZ(Ln;Q)M Fn ! Hn(C�)! 0 (2:12)of (2.3) is �1 times the extension class � of sequence (2.2). Therefore splicing together(2.11) and (2.12) gives an exact sequence0! Zn�1 ! Dn�1 ! HomZ(Ln;Q)MFn ! Hn(C�)! 0 (2:13)which has extension class ��. Since Cn�1 and Cn in (2.2) were assumed to be cohomo-logically trivial for G, cup product with �� induces isomorphisms in Tate cohomologyHj(�;Hn(C�))! Hj+2(�; Zn�1) (2:14)for all integers j and all subgroups � � G. Since HomZ(Ln;Q)L Fn in sequence (2.13)is cohomologically trivial for G, this implies Dn�1 in (2.13) must also be cohomologi-cally trivial for G.Corollary 2.8. Suppose n =m in sequence (2.1), so that Ci = 0 if i 6= n and Cn 6= 0.Then Di = 0 unless i = n � 1. The module Dn�1 = Mn is a �nitely generatedprojective ZG-module, andrankZG(Dn�1) = rankZG(Fn) + 1#G � (rankZ(Ln) � rankZ(Hn(C�)codiv)) (2:15)where Cn = Hn(C�).Proof: If m = n then Zn�1 = 0. Hence sequence (2.11) shows Dn�1 = Mn. ByLemma 2.4, Mn is a �nitely generated torsion-free ZG-module, and Dn�1 is cohomo-logically trivial by Proposition 2.7. Hence Dn�1 = Mn must be a �nitely generatedprojective ZG-module. The equality (2.15) follows from Dn�1 =Mn, the top row andright column of (2.3), rankZ(HomZ(Ln;Z)) = rankZ(Ln), and the fact that �nitelygenerated projective ZG-modules are locally free.8

Lemma 2.9. If i < n � 1 then Hi(D�) = Hi(C�). If i > n � 1 then Hi(D�) = 0.Finally, there is an exact sequence0! Hn�1(C�)! Hn�1(D�)!Mn ! 0 (2:16)Proof: In view of (2.11), the exact sequence (2.16) is0! Zn�1=Bn�1 ! Dn�1=Bn�1 ! Dn�1=Zn�1 ! 0 (2:17)where Bn�1 � Zn�1 is the group of n� 1 boundaries of both C� and D�. The rest ofthe Lemma is clear.Corollary 2.10. Suppose (C�; fLigi; f�igi) is a nearly perfect complex, as in Def-inition 2.1. Let L0i = Li if i � n � 1, and let L0i = 0 if i > n � 1. De�ne� 0i = �i : HomZ(Li;Q=Z) ! Hi(C�)div = Hi(D�)div if i � n � 1, and let � 0i = 0if i > n� 1. Then (D�; fL0igi; f� 0ig) is a nearly perfect complex. If n > m, thenD� : � � � ! 0! Dm ! � � � ! Dn�1 ! 0! � � � (2:18)has length(D�) � n�m < length(C�) = n�m+ 1.Proof: Since Mn is �nitely generated by Lemma 2.4, sequence (2.16) gives an isomor-phism Hn�1(C�)div ! Hn�1(D�)div:The Corollary is now clear from Lemma 2.9.De�nition 2.11. If length(C�) = n �m + 1 = 0, then C� and D� are both the zerocomplex, and we let �(C�; fLigi; f�igi) = 0: (2:19)Suppose now that length(C�) = n �m + 1 = 1. By Corollary 2.8, Dn�1 has a class[Dn�1] in K0(ZG), so we can let�(C�; fLigi; f�igi) = (�1)n�1 � [Dn�1] + (�1)n � [Fn]: (2:20)We now assume by induction that n0 � 1 is an integer such that �(C�; fLigi; f�igi)has been de�ned whenever length(C�) = n�m+ 1 � n0. If length(C�) = n0 + 1 thenusing Corollary 2.10 we can de�ne�(C�; fLigi; f�igi) = �(D�; fL0igi; f� 0igi) + (�1)n � [Fn]: (2:21)9

Proposition 2.12. The nearly perfect complex (D�; fL0igi; f� 0igi) depends on(C�; fLigi; f�igi), the projective module Fn and the homomorphism � : Fn ! Hn(C�)induced by the middle column of diagram (2.3). The class �(D�; fL0igi; f� 0igi)+(�1)n �[Fn] does not depend on the choice of Fn or � : Fn ! Hn(C�). Therefore in de�nition2.11, �(C�; fLigi; f�igi) depends only on the nearly perfect complex (C�; fLigi; f�igi)and not on further choices.To begin the proof of this Proposition, we observe that the choice of Fn and � :Fn ! Hn(C�) determines the diagram (2.3). This �xes the long exact sequence (2.4),so from Corollary 2.5 and De�nition 2.6 we see that (D�; fL0igi; f� 0igi) is also determinedby these choices together with the original nearly perfect complex (C�; fLigi; f�igi).If C� has length 0, then �(C�; fLigi; f�igi) = 0 so Proposition 2.12 holds. We nowassume C� has positive length, and that by induction, Proposition 2.12 is true for allnearly perfect complexes of length less than that of C�.In the course of the proof, we will show by induction the followingHypothesis 2.13. Suppose (C�1 ; fL1;igi; f�1;igi) and (C�2 ; fL2;igi; f�2;igi) are nearlyperfect complexes having the following properties.a. The length of each of these complexes is less than that of C�.b. There is a morphism of complexes C�1 ! C�2 which is a term by term isomorphism.c. There are isomorphisms L1;i ! L2;i which are compatible with the cohomologyisomorphismsHi(C�1 )! Hi(C�2 ) induced by the morphism in (b), the �1;i and the�2;i.Then �(C�2 ; fL2;igi; f�2;igi) = �(C�1 ; fL1;igi; f�1;igi) (2:22)where the two sides of this equality are well de�ned because by assumption, Proposition2.12 holds for complexes of length less than that of C�.This induction hypothesis is clearly true if C� has length 1, since then C�1 and C�2have length 0.Lemma 2.14. Let ~� : Fn ! Hn(C�)codiv be the homomorphism induced by � : Fn !Hn(C�), so that ~� appears in the right column of diagram (2.3). Suppose �0 : Fn !Hn(C�) is another homomorphism such that ~� = ~�0. Then assuming Hypothesis 2.13,� and �0 lead to the same value for �(C�; fLigi; f�igi). Therefore �(C�; fLigi; f�igi)depends only on the (surjective) homomorphism ~� : Fn ! Hn(C�)codiv.Proof: From diagram (2.3), we see that there is a homomorphism � : Fn !HomZ(Ln;Q=Z) such that � = �0 + i � �, where i : HomZ(Ln;Q=Z) ! Hn(C�) is10

