Extensions of filtered Ogus structures
aa r X i v : . [ m a t h . K T ] D ec EXTENSIONS OF FILTERED OGUS STRUCTURES
BRUNO CHIARELLOTTO AND NICOLA MAZZARI
Abstract.
We compute the Ext group of the (filtered) Ogus category over anumber field K . In particular we prove that the filtered Ogus realisation ofmixed motives is not fully faithful. Introduction
Recently Andreatta, Barbieri-Viale and Bertapelle [1] have defined the filteredOgus realisation T FOg for 1-motives over a number field K . In fact by [5] thereexists a cohomology theory for K -varieties with values in FOg ( K ) compatible with T FOg . More precisely let DM gm ( K ) be the Voevodsky’s category of geometricmotives over K , then there exists a (homological) realisation functor R FOg : DM gm ( K ) → D b ( FOg ( K ))compatible with T FOg .The aim of this paper is to compute the Ext group in
FOg (Proposition 3.2).We follow the method of Beilinson [3, 2].It follows (see remark 3.6) that the filtered Ogus realisation of mixed motives isnot fully faithful in general, even though T FOg is fully faithful.1.1.
Notations and conventions.
Throughout this article, K will denote a num-ber field. A place of K will always mean a finite place (we will never need to considerreal or complex places). For every such place v of K , let K v denote the completion, O v the ring of integers, k v the residue field, p v its characteristic, and q v = p n v v itsorder. For all v which are unramified over Q , let σ v denote the lift to K v of theabsolute Frobenius of k v . 2. The categories
The Ogus category.
Let P be a cofinite set of absolutely unramified places of K . We define C P to be the category whose objects are systems M = ( M dR , ( M v , φ v , ǫ v ) v ∈ P )such that:(1) M dR is a finite dimensional K -vector space;(2) ( M v , φ v ) is a F - K v -isocrystal, that is, M v is equipped with a σ v -linearautomorphism φ v ;(3) ǫ = ( ǫ v ) v ∈ P is a system of K v -linear isomorphisms ǫ v : M dR ⊗ K v → M v . A morphism f : M → M ′ is then a collection ( f dR , ( f v ) v ∈ P ) where:(1) f dR : M dR → M ′ dR is a K -linear map; Mathematics Subject Classification.
Key words and phrases.
Filtered Ogus structures, Extensions. (2) f v : M v → M ′ v is K v -linear morphism compatible with Frobenius and suchthat ǫ − v ◦ f v ◦ ǫ v = f dR ⊗ K v .Note that by the second criterion, to specify a morphism it is enough to specify f dR . There are obvious ‘forgetful’ functors C P → C P ′ whenever P ′ ⊂ P and we canform the Ogus category Og ( K ) as the 2-colimit Og ( K ) = 2 colim P C P where P varies over all cofinite sets of unramified places of K . For an object M ∈ Og ( K ) and n ∈ Z we denote by M ( n ) the Tate twist of M , that is whereeach Frobenius φ v is multiplied by p − nv .2.2. Weights. A weight filtration on an object M = ( M dR , ( M v , φ v , ǫ v ) v ∈ P ) ∈ C P is an increasing filtration W • M by sub-objects in C P such that for all v ∈ P thegraded pieces Gr Wi M v are pure of weight i . That is, all eigenvalues of the linearmap φ n v v are Weil numbers of q v -weight i (i.e. all their conjugates have absolutevalue q i/ v [4]). Again, to give a weight filtration on M it suffices to give a filtrationon M dR which induces a weight filtration on all M v .2.3. The filtered Ogus category.
We can therefore consider the filtered Oguscategory
FOg ( K ) whose objects are objects of Og ( K ) equipped with a weightfiltration, and morphisms are required to be compatible with this filtration. Lemma 2.1 ([1], Lemma 1.3.2) . The filtered Ogus category
FOg ( K ) is a Q -linearabelian category, and the forgetful functor FOg ( K ) → Og ( K ) is fully faithful. Internal Hom. If M, N are two objects in
FOg then we can define the in-ternal Hom, denoted by Hom
FOg ( M, N ) as follows:(1) Hom
FOg ( M, N ) dR := Hom K ( M dR , N dR ) is just the usual Hom of K -vectorspaces.(2) for all v , Hom FOg ( M, N ) v := Hom K v ( M v , N v ) and for almost all v this K v -vector space is endowed with the Frobenius f φ Nv ◦ f ◦ ( φ Mv ) − (3) W r Hom
FOg ( M, N ) := { f ∈ Hom
FOg ( M, N ) : f ( W i M ) ⊂ W i + r N } .3. Ext computation
Let
M, N be two objects in C b ( FOg ) (the category of bounded complexes of
FOg ) and consider the following complexes A ( M, N ) = W Hom • ( M, N ) dR = W Hom • K ( M dR , N dR ) B ( M, N ) = ′ Y v W Hom • ( M, N ) v (restricted product)and the morphism ξ M,N : A ( M, N ) → B ( M, N ) ξ ( x ) = ( xφ M − φ N x ) , . We want to prove that the cone of this map compute the ext-groups of
FOg , i.e.Ext i FOg ( M, N ) ∼ = H i − (Cone( ξ M,N )) . XTENSIONS OF FILTERED OGUS STRUCTURES 3
Lemma 3.1.
