ℓ On the action of the Weil group on the -adic cohomology of rigid spaces over local fields 6 0 Yoichi Mieda 0 2 Abstract. We investigate the action of the Weil group on the com- n a pactly supported ℓ-adic cohomology groups of rigid spaces over local J fields. We prove that every eigenvalue of the action is a Weil number 5 when either a rigid space is smooth or the characteristic of the base field is equal to 0. Since a smooth rigid space is locally isomorphic to ] T the Raynaud generic fiber of an algebraizable formal scheme, we can N use alterations and an analogue of the weight spectral sequence in the . h smooth case. Inthe general case, we usethe continuity theorem of Hu- at ber ([Hu4], [Hu5]), which requires the restriction on the characteristic m of the base field. [ 2 1. Introduction v 9 0 Let K be a complete discrete valuation field with finite residue field F and K a separable q 5 closure of K. We denote by Fr the geometric Frobenius element (the inverse of the qth power 8 q 0 map) in Gal(F /F ). The Weil group W of K is defined as the inverse image of the subgroup q q K 5 hFr i ⊂ Gal(F /F )bythecanonicalmapGal(K/K) −→ Gal(F /F ). Forσ ∈ W ,letn(σ)be 0 q q q q q K h/ the integer such that the image of σ in Gal(Fq/Fq) is Frnq(σ). Put WK+ = {σ ∈ WK | n(σ) ≥ 0}. Let X be a rigid space over K. In this paper, we consider the action of W on the t K a compactly supported ℓ-adic cohomology group Hi(X⊗ K,Q ) (cf. [Hu3], [Hu5]), where ℓ is m c K ℓ a prime number which does not divide q. Our main theorem is the following: : v i X Theorem 1.1 (Theorem 4.2, Theorem 5.5) r Let X be a quasi-compact separated rigid space over K. Assume one of the following a conditions: • The rigid space X is smooth over K. • The characteristic of K is equal to 0. Then for any σ ∈ W+, every eigenvalue α ∈ Q of its action on Hi(X ⊗ K,Q ) is an alge- K ℓ c K ℓ braic integer. Moreover, there exists a non-negative integer m such that for any isomorphism ι: Q −−∼→ C, the absolute value |ι(α)| is equal to qn(σ)·m/2. ℓ Graduate School of Mathematical Sciences, the University of Tokyo, 3–8–1 Komaba, Meguro-ku, Tokyo 153–8914,Japan. E-mail address: [email protected] 2000 Mathematics Subject Classification. Primary: 14F20; Secondary: 14G20, 14G22. 1 Yoichi Mieda Note that Hi(X⊗ K,Q ) is known to be a finite-dimensional Q -vector space when one c K ℓ ℓ of the above conditions is satisfied ([Hu5, Theorem 3.1]). When X is a scheme separated of finite type over K, the corresponding property is proven by Ochiai ([O, Proposition 2.1]). Furthermore the ℓ-independence of the alternating sum of the traces 2dimX (−1)iTr σ ;Hi(X ⊗ K,Q ) ∗ c K ℓ Xi=0 (cid:0) (cid:1) is obtained there ([O, Theorem 2.4]). See also [Sa], which treats the composite action of an element of W and an algebraic correspondence. K We sketch the outline of the proof. When X is smooth over K, it is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme. In §2, we briefly recall this fact. By using the techniques of alterations and cohomological descent described in §3, the smooth case is reduced to the case where X is the Raynaud generic fiber of the completion of a strictly semistable scheme over O along its special fiber. In