the inclusion in the bottom row of (2.3). Since Fn is projective and the homomor-phism � : HomZ(Ln;Q)! HomZ(Ln;Q=Z) in the left column of (2.3) is surjective, wecan lift � to a homomorphism �0 : Fn ! HomZ(Ln;Q) such that� = �0 + i � � � �0: (2:23)Let T be the automorphism of HomZ(Ln;Q)L Fn de�ned byT (aM b) = (a+ �0(b))M b: (2:24)De�ne (2:3)0 to be the diagram (2.3) when �0 is used instead of �. Then we see that thereis a unique isomorphism ~T from diagram (2.3) to diagram (2:3)0 which �xes the left andright columns and which is the automorphism T on the module HomZ(Ln;Q)LFnin the middle of the diagram. This ~T induces an automorphism of Hn(C�) and anisomorphismMn !M 0n fromMn to the corresponding moduleM 0n in diagram (2:3)0.Following through the construction in De�nition 2.6, we see that there is a commutativediagram 0 �! Zn�1 �! Dn�1 �! Mn �! 0??y ??y ??y0 �! Zn�1 �! D0n�1 �! M 0n �! 0 (2:25)in which the left vertical homomorphism is the identity map, the right vertical homo-morphism is the above isomorphismMn !M 0n, and the middle homomorphism is anisomorphism.Let (D�; fL1;igi; f�1;igi) and (D0�; fL2;igi; f�2;igi) be the nearly perfect complexstructures resulting from De�nition 2.6 when one uses � and �0, respectively. Via (2.25)we get a morphism of complexes D� ! D0� which is the identity on all terms exceptDn�1, and which is the middle vertical isomorphism of (2.25) on Dn�1. This morphisminduces a commutative diagram0 �! Hn�1(C�) �! Hn�1(D�) �! Mn �! 0??y ??y ??y0 �! Hn�1(C�) �! Hn�1(D0�) �! M 0n �! 0 (2:26)in which the left and right isomorphisms are those induced by ~T . By construction ~Tinduces the identity isomorphism of Hn�1(C�)div = Hn�1(D�)div = Hn�1(D0�)div.11

Hence the identity isomorphism L1;i ! L2;i for each i is compatible with D� ! D0�,�1;i and �2;i. It follows now from our induction hypothesis (2.22) that�(D�; fL2;igi; f�2;igi) + (�1)n � [Fn] = �(D0�; fL1;igi; f�1;igi) + (�1)n � [Fn]: (2:27)Therefore we get the same value for �(C�; fLigi; f�igi) whether we use � or �0. Thisproves Lemma 2.14.Proof of Proposition 2.12 and Hypothesis 2.13Let Fn1 be another choice of a projective ZG-module and a surjective homomor-phism �1 : Fn1 ! Hn(C�). Since both of the induced homomorphisms ~� : Fn !Hn(C�)codiv and ~�1 : Fn1 ! Hn(C�)codiv are surjective, we can form a pull backsquare FnLFn1 �! Fn1??y ??y~�1Fn �!~� Hn(C�)codiv (2:28)It follows from Lemma 2.14 that to show we get the same value for �(C�; fLigi; f�igi)whether we use ~� or ~�1, we can reduce to the case in which ~�1 is the composition of asurjection h : Fn1 ! Fn and ~�. Since the value of �(C�; fLigi; f�igi) does not dependon how ~� is lifted to �, we can furthermore assume that �1 : Fn1 ! Hn(C�) is thecomposition of h and � : Fn ! Hn(C�).De�ne (2:3)1 to be the diagram (2.3) which results from using �1 instead of �, andlet Mn1 and Nn1 be the modules appearing in (2:3)1 at the positions corresponding toMn and Nn. We have a commutative exact diagram0 �! 0 �! 0??y ??y ??y0 �! F2 �! Nn1 �! Nn �! 0??y ??y ??y0 �! F2 �! Fn1 �! Fn �! 0??y ??y~�1 ??y~�0 �! 0 �! Hn(C�)codiv �! Hn(C�)codiv �! 0??y ??y ??y0 �! 0 �! 0 (2:29)12

