Tate motives and the vanishing conjectures for algebraic K -theory
Marc Levine
Northeastern University
Boston, MA 02115
USA
partially supported by the NSF
Introduction
The existence of an abelian category TMk of -mixed Tate motives over a field k, whose Ext-groups
compute the weight graded pieces of the rational algebraic K-theory of k, implies the following weak form
of the vanishing conjectures of Soulé and Beilinson:
Conjecture. K2q−p (k)(q) = 0 for p < 0
If TMk has as well a weight filtration structure, the stronger version of Soulé-Beilinson vanishing:
Conjecture. K2q−p (k)( q) = 0 for p ≤ 0 and q > 0
is true. On the other hand, we have constructed in [L] a triangulated category DMk with objects (n), and
with a natural isomorphism
HomDMk ( (0), (q)[p]) ⊗
→ K2q−p (k)(q) .
Letting DTMk be the triangulated subcategory of DMk ⊗ generated by the objects (n), it is natural to
attempt to apply the methods of Faisceaux pervers [BBD] to construct the category TMk as a subcategory
of DTMk . This is the main purpose of this paper.
We will show that, without any assumptions on k, the category DTMk has a natural weight structure.
Furthermore, if we assume the strong version of the vanishing conjecture above for k, there is a t-structure
on DTMk with heart TMk generated as an abelian category by the Tate objects. In addition, there is a
natural duality on TMk , making TMk a Tannakian category. The weight structure on DTMk gives rise to
W
a canonical weight filtration on TMk ; the functor ⊕gr2a
is then a fiber functor from TMk to the Tannakian
category of finite dimensional graded –vector spaces.
What is missing from this picture is the desired relationship between the Ext-groups in TMk and the
shifted Hom-groups in DTMk . By our Theorem 4.2, there is, for each p ≥ 1, a natural map
φp : ExtpTMk (M, N ) → HomDTMk (M, N [p])
(assuming the vanishing conjecture for k). φ1 is an isomorphism, and φ2 is injective. Taking M =
N = (q), and combining with the isomorphism above, we have the maps
(0),
τq,p : ExtpTMk ( (0), (q)) → K2q−p (k)(q) .
If k is a number field, the vanishing conjecture is true; in addition the maps τq,p are isomorphisms (see Cor.
4.3). Presumably, the conjecture of Suslin:
For F an algebraically closed field with subfield of constants F0 , K∗ (F ) is generated by K∗M (F ) and K∗ (F0 ),
would suffice to show that the τq,p are surjective.. Similarly, it would probably suffice that Goncharov’s
complexes [G] compute the weight graded pieces of K∗ (k) ⊗ . It would be interesting to find a criterion for
the maps τq,p to be isomorphisms, without directly refering to Suslin’s conjecture or Goncharov’s complexes.
i
§1. Categories of Tate type
We begin with a discussion of weight filtrations on certain triangulated -categories. This is very much in
the spirit of [BGS]; as many of the results we require do not appear explictly in that paper, we include this
discussion for the reader’s convenience. For the basic notions concerning triangulated categories, we refer
the reader to Verdier [V]. For the foundational aspects of tensor categories, we will use Deligne [D] and
Saavedra-Rivano [S]. By a triangulated tensor category D over a commutative ring A, we mean a category
which is both a triangulated category, and a tensor category over A, with the property:
Let (X, Y, Z, u, v, w) be an exact triangle. Then, for all W in D, (X ⊗ W, Y ⊗ W, Z ⊗ W, u ⊗ id, v ⊗ id, w ⊗ id)
is an exact triangle.
Definition 1.1. A triangulated tensor category of Tate type (over ) is a triangulated tensor category T over
, generated by objects (n), n ∈ , together with isomorphisms
(n) ⊗ (m) →
(n + m),
such that
i) HomT ( (n)[a], (m)[b]) = 0; if n > m
ii) HomT ( (n)[a], (n)[b]) = 0; if a '= b
iii) HomT ( (n), (n)) =
vi) the isomorphisms
· id.
(n) ⊗ (m) →
(n + m)
satisfy the usual compatiblities of associativity and commutativity.
Let T be a triangulated -tensor category of Tate type. We let T[a,b] be the strictly full triangulated
subcategory of T generated by the objects (n), for a ≤ −2n ≤ b; we allow a = −∞, b = ∞, and we denote
T[a,a] by Ta . The axioms (ii) and (iii) above readily imply that, for a even, the category Ta is equivalent to
the derived category of the category of finite dimensional -vector spaces (and is zero for a odd).
Definition 1.2. ([BBD], Def. 1.1.1) A t-structure (T ≤0 , T ≥0 ) on a triangulated category T consists of strictly
full subcategories T ≤0 , T ≥0 of T such that
i) T ≤0 [1] ⊂ T ≤0 and T ≥0 [−1] ⊂ T ≥0
ii) For X in T ≤0 , Y in T ≥0 [−1], we have HomT (X, Y ) = 0
iii) each object X of T fits into an exact triangle
X ≤0 → X → X >0 → X ≤0 [1]
with X ≤0 in T ≤0 and X >0 in T ≥0 [−1]
The t-structure is called non-degenerate if
iv) The intersections ∩n T ≥0 [n] and ∩n T ≤0 [n] consist only of zero objects.
We denote T ≤0 [−n] by T ≤n , T ≥0 [−n] by T ≥n , T ≤n−1 by T <n and T ≥n+1 by T >n . The heart of a
t-structure (T ≤0 , T ≥0 ) on T is the full subcategory T ≤0 ∩ T ≥0 . If T is a triangulated category, A an abelian
subcategory of T , we say that A is admissible if a sequence
i
j
0 → M % → M → M %% → 0
is exact in A if and only if there is an exact triangle
j
i
M % → M → M %% → M % [1]
in T .