this case, we use an analogue of the K weight spectral sequence to show the theorem. The smooth case is treated in §4. Assuming that the characteristic of K is 0, we prove the theorem for a general X by induction on dimX. In this process, we need the continuity theorem of Huber ([Hu4], [Hu5]). The general case is treated in §5. Acknowledgements TheauthorwouldliketothankTetsushiItoforstimulating discussions, TakeshiSaitoforpointingoutamistakeinthepreviousversionofthispaper. Heisalsograteful to his advisor Tomohide Terasoma. He was supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists. Notation and Conventions Throughout this paper, K is a complete discrete valuation field with finite residue field F . We denote the ring of integers of K by O . Fix a separable q K closure K of K and write X for X ⊗ K. K K WedenoteaschemebyanordinaryitaliclettersuchasX,aformalscheme byacalligraphic letter such as X, and a rigid space by a sans serif letter such as X. For a scheme X of finite type over O , X (resp. X ) denotes its special fiber (resp. geometric special fiber) and X∧ K s s the completion of X along its special fiber. For a formal scheme X over SpfO , we write Xrig for its Raynaud generic fiber. It is the K analytic adic space d(X) in [Hu3, 1.9]. We consider rigid spaces as adic spaces of finite type over Spa(K,O ). We write SpA for K the adic space Spa(A,A◦), where A is a topologically finitely generated K-algebra. 2. Algebraization (2.1) In this section we see that a smooth rigid space over K is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme, i.e., the completion of a scheme of finite type over O along its special fiber. K (2.2) Here we recall some definitions about topological rings introduced in [Hu1], [Hu2]. A topological ring is said to be adic if it has an ideal of definition, i.e., an ideal I such that 2 On the action of the Weil group on the ℓ-adic cohomology of rigid spaces over local fields {In} is a fundamental system of neighborhoods of 0. A topological ring A is said to be n≥1 f-adic if it has an open subring A which has a finitely generated ideal of definition. Such 0 an open subring A is called a ring of definition. An f-adic ring is said to be Tate if it has a 0 topologically nilpotent unit. Let A be a complete Tate ring. For a subset U of A, we write UhT ,...,T i for the subset 1 n (cid:26) aITI ∈ A[[T1,...,Tn]] (cid:12) aI ∈ U,(aI)I∈(Z≥0)n converges to 0 in A(cid:27) I∈X(Z≥0)n (cid:12)(cid:12) (cid:12) of A[[T ,...,T ]]. We equip AhT ,...,T i with the group topology such that UhT ,...,T i 1 n 1 n 1 n for neighborhoods U of 0 ∈ A form a fundamental system of neighborhoods of 0. Then AhT ,...,T i is a complete Tate ring. It is easy to see that AhT ,...,T ihT ,...,T i is 1 n 1 n n+1 m naturally isomorphic to AhT ,...,T i as a topological ring. 1 m Example 2.3 A complete discrete valuation field K with its natural topology is a complete Tate ring. TheringofintegersO isaringofdefinitionandauniformizerofK isatopologicallynilpotent K unit of K. The topological ring KhT ,...,T i is the same one as that in [BGR]. 