aa r X i v : . [ m a t h . AG ] J un A Tauberian Theorem for ℓ -adic Sheaves on A To Wang Yuan on his 80th birthday ∗ Lei Fu
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, P. R. [email protected]
Abstract
Let K ∈ L ( R ) and let f ∈ L ∞ ( R ) be two functions on R . The convolution( K ∗ f )( x ) = Z R K ( x − y ) f ( y ) dy can be considered as an average of f with weight defined by K . Wiener’s Tauberian theoremsays that under suitable conditions, iflim x →∞ ( K ∗ f )( x ) = lim x →∞ ( K ∗ A )( x )for some constant A , then lim x →∞ f ( x ) = A. We prove the following ℓ -adic analogue of this theorem: Suppose K, F, G are perverse ℓ -adicsheaves on the affine line A over an algebraically closed field of characteristic p ( p = ℓ ). Undersuitable conditions, if ( K ∗ F ) | η ∞ ∼ = ( K ∗ G ) | η ∞ , then F | η ∞ ∼ = G | η ∞ , where η ∞ is the spectrum of the local field of A at ∞ . Key words:
Tauberian theorem, ℓ -adic Fourier transformation. Mathematics Subject Classification:
Introduction
A Tauberian theorem is one in which the asymptotic behavior of a sequence or a function is deducedfrom the behavior of some of its average. The ℓ -adic Fourier transform was first introduced byDeligne in the study of exponential sums using ℓ -adic cohomology theory. It was further developedby Laumon [5]. In this paper, using the ℓ -adic Fourier transform, we prove an ℓ -adic analogueof Wiener’s Tauberian theorem in the classical harmonic analysis. Our study shows that many ∗ The research is supported by the NSFC. ℓ -adic analogues and this area has not been fullyexplored. The result in this paper is absolutely not in its final form.For any f , f ∈ L ( R ), their convolution f ∗ f ∈ L ( R ) is defined to be( f ∗ f )( x ) = Z R f ( x − y ) f ( y ) dy. If we define the product of two functions to be their convolution, then L ( R ) becomes a Banachalgebra. A function f ∈ L ∞ ( R ) is called weakly oscillating at ∞ if for any ǫ >
0, there exist
N > δ > x , x ∈ R with the properties that | x | , | x | > N and | x − x | < δ ,we have | f ( x ) − f ( x ) | ≤ ǫ. Recall the following theorem ([3] VIII 6.5).
Theorem 0.1 (Wiener’s Tauberian theorem) . Let K ∈ L ( R ) and f ∈ L ∞ ( R ) .(i) If lim x →∞ f ( x ) = A, then lim x →∞ Z R K ( x − y ) f ( y ) dy = A Z R K ( x ) dx. (ii) Suppose the Fourier transform ˆ K ( ξ ) = Z R K ( x ) e iξx dx of K has the property ˆ K ( ξ ) = 0 for all ξ ∈ R and suppose lim x →∞ Z R K ( x − y ) f ( y ) dy = A Z R K ( x ) dx. Then lim x →∞ Z R K ( x − y ) f ( y ) dy = A Z R K ( x ) dx for all K ∈ L ( R ) . Suppose furthermore that f is weakly oscillating at ∞ . Then we have lim x →∞ f ( x ) = A. We quickly recall a proof of (ii). Let I = { K ∈ L ( R ) | lim x →∞ Z R K ( x − y ) f ( y ) dy = A Z R K ( x ) dx } . Then I is a closed linear subspace of L ( R ). If K ∈ I , then for any y ∈ R , the translation K y of K defined by K y ( x ) = K ( x − y ) lies in I . This implies that I is a closed ideal of the Banach2lgebra L ( R ). Since ˆ K ( ξ ) = 0 for all ξ , by a theorem of Wiener ([3] VIII 6.3), for any g ∈ L ( R )such that ˆ g has compact support, their exists g ∈ L ( R ) such that ˆ g = ˆ g ˆ K , which implies that g = g ∗ K . So the closure of the ideal generated by K is L ( R ). We have K ∈ I , so we have I = L ( R ). Hence for any K ∈ L ( R ), we havelim x →∞ Z R K ( x − y ) f ( y ) dy = A Z R K ( x ) dx. For any h >