Let (T ≤0 , T ≥0 ) be a t-structure on a triangulated category T . Theorem 1.3.6 of [BBD] states that the
heart A of (T ≤0 , T ≥0 ) is a full admissible abelian subcategory of T . In addition, by ([BBD], Prop. 1.3.3),
the triangle X ≤0 → X → X >0 → X ≤0 [1] of Definition 1.2(iii) is uniquely determined by X, up to unique
isomorphism. Sending X to X >0 determines an exact functor
τ>0 : T → T >0 ,
left adjoint to the inclusion T >0 → T ; sending X to X ≤0 determines an exact functor
τ≤0 : T → T ≤0 ,
right adjoint to the inclusion T ≤0 → T .
Lemma 1.1. Let D be a triangulated category, with exact triangles
h
fi
gi
i
Yi →Z
i →Xi →Yi [1],
i=1,2, and
f
Z1 →Z2 →Z→Z1 [1].
Suppose HomD (Y1 , X2 ) = 0. Then there are exact triangles
Y1 → Y2 → Y3 → Y1 [1]
X1 → X2 → X3 → X1 [1]
and
Y3 → Z → X3 → Y3 [1].
Proof. Since HomD (Y1 , X2 ) = 0, we have a commutative diagram
h
→1
Y1
α↓
h
→2
Y2
Z1
f↓
Z2 .
This we can complete (cf. Verdier [V] or [BBD], Prop. 1.1.11) to a diagram with exact rows and columns,
and with all the squares commutative, except the square marked with a ∗, which is anti-commutative:
Y1
α↓
Y2
h
→1
h2
→
β↓
Y3
γ↓
Y1 [1]
Z1
f↓
Z2
f1
→
X1
↓
f2
→
X2
↓
→
h1 [1]
→
Z3
↓
Z1 [1]
g1
→
g2
→
↓
→
X3
f1 [1]
→
2
↓
X1 [1]
Y1 [1]
α[1] ↓
Y2 [1]
β[1] ↓
→
∗
g1 [1]
→
Y3 [1]
γ[1] ↓
Y1 [2].
The identity maps on Z1 and Z2 give the isomorphism of triangles
Z1
||
Z1
f
→ Z2
f
→
Z
||
→
Z1 [1]
g↓
→ Z2
→
Z3
||
→
Z1 [1],
completing the proof.
Lemma 1.2. Let T be a triangulated -tensor category of Tate type, and let a ≤ b ≤ c be integers (we
also allow a = −∞, c = ∞). Then (T[a,b−1] , T[b,c] ) is a t-structure on T[a,c] .
Proof. Since
T(−∞,∞) = ∪−∞<a≤c<∞ T[a,c]
and
T(−∞,b−1] = ∪−∞<a<b T[a,b−1]
T[b,∞) = ∪b≤c<∞ T[b,c] ,
it suffices to prove the lemma for a > −∞ and c < ∞.
We proceed by induction on b − a − 1 and c − b. We first prove (again, by induction on b − a − 1 and c − b)
that HomT (Y, X) = 0 for X in T[b+m,c+m] , Y in T[a−n,b−n−1] and n, m ≥ 0. Indeed, if b = c = a + 1, this is
just Def. 1.1(i). By induction, (T[a,b−2] , Tb−1 ) is a t-structure on T[a,b−1] and (T[b,c−1] , Tc ) is a t-structure on
T[b,c] . Now let X be in T[a,b−1] , Y in T[b,c] . Thus we have exact triangles
and
X ≤b−2 → X → X >b−2 → X ≤b−2 [1]
Y ≤c−1 → Y → Y >c−1 → Y ≤c−1 [1],
with X ≤b−2 in T[a,b−2] , X >b−2 in Tb−1 , etc. The vanishing of HomT (Y, X) follows from the long exact
sequences of Hom’s associated to the two triangles above, together with our induction assumption. In
particular, we have verified Definition 1.2(ii)
We now verify Definition 1.2(iii). Let W be the strictly full additive subcategory of T[a,c] generated by
objects Z of T[a,c] which fit into an exact triangle
Y → Z → X → Y [1],
with Y in T[a,b−1] and X in T[b,c] . Clearly W is graded and contains T[a,b−1] and T[b,c] . Thus, we need only
show that if two members of a triangle in T[a,c] are in W , then so is the third.
Suppose then we have Yi in T[a,b−1] , Xi in T[b,c] , exact triangles
h
fi
gi
i
Yi →Z
i →Xi →Yi [1],
i = 1, 2, and an exact triangle
f
Z1 →Z2 →Z→Z1 [1]
3
By the first part of the proof, we have HomT (Y1 [n], X2 [m]) = 0 for all n, m. By Lemma 1.1, we have triangles
Y1 → Y2 → Y3 → Y1 [1]
X1 → X2 → X3 → X1 [1]
and
Y3 → Z → X3 → Y3 [1].
Since T[a,b−1] and T[b,c] are strictly full triangulated subcategories of T , Y3 is in T[a,b−1] and X3 is in T[b,c] .
Thus, Z is in W , as desired.
Since T[a,b−1] and T[b,c] are graded subcategories of T , Definition 1.2(i) is immediately verified. This
completes the proof.
Definition 1.3. Denote the truncation functors τ ≤0 , τ >0 for the t-structure (T(−∞,b] , T[b+1,∞) ) on T(−∞,∞)
by
W≤b : T(−∞,∞) → T(−∞,b]
and
W >b : T(−∞,∞) → T[b+1,∞) .
For each X in T , we have the exact triangle in T :
(1.1)
W≤b (X) → X → W >b (X) → W≤b (X)[1];
this gives us the functor from T to exact triangles in T :
(1.2)
W≤b (?) → (?) → W >b (?) → W≤b (?)[1];
By the uniqueness of the triangle (1.1), the functors W≤b and W >b map T[a,c] into T[a,b] and T[b+1,c] ,
respectively. For a < b, we have the canonical isomorphisms
(1.3)
W≤a (W≤b (?)) → W≤a (?);
W >b (W >a (?)) → W >b (?),
and the map of triangles
(1.4)
W≤a (?) →
(?)
