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An effective 1-motive M =[Γ
f
→ G] over k consists of a locally finite
commutative group scheme Γ/k and a semiabelian variety G/k together with
a morphism of k-group schemes f :Γ→ G such that Γ(
k)isafinitely generated
abelian group. Note that Γ is identified with Γ(
k) endowed with Galois action
because k is a perfect field. Sometimes an effective 1-motive is simply called
a 1-motive, since the category of 1-motives will be defined by modifying only
morphisms. A locally finite commutative group scheme Γ/k and a semiabelian
variety G/k are identified with 1-motives [Γ → 0] and [0 → G] respectively.
An effective morphism of 1-motives
u =(u
lf
,u
sa
):M =[Γ
f
→ G] → M
=[Γ
f
→ G
]
consists of morphisms of k-group schemes u
lf
:Γ→ Γ
and u
sa
: G → G
forming a commutative diagram (together with f,f
). We will denote by
Hom
eff
(M,M
)
the abelian group of effective morphisms of 1-motives.
An effective morphism u =(u
lf
,u
sa
)iscalled strict,ifthe kernel of u
sa
is
connected. We say that u is a quasi-isomorphism if u
sa
is an isogeny and if we
have a commutative diagram with exact rows
(1.1.1)
0 −−→ E −−→ Γ −−→ Γ
−−→ 0
0 −−→ E −−→ G −−→ G
−−→ 0
(i.e. if the right half of the diagram is cartesian).
We define morphisms of 1-motives by inverting quasi-isomorphisms from
the right; i.e. a morphism is represented by u
◦
v
−1
with v a quasi-isomorphism.
More precisely, we define
(1.1.2) Hom(M,M
)=lim
−→
Hom
eff
(
M,M
),
where the inductive limit is taken over isogenies
G → G, and
M =[
Γ →
G]
with
Γ=Γ×
G
G. (This is similar to the localization of a triangulated category
in [33].) Here we may restrict to isogenies n : G → G for positive integers n,
because they form a cofinal index subset. Note that the transition morphisms
of the inductive system are injective by the surjectivity of isogenies together
with the property of fiber product. By (1.2) below, 1-motives form a category
which will be denoted by M
1
(k).
Let Γ
tor
denote the torsion part of Γ, and put M
tor
=Γ
tor
∩ Ker f. This is
identified with [M
tor
→ 0], and is called the torsion part of M .Wesay that M
is reduced if f(Γ
tor
)=0,torsion-free if M
tor
=0,free if Γ
tor
=0,and torsion
DELIGNE’S CONJECTURE ON 1-MOTIVES 597
if Γ is torsion and G =0(i.e. if M = M
tor
). Note that M is free if and only if
it is reduced and torsion-free. We say that M has split torsion,ifM
tor
⊂ Γ
tor
is a direct factor of Γ
tor
.
We define M
fr
=[Γ/Γ
tor
→ G/f(Γ
tor
)]. This is free, and is called the free
part of M.IfM is torsion-free, M
fr
is naturally quasi-isomorphic to M . This
implies that [Γ/M
tor
→ G]isquasi-isomorphic to M
fr
in general, and (1.3)
gives a short exact sequence
0 → M
tor
→ M → M
fr
→ 0.
Remark.IfM is free, M is a 1-motive in the sense of Deligne [10]. We
can show
(1.1.3) Hom
eff
(M,M
)=Hom(M,M
)
for M,M
∈M
1
(k) such that M
is free. This is verified by applying (1.1.1)
to the isogenies
G → G in (1.1.2). In particular, the category of Deligne 1-
motives, denoted by M
1
(k)
fr
,isafull subcategory of M
1
(k). The functoriality
of M → M
fr
implies
(1.1.4) Hom(M
fr
,M
)=Hom(M,M
)
for M ∈M
1
(k), M
∈M
1
(k)
fr
.Inother words, the functor M → M
fr
is left
adjoint of the natural functor M
1
(k)
fr
→M
1
(k).
1.2. Lemma. For any effective morphism u :
M → M
and any quasi -
isomorphism
M
→ M
, there exists a quasi-isomorphism
M →
M together
with a morphism v :
M →
M
forming a commutative diagram. Furthermore,
v is uniquely determined by the other morphisms and the commutativity. In
particular, we have a well-defined composition of morphisms of 1-motives (as
in [33])
(1.2.1) Hom(M,M
) × Hom(M
,M
) → Hom(M,M
).
