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Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)

Published online by Cambridge University Press:  17 December 2010

M.V. Bondarko
M.V. Bondarko
Affiliation:
St. Petersburg State University, Department of Mathematics and Mechanics, Bibliotechnaya Pl. 2, 198904, St. Petersburg, Russia, [email protected]
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Abstract

In this paper we introduce a new notion of weight structure (w) for a triangulated category C; this notion is an important natural counterpart of the notion of t-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors.

The heart of w is an additive category Hw ⊂ C. We prove that a weight structure yields Postnikov towers for any X ∈ ObjC (whose 'factors' Xi ∈ ObjHw). For any (co)homological functor H : C → A (A is abelian) such a tower yields a weight spectral sequenceT : H(Xi [j]) ⇒ H(X[i + j]); T is canonical and functorial in X starting from E2. T specializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motives DMeffgm (if we consider w = wChow with Hw = Choweff ) and to Atiyah-Hirzebruch spectral sequences in topology.

We prove that there often exists an exact conservative weight complex functor C → K(Hw). This is a generalization of the functor t : DMeffgm → Kb(Choweff) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove that K0(C) ≅ K0(Hw) under certain restrictions.

We also introduce the concept of adjacent structures: a t-structure is adjacent to w if their negative parts coincide. This is the case for the Postnikov t-structure for the stable homotopy category SH (of topological spectra) and a certain weight structure for it that corresponds to the cellular filtration. We also define a new (Chow) t-structure tChow for DMeff_ ⊃ DMeffgm which is adjacent to the Chow weight structure. We have HtChow ≅ AddFun(Choweffop,Ab); tChow is related to unramified cohomology. Functors left adjoint to those that are t-exact with respect to some t-structures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging to C(X,Y): the one that comes from weight truncations of X with the one coming from t-truncations of Y (for X,Y ∈ ObjC). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the étale cohomology of their 'submotives'. This is an extension of the calculation of E2 of coniveau spectral sequences (by Bloch and Ogus).

Keywords

weight structurest-structurestriangulated categoriesmotivescohomologystable homotopy category
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 3 , December 2010 , pp. 387 - 504
Copyright
Copyright © ISOPP 2010