W≤b (?)
(?)
↓
→
↓
→ W >a (?)
→
W≤a (?)[1]
(?)
→
W≤b (?)[1]
→
W
↓
>b
↓
We write W ≥b for W >b−1 . Let W[a,b] (Z) denote W ≥a (W≤b (Z)), and write graW (Z) for W[a,a] (Z). This
determines functors
W[a,b] : T → T[a,b] ;
the restriction of W[a,b] to T[a,b] is isomorphic to the identity. If a is odd, then we set
zero object.
4
(−a/2) equal to the
Lemma 1.3. The functor
⊕bi=a griW : T[a,b] → ⊕bi=a Ti
is an exact tensor functor.
Proof. We have already remarked that the functors griW are exact. The isomorphisms
(n) ⊗ (m) →
(n + m)
give rise to functorial isomorphisms
W
W
W
gr−2n
(X) ⊗ gr−2m
(X) → gr−2(n+m)
(X),
showing that ⊕bi=a griW is a tensor functor.
Definition 1.4. Let T be a triangulated category of Tate type. Let a be even, and let Ta≥0 be the full
subcategory of Ta generated by objects (−a/2)[n] for n ≤ 0. Similarly, let Ta≤0 be the full subcategory of
≥0
Ta generated by objects (−a/2)[n] for n ≥ 0. For a ≤ b, let T[a,b]
be the full subcategory of T[a,b] generated
≤0
by objects X of T[a,b] with grcW (X) in Tc≥0 for a ≤ c ≤ b. Let T[a,b]
be the full subcategory of T[a,b] generated
W
≤0
by objects X of T[a,b] with grc (X) in Tc for a ≤ c ≤ b.
We note that the two definitions of Ta≤0 and Ta≥0 are the same, and that (Ta≤0 , Ta≥0 ) is the standard
t-structure on Ta , under the equivalence
V /→ V ⊗ (−a/2)
of Ta with the bounded derived category of the category V of finite dimensional -vector spaces.. The
≤0
≤0
≥0
≥0
functors W[c,d] map T[a,b]
to T[c,d]
and T[a,b]
to T[c,d]
. We let GrV denote the tensor category of graded,
finite dimensional -vector spaces.
Theorem 1.4. Suppose T satisfies the vanishing condition:
(1.5)
HomT ( (r), (s)[n]) = 0 for r < s and n ≤ 0.
Then
≤0
≥0
i) (T[a,b]
, T[a,b]
) is a non-degenerate t-structure on T[a,b] for all a ≤ b.
≤0
≥0
ii) The heart A[a,b] of (T[a,b]
, T[a,b]
) contains the objects
category by the (−c/2).
iii) Each object X of A[a,b] has a functorial filtration
(−c/2), a ≤ c ≤ b, and is generated as an abelian
graW (X) ⊂ W[a,a+1] (X) ⊂ . . . ⊂ W[a,b−1] (X) ⊂ X,
with quotients grcW (X) in Ac , a ≤ c ≤ b. We call this filtration the weight filtration on X.
v) The functor
b
!
griW : A[a,b]
i=a
5
→
b
!
i=a
Ai
is a faithful exact tensor functor.
≤0
≥0
Proof. We first claim that HomT (T[a+1,b]
, Tc≥0 ) = 0 for c ≤ a ≤ b. Indeed, let X be in T[a,b]
. We have the
exact triangle
W
gra+1
(X) → X → W[a+2,b] (X)
≤0
with graW (X) in Ta≤0 and W[a+2,b] (X) in T[a+2,b]
. By induction, HomT (W[a+2,b] (X), Tc≥0 ) = 0; since
W
HomT ( (−(a + 1)/2)[n], (−c/2)[m]) = 0 for n ≥ m by the vanishing hypothesis, and since gra+1
(X)
≤0
≤0
W
≥0
≥−1
is in Ta+1 , we have HomT (gra+1 (X), Tc ) = 0. Thus HomT (T[a+1,b] , Tc ) = 0 as claimed.
We now check Def. 1.2(iii). Let X be in T[a,b] . We have the exact triangle
graW (X) → X → W[a+1,b] (X) → graW (X)[1]
and the exact triangle
graW (X)− → graW (X) → graW (X)+ ,
with graW (X)− in Ta≤0 and graW (X)+ in Ta≥1 By induction, we have the exact triangle
W[a+1,b] (X)− → W[a+1,b] (X) → W[a+1,b] (X)+ ,
≤0
≥1
with W[a+1,b] (X)− in T[a+1,b]
and W[a+1,b] (X)+ in T[a+1,b]
. Since HomT (W[a+1,b] (X)− , graW (X)+ [1]) = 0,
we may apply Lemma 1.1, giving us exact triangles
X − → W[a+1,b] (X)− → graW (X)− [1] → X − [1]
and
X + → W[a+1,b] (X)+ → graW (X)+ [1] → X + [1]
X − → X → X + → X − [1]
≤0
≥1
Thus, X − is in T[a,b]
and X + is in T[a,b]
, verifying Def. 1.3(iii).
Def. 1.2(ii) follows from Def. 1.2(iii), and induction, beginning with the result proved in the first
≥0
paragraph of this proof. Def. 1.2(i) follows directly from Definition 1.4: since (−a/2)[n] is in T[a,b]
if n ≤ 0
≤0
and is in T[a,b]
if n > 0. The t-structure is non-degenerate since the induced t-structures on the categories
Tc are all non-degenerate. This completes the proof of (i).
