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The homotopy coniveau tower. (English) Zbl 1154.14005
The article under review is a major contribution to motivic homotopy theory, as it solves several of Voevodsky’s open problems stated in [V. Voevodsky, Open problems in the motivic stable homotopy theory. I. Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. Int. Press Lect. Ser. 3, No. I, 3–34 (2002; Zbl 1047.14012)]. As the author mentions, his original aim was to “give an alternative argument for the technical underpinnings of ... the construction of a spectral sequence from motivic cohomology to $$K$$-theory”. The spin-off is an identification of Voevodsky’s slice filtration, a proof of Voevodsky’s connectivity conjecture, a computation of the slices of $$K$$-theory (which implies the desired spectral sequence), and an identification of the zero slice of the sphere spectrum with the Eilenberg-MacLane spectrum. All of this works over a perfect and sometimes infinite field $$k$$, with no restriction on the characteristic. Here are some details. For the purpose of this review, a homotopy invariant, Nisnevich-excisive presheaf $$E\colon {\mathbf{Sm}}/k^{\mathrm{op}} \to {\mathbf{Spt}}$$ from the category of smooth $$k$$-schemes with values in the category of spectra will be called a good theory. The main example of a good theory is $$K$$-theory. Relying heavily on his work in [M. Levine, $$K$$-Theory 37, No. 1–2, 129–209 (2006; Zbl 1117.19003) and J. Algebr. Geom. 10, No. 2, 299–363 (2001; Zbl 1077.14509)], the author constructs, for any good theory $$E$$, the tower $\dotsm \to E^{(p)} \to E^{(p-1)} \to \dotsm \to E^{(0)}\sim E$ of good theories mentioned in the title of the article under review. The spectrum $$E^{(p)}(X)$$ is the realization of a simplicial spectrum whose $$n$$-simplices are the homotopy colimit of homotopy fibers $\mathrm{hofib}\bigl(E(X\times \Delta^{n}) \to E(X\times \Delta^{n} - W)\bigr)$ where $$W$$ runs through the closed subsets of $$X\times \Delta^n$$ of codimension at least $$p$$ which are in good position with respect to the faces. A localization theorem implies that the homotopy coniveau tower is compatible with taking $$\mathbb{P}^1$$-loops in the sense that there is a natural zig-zag of weak equivalences connecting $$(\Omega_{\mathbb{P}^1} E)^{(p-1)}$$ and $$\Omega_{\mathbb{P}^1} \bigl(E^{(p)}\bigr)$$. As a consequence, the good theory $$E^{(0/1)}$$ obtained as the homotopy fiber of $$E^{(1)} \to E^{(0)}$$ is birational and rationally invariant. Here are the applications. In the case of $$K$$-theory, the author identifies the layer $$K^{(p/p+1)}(X)$$ with the Eilenberg-MacLane spectrum corresponding to Bloch’s cycle complex, which in turn produces the desired spectral sequence. Over an infinite perfect field, the homotopy coniveau tower is shown to coincide with Voevodsky’s slice tower for $$S^1$$-spectra, thereby proving Voevodsky’s connectivity conjecture in this case. In the case of $$\mathbb{P}^1$$-spectra, the comparison works over any perfect field. The author finally proves that the zero slice of the $$\mathbb{P}^1$$-sphere spectrum and the integral Eilenberg-MacLane spectrum are equivalent via a “reverse cycle map”, whose construction is fairly involved. Voevodsky obtained the last result for fields of characteristic zero by a completely different method [V. Voevodsky, Proc. Steklov Inst. Math. 246, 93–102 (2004; Zbl 1182.14012)]. Modulo a conjecture which is proven by P. Pelaez [Multiplicative properties of the slice filtration, Thesis, http://www.math.uiuc.edu/K-theory/0898/], the last comparison supplies the slices of any $$\mathbb{P}^1$$-spectrum over a perfect field with a module structure over the integral Eilenberg-MacLane spectrum. If again the field has characteristic zero, or one uses the rational Eilenberg-MacLane spectrum instead, the slices of any $$\mathbb{P}^1$$-spectrum are thus motives [O. Röndigs, P. A. Østvær, C. R., Math., Acad. Sci. Paris 342, No. 10, 751–754 (2006; Zbl 1097.14016)] (see also [O. Röndigs, P. A. Østvær, Adv. Math. 219, 689–727 (2008; Zbl 1180.14015)]).