in which the right columns are the right columns of diagrams (2:3)1 and (2.3). Themiddle row of (2.29) splits because Fn is projective. Hence the top row of (2.29) alsosplits, and F2 is projective because Fn and Fn1 are. We thus have compatible isomor-phisms Nn1 = NnLF2 and Mn1 =MnLF2. Using the fact that F2 is projective, wesee that the construction of De�nition 2.6 leads to an isomorphismDn�11 = Dn�1LF2,where D�1 is the complex which results from using �1 instead of �. The homomorphismDn�21 = Dn�2 = Cn�2 ! Dn�11 is the composition of Dn�2 ! Dn�1 and the inclusionDn�1 ! Dn�11 = Dn�1LF2 which is the identity map onto the �rst summand ofDn�1. If F2 = 0, then ~� = ~�1 and there is nothing to prove because of Lemma 2.14.So we assume in what follows that F2 is non-trivial.We see from this description that Hn�1(D�1) = Hn�1(D�)LF2, and that thenearly perfect complex structure (D�1 ; fL0igi; f� 0igi) of D�1 is the one which results from(D�; fL0igi; f� 0igi) by simply adding the projective module F2 to Dn�1.In view of De�nition 2.11 and (2.29), the expressions for �(C�; fLigi; f�igi) whichresult from using ~�1 and ~�, respectively, are�(D�1 ; fL0igi; f� 0igi) + (�1)n � ([Fn] + [F2]) (2:30)and �(D�; fL0igi; f� 0igi) + (�1)n � ([Fn]): (2:31)We will now show (2.30) and (2.31) are equal.Choose a free module F3 and a G-morphism �3 : F3 ! Hn�1(D�) which inducesa surjection F3 ! Hn�1(D�)codiv. We further require F3 = f0g if Hn�1(D�)codiv = 0.We compute (D�1 ; fL0igi; f� 0igi) by choosing the projective module F3LF2 togetherwith the homomorphism�4 : F3MF2�!Hn�1(D�1) = Hn�1(D�)MF2which is the identity map on the second summand and �3 on the �rst. >From thediagram of the form (2.3) which results and from De�nitions 2.6 and 2.11, we see thatthere is a single nearly perfect complex (D�2 ; fL00i gi; f� 00i gi) of length less than length(C�)such that �(D�; fL0igi; f� 0igi) = �(D�2 ; fL00i gi; f� 00i gi) + (�1)n�1 � [F 03] (2:32)and �(D�1 ; fL0igi; f� 0igi) = �(D�2 ; fL00i gi; f� 00i gi) + (�1)n�1 � [F 03MF2]: (2:33)13

where F 03 = Dn�1 is projective if length(C�) = 1, and F 03 = F3 if length(C�) > 1. Herewe may use Hypothesis 2.13 to assert that �(D�2 ; fL00i gi; f� 00i gi) has the same value onthe right sides of (2.32) and (2.33). It follows easily from (2.32) and (2.33) that (2.30)and (2.31) are equal.The proof of Proposition 2.12 will now be complete once we show that our in-duction hypothesis (2.22) holds for nearly perfect complexes (C�1 ; fL1;igi; f�1;igi) and(C�2 ; fL2;igi; f�2;igi) of the same length as C�. This may be proved by using the induc-tive de�nition of �(C�j ; fLj;igi; f�j;igi) to reduce to the case of complexes of length lessthan that of C�; we will leave the details to the reader.The proof of part(a) of Theorem 2.3 is now completed byProposition 2.15. Suppose (C�; fLigi; f�igi) and (C 0�; fL0igi; f� 0igi) are quasi-isomorphic in the sense of De�nition 2.2. Then �(C�; fLigi; f�igi) =�(C 0�; fL0igi; f� 0igi).Proof: By [HRD, xI.3], we can reduce to the case in which there is a morphism ofcomplexes C� ! C 0� which gives rise to the quasi-isomorphism in the derived categorywhich is referred to in De�nition 2.2(a).A shift by 1 to either the left or the right of the terms of (C�; fLigi; f�igi) multiplies�(C�; fLigi; f�igi) by �1. Thus we can reduce to the case in which at least one of C�or C 0� is not the zero complex, Ci = C 0i = 0 if i < 0, and C 00 6= 0 or C0 6= 0. Letn0 � 0 ( resp. n � 0 ) be the largest non-negative integer for which C 0n 6= 0 ( resp.Cn 6= 0 ). We wish to reduce to the case n0 = n, by replacing C� (resp. C 0� ) by aquasi-isomorphic longer complex if n0 > n (resp. if n > n0). We will treat only thecase n0 > n, since the case n > n0 is similar. If n0 > n we can add a non-zero �nitelygenerated free module F to both Cn and Cn+1 = 0 and use the identity map F ! F toconstruct a perfect complex (C�2 ; fLigi; f�igi) from (C�; fLigi; f�igi) which has lengthone greater than (C�; fLigi; f�igi). Since Hn+1(C�2 ) = 0, we can use the trivial freemodule surjecting onto Hn+1(C�2 ) to compute �(C�2 ; fLigi; f�igi) via the counterpartof diagram (2.3). This leads to �(C�2 ; fLigi; f�igi) = �(C�; fLigi; f�igi). In this waywe can increase the length of C� to be able to assume n0 = n.We will prove Proposition 2.15 by induction on n = n0. Suppose n = n0 = 0. ThenC 00 and C0 are the only non-zero terms of C 0� and C�, respectively. The morphismC� ! C 0� must induce an isomorphism C0 = H0(C�) ! H0(C 0�) = C 00. Hence thecase n = n0 = 0 is treated by Hypothesis 2.13, which was proved in the course of theproof of Proposition 2.12.We now suppose n = n0 > 0 and that Proposition 2.15 is true for all pairs of com-14