For (ii), we have already seen that each (−c/2) is in A[a,b] . If X is in A[a,b] , then the exact triangles
(1.1) give rise to the functorial filtration
grbW (X) ⊂ W[b−1,b] (X) ⊂ . . . ⊂ W[a+1,b] (X) ⊂ X,
with quotients grcW (X), a ≤ c ≤ b. Since grcW (X) is in A[c,c] , which is generated as an additive category by
(−c/2), we see that the (−c/2) generate A[a,b] . This proves (ii) and (iii).
To prove (iv), it suffices by Lemma 1.3 to show that ⊕grcW is faithful. It suffices by induction to show
that the functor
W[a,b−1] ⊕ grbW : A[a,b] → A[a,b−1] ⊕ Ab
is faithful. For this, we first note that HomA[a,b] (X, Y ) = 0 for X in Ab , Y in A[a,b−1] . Indeed, this follows
from the vanishing hypothesis (1.5), together with the exact sequences of Hom’s arising from the weight
filtration on Y . Let f : X → Y be a map in A[a,b] and suppose grbW (f ) = W[a,b−1] (f ) = 0. Then we can
factor f as a composition
α
X → grbW (X)→W[a,b−1] (Y ) → Y ;
since α = 0, we have f = 0, completing the proof of (iv) and the theorem.
6
Sending (i) to the one dimensional vector space over
defines an equivalence of Ai with V , so
W
gr
defines
an
equivalence
of
A
with
a
subcategory
of
GrV .
[a,b]
i
i=a
"b
Let A be a full, admissible abelian subcategory of a triangulated category D. We will now examine the
relationship between ExtpA (M, N ) and HomD (M, N [p]), for objects M and N of A. This has been carried
out in [BBD]; we include the discussion here for the reader’s convenience.
Each short exact sequence 0 → M % → M → M %% → 0 in A extends uniquely to an exact triangle
%
M → M → M %% → M % [1] in D. Indeed, if we have two extensions, say
α
M % → M → M %% → M % [1]
and
β
M % → M → M %% → M % [1],
the identity maps on M % and M extend to a map of triangles
M%
||
M%
→
→
M
||
M
→
M %%
g↓
M %%
→
α
→
M % [1]
β
M % [1],
→
||
for some map g. Since A is a full subcategory, and the map M → M %% is surjective, we have g = id. Thus
α = β, as claimed. More generally, let
0 → M0 → M1 → . . . → Mn+1 → 0
be a long exact sequence in A. Breaking this sequence up into a series of short exact sequences, we get a
uniquely defined map Mn+1 → M0 [n]. Letting SeqnA (M, N ) denote the set of long exact sequences as above,
with M0 = N and Mn+1 = M , we have defined a map
φ̂n : SeqnA (M, N ) → HomD (M, N [n]).
Lemma 1.5. Suppose we have a commutative ladder
0→
0→
M0
f0 ↓
N0
→
→
M1
f1 ↓
N1
→ ... →
→ ... →
Mn+1
fn+1 ↓
Nn+1
→0
→ 0,
with exact rows. Let M∗ denote the first row, and N∗ the second. Then
φ̂n (N∗ ) ◦ fn+1 = f0 [n] ◦ φ̂n (M∗ ).
Proof. Let M = cok(M0 → M1 ) and N = cok(N0 → N1 ). Then we have
φ̂n (M∗ ) = φ̂1 (M0 → M1 → M )[n − 1] ◦ φ̂n−1 (M → M2 → . . . → Mn+1 ),
and similarly for φ̂n (N∗ ). By induction, this reduces us to the case n = 1.
7
Each ladder
0→
0→
M0
f0 ↓
N0
→
→
M1
→0
→
M1
f1 ↓
→
N1
M2
f2 ↓
→
N2
→0
extends to the map of triangles
0→
M0
f0 ↓
0→
→
f1 ↓
→
N0
M2
φ̂1 (M∗ )
M0 [1]
→
f0 [1] ↓
g↓
→
N1
N2
φ̂1 (N∗ )
N0 [1],
→
for some map g. Since the maps M1 → M2 and N1 → N2 are surjective, we have g = f2 , proving the case
n = 1, and completing the proof.
Proposition 1.6. Let A be a full admissible abelian subcategory of a triangulated category D. Let M and
N be in A. Then the map φ̂n descends to a homomorphism
φn : ExtnA (M, N ) → HomD (M, N [n]).
In addition, if A is closed under extensions in D, then
φ1 : Ext1A (M, N ) → HomD (M, N [1]).
is an isomorphism, and
φ2 : Ext2A (M, N ) → HomD (M, N [2])
is an injection. Here Ext is the Yoneda Ext.
Proof. Suppose we have a commutative ladder
0→
N
idN ↓
0→
N
→
M1
f1 ↓
→
N1
→
→
... →
Mn
fn ↓
... →
Nn
→
M
idM ↓
→
M
→0
→ 0,
with exact rows M∗ and N∗ . By Lemma 1.5, φ̂n (M∗ ) = φ̂n (N∗ ). As ExtnA (M, N ) is the quotient of the set
SeqnA (M, N ) by the relations given by commutative ladders as above, we have shown that φ̂n descends to
the map φn , as claimed. The addition in ExtnA (M, N ) is gotten by taking direct sums of sequences, pushing
out by the sum N ⊕ N → N and pulling back by the diagonal M → M ⊕ M . As f + g : M → N [n] is the
composition
∆
f ⊕g
Σ
M → M ⊕ M → N [n] ⊕ N [n] → N [n],
applying Lemma 1.5 shows that φn is a homomorphism.
To show φ1 is an isomorphism, let α: M → N [1] be a map in D. If
α
N → E → M → N [1]
8
α
N → E % → M → N [1]
are two triangles in D, then we have the map of triangles
→
N
||
E
↓f
E%
→
N
→
M
||
→
M
α
→
N [1]
α
N [1]
|| .