0, taking K ( x ) = (cid:26) h if x ∈ [0 , h ] , x [0 , h ] , we get lim x →∞ h Z xx − h f ( y ) dy = A. If f is weakly oscillating at ∞ , this implies that lim x →∞ f ( x ) = A. In this paper, we study an analogue of the above result for ℓ -adic sheaves on the affine line.Throughout this paper, p is a prime number, k is an algebraically closed field of characteristic p , F p is the finite field with p elements contained in k , ℓ is a prime number distinct from p , and ψ : F p → Q ∗ ℓ is a fixed nontrivial additive character. Let A = Spec k [ x ] be the affine line. TheArtin-Schreier morphism ℘ : A → A corresponding to the k -algebra homomorphism k [ t ] → k [ t ] , t t p − t is a finite Galois ´etale covering space, and it defines an F p -torsor0 → F p → A ℘ → A → . Pushing-forward this torsor by ψ − , we get a lisse Q ℓ -sheaf L ψ of rank 1 on A . Let A ′ = Spec k [ x ′ ]be another copy of the affine line, let π : A × k A ′ → A , π ′ : A × k A ′ → A ′ be the projections, and let L ψ ( xx ′ ) be the inverse image of L ψ under the k -morphism A × k A ′ → A , ( x, x ′ ) xx ′ k -algebra homomorphism k [ t ] → k [ x, x ′ ] , t xx ′ . For any object K in the triangulated category D bc ( A , Q ℓ ) defined in [2] 1.1, the Fourier transform F ( K ) ∈ ob D bc ( A , Q ℓ ) of K is defined to be F ( K ) = Rπ ′ ! ( π ∗ K ⊗ L ψ ( xx ′ ))[1] . Let s : A × k A → A , ( x, y ) x + y be the k -morphism corresponding to the k -algebra homomorphism k [ t ] → k [ x, y ] , t x + y, and let p , p : A × k A → A be the projections. For any K , K ∈ ob D bc ( A , Q ℓ ), define their convolution K ∗ K ∈ ob D bc ( A , Q ℓ )to be K ∗ K = Rs ! ( p ∗ K ⊗ p ∗ K ) . Let F ∈ D bc ( A , Q ℓ ). We say F is a perverse sheaf (confer [1]) if H ( F ) has finite support, H − ( K ) has no sections with finite support, and H i ( K ) = 0 for i = 0 ,
1. The Fourier transformof a perverse sheaf on A is a perverse sheaf on A ′ .Let P = A ∪{∞} and P ′ = A ′ ∪{∞ ′ } be the smooth compactifications of A and A ′ , respectively.They are projective lines. For any Zariski closed point x (resp. x ′ ) in P (resp. P ′ ), let η x (resp. η x ′ ) be the generic point of the henselization of P (resp. P ′ ) at x (resp. x ′ ), and let ¯ η x (resp. ¯ η x ′ )be a geometric point above η x (resp. η x ′ ). On Gal(¯ η x /η x ) (resp. Gal(¯ η x ′ /η x ′ )), we have a filtrationby ramification subgroups in upper numbering. We can use this filtration to define the breaks of Q ℓ -representations of Gal(¯ η x /η x ) (resp. Gal(¯ η x ′ /η x ′ )). For any perverse sheaf F on A , H − ( F ) ¯ η x is a Q ℓ -representation of Gal(¯ η x /η x ). Confer [5] for the definition of the local Fourier transform F ( x,x ′ ) . 4 heorem 0.2 (Tauberian theorem) . Let K ∈ ob D bc ( A , Q l ) be a perverse sheaf on A . Suppose theFourier transform F ( K ) is of the form L [1] for some lisse Q ℓ -sheaf L on A ′ . Let M, N be lisse Q l -sheaves on A . Then K ∗ ( M [1]) and K ∗ ( N [1]) are perverse.