Let ξ M,N as above, then for any i and for any element b ∈ B i ( M, N ) there exist a quasi-isomorphism N → E of complexes such that the image of b in Coker( ξ M,E ) is zero.Proof. Take b ∈ B ( M, N ), so that b = ( b i ) with b i ∈ Q ′ v W Hom( M i , N i ) v . Thenwe construct E as follows E := Cone((0 , id) : M [ − → N ⊕ M [ − φ E on E i = N i ⊕ M i − ⊕ M i is given by φ E ( x, ,
0) =( φ N ( x ) , , φ E (0 , y,
0) =( b i d M y − d N b i − , φ M ( y ) , φ E (0 , , z ) =( − b i z, , φ M z )By construction N → E is a quasi-isomorphism and there is a short exact se-quence 0 → N → E → Cone(id M )[ − → . Finally we remark that the natural map B ( M, N ) → B ( M, E ) sends b to( b, ,
0) and we can explicitly compute ξ M,E (0 , , id) = (0 , , id) φ M − φ E (0 , , id) = (0 , , φ M ) − ( − b, , φ M ) = ( b, , (cid:3) Proposition 3.2.
Let
M, N be two complexes in C b ( FOg )Ext i FOg ( M, N ) ∼ = H i − (Cone( ξ M,N )) . Proof.
The proof is similar to [2, Proposition 1.7]. We have by definitionExt i FOg ( M, N ) = Hom D b ( FOg )( M,N [ i ]) = colim I Hom K b ( FOg ) ( M, L [ i ])where I is the category of quasi-isomorphisms s : N → L in the homotopy category K b ( FOg ).By the octahedron axiom and the exactness of A ( M, − ) , B ( M, − ) there is a longexact sequence H i (ker ξ M,N ) → H i (Cone( ξ M,N )[ − → H i (coker( ξ M,N )[ − → + . Note that H i (ker ξ M,N ) = Hom K b ( FOg ) ( M, N [ i ]). By the previous lemmacolim I H i (coker( ξ M,L )[ − . Thus we obtain the expected result by taking the colimit over I of the above longexact sequence.We can also give a direct proof in the case of chain complexes concentrated indegree zero, as explained in the following remark. (cid:3) Remark . When
M, N ∈ FOg we can derive the above formula as follows. Let0 → N → E π −→ M → FOg . Choose a section s dR ∈ W Hom(
M, E ) dR of π d R . Afterbase change to K v we get sections s v ∈ W Hom(
M, E ) v and we can define (foralmost all v ) x v := s v ◦ φ M v − φ E v ◦ s v . BRUNO CHIARELLOTTO AND NICOLA MAZZARI
It follows that x v ∈ W Hom(
M, N ) v so that x = ( x v ) v is an element of Q ′ v W Hom(
M, N ) v .Starting with another section s ′ dR we will get another x ′ and the difference x − x ′ lies in ( ◦ φ M − φ N ◦ ) W Hom(
M, N ) dR by construction. Then we easily get a mapΦ : Ext FOg ( M, N ) → Q ′ v W Hom(
M, N ) v ( ◦ φ M − φ N ◦ ) W Hom(
M, N ) dR , Φ( E ) = ( x v ) v Moreover given a family x = ( x v ) v as above we can define the extension E x to bethe direct sum N ⊕ M except for the fact that we set the Frobenius to be φ E,v ( n, m ) := ( φ N,v ( n ) − x v ( m ) , φ M,v ( m )) . By construction we have Φ( E x ) = x and we prove that Φ is an isomorphism. Proposition 3.4.
Let
M, N ∈ FOg there is a short exact sequence → Ext FOg ( M, N ) → Ext Og ( M, N ) → Q ′ v W ≥ Hom(
M, N ) v ( ◦ φ M − φ N ◦ ) Hom( M, N ) dR → . Proof.
The methods we have introduced to compute the extension groups in
FOg K work also for Og K . In fact we consider the above construction forgetting aboutweights A ′ ( M, N ) = Hom • ( M, N ) dR B ′ ( M, N ) = ′ Y v Hom • ( M, N ) v ξ ′ M,N : A ′ ( M, N ) → B ′ ( M, N ) ξ ( x ) = ( xφ M − φ N x )so that we have Ext i Og ( M, N ) ∼ = H i − (Cone( ξ ′ M,N )) . In particular if
M, N ∈ FOg there is an exact sequence of complexes0 → Cone( ξ M,N ) = W Cone( ξ ′ M,N ) → Cone( ξ ′ M,N ) → W ≥ Cone( ξ ′ M,N ) → (cid:3) Remark . Let us consider an intermediate category
FOg ⊂ FOg ′ ⊂ Og whoseobjects are M ∈ Og ( K ) endowed with an increasing filtration M i ⊂ M i +1 (withoutany condition on Frobenius eigenvalues). This is just an exact category and it isnot full in Og . Nevertheless FOg ⊂ Og is full and for two objects M, N ∈ FOg we have Ext FOg ( M, N ) ∼ = Ext FOg ′ ( M, N )just following the previous proof.
Remark . It follows from the previous proposition that the Ext FOg ( M, N ) arenot countable in general and in particular different from motivic cohomology. Forinstance already for K = Q we getExt FOg ( Q , Q (1)) ∼ = { ( a p ) p ∈ Q ′ p Q p }{ ( b − p − b ) p : b ∈ Q } , which is uncountable and so different from Ext DM ( Q , Q (1)) = Q ∗ ⊗ Q . Hence thefiltered Ogus realisation of mixed motives in not full. XTENSIONS OF FILTERED OGUS STRUCTURES 5
References [1] F. Andreatta, L. Barbieri-Viale, and A. Bertapelle. Ogus realization of 1-motives.
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E-mail address : [email protected] Universit´e de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251
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