Algebraic cycles and the triangulated category of mixed motives

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by  Kazuma Morita

ALGEBRAIC CYCLES AND THE TRIANGULATED
CATEGORY OF MIXED MOTIVES
KAZUMA MORITA
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n Abstract. In this paper, we shallgive a candidate for the t-structure on the
a triangulated category of mixed motives due to Voevodsky.
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1. Introduction

]
T
N Let k be a field of characteristic 0 and L(k) denote the category of the smooth
. schemes over k. Voevodsky constructsthetriangulatedcategoryofmixedmotives
h
t with coefficients in Q (denoted by DMgm(k)Q) such that we have a canonical
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m functor
M : L(k) → DM (k) , X 7→ M(X).
[ gm Q

1 It has been believed that a good t-structure on DMgm(k)Q would capture the
v mixed motives of Grothendieck and this is one of the most significant problems
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in the field of algebraic cycles.
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Remark 1.1. Although other triangulated categories of mixed motives are con-
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. structed ([H],[L]), weshall usethecategoryofVoevodsky since itiswidely spread
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0 and more accessible than others.
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v 2. Review of t-structures
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r For a triangulated category D, let D≤0 and D≥0 be full subcategories of D. We
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say that the couple (D≤0,D≥0) is a t-structure on D if the following conditions
are satisfied:
(1) D≤0[1] ⊂ D≤0 and D≥0[−1] ⊂ D≥0
(2) Hom (X,Y) = 0 for X ∈ D≤0 and Y ∈ D≥0[−1]
D
(3) For any X ∈ D, there exists a distinguished triangle X → X → X → in
0 1
D such that we have X ∈ D≤0 and X ∈ D≥0[−1].
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The full subcategory A = D≤0 ∩ D≥0 is called the heart of the t-structure. In
the next section, we shall give a t-structure on D = DM (k) which reflects the
gm Q
Date: January 22, 2013.
1991 Mathematics Subject Classification. 14C15, 14C25 .
Key words and phrases. Mixed motives, Algebraic cycles.
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intersections of algebraic cycles and we expect that the heart of this t-structure
is the mixed motives which Grothendieck has dreamed of.
3. Definition of a t-structure
Keep the notation as in the Introduction and denote D = DM (k) for sim-
gm Q
plicity. Let us define a t-structure (D≤0,D≥0) on D as follows.
Preliminary
Let φ : A → M(X) and ψ : M(Y) → B be two morphisms in D. If we
convert the orientations of X and Y, we denote the corresponding morphisms by
φX : A → M(X) and ψ : M(Y) → B.
Y
Definition of D≤0
The full subcategory D≤0 is consisted of objects A of D such that we have
φ+φX = 0 : A → M(X)[i] (∃X ∈ L(k), ∃i ∈ Z ).
≥0
Definition of D≥0
The full subcategory D≥0 is consisted of objects B of D such that we have
ψ −ψ = 0 : M(Y)[j] → B (∀Y ∈ L(k), ∀j ∈ Z ).
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Acknowledgments. The author would like to thank Professor Masanori
Asakura, Kazuya Kato and Iku Nakamura for the steadfast kindness. This re-
search is partially supported by Grant-in-Aid for Young Scientists (B).
References
[H] Hanamura, M.:Mixed motives and algebraic cycles. II. Invent. Math. 158 (2004), no. 1,
105-179.
[L] Levine,M.:Mixedmotives.Mathematical SurveysandMonographs, 57.AmericanMath-
ematical Society, Providence, RI, 1998.
[M] Morita, K.:Generalization of the theory of mixed Hodge structures and its application.
[VSF] Voevodsky, V.; Suslin, A.; Friedlander, E.: Cycles, transfers, and motivic homology
theories. Annals of Mathematics Studies, 143. Princeton University Press, Princeton,
NJ, 2000.
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
E-mail address: [email protected]