1 n Lemma 2.4 Let A be a complete Tate ring and f ,...,f elements of AhT ,...,T i. Assume that the 1 n 1 n image of det(∂f /∂T ) in AhT ,...,T i/(f ,...,f ) is a unit. Then there exists an open i j 1≤i,j≤n 1 n 1 n neighborhood V of 0 ∈ AhT ,...,T i such that any g ,...,g with g ∈ f + V satisfy the 1 n 1 n i i following two conditions: • The two topological rings AhT ,...,T i/(f ,...,f ) and AhT ,...,T i/(g ,...,g ) are 1 n 1 n 1 n 1 n (topologically) A-isomorphic. • The image of det(∂g /∂T ) in AhT ,...,T i/(g ,...,g ) is a unit. i j 1≤i,j≤n 1 n 1 n Proof. This Lemma is due to [Hu3, Proposition 1.7.1 ii) =⇒ iii)]. Note that the noetherian assumption loc. cit. is not necessary when A is Tate. Indeed, in the proof given there, (3) is the only step where the noetherian assumption is used. In our case we may derive the bijectivity of f: B⊲ −→ B⊲ from that of f: B −→ B and the equality B⊲ = B [1/π], where 0 0 0 π is a topologically nilpotent unit of A⊲. Nevertheless we use this lemma only for strongly noetherian rings A. Corollary 2.5 Let X be a smooth rigid space over K and x a point of X. Then there exist an open neighborhood U of x, a scheme X of finite type over O with smooth generic fiber, and a K K-isomorphism U ∼= (X∧)rig. Proof. Since X is smooth over K, there exists an open neighborhood U of x which is iso- morphic to SpKhT ,...,T i/(f ,...,f ) for some m ≤ n and f ,...,f ∈ KhT ,...,T i 1 n 1 m 1 m 1 n 3 Yoichi Mieda such that the image of det(∂f /∂T ) in KhT ,...,T i/(f ,...,f ) is a unit. Put i j 1≤i,j≤m 1 n 1 m A = KhT ,...,T i and apply Lemma 2.4. Then we have an open neighborhood V of m+1 n 0 ∈ AhT ,...,T i = KhT ,...,T i satisfying the two conditions above. Since K[T ,...,T ] 1 m 1 n 1 n is dense in KhT ,...,T i, there exist g ,...,g ∈ K[T ,...,T ] such that the two topologi- 1 n 1 m 1 n cal rings KhT ,...,T i/(f ,...,f ) and KhT ,...,T i/(g ,...,g ) are A-isomorphic (a for- 1 n 1 m 1 n 1 m tiori K-isomorphic) and ∆ = det(∂g /∂T ) is invertible in KhT ,...,T i/(g ,...,g ). i j 1≤i,j≤m 1 n 1 m Clearly we may assume that g ∈ O [T ,...,T ]. Let π be a uniformizer of K. Since i K 1 n x(∆) 6= 0 for every x ∈ U = SpKhT ,...,T i/(g ,...,g ), there exists an integer k such 1 n 1 m that x(∆) ≥ x(πk) for every x ∈ U ([Hu1, Lemma 3.11]). On the other hand, we have an isomorphism O [T ,...,T ]/(g ,...,g ,T ∆−πk) ∧[1/π] ∼= KhT ,...,T i/(g ,...,g ,T ∆−πk), K 1 n+1 1 m n+1 1 n+1 1 m n+1 (cid:0) (cid:1) where ∧ denotes the π-adic completion. Thus we conclude that U ∼= (X∧)rig, where X = SpecO [T ,...,T ]/(g ,...,g ,T ∆−πk). Since K 1 n+1 1 m n+1 SpecK[T ,...,T ]/(g ,...,g ,T ∆−πk) = SpecK[T ,...,T ,∆−1]/(g ,...,g ) 1 n+1 1 m n+1 1 n 1 m is smooth over K, the corollary is proven. (2.6) When X ∼= (X∧)rig, the compactly supported ℓ-adic cohomology groups of X can be calculated by the nearby cycle of X: Theorem 2.7 Let X be a scheme which is separated of finite type over O and X the rigid space (X∧)rig. K Then Hi(X ,Q ) is canonically isomorphic to Hi(X ,RψQ ). Moreover, the isomorphism is c K ℓ c s ℓ W -equivariant. K Proof. Apply [Hu3, Theorem 5.7.6] to X ⊗O , where O is the integral closure of O in K K K K. The W -equivariantness is obvious. K 3. Cohomological descent (3.1) In the present section, we provide several results on the technique of cohomological descent for ℓ-adic cohomology of rigid spaces. Here we will use the terminology in [De2, Section 5] (see also [SGA4-II, Expos´e Vbis]). Proposition 3.2 Let X be a quasi-compact separated rigid space which is smooth over K. Then we can construct an´etale hypercovering U −→ X such that U ∼= (U∧)rig for some schemes U which • n n n is separated of finite type over O . Moreover we have a following spectral sequence: K E−i,j = Hj(U ,Q ) =⇒ H−i+j(X ,Q ). 