→
α
This shows that filling in α: M → N [1] to a triangle N → E → M → N [1], and taking the extension class
of the short exact sequence 0 → N → E → M → 0 in A defines an inverse to φ1 .
Finally, to show that φ2 is injective, we first note that, if N ⊂ B ⊂ X is a filtration in A, then the class
of the exact sequence
0 → N → B → X/N → X/B → 0
in Ext2A (X/B, N ) is zero. Now suppose
M ∗ = 0 → N → M1 → M2 → M → 0
is in Seq2A (M, N ), with φ2 (M∗ ) = 0. Let C = cok(N → M1 ). Then we have the sequence
β
α
M [−1] → C → N [1],
with α = φ1 (C → M2 → M )[−1], β = φ1 (N → M1 → C), and β ◦ α = 0. This gives the commutative square
M [−1]
↓
0
α
→
C
→
N [1] ,
↓β
which we fill in to a map of triangles
M [−1]
(1.6)
↓
0
Filling in γ: M2 → N [1] to a triangle
→
α
C
→
N [1]
↓β
→
M2
=
N [1].
↓γ
X → M2 → N [1],
we have the short exact sequence in A
0 → N → X → M2 → 0.
The octahedral axiom applied to the map of triangles (1.6) gives the triangle
M [−1] → M1 → X,
9
giving the short exact sequence in A
0 → M1 → X → M → 0,
and the commutative diagram
0→
0→
N
↓
M1
→ X
||
→ X
→
→
M2 → 0
↓
M →0
This gives the filtration N ⊂ M1 ⊂ X trivializing the element of Ext2 (M, N ) defined by M∗ , which shows
that φ2 is injective.
10
§2. Duality
Our object in this section is to show that there is a canonical duality on the heart of a triangulated category
of Tate type, assuming the vanishing condition (1.5) of Theorem 1.4. We begin with some generalities on
duality in tensor categories.
Let A be a small tensor category with unit 1. A duality D on A is a map D: Obj(A) → Obj(A),
D(X) = X D , together with maps
%
δ
X
X
1→
X ⊗ XD →
1,
such that the composition
δ⊗id
id⊗τ
id⊗%
X → 1 ⊗ X → X ⊗ XD ⊗ X → X ⊗ X ⊗ XD → X ⊗ 1 → X
is the identity. We assume in addition that δ1 and '1 are the isomorphisms given by the product 1 ⊗ 1 → 1.
A duality D on A gives rise to homomorphisms
δ(M,N,Z) : HomA (Z ⊗ M, N ) → HomA (Z, N ⊗ M D ); '(M,N,Z) : HomA (M, Z ⊗ N ) → HomA (M ⊗ N D , Z)
by taking the compositions
HomA (Z ⊗ M, N ) → HomA (Z ⊗ M ⊗ M D , N ⊗ M D )
∗
idZ ⊗δM
−→
HomA (Z, N ⊗ M D )
and
HomA (M, Z ⊗ N ) → HomA (M ⊗ N D , Z ⊗ N ⊗ N D )
idZ ⊗%N ∗
−→
HomA (M ⊗ N D , Z).
∗
We let DM,N,Z : HomA (Z ⊗M, N ) → HomA (Z ⊗N D , M D ) be the map 'Z,N,M D ◦τN,M
D ◦δM,N,Z . We have as
well the canonical map M → M D
D
given by δM D ,1,M ◦ 'M,M,1 (idM ). We call D a perfect duality if δ(M,N,Z)
D
and '(M,N,Z) are isomorphisms for all M , N and Z. The maps DM,N,Z and the map M → M D are then
isomorphisms for all M , N and Z.
If A has a perfect duality D, we can define an internal Hom on A by Hom(M, N ) = N ⊗ M D . This
gives A the structure of a rigid Tannakian category.
If A is a graded tensor category, a duality D is a graded duality on A if M [n]D = M D [−n], and the
maps δM [n] and 'M [n] are the maps δM and 'M composed with the isomorphism M [n]⊗M D [−n] ∼
= M ⊗M D .
Let T be a triangulated category of Tate type, satisfying the vanishing conditions of Theorem 1.4, A the
heart of (T ≤0 , T ≥0 ). The graded category gr(A) := ⊕c Ac is equivalent to the category of finite dimensional
(even) graded -vector spaces; fixing an equivalence, we denote the object corresponding to a vector space V
in degree −2a (Va ) by V ⊗ (a). Similarly, the category ⊕c Tc is equivalent to the bounded derived category
Db GrV , we denote the object corresponding to V [n] in degree −2a by V ⊗ (a)[n]. For a -vector space
V , denote the dual by V D ; this gives the usual duality in GrV by (Va )D = (V D )−a . This defines the
duality in ⊕c Ac by
(V ⊗ (a))D = V D ⊗ (−a),
where the maps
1 → (V ⊗ (a)) ⊗ (V ⊗ (a))D → 1
defining the duality isomorphisms
and
Homgr(A) (V ⊗ (a), W ⊗ (a)) → Homgr(A) (1, (W ⊗ (a)) ⊗ (V ⊗ (a))D )
11
Homgr(A) (V ⊗ (a), W ⊗ (a)) → Homgr(A) ((W ⊗ (a))D ⊗ (V ⊗ (a)), 1)
are given by the duality maps
→V ⊗VD →
combined with the isomorphism
(V ⊗ (a)) ⊗ (V D ⊗ (−a)) → (V ⊗ V D ) ⊗ ( (a) ⊗ (−a)) → (V ⊗ V D ) ⊗ (0)
and the isomorphism
⊗ (0) →
(0) → 1.