Proof. For the existence of
M,itissufficient to consider the semiabelian
part
G by the property of fiber product. Then it is clear, because the isogeny
n : G
→ G
factors through
G
→ G
for some positive integer n, and it is
enough to take n :
G →
G.Wehave the uniqueness of v for
G since there is
no nontrivial morphism of
G to the kernel of the isogeny
G
→
G which is a
torsion group. The assertion for
Γ follows from the property of fiber product.
Then the first two assertions imply (1.2.1) using the injectivity of the transition
morphisms.
598 L. BARBIERI-VIALE, A. ROSENSCHON, AND M. SAITO
1.3. Proposition. Let u : M → M
be an effective morphism of
1-motives. Then there exists a quasi-isomorphism
M
→ M
such that u is
lifted to a strict morphism u
: M →
M
(i.e. Ker u
sa
is connected ). In partic-
ular, M
1
(k) is an abelian category.
Proof.Itisenough to show the following assertion for the semiabelian
variety part: There exists an isogeny
G
→ G
with a morphism u
sa
: G →
G
lifting u
sa
such that Ker u
sa
is connected. (Indeed, the first assertion implies the
existence of kernel and cokernel, and their independence of the representative
of a morphism is easy.)
For the proof of the assertion, we may assume that Ker u
sa
is torsion,
dividing G by the identity component of Ker u
sa
. Let n beapositive integer
annihilating E := Ker u
sa
(i.e. E ⊂
n
G). We have a commutative diagram
(1.3.1)
E
n
−−→ E
n
G
ι
−−→ G
n
−−→ G
u
sa
u
sa
n
G
ι
−−→ G
n
−−→ G
.
Let
G
be the quotient of G
by u
sa
ι(
n
G), and let q : G
→
G
denote the
projection. Since u
sa
ι(
n
G) ⊂ ι
(
n
G
), there is a canonical morphism q
:
G
→ G
such that q
q = n : G
→ G
. Then the u
sa
in the right column
of the diagram is lifted to a morphism u
sa
: G →
G
(whose composition with
q
coincides with u
sa
), because G is identified with the quotient of G by
n
G.
Furthermore, Im u
sa
is identified with the quotient of G by
n
G + E, and the
last term coincides with
n
G by the assumption on n.Thusu
sa
is injective, and
the assertion follows.
Remark.Anisogeny of semiabelian varieties G
→ G with kernel E cor-
responds to an injective morphism of 1-motives
[0 → G
] → [E → G
]=[0→ G].
1.4. Lemma. Assume k is algebraically closed. Then, for a 1-motive M,
there exists a quasi-isomorphism M
→ M such that M
=[Γ
f
→ G
] has split
torsion.
Proof. Let n be apositive integer such that E := Γ
tor
∩Ker f is annihilated
by n. Then G
is given by G with isogeny G
→ G defined by the multiplication
DELIGNE’S CONJECTURE ON 1-MOTIVES 599
by n. Let Γ
=Γ×
G
G
.Wehave a diagram of the nine lemma
(1.4.1)
n
G
−−−−
−−−−
n
G
E −−→ Γ
tor
−−→ f
(Γ
tor
)
E −−→ Γ
tor
−−→ f(Γ
tor
).
The l-primary torsion subgroup of G is identified with the quotient of V
l
G :=
T
l
G ⊗
l
l
by M := T
l
G. Let M
be the
l
-submodule of V
l
G such that
M
⊃ M and M
/M is isomorphic to the l-primary part of f(Γ
tor
). Then
there exists a basis {e
i
}
1≤i≤r
of M
together with integers c
i
(1 ≤ i ≤ r) such
that {l
c
i
e
i
}
1≤i≤r
is a basis of M .Sothe assertion is reduced to the following,
because the assumption on the second exact sequence
0 →
n
G → f
(Γ
tor
) → f(Γ
tor
) → 0
is verified by the above argument.
Sublemma. Let 0 → A
i
→ B
i
→ C → 0 be short exact sequences of finite
abelian groups for i =1, 2. Put B = B
1
×
C
B
2
. Assume that the second exact
sequence (i.e., for i =2)is the direct sum of
0 →
/n → /nb
j
→ /b
j
→ 0,
such that A
1
is annihilated by n. Then the projection B → B
2
splits.