1 c iK ℓ c K ℓ 4 On the action of the Weil group on the ℓ-adic cohomology of rigid spaces over local fields Proof. This is an easy consequence of Corollary 2.5. The existence of spectral sequence can be shown as in [Hu3, Remark 5.5.12 ii)]. Proposition 3.3 Let X and Y be rigid spaces and f: Y −→ X a proper surjection. Then it is a morphism universally of cohomological descent relative to torsion sheaves. Proof. This is due to the proper base change theorem for torsion sheaves, which is proven in [Hu3, Theorem 4.4.1] (cf. [SGA4-II, Expos´e Vbis, Corollary 4.1.6] for the Betti cohomology and loc. cit. Proposition 4.3.2 for the ´etale cohomology of schemes). Proposition 3.4 Let X be a scheme of finite type over O and n a positive integer. Then we can construct K a system of field extensions K ⊂ K ⊂ ··· ⊂ K and an n-truncated proper hypercovering 0 n Y −→ X ⊗ O satisfying the following conditions: • OK Kn • The scheme Y is an O -scheme and every structure morphism is an O -morphism. i Kn Kn • For every i there exists a scheme Y′ which is strictly semistable (cf. [Sa, section 1]) over i O such that Y ∼= Y′ ⊗ O . Ki i i OKi Kn Moreover, the associated semi-simplicial rigid space (Y∧)rig is an n-truncated proper hyper- • covering of (X∧ ⊗ O )rig = (X∧)rig ⊗ K . OK Kn K n Proof. All the assertions except the last are well-known consequences of [dJ]. The last one follows from Proposition 3.3 and the two lemmas below. Lemma 3.5 The functor X 7−→ (X∧)rig from the category of schemes of finite type over O to the K category of rigid spaces over K preserves any finite projective limit (therefore the functor commutes with cosq ). n Proof. Since each of the categories has fiber products and the final object, it is sufficient to provethatthefunctorcommuteswithfiberproducts. Itisknownthatthefunctorrig commutes with fiber products ([BL, Corollary 4.6]). We will show that the functor ∧ commutes with fiber products. We may work locally: let A be a finitely generated O -algebra and B, C be K finitely generated A-algebras. We should prove (B ⊗ C)∧ ∼= B∧⊗ C∧, which can be found A A∧ in [EGA1, Chap. 0 (7.7.1)]. b Lemma 3.6 Let X and Y be schemes of finite type over O and f: Y −→ X a proper surjection. K Then the morphism (f∧)rig: (Y∧)rig −→ (X∧)rig is also a proper surjection. Proof. The lemma seems well-known, but we include its proof in the context of adic spaces. First recall some general facts on adic spaces. Let L be a non-archimedean field ([Hu3, 5 Yoichi Mieda Definition 1.1.3]) and L+ be a valuation ring of L. For a scheme X of finite type over O , K consider a Spa(K,K+)-morphism i: Spa(L,L+) −→ (X∧)rig. Then we have the morphism of formal schemes i : SpfL+ −→ X∧ and that of schemes i : SpecL+ −→ X. The maps 1 2 i 7−→ i and i 7−→ i are clearly injective. Since every point of (X∧)rig is analytic, the image 1 1 2 of the generic point of SpecL+ under i lies on the generic fiber of X. Furthermore, for every 2 O -morphism f: Y −→ X, the following three sets are naturally identified: K • the set of morphisms of adic spaces i′: Spa(L,L+) −→ (Y∧)rig such that i = (f∧)rig◦i′; • the set of morphisms of formal schemes i′ : SpfL+ −→ Y∧ such that i = f∧ ◦i′; 1 1 1 • the set of morphisms of schemes i′ : SpecL+ −→ Y such that i = f ◦i′. 