(i) If M ¯ η ∞ ∼ = N ¯ η ∞ , then H − ( K ∗ ( M [1])) ¯ η ∞ ∼ = H − ( K ∗ ( N [1])) ¯ η ∞ .(ii) Suppose L has rank , and all the breaks of L ¯ η ∞′ ⊗ F ( ∞ , ∞ ′ ) ( M ¯ η ∞ ) and L ¯ η ∞′ ⊗ F ( ∞ , ∞ ′ ) ( N ¯ η ∞ ) lie in (1 , ∞ ) . If H − ( K ∗ ( M [1])) ¯ η ∞ ∼ = H − ( K ∗ ( N [1])) ¯ η ∞ , then M ¯ η ∞ ∼ = N ¯ η ∞ .Remark . In Wiener’s Tauberian Theorem 0.1, we have K ∈ L ( R ). This implies that ˆ K is auniformly continuous function on R . This corresponds to the condition in Theorem 0.2 that F ( K )is of the form L [1] for a lisse sheaf L on A ′ . There are many perverse sheaves K on A satisfying thiscondition. For example, we can start with a lisse sheaf L on A ′ , and then take K = a ∗ F ′ ( L [1])(1),where F ′ is the Fourier transform operator defined as above but interchanging the roles of A and A ′ , a : A → A is the k -morphism corresponding to the k -algebra homomorphism k [ x ] → k [ x ] , x
7→ − x, and (1) denotes the Tate twist. Remark . As one can see from the proof of Wiener’s Tauberian Theorem 0.1, the conditionˆ K ( ξ ) = 0 for all ξ ensures that for any g ∈ L ( R ) such that ˆ g has compact support, there exists g ∈ L ( R ) such that ˆ g = ˆ g ˆ K and g = g ∗ K . So the closure of the ideal generated by K in L ( R ). This corresponds to the condition in Theorem 0.2 that F ( K ) = L [1] for a lisse sheaf L ofrank 1 on A ′ . Indeed, for any G ∈ ob D bc ( A , Q ℓ ), we have F ( G ) ∼ = ( F ( G ) ⊗ L − ) ⊗ L ∼ = ( F ( G ) ⊗ L − [ − ⊗ F ( K ) . It follows that G ∼ = G ∗ K, where G = a ∗ F ′ ( F ( G ) ⊗ L − )(1). Remark . It is interesting to find a Tauberian theorem in the case where k is of characteristic 0.In this case, the Fourier transform is not available. We need to find a convenient condition on K which ensures that for any G ∈ ob D bc ( A , Q ℓ ), their exists G ∈ ob D bc ( A , Q ℓ ) such that G ∼ = G ∗ K.
5y [4] Theorem II 8.1, the condition ˆ K ( ξ ) = 0 for all ξ in Wiener’s Theorem 0.1 is equivalentto the condition that if K ∗ f = 0 for some f ∈ L ∞ ( R ), then we have f = 0. So to obtain aTauberian theorem for ℓ -adic sheaves, we may try to find a condition on a perverse sheaf K on A which ensures that for any G ∈ ob D bc ( A , Q ℓ ) such that K ∗ G = 0, we have G = 0. Keep the notations in the introduction. Denote by¯ π : P × k P ′ → P , ¯ π ′ : P × k P ′ → P ′ the projections, by α : A ֒ → P and α ′ : A ′ ֒ → P ′ the immersions, and by L ψ ( xx ′ ) the sheaf( α × α ′ ) ! L ψ ( xx ′ ) on P × k P ′ . For any Q ℓ -representation V of Gal(¯ η x /η x ) or Gal(¯ η x ′ /η x ′ ) and anyinterval ( a, b ) in R , denote by V ( a,b ) the largest subspace of V with breaks lying in ( a, b ). Lemma 1.1.
Let L , U an V be Q ℓ -representations of Gal(¯ η x /η x ) . Suppose either L (1 , ∞ ) = 0 or U [0 , = V [0 , = 0 . (i) If U (1 , ∞ ) ∼ = V (1 , ∞ ) , then ( L ⊗ U ) (1 , ∞ ) ∼ = ( L ⊗ V ) (1 , ∞ ) . (ii) Suppose furthermore that L has rank , and all the breaks of L ⊗ U (1 , ∞ ) and L ⊗ V (1 , ∞ ) lie in (1 , ∞ ) . If ( L ⊗ U ) (1 , ∞ ) ∼ = ( L ⊗ V ) (1 , ∞ ) , then U (1 , ∞ ) ∼ = V (1 , ∞ ) . Proof.