Proof. We see that B corresponds to (e
1
,e
2
) ∈ Ext
1
(C, A
1
× A
2
), where
the e
i
∈ Ext
1
(C, A
i
) are defined by the exact sequences. Then it is enough to
construct a morphism u : A
2
→ A
1
such that e
1
is the composition of e
2
and
u,because this implies an automorphism of A
1
× A
2
over A
2
which is defined
by (a
1
,a
2
) → (a
1
− u(a
2
),a
2
)sothat (e
1
,e
2
) corresponds to (0,e
2
). (Indeed,
it induces an automorphism of B over B
2
so that e
1
becomes 0.) But the
existence of such u is clear by hypothesis. This completes the proof of (1.4).
The following is a generalization of Deligne’s construction ([10, 10.1.3]).
1.5. Proposition. If k =
, we have an equivalence of categories
(1.5.1) r
H
: M
1
( )
∼
→ MHS
1
,
where MHS
1
is the category of mixed -Hodge structures H of type
{(0, 0), (0, −1), (−1, 0), (−1, −1)}
such that Gr
W
−1
H is polarizable.
600 L. BARBIERI-VIALE, A. ROSENSCHON, AND M. SAITO
Proof. The argument is essentially the same as in [10]. For a 1-motive
M =[Γ
f
→ G], let Lie G → G be the exponential map, and Γ
1
be its kernel.
Then we have a commutative diagram with exact rows
(1.5.2)
0 −−→ Γ
1
−−→ H −−→ Γ −−→ 0
0 −−→ Γ
1
−−→ Lie G −−→ G −−→ 0
which defines H
, and F
0
H is given by the kernel of the projection
H
:= H ⊗ → Lie G.
We get W
−1
H from Γ
1
, and W
−2
H from the corresponding exact sequence
for the torus part of G. (See also Remark below.)
We can verify that H
and F
0
are independent of the representative of M
(i.e. a quasi-isomorphism induces isomorphisms of H
and F
0
). Indeed, for an
isogeny M
→ M ,wehaveacommutative diagram with exact rows
(1.5.3)
0 −−→ Γ
1
−−→ Lie G
−−→ G
−−→ 0
0 −−→ Γ
1
−−→ Lie G −−→ G −−→ 0
and the assertion follows by taking the base change by Γ → G.Soweget the
canonical functor (1.5.1). We show that this is fully faithful and essentially
surjective. (To construct a quasi-inverse, we have to choose a splitting of the
torsion part of H
for any H ∈ MHS
1
.)
For the proof of the essential surjectivity, we may assume that H is either
torsion-free or torsion. Note that we may assume the same for 1-motives by
(1.4). But for these H we have a canonical quasi-inverse as in [10]. Indeed, if
H is torsion-free, we lift the weight filtration W to H
so that the Gr
W
k
H are
torsion-free. Then we put
Γ=Gr
W
0
H ,G= J(W
−1
H)(=Ext
1
MHS
( ,W
−1
H)),
(see [8]), and f :Γ→ G is given by the boundary map
Hom
MHS
( , Gr
W
0
H) → Ext
1
MHS
( ,W
−1
H)
associated with 0 → W
−1
H → H → Gr
W
0
H → 0. It is easy to see that this is
a quasi-inverse. The quasi-inverse for a torsion H is the obvious one.
As a corollary, we have the full faithfulness of r
H
for free 1-motives using
(1.1.3). So it remains to show that (1.5.1) induces
(1.5.4) Hom(M,M
)=Hom(r
H
(M),r
H
(M
))
when M =[Γ→ G]isfree and M
is torsion. Put H = r
H
(M). We will
identify both M
and r
H
(M
) with a finite abelian group Γ
.
DELIGNE’S CONJECTURE ON 1-MOTIVES 601
Let W
−1
M =[0→ G], Gr
W
0
M =[Γ→ 0]. Then we have a short exact
sequence
0 → Hom(Gr
W
0
M,M
) → Hom(M,M
) → Hom(W
−1
M,M
) → 0,
because Ext
1
(Gr
W
0
M,M
)=Ext
1
(Γ, Γ
)=0. Since we have the corresponding
exact sequence for mixed Hodge structures and the assertion for Gr
W
0
M is
clear, we may assume M = W
−1
M, i.e., Γ = 0.
Let T (G) denote the Tate module of G. This is identified with the com-
pletion of H
using (1.5.3). Then
Hom(M,M
)=Hom(T (G), Γ
)=Hom(H , Γ
),
and the assertion follows.
Remark. Let T be the torus part of G. Then we get in (1.5.2) the integral
weight filtration W on H := r
H
(M)by
(1.5.5) W
−1
H =Γ
1
,W
−2
H =Γ
1
∩ Lie T.