2 2 2 Indeed, by local consideration, we can easily see that the two maps i′ 7−→ i′ and i′ 7−→ i′ are 1 1 2 bijective. The adicness of the morphism f∧: Y∧ −→ X∧ is crucial for the second bijectivity. Now we will show the properness of (f∧)rig by the valuative criterion ([Hu3, Corollary 1.3.9, Lemma 1.3.10]). Let L+ be another valuation ring of L. Suppose that a commutative 1 diagram Spa(L,L+) i′ // (Y∧)rig 1 (f∧)rig (cid:15)(cid:15) (cid:15)(cid:15) Spa(L,L+) i //(X∧)rig is given. We should prove that there exists a unique morphism i′′: Spa(L,L+) −→ (Y∧)rig that makes the following diagram commutative: Spa(L,L+) i′ // (Y∧)rig 1 q88 q i′′q q (f∧)rig (cid:15)(cid:15) q (cid:15)(cid:15) Spa(L,L+) i //(X∧)rig. By the consideration above, this is equivalent to proving that there exists a unique morphism i′′: SpecL+ −→ Y that makes the following diagram commutative: 2 i′ SpecL+1 2 w;;//Y i′′ w 2w w f Spec(cid:15)(cid:15) Lw+ i2 //X(cid:15)(cid:15) . This immediately follows from the valuative criterion for schemes. Next we will give a proof of the surjectivity of (f∧)rig. Let x be a point of (X∧)rig and i: Spa(L,L+) −→ (X∧)rig be a morphism which maps the closed point of Spa(L,L+) to x. We may assume that the field L is algebraically closed. We get the morphism of schemes i : SpecL+ −→ X. Since f is surjective and L is algebraically closed, we can find a morphism 2 k: SpecL −→ Y such that the composite f ◦ k coincides with the restriction of i . By the 2 properness of f, there exists a unique morphism i′ : SpecL+ −→ Y such that i = f ◦i′ and 2 2 2 6 On the action of the Weil group on the ℓ-adic cohomology of rigid spaces over local fields i′| = k. Let i′: Spa(L,L+) −→ (Y∧)rig be the corresponding morphism and y be the 2 SpecL image of the closed point of Spa(L,L+) under i′. It is clear that the image of y under (f∧)rig coincides with x. 4. Smooth case (4.1) In this section, we will give a proof of the main theorem for smooth rigid spaces. Theorem 4.2 Let X be a quasi-compact separated rigid space which is smooth over K. Then for any σ ∈ W+,every eigenvalue α ∈ Q ofitsactiononHi(X ,Q )isanalgebraicinteger. Moreover, K ℓ c K ℓ there exists a non-negative integer m such that for any isomorphism ι: Q −−∼→ C, the absolute ℓ value |ι(α)| is equal to qn(σ)·m/2. Lemma 4.3 To prove Theorem 4.2, we have only to show that for some finite extension L of K the action of W+ on Hi(X ,Q ) satisfies the assertion in Theorem 4.2. L c L ℓ Proof. SinceaninseparableextensionaffectsneithertheWeilgroupnorthe´etalecohomology, we may assume that the extension L/K is separable and that L is a subfield of K. Write e for the ramification index of L/K and f for the degree of the extension of the residue field of L/K. Let n′: W −→ Z be the map n for L defined as in §1. Then it is equal to the L restriction of n: W −→ Z multiplied by 1/f. Take an element σ of W+ and an eigenvalue K K α of the action of σ on Hi(X ,Q ). Then we have σef ∈ W+. By the assumption, the c K ℓ L eigenvalue αef of the actionof σef on Hi(X ,Q ) is an algebraic integer and there exists a non- c K ℓ negativeintegermsuchthat|ι(αef)| = (qf)n′(σef)·m/2 foreveryisomorphismι: Q −−∼→ C. Since ℓ n′(σef) = n(σef)/f = e·n(σ),theeigenvalue αisalsoanalgebraicintegerand|ι(α)| = qn(σ)·m/2 for every ι. Lemma 4.4 To prove Theorem 4.2, we may assume that X ∼= (X∧)rig for some scheme X which is separated of finite type over O . K Proof. Take an ´etale hypercovering U −→ X as in Proposition 3.2. Then we have the • following spectral sequence: E−i,j = Hj(U ,Q ) =⇒ H−i+j(X ,Q ). 1 c iK ℓ c K ℓ Every eigenvalue α of the action of σ ∈ W+ on Hν(X ,Q ) occurs as an eigenvalue of the K c K ℓ action of σ on Hj(U ,Q ) for some non-negative integers i, j satisfying −i + j = ν. Thus c iK ℓ we have to show Theorem 4.2 only for U , which is isomorphic to (U∧)rig for some scheme U . i i i This completes the proof. Lemma 4.5 7 Yoichi Mieda To prove Theorem 4.2, we may assume that X ∼= (X∧)rig for some scheme X which is strictly semistable over O . K Proof. By Lemma 4.4, we may assume that X = (X∧)rig for some scheme X which is separated of finite type over O . It is sufficient to consider the action of W on Hν(X ,Q ) K K c K ℓ for 0 ≤ ν ≤ 2dimX, since Hν(X ,Q ) = 0 otherwise ([Hu3, Corollary 1.8.8 and Proposition c K ℓ 5.5.8]). Put n = 2dimX and take an n-truncated proper hypercovering Y −→ X⊗ O as • OK Kn in Proposition 3.4. Then we have the proper hypercovering Y = cosq (Y ) and the following • n • W -equivariant spectral sequence: Kn Ei,j = Hj(Y ,Q ) =⇒ Hi+j(X ,Q ), 1 c iK ℓ c K ℓ where Y = (Y∧)rig. Every eigenvalue α of the action of σ ∈ W+ on Hν(X ,Q ) occurs as i i Kn c K ℓ an eigenvalue of the action of σ on Hj(Y ,Q ) for some non-negative integers i, j satisfying c iK ℓ i+j = ν. Sincei = ν−j ≤ ν ≤ 2dimX,thereexistaschemeY′ whichisstrictlysemistableover i O and an isomorphism Y ∼= Y′ ⊗ O . Then we have a W -equivariant isomorphism Ki i i OKi Kn Kn Hj(Y ,Q ) ∼= Hj(Y′ ,Q ), where Y′ = (Y′∧)rig. Since Y′ is strictly semistable over O , the c iK ℓ c iK ℓ i i i Ki eigenvalue α satisfies the property stated in Theorem 4.2 by the assumption. Combining this with Lemma 4.3, we may conclude the present lemma. (4.6) By Theorem 2.7 and Lemma 4.5, we reduce Theorem 4.2 to the following proposition: Proposition 4.7 Let X be a strictly semistable scheme over O . Then for any σ ∈ W+, every eigenvalue K K α ∈ Q of its action on Hi(X ,RψQ ) is an algebraic integer. Moreover, there exists a non- ℓ c s ℓ ∼ negative integer m such that for any isomorphism ι: Q −−→ C, the absolute value |ι(α)| is ℓ equal to qn(σ)·m/2. Proof. Denote the irreducible components of X by D ,...,D . For a non-empty subset s 1 m I of {1,...,m} and a non-negative integer k, we put D = D and D(k) = D . I i∈I i |I|=k+1 I These are smooth over Fq. We use the spectral sequence T ` Ei,j = Hj−2k D(i+2k),Q (−k) =⇒ Hi+j(X ,RψQ ). 