plexes which have trivial terms in negative dimension and highest terms in dimensionless than n. As in the de�nition of �(C�; fLigi; f�igi), we choose a free module Fntogether with a morphism Fn ! Hn(C�) inducing a surjection Fn ! Hn(C�)codiv.We now use the isomorphisms Hn(C�)! Hn(C 0�) and Ln ! L0n in De�nition 2.2 totake the diagram (2.3) used to de�ne �(C�; fLigi; f�igi) to the diagram (2:3)0 used tode�ne �(C 0�; fL0igi; f� 0igi). In particular, we have an induced isomorphismMn !M 0n.Via Lemma 2.4, Corollary 2.5 and De�nition 2.6, this leads to a diagram0 �! Zn�1 �! Dn�1 �! Mn �! 0??y ??y ??y0 �! Z 0n�1 �! D0n�1 �! M 0n �! 0 (2:34)in which the right vertical arrow is an isomorphism. The morphism of complexesC� ! C 0� together with (2.34) now gives a morphism of complexes D� ! D0� whichinduces cohomology isomorphisms Hi(D�) ! Hi(D0�) if i 6= n � 1. When i = n � 1,one has a commutative diagram0 �! Hn�1(C�) �! Hn�1(D�) �! Mn �! 0??y ??y ??y0 �! Hn�1(C 0�) �! Hn�1(D0�) �! M 0n �! 0 (2:35)in which the left and right vertical homomorphisms are isomorphisms, so the middlevertical homomorphism is as well. This proves D� ! D0� is a quasi-isomorphism. Thediagram (2.35) also shows that the nearly perfect complex structures on D� and D0�which are constructed in Corollary 2.10 are quasi-isomorphic in the sense of De�nition2.2. Since D� and D0� have highest non-zero terms in degree less than n = n0, theproof of Proposition 2.15 now follows by induction.Completion of the proof of Theorem 2.3As noted before, Proposition 2.15 shows part (a) of Theorem 2.3. We now showparts (b) and (c) of Theorem 2.3 by induction on length(C�) = n�m+ 1.If length(C�) < 0 then C� is the zero complex and �(C�; fLigi; f�igi) = 0 by(2.19), so we are done.>From diagram (2.3) we have[Mn] = [HomZ(Ln;Z)] + [Fn]� [Hn(C�)codiv] in G0(ZG) (2:36)15

Suppose length(C�) = 1. By Corollary 2.8, Ci = 0 if i 6= n, andHn(C�) = Cn 6= 0.Furthermore, Dn�1 =Mn is a projective Z[G]-module, and�(C�; fLigi; f�igi) = (�1)n�1 � [Mn] + (�1)n � [Fn] (2:37)by (2.20). Combining (2.36) and (2.37) shows �(C�; fLigi; f�igi) has image(�1)n�1 � [Mn] + (�1)n � [Fn] = (�1)n � ([Hn(C�)codiv]� [HomZ(Ln;Z)])in G0(ZG) as asserted in Theorem 2.3(b). Suppose Hn(C�) = Cn is �nitely generatedas a ZG-module. Then Ln = 0, and diagram (2.3) shows [Fn] � [Mn] = [Hn(C�)] =[Cn] in K0(ZG) when we identify K0(ZG) with the Grothendieck group of all �nitelygenerated cohomologically trivial ZG-modules (c.f. [C, Prop. 4.1(b)]). Thus (2.37)gives �(C�; fLigi; f�igi) = (�1)n � [Cn] in this case, which by Schanuel's Lemma isequivalent to the assertion of Theorem 2.3(c).Finally, suppose length(C�) > 1, and let (D�; fL0igi; f� 0igi) be a nearly perfectcomplex of the kind constructed in Corollary 2.10. By De�nition 2.11,�(C�; fLigi; f�igi) = �(D�; fL0igi; f� 0igi) + (�1)n � [Fn]: (2:38)By induction, the image of �(D�; fL0igi; f� 0igi) in G0(ZG) isXi (�1)i � ([Hi(D�)codiv]� [HomZ(L0i;Z)]):In view of the construction of (D�; fL0igi; f� 0igi), this equals(�1)n�1 � ([Hn�1(D�)codiv]� [HomZ(Ln�1;Z)])+ Xi<n�1(�1)i � ([Hi(C�)codiv]� [HomZ(Li;Z)]) (2:39)By the exact sequence (2.16), we have[Hn�1(D�)codiv] = [Hn�1(C�)codiv]� [Mn] (2:40)since Mn is �nitely generated. Hence on combining (2.38), (2.39) and (2.40), we seethat �(C�; fLigi; f�igi) has image in G0(ZG) equal toXi (�1)i � ([Hi(C�)codiv]� [HomZ(Li;Z)])16

as asserted in Theorem 2.3(b).Suppose now that all the Hi(C�) are �nitely generated. Then the groups Hi(D�)are as well by Lemmas 2.4 and 2.9, and Ln = 0. By composing the inclusion mapZn�1 ! Dn�1 with multiplication by �1, if necessary, we conclude from (2.13) thatthere is an exact sequence0! Zn�1 ! Dn�1 ! Fn ! Hn(C�)! 0 (2:41)which has the same extension class in Ext2ZG(Hn(C�); Zn�1) as the exact sequence0! Zn�1 ! Cn�1 ! Cn ! Hn(C�)! 0:Let D0� be the complex obtained from D� by putting Fn in the n-th position and usingthe morphism Dn�1 ! Fn of (2.41). It follows that D0� is isomorphic in the derivedcategory to C�.Using [H, Lemma III.12.3] and [M, p. 263], we now construct perfect complexes Q�and P � together with quasi-isomorphismsQ� ! D� and P � ! C�. We can furthermoreassume that Qi = 0 if i > n� 1, since Di = 0 for such i. De�ne Q0� to be the complexobtained fromQ� by adjoining Fn in degree n, and by letting the morphismQn�1 ! Fnbe the one induced by the isomorphism Qn�1=Zn�1(Q�) = Hn�1(Q�) ! Hn�1(D�)followed by the map Hn�1(D�) = Dn�1=Zn�1 ! Fn induced by the boundary mapDn�1 ! Fn of D0�. Then Q0� is isomorphic to D0� in the derived category, and hencealso to C� and to P �. It follows thatXi (�1)i[Q0i] =Xi (�1)i[P i] (2:42)in K0(ZG). However, since D� has smaller length than C�, we know by induction onthe statement of Theorem 2.3(c) that�(D�; fL0igi; f� 0igi) =Xi (�1)i[Qi] =Xi (�1)i[Q0i]� (�1)n[Fn]: (2:43)Now combining (2.38) with (2.43) and (2.42) shows�(C�; fLigi; f�igi) =Xi (�1)i[P i]and this is the assertion of Theorem 2.3(c).17