2. Geometric resolution
Using the notion of a complex of varieties together with some arguments
from [17] (see also [14], [16]), we show the existence of a canonical integral
weight filtration on cohomology.
2.1. Let V
k
denote the additive category of k-varieties, where a morphism
X
→ X
is a (formal) finite
-linear combination
i
[f
i
] with f
i
a morphism
of connected component of X
to X
.Itisidentified with a cycle on X
×
k
X
by taking the graph. (This is similar to a construction in [14].) We say that a
morphism
i
n
i
[f
i
]isproper, if each f
i
is. The category of k-varieties in the
usual sense is naturally viewed as a subcategory of the above category. For a
k-variety X,wehave similarly the additive category V
X
consisting of proper
k-varieties over X, where the morphisms are assumed to be defined over X in
the above definition.
Since these are additive categories, we can define the categories of com-
plexes C
k
, C
X
, and also the categories K
k
, K
X
where morphisms are considered
up to homotopy as in [33]. We will denote an object of C
X
, K
X
(or C
k
, K
k
)
by (X
•
,d), where d : X
j
→ X
j−1
is the differential, and will be often omitted
to simplify the notation. The structure morphism is denoted by π : X
•
→ X.
(This lower index of X
•
is due to the fact that we consider only contravariant
functors from this category.) For i ∈
,wedefine the shift of complex by
(X
•
[i])
p
= X
p+i
.Wesay that Y
•
is a closed subcomplex of X
•
if the Y
i
are
closed subvarieties of X
i
, and are stable by the morphisms appearing in the
differential of X
•
.
602 L. BARBIERI-VIALE, A. ROSENSCHON, AND M. SAITO
We will denote by C
b
X
the full subcategory of C
X
consisting of bounded
complexes, and by C
b
X
nsqp
the full subcategory of C
b
X
consisting of complexes
of smooth quasi-projective varieties. (Here nsqp stands for nonsingular and
quasiprojective.) Let D be a closed subvariety of X.Wedenote by C
b
XD
nsqp
the full subcategory of C
b
X
nsqp
consisting of X
•
such that D
j
:= π
−1
(D) ∩
X
j
is locally either a connected component or a divisor with simple normal
crossings for any j. Here simple means that the irreducible components of
D
j
are smooth. For an integer j, let C
b,≥j
X
denote the full subcategory of C
b
X
consisting of complexes X
•
such that X
i
= ∅ for i<j, and similarly for C
b,≥j
X
nsqp
,
C
b,≥j
XD
nsqp
. Replacing C with K,wedefine similarly K
b
X
nsqp
, K
b
XD
nsqp
, etc.
We say that X
•
∈K
b
X
is strongly acyclic if there exist X
•
∈K
b
X
isomorphic
to X
•
in K
b
X
and a finite filtration G on X
•
such that the restriction of G to
each component X
j
is given by direct factors, and for each integer i there exists
a birational proper morphism of k-varieties g : Y
→ Y together with a closed
subvariety Z of Y satisfying the following condition: Letting Z
=(Y
×
Y
Z)
red
,
the morphism g : Y
\ Z
→ Y \ Z is an isomorphism and the graded piece
Gr
G
i
X
•
is isomorphic in K
b
X
to the single complex associated to
(2.1.1)
Z
−−→ Y
Z −−→ Y
up to a shift of complex. Clearly, this condition is stable by mapping cone.
We say that a morphism X
•
→ X
•
in C
b
X
or K
b
X
is a strong quasi-isomorphism
if its mapping cone is strongly acyclic in K
b
X
. This condition is stable by com-
positions, using the octahedral axiom of the triangulated category. Similarly,
if vu and u or v are strongly acyclic, then so is the remaining. (It is not clear
whether the strongly acyclic complexes form a thick subcategory in the sense
of Verdier.)
We say that a proper morphism of k-varieties X
→ X has the lifting
property if it induces a surjective morphism
X
(K) → X(K)
for any field K (see [14]), or equivalently, if any irreducible subvariety of X
can be lifted birationally to X
.Wesay that a morphism u : X
→ X in V
k
has the lifting property,iffor any connected component X
i
of X, there exists
a connected component X
i
of X
such that the restriction of u to X
i
is given
byaproper morphism
f
i
: X
i
→ X
i
with coefficient ±1 and f
i
has the lifting property. We say that a morphism
u : X
→ X in V
k
is of birational type if for any irreducible component X
i
DELIGNE’S CONJECTURE ON 1-MOTIVES 603
of X, there exists uniquely a connected component X
i
of X
such that the
restriction of u to X
i
is defined by a birational proper morphism
f
i
: X
i
→ X
i
with coefficient ±1, and this gives a bijection between the irreducible compo-
nents of X
and X.