We have decompositions L ∼ = L [0 , M L (1 , ∞ ) , U ∼ = U [0 , M U (1 , ∞ ) . It follows that L ⊗ U ∼ = ( L [0 , ⊗ U [0 , ) M ( L (1 , ∞ ) ⊗ U [0 , ) M ( L ⊗ U (1 , ∞ ) ) . Note that the breaks of L [0 , ⊗ U [0 , lie in [0 , L (1 , ∞ ) ⊗ U [0 , lies in (1 , ∞ ).It follows that ( L ⊗ U ) (1 , ∞ ) ∼ = ( L (1 , ∞ ) ⊗ U [0 , ) M ( L ⊗ U (1 , ∞ ) ) (1 , ∞ ) . Since either L (1 , ∞ ) = 0 or U [0 , = 0, we have( L ⊗ U ) (1 , ∞ ) ∼ = ( L ⊗ U (1 , ∞ ) ) (1 , ∞ ) . We have a similar equation for V . Our assertion follows immediately.6 emma 1.2. Let H be a perverse sheaf on A and let S ⊂ A be the set of those closed points s in A such that either H ( H ) ¯ s = 0 or H − ( H ) is not a lisse sheaf near s . Then we have (cid:16) H − ( F ( H )) ¯ η ∞′ (cid:17) (1 , ∞ ) ∼ = F ( ∞ , ∞ ′ ) ( H − ( H ) ¯ η ∞ ) , (cid:16) H − ( F ( H )) ¯ η ∞′ (cid:17) [0 , ∼ = M s ∈ S R Φ ¯ η ∞′ (cid:0) ¯ π ∗ α ! H ⊗ L ψ ( xx ′ ) (cid:1) ( s, ∞ ′ ) . Proof.
Let j : A − S → A be the open immersion, and let ∆ be the mapping cone of the canonicalmorphism j ! j ∗ H → H . Then ∆ has finite support. Hence H i ( F (∆)) ¯ η ∞′ are extensions of L ψ ( ax ′ ) | ¯ η ∞′ for some a ∈ k . In particular, they have no subspace with breaks lying in (1 , ∞ ). Wehave a distinguished triangle F ( j ! j ∗ H ) → F ( H ) → F (∆) → . It follows that (cid:16) H − ( F ( H )) ¯ η ∞′ (cid:17) (1 , ∞ ) ∼ = (cid:16) H − ( F ( j ! j ∗ H )) ¯ η ∞′ (cid:17) (1 , ∞ ) . By [5] 2.3.3.1, we have H − ( F ( j ! j ∗ H )) ¯ η ∞′ ∼ = M s ∈ S F ( s, ∞ ′ ) ( H − ( H ) ¯ η s ) M F ( ∞ , ∞ ′ ) ( H − ( H ) ¯ η ∞ ) . (1)We have F ( s, ∞ ′ ) ( H − ( H ) ¯ η s ) ∼ = F (0 , ∞ ′ ) ( H − ( H ) ¯ η s ) ⊗ L ψ ( sx ′ ) | ¯ η ∞′ . So by [5] 2.4.3 (i) (b), F ( s, ∞ ′ ) ( H − ( H ) ¯ η s ) has breaks lying in [0 , F ( ∞ , ∞ ′ ) ( H − ( H ) ¯ η ∞ ) has breaks lying in (1 , ∞ ). Taking the part with breaks lying in (1 , ∞ )on both sides of the equation (1), we get the first equation in the lemma. By [5] 2.3.3.1, we have H − ( F ( H )) ¯ η ∞′ ∼ = M s ∈ S R Φ ¯ η ∞′ (cid:16) ¯ π ∗ α ! H ⊗ L ψ ( xx ′ ) (cid:17) ( s, ∞ ′ ) M F ( ∞ , ∞ ′ ) ( H − ( H ) ¯ η ∞ ) . (2)Taking the part with breaks lying in [0 ,
1] on both sides of the equation (2), we get the secondequation in the lemma.The following proposition apparently looks more general than Theorem 0.2.
Proposition 1.3.