3. Arithmetic applications.In this section we will suppose that X is a regular two-dimensional scheme which isproper over Spec(Z), geometrically connected, and for which the Brauer group Br(X)is �nite. Let G be a �nite group acting freely on X, which by de�nition means thatthe inertia group in G of each point of X is trivial. This implies that the quotient mapX ! X=G is �etale. Let K(X) be the function �eld of X. There are two cases; thegeometric one, when char(K(X)) > 0 and the arithmetic one, when char(K(X)) = 0.Let G0m = Gm (resp. G0m = Gm Z Z[1=2]) in the geometric (resp. arithmetic)case. The cohomology groups of G0m have been computed by Lichtenbaum [L1] in thegeometric case, and by Saito [Sa] in the arithmetic case. In x3.A and x3.B we will showhow their results lead naturally to a nearly perfect complex structure associated to thehypercohomology of G0m. One may thus use the Euler characteristic construction ofthe previous section to de�ne an Euler characteristic �(X;G0m) for G0m on X which liesin K0(ZG) (resp. K0(Z[1=2][G])) in the geometric (resp. arithmetic) case.Let f : K0(ZG) ! G0(ZG) be the forgetful map. In x3.C we will use work ofLichtenbaum to prove:Theorem 3.1. Suppose K(X) has positive characteristic. Thenf(�(X;Gm)) = LX;1where LX;1 is de�ned in Proposition 3.11. The class LX;1 has order 1 or 2, and isdetermined by the signs of all the conjugates of the totally real algebraic numberswhich are the leading terms at s = 1 of the Artin L-series associated to symplecticrepresentations of the Galois group of the cover X ! X=G.As mentioned in x1, Theorem 3.1 is deduced from a result (Proposition 3.11)which involves Euler characteristics of both Gm and Ga. In a later paper we will provegeneralizations of these results to tame actions of G on X. In the arithmetic case, onecould presumably develop a counterpart to Theorem 3.1 using a form of the conjectureof Birch and Swinnerton-Dyer.3.A. The cohomological triviality of hypercohomologyProposition 3.2. Let U be an integral scheme, and let G be a �nite group acting freelyon U . The quotient map U ! V = V=G is �etale. Suppose R is a ring and that F � is acomplex of sheaves of R-modules for the �etale topology of V which is bounded below.Suppose that for each subgroup H of G, only �nitely many of the cohomology groups18

Hi(U=H;F �) are non-trivial. Then the hypercohomology H�(U;F �) is isomorphic inthe derived category of the homotopy category of R-modules to a bounded complex C�of RG-modules which are cohomologically trivial for G.Proof: Let I� : I0 ! I1 ! I2 ! I3 ! : : : (3:1)be a complex of injective sheaves of R-modules for the �etale topology on V for whichthere is a quasi-isomorphism F � ! I�. Then�(U; I�) : �(U; I0)! �(U; I1)! �(U; I2)! : : : (3:2)is a complex of RG-modules which is isomorphic to H�(U;F �) in the derived categoryof the homotopy category of RG-modules.LetK be the function �eld of V , and let j : Spec(K)! V be the natural morphism.Fix a separable closure Ks of K containing the function �eld N of U . Because G actsfreely on U , N=K is a �nite Galois extension, and we have a natural identi�cation ofG with Gal(N=K). For each RG-module M , let Ms be the in ation of M from G toGal(Ks=K). ViewingMs as a sheaf on the �etale topology of Spec(K), we have an �etalesheaf j�(Ms) on V . The functor which sends M to j�(Ms) is an exact fully faithfulfunctor from the category of RG-modules to that of �etale sheaves of R-modules on V .Furthermore, if J is any �etale sheaf of R-modules on V , �(U; J) is an RG-module, andHomR(j�(Ms); J) = HomRG(M;�(U; J)). It follows that if J is injective, then �(U; J)is an injective RG-module. In particular, the terms of (3.2) are injective RG-modules.Let n be a positive integer such that Hi(U=H;F �) = 0 if i > n andH is a subgroupof G. De�ne Mn = ker(�(U; In)! �(U; In+1)). Then�(U; I0)! � � ��(U; In�1)!Mn (3:3)is isomorphic to H�(U;F �) in the derived category, and0!Mn ! �(U; In)! �(U; In+1)! : : : (3:4)is exact. It will su�ce to show that Mn is a cohomologically trivial G-module.An injective G-module M is cohomologically trivial, since we can split the inclu-sion of such M into the induced G-module M Z ZG, and induced G-modules arecohomologically trivial. Hence (3.4) gives a resolution of Mn by cohomologically triv-ial G-modules. Therefore for each subgroup H of G, the cohomology groups of thecomplex �(U; In)H ! �(U; In+1)H ! : : : (3:5)19