This extends in the obvious way to a graded duality on ⊕c Tc . We let A[∗] denote the full graded subcategory
of T generated by A
Proposition 2.1. There is a perfect graded duality D: A[∗] → A[∗] on A[∗] such that the functor
⊕a gra : A → gr(A)
is compatible with the duality functors. The duality D is unique up to natural isomorphism.
Proof. We may assume by induction that we have defined a perfect duality D on the full additive subcategory
A(n) of A generated by objects those X in A having a weight filtration of length less than n; the induction
starts by defining D on A(1) via the isomorphism A(1) → gr(A), and using the duality we have defined
above on gr(A). D extends canonically to a duality on the graded category A(n)[∗] generated by A(n). We
may also assume that the maps δM,N,Z are isomorphisms for all M in A(n)[∗], and for all N and Z in A[∗].
Similarly, we may assume that the maps 'M,N,Z are isomorphisms for all N in A(n)[∗], and for all M and Z
in A[∗]; in particular, D defines a perfect duality on A(n)[∗].
Let X be in A(n + 1), and let a be the minimum integer for which W≤a (X) is not zero. We have the
exact sequence
(2.1)
0 → gra (X) → X → W >a (X) → 0;
clearly W >a (X) is in A(n). Thus, we have maps
1
δW >a (X)
−→
D %W >a (X)
W >a (X) ⊗ W >a (X)
δgra (X)
−→
1
D %gra (X)
1 −→ gra (X) ⊗ gra (X)
−→ 1
giving the duality isomorphisms on the appropriate Hom groups.
The exact sequence (2.1) defines the map α: W >a (X) → gra (X)[1], which in turn defines the map
D
α : gra (X)D [−1] → W >a (X)D . Let X D be an object of A fitting into the exact triangle
αD [1]
W >a (X)D → X D → gra (X)D −→ W >a (X)D [1].
The exact sequence
0 → W >a (X)D → X D → gra (X)D → 0
12
is uniquely isomorphic to the sequence
0 → W<−a (X D ) → X D → W ≥−a (X) → 0,
hence X D is determined by X up to unique isomorphism.
We have the filtration
W<0 (X ⊗ X D ) ⊂ W≤0 (X ⊗ X D ) ⊂ X ⊗ X D ,
and isomorphisms
W<0 (X ⊗ X D ) ∼
= graW (X) ⊗ W >a (X)D ; W≤0 (X ⊗ X D ) ∼
= graW (X) ⊗ X D + X ⊗ W >a (X)D
and
gr0W (X ⊗ X D ) ∼
= graW (X) ⊗ (graW (X))D ⊕ W >a (X) ⊗ W >a (X)D .
In addition, via these isomorphisms, we can identify the map γ in the triangle
γ
W<0 (X ⊗ X D ) → W≤0 (X ⊗ X D ) → gr0W (X ⊗ X D ) →W<0 (X ⊗ X D )[1]
with the map
1 ⊗ αD [1] + ω ◦ α ⊗ 1: graW (X) ⊗ (graW (X))D ⊕ W >a (X) ⊗ W >a (X)D → graW (X) ⊗ W >a (X)D [1],
where ω: graW (X)[1] ⊗ W >a (X)D → graW (X) ⊗ W >a (X)D [1] is the canonical isomorphism.
One computes directly that
(1 ⊗ αD [1]) ◦ δgra (X) = −(ω ◦ α ⊗ 1) ◦ δW >a (X) ,
the minus sign coming from the shift by 1. Thus the map
δgra (X) ⊕ δW >a (X) : 1 → gr0 (X ⊗ X D )
lifts to a map δ: 1 → W≤0 (X ⊗ X D ); since HomA (1, W<0 (A)) = 0, the lifting is unique. This defines the map
δX : 1 → X ⊗ X D .
The constuction of 'X : X ⊗ X D → 1 is similar, and is left to the reader.
The above construction is clearly compatible, via the functor gr: A → gr(A), with the perfect dualtity
on gr(A). As gr is faithful, this implies that we have extended D to a duality on A(n + 1), which then
extends canonically to a duality on A(n + 1)[∗]. If M is in A(n + 1), we have the short exact sequences
(2.2)
0 → graW (M ) → M → W >a (M ) → 0
0 → W >a (M )D → M D → graW (M )D → 0
where a is the minimal integer such that graW (M ) '= 0. Using the long exact sequences of Hom’s coming
from the appropriate short exact sequences, and using induction, we find that the maps δN,M,Z and 'N,M,Z
are isomorphisms for all N and Z in A[∗]. Thus, D defines a perfect duality on A(n + 1)[∗]. The induction
therefore goes through, completing the proof.
13
§3. The triangulated category of Tate motives
In this section we recall some aspects of the construction of the motivic triangulated category DM given in
[L], and describe the triangulated Tate category DTM as a subcategory of DM.
Fix a base field k, and let Schk denote the category of smooth, quasi-projective schemes over k. Let
PSchk denote the category of pairs (X, F ), where X is in Schk and F is a closed subset of X; a map of
pairs f : (X, F ) → (Y, G) is a map f : X → Y with f −1 (G) ⊂ F . The category DMk is a triangulated tensor
category, equipped with a functor
mot
: PSchop
k ×
→ DMk .
mot
We write mot
((X, F ), n), and for a morphism p in PSchk , we denote mot (p) by p∗ . We denote
X,F (n) for
mot
mot
mot
X,X (n) by X (n), and Spec(k) (n) by (n); (0) is the unit for the tensor structure on DMk .
mot
The functor
satisfies the axioms for a twisted duality theory (see Gillet [G]) in the following sense:
a) (Homotopy) Let p:
1
X
→ X be the projection. Then
p∗ :
mot
X,F (n)
mot
1 ,
X
→
1
F
(n)
is an isomorphism for every n.
