×

zbMATH — the first resource for mathematics

Modules over motivic cohomology. (English) Zbl 1180.14015
In analogy to the stable homotopy category in Topology, the introduction of the motivic stable homotopy category by Voevodsky provided a very convenient framework for defining and understanding cohomology theories for algebraic varieties. For example the motivic Eilenberg-MacLane spectrum represents motivic cohomology and is hence of fundamental interest in motivic homotopy theory. Furthermore, as in the classical situation, there are highly structured models for the motivic stable homotopy category such as the category of motivic symmetric spectra. In light of such a model the Eilenberg-MacLane spectrum \(M\mathbb{Z}\) becomes an even more interesting object. The main theorem of the present paper states that over a field \(k\) of characteristic zero, the homotopy category of \(M\mathbb{Z}\)-modules in the category of motivic symmetric spectra is equivalent to Voevodsky’s big category of motives over \(k\). In fact this holds for more general objects than \(M\mathbb{Z}\) and more generally over a noetherian and separated base scheme of finite Krull dimension for which the motivic stable homotopy category is generated by a dualizable object. One of the key point in the proof is the dualizability of varieties. This is also the moment where the assumption of characteristic zero enters the stage. Over a field \(k\) of characteristic zero every smooth quasi-projective scheme is dualizable in the motivic stable homotopy category over \(k\). To prove this generalization of a result of P. Hu [Topology 44, No. 3, 609–640 (2005; Zbl 1078.14025)] the authors use resolution of singularities. This hypotheses can be removed if one is interested in rational coefficients by using de Jong’s theorem on alterations [Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996; Zbl 0916.14005)] as the authors show in the last section. But before explaining the case of rational coefficients, the authors prove the main theorem by a zig-zag of Quillen equivalences to relate Voevodsky’s motives to \(M\mathbb{Z}\)-modules. They discuss various model structures on different categories and show that their corresponding homotopy categories are isomorphic to the motivic stable homotopy category. This provides a very interesting and complete account on the properties of motivic spaces, motivic (symmetric) spectra, motivic spectra of chain complexes and their analogs with transfers. Finally, the authors also show that the category of mixed Tate motives over a noetherian and separated base scheme of finite Krull dimension is equivalent to the associated homotopy category of the module category of a differential graded algebra with many objects.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
55U35 Abstract and axiomatic homotopy theory in algebraic topology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adámek, J.; Rosický, J., Locally presentable and accessible categories, LMS lecture note ser., vol. 189, (1994), Cambridge University Press Cambridge · Zbl 0795.18007
[2] Bloch, S.; Kriz, I., Mixed Tate motives, Ann. of math., 140, 557-605, (1994) · Zbl 0935.14014
[3] Day, B., On closed categories of functors. reports of the midwest category seminar, IV, Lecture notes in math., 137, 1-38, (1970)
[4] P. Deligne, Voevodsky’s lectures on cross functors, preprint, available via http://www.math.ias.edu/ vladimir/
[5] Dugger, D., Replacing model categories with simplicial ones, Trans. amer. math. soc., 353, 5003-5027, (2001) · Zbl 0974.55011
[6] Dugger, D., Spectral enrichments of model categories, Homology homotopy appl., 8, 1, 1-30, (2006) · Zbl 1084.55011