beginning in degree 0 give the H-cohomology groups of Mn. However, since U ! Vis an �etale G-cover, �(U; Ij )H = �(U=H; Ij ). Hence if i � n, the (i � n)th cohomologygroup of (3.5) equals Hi(U=H;F �), and this is 0 if i > n by assumption. Thus (3.5)has trivial cohomology in dimensions greater than 0, and we conclude that Mn is acohomologically trivial G-module.3.B. The cohomology of Gm on surfacesThe following result is shown in [L1,x3.4 and x4] in the geometric case and in [Sa]in the arithmetic case.Theorem 3.3. (Lichtenbaum - Saito) Let X be a regular two-dimensional schemewhich is proper over Spec(Z), geometrically connected, and for which the Brauer groupBr(X) is �nite. De�ne R = Z in the geometric case, and let R = Z[1=2] in thearithmetic case. Let G0m = R Z Gm. One has Hi(X;G0m) = 0 if i > 4. The groupH0(X;G0m) if �nite (resp. �nitely generated overR) in the geometric (resp. arithmetic)case by the Dirichlet unit Theorem. The group H2(X;G0m) is �nite by assumption.The group H1(X;G0m) is �nitely generated over R by the Mordell-Weil Theorem. Fori = 0; 1, there is a natural pairing Hi(X;G0m) �H4�i(X;G0m) ! Q=R which inducesan isomorphism between H4�i(X;G0m) and HomR(Hi(X;G0m);Q=R).We now apply Proposition 3.2 to X = U and F � the complex having G0m indimension 0 and the zero sheaf in other dimensions. This showsH�(U;F �) is isomorphicto a complex of the kind in De�nition 2.1(a). Thus Theorems 3.3 and 2.3 show:Corollary 3.4. Let Li = H4�i(X;G0m)=H4�i(X;G0m)tor for i 2 f3; 4g. Let �i :HomR(Li;Q=R)! Hi(X;G0m)div be the isomorphism induced by Theorem 3.3. De�neLi = 0 and �i = 0 if i =2 f3; 4g. Then (C�; fLigi; f�igi) is a nearly perfect complex ofR[G]-modules. By Theorem 2.3 of x2, and the Remark following it, we have an Eulercharacteristic �(C�; fLigi; f�igi) 2 K0(RG); call this invariant �G(X;G0m). The imageof �G(X;G0m) in G0(RG) is the class[H0(X;G0m)] � [H1(X;G0m)] + [H2(X;G0m)]+ [HomR(H1(X;G0m); R)] � [HomR(H1(X;G0m)tor;Q=R)]� [HomR(H0(X;G0m); R)] + [HomR(H0(X;G0m)tor;Q=R)]20

3.C. Leading terms of L-functions at s = 1.In this section we will suppose that X is a smooth projective geometrically con-nected surface over a �nite �eld, and that Br(X) is �nite. As before, G will be a�nite group acting freely on X. Our goal is to prove Theorem 3.1 relating the imageof �(X;Gm) in G0(ZG) to the signs at in�nity of the leading terms at s = 1 of theL-functions of symplectic representations of G.The strategy of the proof is to work in the �ner Grothendieck group G0T (ZG) all�nite ZG-modules, in which one can de�ne re�ned Euler characteristics �GT (X;Gm) and�GT (X;Ga) of Gm and the additive sheaf Ga (c.f. De�nition 3.5). We will use work ofLichtenbaum to show in Theorem 3.9 that the di�erence of �GT (X;Ga)��GT (X;Gm) isdetermined by the leading terms of L-functions at s = 1. To carry out this calculation,we use a \Hom-description" of G0T (ZG) due to Queyrut (c.f. Proposition 3.7) whichmakes it possible to iden�ty classes in G0T (ZG) via suitable functions on the charactersof G. The proof of Theorem 3.1 then follows from Theorem 3.9, Queyrut's \Hom-description" of the forgetful homomorphism G0T (ZG) ! G0(ZG), and a result ofNakajima which shows �GT (X;Ga) has trivial image in G0(ZG).As in [L1], H1(X;Gm) = Pic(X) and the intersection pairing on divisors induces apairing H1(X;Gm)�H1(X;Gm)! Z which is non-degenerate when tensored with Q.Let � : H1(X;Gm)! HomZ(H1(X;Gm);Z) = H1(X;Gm)D be the G-homomorphisminduced by this pairing. Then � has �nite kernel and cokernel. Since H1(X;Gm)D istorsion-free, this implies Ker(�) = H1(X;Gm)tor. Recall that if A is an abelian group,then Acodiv = A=Adiv , where Adiv is the subgroup of divisible elements of A.De�nition 3.5. De�ne classes �GT (X;Gm) and �GT (X;Ga) in G0T (ZG) by�GT (X;Gm) =[H0(X;Gm)]� [H1(X;Gm)tor] + [H2(X;Gm)]+ [H1(X;Gm)D=�H1(X;Gm)] � [H3(X;Gm)codiv] + [H4(X;Gm)]and �GT (X;Ga) = [H0(X;OX)] � [H1(X;OX )] + [H2(X;OX)]:De�nition 3.6. Suppose v is a place of Q. De�ne Qv to be an algebraic closure ofQv containing an algebraic closure Q of Q. Let RG (resp. RG;v) be the charactergroup of G over Q (resp. Qv). De�ne F = Gal(F=F ) for F = Q and F = Qv.If v is a �nite place of Q, let Zv be the integral closure of the v-adic integers Zv inQv. Let Hv be the group of character functions f 2 HomQv (RG;v;Q�v) such thatf(�) is a unit if � is the character of a projective ZvG-module. If v is the in�nite21