b) (Localization) Let F ⊂ G be closed subsets of some X in Schk , let j: U → X be the inclusion of X\F ,
mot
let H = G\F , and let i∗ : mot
X,F (n) →
X,G (n) be the map induced by the identity on X. Then the
sequence
i∗
mot
X,F (n) →
j∗
mot
X,F (n) →
mot
U,H (n)
extends canonically to an exact triangle
j∗
mot
X,F (n) →
i∗
mot
X,G (n) →
mot
U,H (n)
→
mot
X,G (n)[1].
c) (Künneth formula) There are functorial exterior products
:
mot
X,F (n)
⊗
mot
Y,G (m)
→
mot
X×Y,F ×G (n
+ m)
which are isomorphisms. This defines the cup product
∪:
mot
X,F (n)
⊗
mot
X,G (m)
→
mot
X,F ∩G (n
+ m)
by ∪ = ∆∗ ◦ .
d) (Poincaré duality) Let i: Z → X be a closed embedding of pure codimension d in Schk , F a closed
subset of Z. There is an isomorphism
i∗ :
mot
Z,F (n)
→
mot
X,F (n
+ d)[2d].
e) (cycle classes) For (X, F ) in PSchk , let ZdF (X) denote the group of codimension d cycles on X, supported on F , ChdF (X) the quotient group of cycles modulo rational equivalence (on F ). There is a
homomorphism
cl: ChdF (X) → HomDM ( (0),
14
mot
Z,F (d)[2d]).
f) (projective bundle formula) Let p: PX → X be a n -bundle over X in Schk , F a closed subset of X,
and PF = p−1 (F ). Let ξ = cl(O(1)). Then the maps
(ξ i ∪?) ◦ p∗ :
mot
X,F (q
− i)[−2i] →
mot
PX ,PF (q)
define an isomorphism
n
!
mot
X,F (q
i=0
g) (projective pushforward) For a
n
mot
PX ,PF (q).
− i)[−2i] →
-bundle as in (f), let
p∗ :
mot
PX ,PF (q)
→
mot
X,F (q
− n)[−2n]
be the inverse of the isomorphism of (f), followed by projection on the factor mot
X,F (q − n)[−2n]. Let
f : Y → X be a projective map in Schk , G a closed subset of Y and F a closed subset of X containing
f (G). Factor f as p ◦ i, with i: Y → P a closed embedding, and p: P → X a projective bundle. Let
d = dim(Y ) − dim(X). Then the composition
p∗ ◦ i∗ :
mot
Y,G (q)
→
mot
X,F (q
− d)[−2d]
is independent of the factorization of f . Defining f∗ as this composition, we have f∗ ◦ g∗ = (f ◦ g)∗ .
h) (projection formula) Let f : Y → X be projective. Then
f∗ ◦ (f ∗ (?) ∪ (?)) = (?) ∪ f∗ (?)
mot
mot
as maps from mot
X,F (n) ⊗ Y,G (m) to X,F ∩f (G) (n + m − d)[−2d]. In addition, the cycle class map cl is
compatible with pullback and pushforward.
For a sub-ring A of , we let DMk ⊗ A denote the category with the same objects as DMk , with
HomDMk ⊗A (X, Y ) = HomDMk (X, Y ) ⊗ A. If A is flat over , this defines a triangulated tensor category
over A, satisfying the properties (a)-(h) above. We denote the object mot
X,F (n), considered as an object of
DMk ⊗ A, by Amot
(n).
X,F
p
We define the motivic cohomology groups, Hmot
(X, F, (q)), by
p
Hmot
(X, F, (q)) = HomDMk ( (0),
For A flat over
mot
X,F (q)[p]).
p
p
as above, set Hmot
(X, F, A(q)) := Hmot
(X, F, (q)) ⊗ A = HomDMk ⊗A (A(0), Amot
X,F (q)[p]).
The category DMk has a universal mapping property for cohomology theories which are constructed in
a certain way, which we won’t spell out here. In particular, the theories of singular and étale cohomology, as
well as Beilinson’s absolute Hodge cohomology, admit realization functors from the category DMk . These
in turn give rise to functorial maps
p
p
ReB,σ : Hmot
(X, F, (q)) → HB
(X σ ( ), F σ ( ), (q))
and
p
p
Reét : Hmot
(X, F, (q)) → Hét
(X, F,
15
l (q))
p
p
ReH,σ : Hmot
(X, F, (q)) → HH
(X σ ( ), F σ ( ), (q)).
p
p
p
Here σ is an embedding of k into and HB
and HH
, Hét
denote the Betti, étale and Hodge cohomology,
p
respectively. We can define the mod-n theory Hmot (X, F, /n(q)) by defining the object mot
X,F (q)/n, fitting
into an exact triangle
mot
X,F (q)/n
→
×n mot
mot
X,F (q) → X,F (q),
and setting
p
Hmot
(X, F, /n(q)) = HomDMk ( (0),
mot
X,F (q)/n[p]).
The realization functors ReB,σ and Reét extend to realization functors
p
p
ReB,σ : Hmot
(X, F, /n(q)) → HB
(X σ ( ), F σ ( ), /n(q))
p
p
Reét : Hmot
(X, F, /n(q)) → Hét
(X, F, /n(q)),
compatible with the appropriate Bockstein sequences.
Let Chq (X, n) denote Bloch’s higher Chow group of codimension q cycles on the “algebraic n-sphere”
over X (see [B] for details). For a closed subset F of X, let Chq (X, F, n) denote the higher Chow group
n
of codimension q cycles on SX
, with support on SFn . Combining the cycle class map cl with the homotopy
property (a), we arrive at the homomorphism
n
SX
p
clq,p : Chq (X, F, 2q − p) → Hmot
(X, F, (q)).
Theorem 1(Theorem 5.2 of [L]). Let A be a sub-ring of , flat over
groups Chq (?, n) ⊗ A satisfy the localization property. Then the map
, such that Bloch’s higher Chow
p
clq,p : Chq (X, F, 2q − p) ⊗ A → Hmot
(X, F, A(q))
is an isomorphism.