1 c s ℓ c s ℓ k≥mMax(0,−i) (cid:0) (cid:1) This is the spectral sequence associated with the monodromy filtration of RψQ ([Sa]) and is ℓ called the weight spectral sequence when X is proper over O ([RZ]). K For every σ ∈ W+, we have the following morphism of spectral sequences (cf. [Sa, proof of K Lemma 3.2]): Ei,j = Hj−2k D(i+2k),Q (−k) +3 Hi+j(X ,RψQ ) 1 k≥max(0,−i) c s ℓ c s ℓ L (cid:0) (cid:1) Frnq(σ) σ∗ (cid:15)(cid:15)(cid:0) (cid:1)∗ (cid:15)(cid:15) Ei,j = Hj−2k D(i+2k),Q (−k) +3 Hi+j(X ,RψQ ). 1 k≥max(0,−i) c s ℓ c s ℓ L (cid:0) (cid:1) 8 On the action of the Weil group on the ℓ-adic cohomology of rigid spaces over local fields Therefore every eigenvalue α of the action of Frn(σ) on Hν(X ,RψQ ) occurs as an eigenvalue q c s ℓ of the action of Frn(σ) on Hj−2k(D(i+2k),Q (−k)) for some i, j, k satisfying i + j = ν, k ≥ q c s ℓ max(0,−i). On the other hand, by the purity theorem of Deligne ([De1, Corollaire 3.3.4]) and loc. cit. Corollaire 3.3.3, every eigenvalue α of the action of Frn(σ) on Hj−2k(D(i+2k),Q (−k)) is q c s ℓ an algebraic integer and there exists an integer m with 2k ≤ m such that |ι(α)| = qn(σ)·m/2 for every isomorphism ι: Q −−∼→ C. Since k ≥ 0, the integer m is non-negative. This completes ℓ the proof. 5. General case (5.1) In this section, we assume that the characteristic of K is equal to 0. Recall the following theorems by Huber, which are used to prove the main theorem. Theorem 5.2 Assume that the characteristic of K is equal to 0. Let X be a quasi-compact separated rigid space over K. Then the cohomology group Hi(X ,Q ) is a finite-dimensional Q -vector c K ℓ ℓ space. Proof. [Hu5, Theorem 3.1] (see also [Hu4, Corollary 2.3]). Theorem 5.3 Let X be a separated rigid space over K which is not necessarily quasi-compact, {U } λ an open covering of X consisting of quasi-compact open subspaces such that for every λ , λ 1 2 there exists λ satisfying U ∪U ⊂ U . Then the canonical homomorphism λ1 λ2 λ limHi(U ,Q ) −→ Hi(X ,Q ) −→ c λK ℓ c K ℓ λ is an isomorphism. Proof. [Hu5,Proposition2.1(iv)]. Notethatthistheoremisalsovalidwhenthecharacteristic of K is positive. (5.4) Now we can give a proof of our main theorem. Theorem 5.5 Assume thatthecharacteristic ofK isequalto0. LetXbeaquasi-compact separatedrigid space over K. Then for any σ ∈ W+, every eigenvalue α ∈ Q of its action on Hi(X ,Q ) K ℓ c K ℓ is an algebraic integer. Moreover, there exists a non-negative integer m such that for any isomorphism ι: Q −−∼→ C, the absolute value |ι(α)| is equal to qn(σ)·m/2. ℓ Proof. We proceed by induction on dimX. We may assume that X is reduced. Let Z be the singular locus of X. It is a closed analytic subspace whose dimension is strictly less than 9 Yoichi Mieda dimX. Thus we have only to show our claim on Hi(U ,Q ), where U = X\Z. Let {U } be the c K ℓ λ set of quasi-compact open subspaces of U (note that U itself is rarely quasi-compact). Then we have the isomorphism limHi(U ,Q ) −−∼→ Hi(U ,Q ) −→ c λK ℓ c K ℓ λ by Theorem 5.3. Since every eigenvalue of the action of σ ∈ W+ on Hi(U ,Q ) occurs as an K c K ℓ eigenvalue of the action of σ on Hi(U ,Q ) for some λ, and each U is quasi-compact and c λK ℓ λ smooth over K, the theorem follows from Theorem 4.2. 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