place of Q, let Hv be the group of character functions f 2 HomQv (RG;v;Q�v) suchthat f(�) is real and positive if � 2 RG;v is the character of a simple QvG-moduleof Schur index 2. For all v, de�ne (Q)v = Q Q Qv. Choosing a place of Q over vgives rise to an isomorphism HomQv (RG;v;Q�v) ! HomQ(RG; (Q)�v). De�ne H(v)to be the image of Hv under this isomorphism. Let J(Q) (resp. Jf (Q) be the groupof ideles (resp. �nite ideles) of Q. De�ne HA = HomQ(RG; J(Q)) \ (QvH(v)) andHA;f = HomQ (RG; Jf (Q)) \ (QvfiniteH(v)).Proposition 3.7. (Queyrut [Q]) Let v be a �nite place of Q with residue �eld k(v).There is a unique isomorphism�v : HomQv (RG;v;Q�v)=Hv ! G0(k(v)G) (3:6)which sends Detv(�v) to [ZvG=(ZvG � �v)] for �v 2 (QvG)� \ ZvG, where Detv :(QvG)� ! HomQv (RG;v;Q�v) is the usual v-adic determinant map. The direct sumof these isomorphisms gives an isomorphism� : HomQ(RG; Jf (Q))=HA;f ! G0T (ZG) = MvfiniteG0(k(v)G) (3:7)Let ~G0(ZG) be the kernel of the homomorphism G0(ZG) ! G0(QG) induced bytensoring ZG-modules with Q over Z. The natural map G0T (ZG)! G0(ZG) togetherwith � give an isomorphism� stab : HomQ(RG; J(Q))HomQ(RG;Q�) �HA;f ! ~G0(ZG): (3:8)De�nition 3.8. Let V be a representation of G over Q. De�ne L(t; V ) to be theArtin L-function of V as a representation of the Galois group of the cover X ! X=Gof smooth surfaces, where t = q�s if q is the order of the �eld of constants of X and sis a complex variable. Let rV be the order of zero of L(t; V ) at t = q�1, correspondingto s = 1. (In fact, rV � 0, since the poles of L(t; V ) arise from terms associatedto H2(X;Ql) in the l-adic formula for L(t; V ) when l is a prime not dividing q andX = Fq Fq X.) De�ne cV = limt!q�1(1� qt)�rV L(t; V )22

Let cX;G 2 Hom(RG;Q�) be the function which sends the character �V of V to cV .De�ne if : Q� ! Jf (Q) to be the diagonal embedding into the �nite ideles.Theorem 3.9. The function cX;G lies in HomQ (RG;Q�). One has� (if (cX;G)) = �GT (X;Ga) � �GT (X;Gm) (3:9)in G0T (ZG).Proof: From the de�nition of Artin L-functions, one has L(t; V �) = L(t; V )� as powerseries in t when � 2 Aut(C=Q). Since L(t; V ) is a rational function in t by the Weilconjectures, it follows that cV � = (cV )�. Hence cX;G lies in HomQ (RG;Q�).Let v be a �nite place of Q corresponding to the rational prime l, so k(v) = Z=l.Fix an embedding iv : Q ! Qv. Let projv : G0T (ZG) ! G0(k(v)G) be the naturalprojection. The composition iv � cX;G lies in HomQv (RG;v;Q�v). To prove Theorem3.9, it will su�ce to show for all v, l and iv as above that�v(iv � cX;G) = projv(�GT (X;Ga)� �GT (X;Gm)) (3:10)where �v is the homomorphism de�ned in (3.6). By the theory of Brauer characters, twoclasses inG0(k(v)G) are equal if they have the same restrictions to every cyclic subgroup� of G of order prime to l. The restriction map res�G : G0(k(v)G) ! G0(k(v)�) isinduced by the induction map indG� : R�;v ! RG;v relative to the isomorphism (3.6).Since Artin L-functions respect induction, we are thus reduced to proving (3.10) whenG = � is a cyclic group of order prime to l, which we assume is the case for the rest ofthe proof.De�ne R to be the ring of integers of the maximal unrami�ed extension of Ql = Qvin Qv. Let : G ! Q� be a one-dimensional character of G, so that 0 = iv � is acharacter of G with values in R�. The group ring R[G] is semisimple; let e 0 be thecentral idempotent associated to 0. If M is a �nite ZG-module, let length 0 (M) bethe length of a composition series for the �nite R-module e 0 (R Z M). There is aunique homomorphism ord 0 : G0T (ZG) ! Q� which sends the class [M ] of a �nitemodule M to llength 0 (M). By applying the exact functor N ! e 0 (R Z N) to eachof the modules appearing in Lichtenbaum's proof of [L1, Theorem 4.8], one �nds anequality of non-zero fractional R-idealsi(c )R = ord 0 (�TG(X;Ga))ord 0 (�TG(X;Gm)) �R (3:11)23

In fact, Theorem 4.8 of [L1] for a projective surface is equivalent to (3.11) when is the trivial character of the trivial group G. (Lichtenbaum's proof of [L1, Theorem4.8] relies on [L2], which in turn depends on work of Tate and Milne on geometriccounterparts of the Birch and Swinnerton-Dyer conjecture.) Unwinding the de�nitionsin Proposition 3.7, we see (3.11) is equivalent to (3.10), so the proof is complete.Lemma 3.10. There is a unique character function h 2 HomQ(RG; J(Q)) with thefollowing properties.a. For all �, the �nite components of h(�) are equal to 1, and the in�nite componentis 1 if � is not-symplectic.b. Suppose � is symplectic, and let 1 be the in�nite place of Q. Let �1 2 (Q)1 =Q Q Q1 be the in�nite component of an idele � 2 J(Q). Under each Q1-algebra map (Q)1 ! Q1 = C, the image of h(�)1 is �1 and the image of(cX;G(�) � h(�))1 is real and positive.Proof: Since symplectic characters are real valued, cX;G(�) must be a totally realalgebraic number if � is symplectic. We then de�ne h(�)1 via the signs of all theconjugates of cX;G(�). One has h 2 HomQ (RG; J(Q)) since cX;G 2 HomQ(RG;Q�).Proposition 3.11. De�ne LX;1 2 G0(ZG) to be the image of the function h of Lemma3.10 under the homomorphism HomQ (RG; J(Q)) ! G0(ZG) resulting from (3.8) ofProposition 3.7. Then LX;1 has order one or two, and is determined by the signs of theconjugates of the totally real algebraic numbers cX;V as V ranges over the symplecticrepresentations of G. One hasLX;1 = z(�GT (X;Ga)� �GT (X;Gm)) (3:12)where z : G0T (ZG)! G0(ZG) is the forgetful homomorphism.Proof: View the �nite ideles Jf (Q) as the subgroup of J(Q) having trivial in�nitecomponents. Then Theorem 3.9 shows that� stab(if (cX;G)) = z(�GT (X;Ga)� �GT (X;Gm)) (3:13)where � stab is the homomorphism resulting from (3.8). Let i : Q ! J(Q) be thediagonal embedding, and let i(1) : Q ! (Q)1 be the natural embedding into ideleswith trivial �nite components. Thenif (cX;G) = i(cX;G) � i(1)(cX;G)�1:24