Presumably, Bloch’s higher Chow groups, (or some suitable modification) satisfy the localization property over , but this is not at present known. Since the higher Chow group Chq (X, F, p) ⊗ agrees with
the weight q portion of Gp (F ) ⊗ , it follows that Bloch’s higher Chow groups Chq (?, n) ⊗ satisfy the
localization property. Thus we have
Corollary 3.1. The map
clq,p : Chq (X, 2q − p) ⊗
p
→ Hmot
(X, (q))
p
defines an isomorphism of K2q−p (X)(q) with Hmot
(X, (q)).
Definition 3.1. Let DTMk , the triangulated category of Tate motives, be the strictly full subcategory of
DMk ⊗ generated by the objects (n) for n ∈ .
16
§4. Tate motives and the vanishing conjectures
In this section, we apply the results of §1 and §2 to the category DTMk .
By Corollary 3.1, we have the isomorphism
(4.1)
K2q−p (k)(q) → HomDTMk ( (n), (n + q)[p]).
Since K2q−p (k)(q) = 0 for q < 0, Kp (k)(0) = 0 for p '= 0 and K0 (k)(0) =
Theorem 4.1. The category DTMk is a
, we have
-triangulated tensor category of Tate type.
We recall the strong version of the vanishing conjectures of Soulé and Beilinson:
Conjecture. K2q−p (k)(q) = 0 if p ≤ 0 and q > 0.
≤0
Define the full subcategories DTM≥0
k and DTMk of DTMk by
W
mn
∼
X is in DTM≥0
[n] for all a
k if and only if gra (X) = ⊕n≤0 (−a/2)
W
mn
∼
X is in DTM≤0
[n] for all a.
k if and only if gra (X) = ⊕n≥0 (−a/2)
Theorem 4.2. Suppose the field k satisfies the vanishing conjecture of Soulé and Beilinson. Then
≥0
i) (DTM≤0
(n), n ∈ .
k , DTMk ) is a t-structure on DTMk , with heart TMk generated by the objects
W
ii) Composing the functor gri with the equivalence TMk,i → V gives a faithful exact tensor functor
b
!
i=a
griW : TMk → GrV
iii) There is a perfect duality on TMk , making
iv) For each p, there is a natural map
"b
i=a
griW into a Tannakian functor.
φp : ExtpTMk (M, N ) → HomDTMk (M, N [p]).
φ1 is an isomorphism, and φ2 is injective.
Proof. The first three assertions follow from Theorem 1.4, the isomorphism (4.1) and Proposition 2.1. Item
(iv) follows from Prop. 1.6.
Take M =
(0), N =
(q) in Thm. 4.2(iv). Composing the map φp with the isomorphism (4.1)
HomDTMk ( (0), (q)[p]) → K2q−p (k)(q) ,
we arrive at the homomorphism
τq,p : ExtpTMk ( (0), (q)) → K2q−p (k)(q) ;
τq,1 is an isomorphism, and τq,2 is injective.
Let k be a number field. It follows from Borel’s computation (see [Bo]) of the rational K-groups of
number fields that, for q > 0, K2q (k) ⊗ = 0 and K2q−1 (k) ⊗ = K2q−p (k)(q) .
17
≥0
Corollary 4.3. Let k be a number field. Then (DTM≤0
k , DTMk ) is a t-structure on DTMk , with heart
W
TMk generated by the objects (n), n ∈ . The functors gri give an equivalence of TMk with a tensor
subcategory of GrV . In addition, for M and N in TMk the maps
φp : ExtpTMk (M, N ) → HomDTMk (M, N [p]).
are isomorphisms for all p (both sides are zero for p > 1). In particular, the maps
τq,p : ExtpTMk ( (0), (q)) → K2q−p (k)(q)
are isomorphisms for all p and q.
Proof. We have HomDTMk ( (0), (q)[p]) = K2q−p (k)(q) . Since K2q−p (k)(q) = 0 if q '= 0 and p '= 1, we may
apply Theorem 4.2 to prove the first two assertions. Also, we have
HomDTMk ( (0), (q)[p]) = 0
for q '= 0 and p '= 1. By Theorem 4.2(iv), this implies that Ext2TMk ( (a), (b)) = 0 for all a and b. Since
each object in TMk has its weight filtration, with quotients direct sums of the (a) for varying a, this
implies that Ext2TMk (M, N ) = 0 for all M and N in TMk . This in turn implies that ExtpTMk (M, N ) = 0
for all M and N in TMk , and for all p ≥ 2. A similar argument shows that HomDTMk (M, N [p]) = 0 for all
all M and N in TMk , and for all p ≥ 2. Since φ1 is an isomorphism, the proof is complete.
18
References
[BBD]
A.A. Beilinson, J.N. Bernstein, P. Deligne, Faisceaux pervers, in Asterisque 100, Soc. Math.
France 1982
[BGS]
A.A. Beilinson, V.A. Ginzberg, V.V. Schechtman, Koszul Duality, J. Geom. Phys. 5(1988) no.
3, 317-350.
[B]
S. Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 No. 3(1986) 267-304.
[Bo]
A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Sup. Ser. 4 7(1974)
235-272.
[D]
P. Deligne, Tannakian Categories, in Hodge Cycles, Motives and Shimura Varieties, LNM
900, Springer 1982.
[G]
A. Goncharov, Polylogarithms and motivic cohomology, preprint (1991).
[L]
M. Levine, The derived motivic category, preprint (1991).
[S]
N. Saavedra Rivano, Catégories Tannakiennes, LNM 265, Springer 1972.
[V]
J.L. Verdier, Catégories triangulées, état 0, in SGA 4 1/2 LNM () 262-308.
19