For X
•
∈C
b
X
,wesay that a morphism u : X
•
→ X
•
of C
b
X
is a quasi-
projective resolution over XD,ifX
•
∈C
b
XD
nsqp
and u is a strong quasi-
isomorphism in K
b
X
.Wesay that u is a quasi-projective resolution of degree
≥ j over XD,iffurthermore X
•
,X
•
∈C
b,≥j
X
and u : X
j
→ X
j
is of birational
type.Wedenote by K
b
XD
nsqp
(X
•
) the category of quasi-projective resolutions
u : X
•
→ X
•
over XD (which are morphisms in C
b
X
). A morphism of u to
v is a morphism w of the source of u to that of v in K
b
X
such that u = vw in
K
b
X
.IfX
•
∈C
b,≥j
X
,wedefine similarly K
b,≥j
XD
nsqp
(X
•
)byassuming further the
condition on degree ≥ j.
For X
•
∈C
b,≥j
X
and a closed subcomplex Y
•
,wesay that
u :(X
•
,Y
•
) → (X
•
,Y
•
)
is a smooth quasi-projective modification of degree ≥ j,ifX
•
∈C
b,≥j
XD
nsqp
,
Y
•
=(X
•
×
Y
•
X
•
)
red
, u : X
•
→ X
•
is a proper morphism inducing an isomor-
phism X
•
\ Y
•
→ X
•
\ Y
•
, and u : X
j
→ X
j
is of birational type.
Remarks. (i) A birational proper morphism f : X
→ X has the lifting
property. Indeed, according to Hironaka [18], there exists a variety X
together
with morphisms g : X
→ X and h : X
→ X
, such that fh = g and
g is obtained by iterating blowing-ups with smooth centers. (Here we may
assume that the centers are smooth using Hironaka’s theory of resolution of
singularities.) This implies that a proper morphism has the lifting property if
the generic points of the irreducible components can be lifted.
(ii) For X
•
∈C
b,≥j
X
and a closed subcomplex Y
•
such that dim Y
•
< dim X
•
,
there exists a smooth quasi-projective modification (X
•
,Y
•
) → (X
•
,Y
•
)ofde-
gree ≥ j by replacing Y
•
with a larger subcomplex of the same dimension if
necessary. This follows from [17, I, 2.6], except the birationality of X
j
→ X
j
,
because there are connected components of X
•
which are not birational to ir-
reducible components of X
•
. Indeed, if we denote by Z
i,a
the images of the
irreducible components of X
k
(k ≤ i)bymorphisms to X
i
which are obtained
by composing morphisms appearing in the differential of X
•
, then the con-
nected components of X
i
are ‘sufficiently blown-up’ resolutions of singularities
of Z
i,a
, and are defined by increasing induction on i, lifting the differential of X
•
(see loc. cit). However, if Z
j,a
is a proper closed subvariety of some irreducible
604 L. BARBIERI-VIALE, A. ROSENSCHON, AND M. SAITO
component X
j,b
of X
j
,wemay replace the resolution of Z
j,a
by its lifting to the
resolution of X
j,b
using the lifting property, because the differential X
j
→ X
j−1
is zero.
2.2. Proposition. For any X
•
∈C
b,≥j
X
, there exists a quasi-projective
resolution X
•
→ X
•
of degree ≥ j over XD, and the category K
b,≥j
XD
nsqp
(X
•
)
is weakly directed in the following sense: For any u
i
∈K
b,≥j
XD
nsqp
(X
•
)
(i =1, 2), there exists u
3
∈K
b,≥j
XD
nsqp
(X
•
) together with morphisms u
3
→ u
i
in
K
b,≥j
XD
nsqp
(X
•
).
Proof. We first show that K
b,≥j
XD
nsqp
(X
•
)isnonempty by induction on
n := dim X
•
. There exists a smooth quasi-projective modification
(X
•
,Y
•
) → (X
•
,Y
•
)
of degree ≥ j as in the above Remark (ii). Then we have a strong quasi-
isomorphism
C(Y
•
→ X
•
⊕ Y
•
) → X
•
where the direct sum means the disjoint union. So it is enough to show by
induction that
X
•
:= C(Y
•
→ Y
•
) has a strong quasi-isomorphism
(2.2.1)
Z
•
→
X
•
in C
b
X
such that
Z
•
∈C
b,≥j
XD
nsqp
and for any irreducible component
Z
j,i
of
Z
j
the restriction of the differential to some irreducible component
Z
j+1,i
of
Z
j+1
is given by an isomorphism onto
Z
j,i
with coefficient ±1, under the inductive
hypothesis:
(2.2.2)
X
j+1
→
X
j
has the lifting property.
