# zbMATH — the first resource for mathematics

Motivic Eilenberg-MacLane spaces. (English) Zbl 1227.14025
In this fundamental paper, the author provides a construction of motivic Eilenberg-MacLane spaces representing motivic cohomology on a large category of schemes, containing smooth schemes, over a field that admits resolution of singularities. This allows the author to compute operations in motivic cohomology which are necessary for the proof of the Bloch-Kato conjecture of the author in [Ann. Math. (2) 174, No. 1, 401–438 (2011; Zbl 1236.14026)].
Since the category $$\mathrm{Sm}/k$$ of smooth schemes over a field $$k$$ does not share all properties necessary for the constructions, the author works with a suitably enlarged subcategory $$C$$ of schemes over $$k$$. He calls $$C$$ $$f$$-admissible if it is closed under the formation of quotients by the action of finite groups. For such an $$f$$-admissible category $$C$$, let $$C_+$$ be the category of objects in $$C$$ with a disjoint basepoint and let $$\mathrm{Cor}(C,R)$$ be the category of finite correspondences over $$C$$ with coefficients in a ring $$R$$. The definitions and computations of the paper take place in the corresponding $$\mathbb{A}^1$$-homotopy categories with respect to the Nisnevich topology $$\mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(C)$$, $$\mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(C_+)$$ and $$\mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(\mathrm{Cor}(C,R))$$. In order to have a uniform presentation of these homotopy categories the author uses the formalism of radditive functors that the author had developed in [J. K-Theory 5, No. 2, 201–244 (2010; Zbl 1194.55021)]. Most of the results of the paper are based on the author’s deep analysis of the properties of the pair of (derived) adjoint functors $\mathbf{L}\Lambda_R^l: \mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(C_+) \leftrightarrows \mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(\mathrm{Cor}(C,R)): \Lambda_R^r$ which is the motivic analog of forgetting and taking free $$R$$-modules on pointed simplicial sets.
The motivic Eilenberg-MacLane space $$K(A,p,q)_C$$ which represents (unstable) motivic cohomology with coefficients in an abelian group $$A$$ for positive integers $$p\geq q$$ can be described by a suitable composition of these functors. Let $$S^q_t:=(S^1_t)^{\wedge q}$$ be the $$q$$th smash product (as simplicial radditive functors) of the motivic sphere $$S^1_t=(\mathbb{A}^1 \setminus \{0\},1)$$. The author sets $$l_q:=\mathbf{L}\Lambda_{\mathbb{Z}}^l(S^q_t)$$ and shows that, for $$p\geq q$$, the motivic Eilenberg-MacLane space $$K(A,p,q)_C$$ in $$\mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(C_+)$$ is given by $K(A,p,q)_C = \Lambda_{\mathbb{Z}}^r(\Sigma^{p-q}(A\otimes_{\mathbf{L}}l_q))$ where $$\Sigma^{p-q}$$ denotes the simplicial suspension.
The author also introduces motivic Moore spaces $$M(A,p,q;R)_C$$ in $$\mathrm{H}_{\mathrm{Nis}, \mathbb{A}^1}(\mathrm{Cor}(C,R))$$ given by $M(A,p,q;R)_C=\mathbf{L}\Lambda^l_{R}(K(A,p,q)_C).$ These spaces encode the crucial information about motivic cohomology operations. For the purposes of the proof of the Bloch-Kato conjecture, the author also analyzes the behavior of these Moore spaces under the change of admissible categories $$C\to D$$ and proves a Dold-Thom theorem expressing $$M(A,p,q;R)_C$$ as a direct sum of symmetric powers.
The main result of the paper is a complete computation of the bistable motivic cohomology operations. If $$k$$ is a field of characteristic zero, the author shows that there is an isomorphism $\lim_n H^{*+2n, *+n}(K(\mathbb{Z}/\ell, 2n,n)_{\mathrm{Sm}/k}, \mathbb{Z}/\ell)= \mathcal{A}^{*,*}(k,\mathbb{Z}/\ell)$ where $$\mathcal{A}^{*,*}(k,\mathbb{Z}/\ell)$$ denotes the motivic Steenrod algebra defined by the author in [Publ. Math., Inst. Hautes Étud. Sci. 98, 1–57 (2003; Zbl 1057.14027)]. The proof relies on the good properties of the topological realization functor and the existence of resolution of singularities in characteristic zero.
Due to the nature of the subject this is a technical and difficult paper. Although the author does his best to provide the reader with the necessary prerequisites, familiarity with the subject is essential for this paper.