[7] Dundas, B.I.; Röndigs, O.; Østvær, P.A., Enriched functors and stable homotopy theory, Doc. math., 8, 409-488, (2003) · Zbl 1040.55002
[8] Dundas, B.I.; Röndigs, O.; Østvær, P.A., Motivic functors, Doc. math., 8, 489-525, (2003) · Zbl 1042.55006
[9] Eilenberg, S.; MacLane, S., On the groups of \(H(\Pi, n)\). I, Ann. of math., 58, 55-106, (1953) · Zbl 0050.39304
[10] Eilenberg, S.; Zilber, J.A., On products of complexes, Amer. J. math., 75, 200-204, (1953) · Zbl 0050.17301
[11] V. Guletskiĭ, Zeta functions in triangulated categories, preprint, arXiv: math/0605040v1
[12] Hironaka, H.; Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of math., Ann. of math., 79, 205-326, (1964) · Zbl 0122.38603
[13] Hirschhorn, P., Model categories and their localizations, Math. surveys monogr., 99, (2003), xvi+457 pages · Zbl 1017.55001
[14] Hovey, M., Model categories, Math. surveys monogr., 63, (1999), xii+209 pages · Zbl 0909.55001
[15] Hovey, M., Spectra and symmetric spectra in general model categories, J. pure appl. algebra, 165, 63-127, (2001) · Zbl 1008.55006
[16] Hovey, M., Model category structures on chain complexes of sheaves, Trans. amer. math. soc., 353, 2441-2457, (2001) · Zbl 0969.18010
[17] Hovey, M.; Shipley, B.; Smith, J., Symmetric spectra, J. amer. math. soc., 13, 149-208, (2000) · Zbl 0931.55006
[18] Hu, P., On the Picard group of the \(\mathbb{A}^1\)-stable homotopy category, Topology, 44, 609-640, (2005) · Zbl 1078.14025
[19] Jardine, J.F., Motivic symmetric spectra, Doc. math., 5, 445-553, (2000) · Zbl 0969.19004
[20] de Jong, J., Smoothness, semi-stability and alterations, Publ. math. IHES, 83, 51-93, (1996) · Zbl 0916.14005
[21] Kriz, I.; May, J.P., Operads, algebras, modules and motives, Astérisque, 233, (1995), 145 pages · Zbl 0840.18001
[22] Levine, M., The homotopy coniveau tower, J. topol., 1, 217-267, (2008) · Zbl 1154.14005
[23] Morel, F., An introduction to \(\mathbb{A}^1\)-homotopy theory, contemporary developments in algebraic K-theory, ICTP trieste lecture note ser., 15, 357-441, (2003)
[24] Morel, F.; Voevodsky, V., \(\mathbb{A}^1\)-homotopy theory of schemes, Publ. math. IHES, 90, 45-143, (1999) · Zbl 0983.14007
[25] O. Röndigs, Functoriality in motivic homotopy theory, preprint
[26] Röndigs, O.; Østvær, P.A., Motives and modules over motivic cohomology, C. R. acad. sci. Paris ser. I, 342, 571-574, (2006)
[27] O. Röndigs, P.A. Østvær, Motivic spaces with transfers, preprint
[28] Schwede, S.; Shipley, B., Algebras and modules in monoidal model categories, Proc. London math. soc., 80, 491-511, (2000) · Zbl 1026.18004
[29] Schwede, S.; Shipley, B., Equivalences of monoidal model categories, Algebr. geom. topol., 3, 287-334, (2003) · Zbl 1028.55013
[30] Schwede, S.; Shipley, B., Stable model categories are categories of modules, Topology, 42, 103-153, (2003) · Zbl 1013.55005
[31] Suslin, A.; Voevodsky, V., Relative cycles and Chow sheaves, cycles, transfers, and motivic homology theories, Ann. of math. stud., 143, 10-86, (2000) · Zbl 1019.14004
[32] Voevodsky, V., \(\mathbb{A}^1\)-homotopy theory, (), 579-604 · Zbl 0907.19002
[33] Voevodsky, V., Triangulated categories of motives over a field, cycles, transfers, and motivic homology theories, Ann. of math. stud., 143, 188-238, (2000) · Zbl 1019.14009
[34] Voevodsky, V., Open problems in the motivic stable homotopy theory I. motives, polylogarithms and Hodge theory, part I, irvine, CA, 1998, Int. press lect. ser., 3, 3-34, (2002)
[35] Voevodsky, V., Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. math. res. not., 7, 351-355, (2002) · Zbl 1057.14026
[36] V. Voevodsky, Cancellation theorem, preprint, arXiv: math/0202012v1
[37] Weibel, C., An introduction to homological algebra, Cambridge stud. adv. math., vol. 38, (1994) · Zbl 0797.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.