Let K ∈ ob D bc ( A , Q l ) be a perverse sheaf on A . Suppose the Fourier transform F ( K ) is of the form L [1] for some lisse Q ℓ -sheaf L on A ′ . Let F, G ∈ ob D bc ( A , Q l ) be perversesheaves on A . Then K ∗ F and K ∗ G are perverse. Suppose furthermore either H − ( F ( F )) [0 , η ∞′ = H − ( F ( G )) [0 , η ∞′ = 0 , r L (1 , ∞ )¯ η ∞′ = 0 . (i) If H − ( F ) ¯ η ∞ ∼ = H − ( G ) ¯ η ∞ , then H − ( K ∗ F ) ¯ η ∞ ∼ = H − ( K ∗ G ) ¯ η ∞ .(ii) Suppose L has rank , and all the breaks of L ¯ η ∞′ ⊗ H − ( F ( F )) (1 , ∞ )¯ η ∞′ and L ¯ η ∞′ ⊗ H − ( F ( G ) ¯ η ∞′ ) (1 , ∞ ) lie in (1 , ∞ ) . If H − ( K ∗ F ) ¯ η ∞ ∼ = H − ( K ∗ G ) ¯ η ∞ , then H − ( F ) ¯ η ∞ ∼ = H − ( G ) ¯ η ∞ .Proof. Denote the Fourier transforms of K and F by b K and b F , respectively. Let a : A → A be the k -morphism corresponding to the k -algebra homomorphism k [ x ] → k [ x ] , x
7→ − x. By [5] 1.2.2.1 and 1.2.2.7, we have K ∗ F ∼ = a ∗ F ′ F ( K ∗ F )(1) ∼ = a ∗ F ′ ( F ( K ) ⊗ F ( F ))[ − ∼ = a ∗ F ′ ( L ⊗ F ( F ))(1) . So by [5] 1.3.2.3, K ∗ F is perverse. Let S ′ ⊂ A ′ be the set of those closed points s ′ in A ′ such thateither H ( F ( F )) ¯ s ′ = 0 or H − ( F ( F )) is not a lisse sheaf near s ′ . By [5] 2.3.3.1, we have H − (cid:16) F ′ ( L ⊗ F ( F )) (cid:17) ¯ η ∞ ∼ = L s ′ ∈ S ′ R Φ ¯ η ∞ (cid:16) ¯ π ′∗ α ′ ! (cid:0) L ⊗ F ( F ) (cid:1) ⊗ L ψ ( xx ′ ) (cid:17) ( ∞ ,s ′ ) L F ( ∞ ′ , ∞ ) ( L ¯ η ∞′ ⊗ H − ( F ( F )) ¯ η ∞′ ) . Since L is lisse on A ′ , we have R Φ ¯ η ∞ (cid:16) ¯ π ′∗ α ′ ! (cid:0) L ⊗ F ( F ) (cid:1) ⊗ L ψ ( xx ′ ) (cid:17) ( ∞ ,s ′ ) ∼ = L ¯ s ′ ⊗ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( F ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) . Denote also by a the morphism η ∞ → η ∞ induced by a . We have H − ( K ∗ F ) ¯ η ∞ ∼ = a ∗ (cid:16) L s ′ ∈ S ′ L ¯ s ′ ⊗ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( F ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) L F ( ∞ ′ , ∞ ) ( L ¯ η ∞′ ⊗ H − ( F ( F )) ¯ η ∞′ ) (cid:17) (1) . By Lemma 1.2, we have H − ( K ∗ F ) (1 , ∞ )¯ η ∞ ∼ = a ∗ (cid:0) F ( ∞ ′ , ∞ ) ( L ¯ η ∞′ ⊗ H − ( F ( F )) ¯ η ∞′ ) (cid:1) (1) , H − ( K ∗ F ) [0 , η ∞ ∼ = a ∗ (cid:16) M s ′ ∈ S ′ L ¯ s ′ ⊗ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( F ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) (cid:17) (1) . H − ( F ) (1 , ∞ )¯ η ∞ ∼ = a ∗ (cid:0) F ( ∞ ′ , ∞ ) ( H − ( F ( F )) ¯ η ∞′ ) (cid:1) (1) , H − ( F ) [0 , η ∞ ∼ = a ∗ (cid:16) M s ′ ∈ S ′ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( F ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) (cid:17) (1) . Let T ′ ⊂ A ′ be the set of those closed points s ′ in A ′ such that either H ( F ( G )) ¯ s ′ = 0 or H − ( F ( G )) is not a lisse sheaf near s ′ . We have similar equations if we replace F by G and S ′ by T ′ .Suppose H − ( F ) ¯ η ∞ ∼ = H − ( G ) ¯ η ∞ . From H − ( F ) [0 , η ∞ ∼ = H − ( G ) [0 , η ∞ . (3)we get a ∗ (cid:16) L s ′ ∈ S ′ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( F ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) (cid:17) (1) ∼ = a ∗ (cid:16) L s ′ ∈ T ′ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( G ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) (cid:17) (1) . (4)Since L is lisse on A ′ , it follows that a ∗ (cid:16) L s ′ ∈ S ′ L ¯ s ′ ⊗ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( F ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) (cid:17) (1) ∼ = a ∗ (cid:16) L s ′ ∈ T ′ L ¯ s ′ ⊗ R Φ ¯ η ∞ (cid:0) ¯ π ′∗ α ′ ! F ( G ) ⊗ L ψ ( xx ′ ) (cid:1) ( ∞ ,s ′ ) (cid:17) (1) . (5)that is, H − ( K ∗ F ) [0 , η ∞ ∼ = H − ( K ∗ G ) [0 , η ∞ . (6)From H − ( F ) (1 , ∞ )¯ η ∞ ∼ = H − ( G ) (1 , ∞ )¯ η ∞ . (7)we get a ∗ (cid:0) F ( ∞ ′ , ∞ ) ( H − ( F ( F )) ¯ η ∞′ ) (cid:1) (1) ∼ = a ∗ (cid:0) F ( ∞ ′ , ∞ ) ( H − ( F ( G )) ¯ η ∞′ ) (cid:1) (1) . (8)So we have F ( ∞ ′ , ∞ ) ( H − ( F ( F )) ¯ η ∞′ ) ∼ = F ( ∞ ′ , ∞ ) ( H − ( F ( G )) ¯ η ∞′ ) . (9)This is equivalent to H − ( F ( F )) (1 , ∞ )¯ η ∞′ ∼ = H − ( F ( G )) (1 , ∞ )¯ η ∞′ (10)9y [5] 2.4.3 (iii) (b) and (c). By Lemma 1.1, we have( L ¯ η ∞′ ⊗ H − ( F ( F )) ¯ η ∞′ ) (1 , ∞ ) ∼ = ( L ¯ η ∞′ ⊗ H − ( F ( G )) ¯ η ∞′ ) (1 , ∞ ) . (11)Hence F ( ∞ ′ , ∞ ) ( L ¯ η ∞′ ⊗ H − ( F ( F )) ¯ η ∞′ ) ∼ = F ( ∞ ′ , ∞ ) ( L ¯ η ∞′ ⊗ H − ( F ( G )) ¯ η ∞′ ) . (12)So we have H − ( K ∗ F ) (1 , ∞ )¯ η ∞ ∼ = H − ( K ∗ G ) (1 , ∞ )¯ η ∞ . (13)By equations (6) and (13), we have H − ( K ∗ F ) ¯ η ∞ ∼ = H − ( K ∗ G ) ¯ η ∞ . The above argument can be reversed. We have the following implications for the above equa-tions: (3) ⇔ (4) ⇒ (5) ⇔ (6) , (7) ⇔ (8) ⇔ (9) ⇔ (10) ⇒ (11) ⇔ (12) ⇔ (13) . Suppose L has rank 1, then we have (5) ⇒ (4). Suppose furthermore that all the breaks of L ¯ η ∞′ ⊗ H − ( F ( F )) (1 , ∞ )¯ η ∞′ and L ¯ η ∞′ ⊗ H − ( F ( G ) ¯ η ∞′ ) (1 , ∞ ) lie in (1 , ∞ ). Then by Lemma 1.1, we have (11) ⇒ (10). If we have H − ( K ∗ F ) ¯ η ∞ ∼ = H − ( K ∗ G ) ¯ η ∞ , then (6) and (13) holds. It follows that (3) and (7) holds. We thus have H − ( F ) ¯ η ∞ ∼ = H − ( G ) ¯ η ∞ . Proof of Theorem 0.2.
Theorem 0.2 follows directly from Proposition 1.3 by taking F = M [1] and G = N [1]. Since M and N are lisse, by [5] 2.3.3.1 (iii), we have H − ( F ( F )) ¯ η ∞′ ∼ = F ( ∞ , ∞ ′ ) ( M ¯ η ∞ ) . By [5] 2.4.3 (iii) (b), the breaks of F ( ∞ , ∞ ′ ) ( M ¯ η ∞ ) lie in (1 , ∞ ). Using this fact, one checks thatthe conditions of Proposition 1.3 hold. 10 emark . Proposition 1.3 is actually not more general than Theorem 0.2. Indeed, if H − ( F ( F )) [0 , η ∞′ = 0 , then F ′ F ( F ) is lisse on A by [5] 2.3.1.3 (ii), and hence F = M [1] for some lisse sheaf M on A .So if we assume the condition H − ( F ( F )) [0 , η ∞′ = H − ( F ( G )) [0 , η ∞′ = 0 , then Proposition 1.3 is exactly Theorem 0.2. If L (1 , ∞ )¯ η ∞′ = 0, then by the formula [5] 2.3.1.1 (i) ′ , K is a perverse sheaf with finite support. In this case, Proposition 1.3 can be proved directly. References [1] A. Beilinson, J. Bernstein and P. Deligne, Faisceaux Pervers, in
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