Indeed, admitting this,
Z
•
is then isomorphic to the mapping cone of
Z
•
→ ⊕
i
C(±id :
Z
j,i
→
Z
j,i
)[−j]
with
Z
•
∈C
b,≥j
XD
nsqp
, and the mapping cone of ±id is isomorphic to zero in K
b
X
.
To show (2.2.1), we repeat the above argument with X
•
replaced by
X
•
,
and get a smooth quasi-projective modification (
X
•
,
Y
•
) → (
X
•
,
Y
•
). By the
lifting property (2.2.2), we may assume that for any irreducible component
X
j,i
of
X
j
, the corresponding irreducible component
X
j,i
of
X
j
has a mor-
phism f
i
to
X
j+1
such that the composition of f
i
and d :
X
j+1
→
X
j
is the
canonical morphism
X
j,i
→
X
j,i
up to a sign. If dim
X
j,i
= dim
X
•
, then f
i
induces a birational morphism to Im f
i
and we may assume that there exists an
irreducible component
X
j+1,i
such that the restriction of d to
X
j+1,i
is given
by the isomorphism
X
j+1,i
→
X
j,i
by replacing
X
•
if necessary, because the
DELIGNE’S CONJECTURE ON 1-MOTIVES 605
differential d of
X
•
is defined by lifting d of
X
•
(see loc. cit). Then we can
modify the morphism
X
j+1
→
X
j+1
by using f
i
for dim
X
j,i
< dim
X
•
, and
replace
X
j
with the union of the maximal dimensional components. So we
may assume that
X
j
is equidimensional, because the modified
X
•
→
X
•
still
induces an isomorphism over the complement of
Y
•
by replacing
Y
•
if necessary.
Here we may assume also that
Y
j+1
→
Y
j
has the lifting property by taking
Y
•
appropriately (due to (2.2.2) and the above Remark (i)). Then, considering
the mapping cone of
Y
•
→
Y
•
, the first assertion follows by induction.
The proof of the second assertion is similar. Consider the shifted mapping
cone (i.e. the first term has degree zero):
(2.2.3) X
•
=[X
1,
•
⊕ X
2,
•
→ X
•
],
where the morphism is given by u
1
− u
2
. Then X
•
→ X
a,
•
is a strong quasi-
isomorphism. Note that the composition of the canonical morphism X
•
→ X
a,
•
and u
a
is independent of a up to homotopy.
By definition, for any irreducible component X
j−1,i
= X
j,i
of X
j−1
= X
j
,
there exist two connected components Z
i
,Z
i
of X
j
such that the restrictions of
d to Z
i
,Z
i
are given by proper morphisms Z
i
→ X
j−1,i
,Z
i
→ X
j−1,i
which have
the lifting property (with coefficient ±1). Then by the same argument as above,
we have a smooth quasi-projective modification u
:(X
•
,Y
•
) → (X
•
,Y
•
)of
degree ≥ j − 1. Here we may assume that the connected component X
j−1,i
of X
j−1
which is birational to X
j−1,i
has morphisms to Z
i
,Z
i
factorizing the
morphisms to X
j−1,i
, and X
j
has two connected components such that the
restriction of d to each of these components is given by an isomorphism onto
X
j−1,i
(with coefficients ±1). We may also assume that Y
j
→ Y
j−1
has the
lifting property as before.
Then, applying the same argument to C(Y
•
→ Y
•
), and using induction
on dimension, we get a strong quasi-isomorphism
X
•
→ X
•
such that
X
•
∈C
b,≥j−1
XD
nsqp
, and for any irreducible component
X
j−1,i
of
X
j−1
,
X
j
has two connected components such that the restrictions of d (resp. of the
morphism to X
j
)tothese components are given by isomorphisms onto
X
j−1,i
(resp. by birational proper morphisms to Z
i
, Z
i
) with coefficients ±1. Thus
X
•
is isomorphic to the mapping cone of
(2.2.4)
X
•
→ ⊕
i
C(±id :
X
j−1,i
→
X
j−1,i
)[−j +1]
where
X
•
∈C
b,≥j−1
XD
nsqp
.Sothe second assertion follows.
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