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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
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