##### MSC:
 14F18 Multiplier ideals
##### MathOverflow Questions:
How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
Full Text:
##### References:
 [1] M. Artin, A. Grothendieck, and J. L. Verdier (eds.), (N. Bourbaki, P. Deligne, and B. Saint-Donat Collaboration), Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos (Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4)), Lecture Notes in Mathematics, vol. 269, Springer, Berlin, 1972. [2] F. Déglise, Finite correspondences and transfers over a regular base, in Algebraic Cycles and Motives. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 343, pp. 138–205, Cambridge Univ. Press, Cambridge, 2007. · Zbl 1149.14014 [3] P. Deligne, Voevodsky’s lectures on motivic cohomology 2000/2001, in Algebraic Topology. Abel Symposia, vol. 4, pp. 355–409, Springer, Berlin, 2009. · Zbl 1183.14028 [4] D. Dugger and D. C. Isaksen, Topological hypercovers and $$\mathbb{A}^{1}$$ -realizations, Math. Z., 246 (2004), 667–689. · Zbl 1055.55016 · doi:10.1007/s00209-003-0607-y [5] E. M. Friedlander and V. Voevodsky, Bivariant cycle cohomology, in Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., vol. 143, pp. 138–187, Princeton Univ. Press, Princeton, 2000. · Zbl 1019.14011 [6] L. Fuchs, Infinite Abelian Groups. Vol. II, Pure and Applied Mathematics, vol. 36-II, Academic Press, New York, 1973. · Zbl 0257.20035 [7] S. Greco and C. Traverso, On seminormal schemes, Compos. Math., 40 (1980), 325–365. · Zbl 0412.14024 [8] A. Grothendieck and J. Dieudonne, Etude Locale des Schemas et des Morphismes de Schemas (EGA 4), Publ. Math. IHES,20,24,28,32, 1964–1967. [9] H. Hironaka, Triangulations of algebraic sets, in Algebraic Geometry (Proc. Sympos. Pure Math., vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 165–185, American Mathematical Society, Providence, 1975. [10] P. S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, 2003. · Zbl 1017.55001 [11] M. Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, 1999. · Zbl 0909.55001 [12] S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, Springer, Berlin, 1971. · Zbl 0705.18001 [13] J. P. May, Weak equivalences and quasifibrations, in Groups of Self-Equivalences and Related Topics (Montreal, PQ, 1988), Lecture Notes in Math., vol. 1425, pp. 91–101, Springer, Berlin, 1990. [14] C. Mazza, V. Voevodsky, and C. Weibel, Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, 2006. · Zbl 1115.14010 [15] J. S. Milne, Etale Cohomology, Princeton Univ. Press, Princeton, 1980. [16] F. Morel and V. Voevodsky, A 1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math., 90 (2001), 45–143. · Zbl 0983.14007 · doi:10.1007/BF02698831 [17] M. Nakaoka, Cohomology mod p of symmetric products of spheres, J. Inst. Polytech. Osaka City Univ., Ser. A, 9 (1958), 1–18. · Zbl 0084.39102 [18] A. Neeman, Triangulated Categories, Ann. of Math. Studies, vol. 148, Princeton Univ. Press, Princeton, 2001. · Zbl 0974.18008 [19] Z. Nie, Karoubi’s construction for motivic cohomology operations, Am. J. Math., 130 (2008), 713–762. · Zbl 1147.14007 · doi:10.1353/ajm.0.0003 [20] D. Puppe, A theorem on semi-simplicial monoid complexes, Ann. Math. (2), 70 (1959), 379–394. · Zbl 0088.39101 · doi:10.2307/1970108 [21] D. Quillen, Homotopical Algebra, Lecture Notes in Math., vol. 43, Springer, Berlin, 1973. [22] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble, 6 (1955–1956), 1–42. [23] N. E. Steenrod and D. B. Epstein, Cohomology Operations, Princeton Univ. Press, Princeton, 1962. [24] A. A. Suslin, Higher Chow groups and etale cohomology, in Cycles, Transfers, and Motivic Homology Theories. Ann. of Math. Stud., vol. 143, pp. 239–254, Princeton Univ. Press, Princeton, 2000. · Zbl 1019.14001 [25] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math., 123 (1996), 61–94. · Zbl 0896.55002 · doi:10.1007/BF01232367 [26] A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, in Cycles, Transfers, and Motivic Homology Theories. Ann. of Math. Stud., vol. 143, pp. 10–86, Princeton Univ. Press, Princeton, 2000. · Zbl 1019.14004 [27] R. G. Swan, On seminormality, J. Algebra, 67 (1980), 210–229. · Zbl 0473.13001 · doi:10.1016/0021-8693(80)90318-X [28] V. Voevodsky, Letter to A. Beilinson. 1993. www.math.uiuc.edu/K-theory/33 . [29] V. Voevodsky, Homology of schemes, Sel. Math. (N.S.), 2 (1996), 111–153. · Zbl 0871.14016 · doi:10.1007/BF01587941 [30] V. Voevodsky, Cohomological theory of presheaves with transfers, in Cycles, transfers, and motivic homology theories. Ann. of Math. Stud., vol. 143, pp. 87–137, Princeton Univ. Press, Princeton, 2000. · Zbl 1019.14010 [31] V. Voevodsky, Triangulated categories of motives over a field, in Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., vol. 143, pp. 188–238, Princeton Univ. Press, Princeton, 2000. · Zbl 1019.14009 [32] V. Voevodsky, Motivic cohomology with Z/l-coefficients, 2003. www.math.uiuc.edu/K-theory/639 . · Zbl 1236.14026 [33] V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci., 98 (2003), 1–57. · Zbl 1057.14027 [34] V. Voevodsky, On the zero slice of the sphere spectrum, Tr. Mat. Inst. Steklova, 246 (2004), 106–115; [Algebr. Geom. Metody, Svyazi i Prilozh.] · Zbl 1182.14012 [35] V. Voevodsky, Cancellation theorem, Doc. Math., to appear. arXiv:math/0202012 , 2009. [36] V. Voevodsky, Motives over simplicial schemes, J. K-theory, to appear. arXiv:0805.4431 , 2009. [37] V. Voevodsky, Motivic cohomology with Z/l-coefficients, Ann. Math., submitted. arXiv:0805.4430 , 2009. · Zbl 1236.14026 [38] V. Voevodsky, Simplicial radditive functors, J. K-theory, to appear. arXiv:0805.4434 , 2009. [39] V. Voevodsky, Unstable motivic homotopy categories in Nisnevich and cdh-topologies, arXiv:0805.4576 , doi: 10.1016/j.jpaa.2009.11.005 , 2009. · Zbl 1187.14025 [40] V. Voevodsky, A. Suslin, and E. M. Friedlander, Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, 2000. · Zbl 1021.14006 [41] C. A. Weibel, Patching the norm residue isomorphism theorem, 2007. www.math.uiuc.edu/K-theory/844 .
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.