##### MSC:
 14C25 Algebraic cycles 14F42 Motivic cohomology; motivic homotopy theory 19E08 $$K$$-theory of schemes 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 55P42 Stable homotopy theory, spectra
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##### References:
 [1] Bloch, Algebraic cycles and higher K-theory, Adv. Math. 61 pp 267– (1986) · Zbl 0608.14004 · doi:10.1016/0001-8708(86)90081-2 [2] S. Bloch S. Lichtenbaum A spectral sequence for motivic cohomology 1995 Preprint http://www.math.uiuc.edu/K-theory/0062/ [3] A. Bousfield D. Kan Homotopy limits, completions and localizations 1972 Berlin Springer Lecture Notes in Mathematics 304 · Zbl 0259.55004 [4] Friedlander, The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 ((6)) pp 773– (2002) · Zbl 1047.14011 · doi:10.1016/S0012-9593(02)01109-6 [5] Goerss, Localization theories for simplicial presheaves, Canad. J. Math. 50 ((5)) pp 1048– (1998) · Zbl 0914.55004 · doi:10.4153/CJM-1998-051-1 [6] M. Hovey Model categories 1999 Providence, RI American Mathematical Society Mathematical Surveys and Monographs 63 [7] Huber, The slice filtration and mixed Tate motives, Compos. Math. 142 ((4)) pp 907– (2006) · Zbl 1105.14022 · doi:10.1112/S0010437X06002107 [8] Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 pp 733– (1987) · Zbl 0645.18006 · doi:10.4153/CJM-1987-035-8 [9] Jardine, Motivic symmetric spectra, Doc. Math. 5 pp 445– (2000) · Zbl 0969.19004 [10] B. Kahn R. Sujatha Birational motives 2002 Preprint http://www.math.uiuc.edu/K-theory/0596/ [11] M. Levine Mixed motives 1998 Providence, RI American Mathematical Society Mathematical Surveys and Monographs 57 [12] Levine, Techniques of localization in the theory of algebraic cycles, J. Algebraic Geom. 10 pp 299– (2001) · Zbl 1077.14509 [13] Levine, Chow’s moving lemma in A1 homotopy theory, K-Theory 37 pp 129– (2006) · Zbl 1117.19003 · doi:10.1007/s10977-006-0004-5 [14] F. Morel A 1 -homotopy theory Lecture Series September 2002 Newton Institute for Mathematics [15] Morel, An introduction to A1-homotopy theory, ICTP Lecture Notes XV, in: Contemporary developments in algebraic K-theory pp 357– (2004) [16] F. Morel Homotopy theory of schemes 2006 Translated from the 1999 French original by James D. Lewis, SMF/AMS Texts and Monographs 12 (American Mathematical Society, Providence, RI; Société Mathématique de France, Paris) [17] Morel, A1-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 pp 45– (1999) · Zbl 0983.14007 · doi:10.1007/BF02698831 [18] A. Neeman Triangulated categories 2001 Princeton, NJ Princeton University Press Annals of Mathematics Studies 148 [19] Østvaer, Motives and modules over motivic cohomology, C. R. Math. Acad. Sci. Paris 342 ((10)) pp 751– (2006) · Zbl 1097.14016 [20] Østvaer, Motives and modules over motivic cohomology, C. R. Acad. Sci. Paris 342 ((10)) pp 751– (2006) · Zbl 1097.14016 [21] Quillen, Higher algebraic K-theory I, Lecture Notes in Mathematics 341, in: Algebraic K-theory I: Higher K-theories pp 85– (1973) · Zbl 0292.18004 · doi:10.1007/BFb0067053 [22] M. Spitzweck Operads, algebras and modules in model categories and motives PhD Thesis 2001 Bonn, Germany Universität Bonn [23] Suslin, On the Grayson spectral sequence, Proc. Steklov Inst. Math. 241 ((2)) pp 202– (2003) · Zbl 1084.14025 [24] Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 2002 pp 351– (2002) · Zbl 1057.14026 · doi:10.1155/S107379280210403X [25] Voevodsky, Open problems in the motivic stable homotopy theory. I, International Press Lecture Series 3, I, in: Motives, polylogarithms and Hodge theory, Part I, Irvine, CA, 1998 pp 3– (2002) [26] Voevodsky, A possible new approach to the motivic spectral sequence for algebraic K-theory, Contemporary Mathematics 293, in: Recent progress in homotopy theory, Baltimore, MD, 2000 pp 371– (2002) · doi:10.1090/conm/293/04956 [27] Voevodsky, On the zero slice of the sphere spectrum, Proc. Steklov Inst. Math. 246 ((3)) pp 93– (2004) · Zbl 1182.14012 [28] V. Voevodsky A. Suslin E. Friedlander Cycles, transfers and motivic homology theories 2000 Princeton, NJ Princeton University Press Annals of Mathematics Studies 143 · Zbl 1021.14006 [29] Vorst, Polynomial extensions and excision for K1, Math. Ann. 244 pp 193– (1979) · Zbl 0426.13005 · doi:10.1007/BF01420342 [30] Weibel, Homotopy algebraic K-theory, Contemporary Mathematics 83, in: Algebraic K-theory and algebraic number theory, Honolulu, HI, 1987 pp 461– (1989) · doi:10.1090/conm/083/991991
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