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References

Alm74.Almkvist, G., The Grothendieck ring of the category of endomorphisms. Journal of Algebra 28 (1974), 375–388.Google Scholar
Alm73.Almkvist, G., Endomorphisms of finetely generated projective modules over a commutative ring. Arkiv for Matematik 11, n. 1 (1973), 263–301.CrossRefGoogle Scholar
BaS01.Balmer, P., Schlichting, M.Idempotent completion of triangulated categories. Journal of Algebra 236, no. 2 (2001), 819–834.Google Scholar
B-VRS03.Barbieri, Viale L., Rosenschon, A., Saito M. Deligne's conjecture on 1-motives. Annals of Mathematics 158, N. 2, (2003) 593–633.Google Scholar
BBD82.Beilinson, A., Bernstein, J., Deligne, P., Faisceaux pervers. Asterisque 100, 1982, 5–171.Google Scholar
BeV08.Beilinson, A., Vologodsky, V.A guide to Voevodsky motives. Geom. Funct. Analysis 17, no. 6, 2008, 1709–1787.CrossRefGoogle Scholar
Bei87.Beilinson, A., On the derived category of perverse sheaves, in: K-theory, Arithmetic and Geometry, Lecture Notes in Mathematics 1289, Springer, 1987, 27–41.Google Scholar
BlE07.Bloch, S., Esnault, H., Künneth projectors for open varieties, in: Algebraic cycles and motives 2, LMS lecture note series 344 (2007), 54–72.Google Scholar
BOg94.Bloch, S., Ogus, A. Gersten's conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Sup. 7 (1994), 181–202.CrossRefGoogle Scholar
BoK90.Bondal, A. I., Kapranov, M. M.Enhanced triangulated categories. (Russian). Mat. Sb. 181 (1990), no. 5, 669–683; translation in Math. USSR-Sb. 70 (1991), no. 1, 93–107.Google Scholar
Bon09.Bondarko, M.V., Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura. J. of the Inst. of Math. of Jussieu 8 (2009), no. 1, 39–97, see also http://arxiv.org/abs/math.AG/0601713CrossRefGoogle Scholar
Bon10a.Bondarko, M.V., Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields. Doc. Math., extra volume: Andrei Suslin's Sixtieth Birthday (2010), 33–117.Google Scholar
Bon10b.Bondarko, M.V., Weights and t-structures: in general, for 1-motives, mixed motives, and for mixed Hodge complexes and modules, preprint, http://arxiv.org/abs/1011.3507Google Scholar
Bon10c.Bondarko, M.V., Weights for relative motives; relation with mixed sheaves, preprint, http://arxiv.org/abs/1007.4543Google Scholar
C-THK.Colliot-Thelène, J.-L., Hoobler, R.T., Kahn, B., The Bloch-Ogus-Gabber Theorem, Algebraic K-theory (Toronto, 1996). Fields Inst. Commun. 16, Amer. Math. Soc., Providence, 1997, 31–94.Google Scholar
Co-T95.Colliot-Thelène, J.-L., Birational invariants, purity and the Gersten conjecture. Proc. Sym. Pure Math 58 (I) (1995), 1–64.Google Scholar
dJo96.de Jong, A. J.Smoothness, semi-stability and alterations. Inst. Hautes Etudes Sci. Publ. Math. 83 (1996), 51–93.CrossRefGoogle Scholar
Deg08.Deglise, F. Around the Gysin triangle, preprint, http://www.math.uiuc.edu/K-theory/0764/Google Scholar
Del71.Deligne, P., Theorie de Hodge. II. Publ. Math Inst. Hautes Etudes Sci. 40 (1971), 5–57.Google Scholar
Del74.Deligne, P., Theorie de Hodge. III. Publications Mathématiques de l'IHÉS 44 (1974), 5–77.CrossRefGoogle Scholar
Gel01.Geisser, T., Levine, M., The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. f. die reine und ang. Math. 530, 2001, 55–103.Google Scholar
GeM03.Gelfand, S., Manin, Yu., Methods of homological algebra. 2nd ed.Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp.Google Scholar
GiS96.Gillet, H., Soulé, C.Descent, motives and K-theory. J. f. die reine und ang. Math. 478, 1996, 127–176.Google Scholar
Han99.Hanamura, M.Mixed motives and algebraic cycles, III. Math. Res. Letters 6, 1999, 61–82.CrossRefGoogle Scholar
Haz78.Hazewinkel, M.Formal groups and applications. Springer-Verlag, Berlin–Heidelberg–New York, 1978.Google Scholar
HKM02.Hoshino, M.; Kato, Y.; Miyachi, J.-I.On t-structures and torsion theories induced by compact objects. J. of Pure and Appl. Algebra 167, n. 1, 2002, 15–35.CrossRefGoogle Scholar
HPS97.Hovey, M., Palmieri, J., Strickland, N., Axiomatic Stable Homotopy Theory, American Mathematical Society, 1997, 114 pp.Google Scholar
HuK06.Huber, A., Kahn, B. The slice filtration and mixed Tate motives. Compos. Math. 142(4), 2006, 907–936.Google Scholar
KaS02.Kahn, B., Sujatha, R., Birational motives, I, preprint, http://www.math.uiuc.edu/K-theory/0596/.Google Scholar
Lev00.Levine, M., Inverting the Motivic Bott Element. K-Theory 19, n. 1, 2000, 1–28.CrossRefGoogle Scholar
Mal06.Maltsiniotis, G, Catégories triangulées supérieures, preprint, www.math.jussieu.fr/˜maltsin/ps/triansup.psGoogle Scholar
Mar83.Margolis, H.R., Spectra and the Steenrod Algebra: Modules over the Steenrod Algebra and the Stable Homotopy Category, North-Holland, Amsterdam-New York, 1983.Google Scholar
MVW06.Mazza, C., Voevodsky, V., Weibel Ch. Lecture notes on motivic cohomology. Clay Mathematics Monographs 2, 2006.Google Scholar
Nee01.Neeman, A. Triangulated Categories. Annals of Mathematics Studies 148 (2001), Princeton University Press, viii+449 pp.Google Scholar
Nee96.Neeman, A., The Grothendieck duality theorem via Bousfield's techniques and Brown representability. J. Amer. Math. Soc. 9 (1996), 205–236.CrossRefGoogle Scholar
Par96.Paranjape, K., Some Spectral Sequences for Filtered Complexes and Applications. Journal of Algebra 186, i. 3, 1996, 793–806.CrossRefGoogle Scholar
Pau08.Pauksztello, D., Compact cochain objects in triangulated categories and co-t-structures. Central European Journal of Mathematics 6, n. 1, 2008, 25–42.CrossRefGoogle Scholar
Pir07.Pirashvili, T., Singular extensions and triangulated categories, preprint, http://arxiv.org/abs/0709.3169Google Scholar
SuV00.Suslin, A., Voevodsky, V. Bloch-Kato conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht, 2000, 117–189.Google Scholar
Swa68.Swan, , Algebraic K-theory. Lecture Notes in Mathematics 76 (1968), Springer.Google Scholar
Vak01.Vaknin, A., Virtual Triangles. K-Theory 22 (2001), no. 2, 161–197.CrossRefGoogle Scholar
Voe00a.Voevodsky, V. Triangulated category of motives, in: Voevodsky V., Suslin A., and Friedlander E., Cycles, transfers and motivic homology theories.Google Scholar
Voe00b.Voevodsky, V. Cohomological theory of presheaves with transfers, same volume, 87–137.CrossRefGoogle Scholar
FrV00.Friedlander, E., Voevodsky, V. Bivariant cycle cohomology, same volume, 138–187.CrossRefGoogle Scholar
Voe03.Voevodsky, V., Motivic Cohomology with ℤ/2-coefficients. Inst. Hautes Études Sci. Publ. Math. 98 (2003), no. 1, 59–104.Google Scholar
Voe10.Voevodsky, V., Cancellation theorem. Doc. Math., extra volume: Andrei Suslin's Sixtieth Birthday (2010), 671–685.Google Scholar
Voe95.Voevodsky, V., A nilpotence theorem for cycles algebraically equivalent to zero. Int. Math. Res. Notices 1995, 187–199.CrossRefGoogle Scholar
Voe08.Voevodsky, V. On motivic cohomology with ℤ=l coefficients, preprint, http://arxiv.org/abs/0805.4430Google Scholar
Wil09c.Wildeshaus, J., Chow motives without projectivity. Compositio Mathematica 145(5), 2009, 1196–1226.CrossRefGoogle Scholar
Wil09at.Wildeshaus, J., Notes on Artin-Tate motives, preprint, http://www.math.uiuc.edu/K-theory/0918/Google Scholar