>From Lemma 3.10 and Proposition 3.7, we see that the character functioni(1)(cX;G)�1 � h�1 lies in the subgroup HA appearing in (3:8), while i(cX;G) lies inHomQ(RG;Q�). Thus� stab(if (cX;G)) = � stab(i(cX;G)) � � stab(i(1)(cX;G)�1 � h�1) � � stab(h)= 1 � 1 � LX;1 (3:14)Combining (3.13) and (3.14) shows Proposition 3.11.Proof of Theorem 3.1We wish to show that f(�G(X;Gm)) = LX;1 (3:15)where �G(X;Gm) 2 K0(ZG) is de�ned in Corollary 3.4, f : K0(ZG)! G0(ZG) is theforgetful homomorphism, and LX;1 is de�ned in Proposition 3.11.By Theorem 3.3,H3(X;Gm)codiv = Hom(H1(X;Gm)tor;Q=Z)and H4(X;Gm) = Hom(H0(X;Gm);Q=Z) = Hom(H0(X;Gm)tor ;Q=Z):Substituting the right sides of these expressions into Corollary 3.4 when R = Z givesf(�G(X;Gm)) = [H0(X;Gm)]� [H1(X;Gm)] + [H2(X;Gm)]+ [H1(X;Gm)D]� [H3(X;Gm)codiv] + [H4(X;Gm)] (3:16)since HomZ(H0(X;Gm);Z) = 0 because H0(X;Gm) is �nite. As noted just prior toDe�nition 3.5, the kernel of the homomorphism � : H1(X;Gm) ! H1(X;Gm)D isH1(X;Gm)tor . Thus[H1(X;Gm)] = [�H1(X;Gm)] + [H1(X;Gm)tor] (3:17)in G0(ZG). On substituting (3.17) into (3.16), we see from the de�nition of �GT (X;Gm)in De�nition 3.5, that f(�G(X;Gm)) = z(�GT (X;Gm)): (3:18)25

By Proposition 3.11,z(�GT (X;Gm)) � z(�GT (X;Ga)) = �LX;1 = LX;1 (3:19)since LX;1 has order 1 or 2. Thus (3.18) and (3.19) show that to prove (3.15), it willsu�ce to show z(�GT (X;Ga)) = 0: (3:20)Since G acts freely on X, it is a Theorem of Nakajima [N] that the class �GT (X;Ga) =[H0(X;OX )] � [H1(X;OX )] + [H2(X;OX )] 2 G0((Z=pZ)G) is the class of a free(Z=pZ)G-module, where p is the characteristic of the function �eld K(X). Hence(3.20) holds, which completes the proof.References[C] Chinburg, T: Galois structure of de Rham cohomology of tame covers of schemes,Annals of Math. 139, 1994, p. 443 - 490.[CE] Chinburg, T. and Erez, B.: Equivariant Euler-Poincar�e Characteristics and Tame-ness, Ast�erisque 209, 1992, p. 179 - 194.[E] Erez, B.: Geometric Trends in Galois module theory, to appear in the Proceedingsof the Durham Conference on arithmetic geometry.[H] Hartshorne, R: Algebraic Geometry, Springer - Verlag, New York, third ed. 1983.[HRD] Hartshorne, R.: Residues and Duality, Springer Lecture Notes in Math. 20,Springer-Verlag, Heidelberg, 1966.[K] Kaplansky, I.: Fields and Rings, Univ. of Chicago Press, Chicago, second ed.1972.[L1] Lichtenbaum, S.: Behavior of the Zeta-Function of Open Surfaces at s = 1, inAdvanced Studies in Pure Mathematics, 17, 1989, p. 271 - 287.[L2] Lichtenbaum, S.: Zeta functions of varieties over �nite �elds at s = 1, inArithmetic and Geometry, Progress in Math 35, Birkhauser, 1983, p. 173 - 194.[M] Milne, J.: �Etale Cohomology, Princeton Univ. Press, Princeton, N. J., 1980.[N] Nakajima, S.: On Galois Module Structure of Cohomology Groups of an AlgebraicVariety, Invent. Math. 75, 1984, p. 1 - 8.[Q] Queyrut, J.: S-Groupes des Classes d'un Ordre Arithm�etique, J. of Algebra 76,1982, p. 234 - 260.[Sa] Saito, S.: Arithmetic theory of arithmetic surfaces, Ann. of Math. 129, 1989, p.547 - 589. 26

T. Chinburg, Dept. of Mathematics, Univ. of Penn., Phila. Pa. 19104, U.S.A.;ted@math.upenn.eduM. Kolster, Dept. of Mathematics and Statistics, McMaster Univ., Hamilton, On-tario, Canada L8S 4K1; kolster@mcmail.CIS.McMaster.CAG. Pappas, Dept. of Mathematics, Princeton Univ., Fine Hall, Washington Road,Princeton, N. J. 08544, U.S.A.; pappas@math.Princeton.EDUV. Snaith, Dept. of Mathematics, University of Southampton, High�eld, Southamp-ton SO17 1BJ United Kingdom; vps@maths.soton.ac.uk

27