Slices and Transfers

arXiv:1003.1830v1 [math.AG] 9 Mar 2010 SLICES AND TRANSFERS MARC LEVINE Abstract. We study the slice filtration for S1-spectra, and raise a number of questions regardings its properties. Contents Introduction 1 1. Infinite P1-loop spectra 4 2. An example 4 3. Co-transfer 6 4. Co-group structure on P1 10 5. Slice localizations and co-transfer 13 6. Higher loops 24 7. Supports and co-transfers 25 8. Slices of loop spectra 29 9. Transfers on the generalized cycle complex 32 10. The Friedlander-Suslin tower 37 References 38 Introduction Voevodsky [21] has defined an analog of the classical Postnikov tower in the setting of motivic stable homotopy theory by replacing the simplicial suspension Σs := −∧S1 with P1-suspension ΣP1 := −∧P1; we call this construction the motivic Postnikov tower. Let SH(k) denote the motivic stable homotopy category of P1-spectra. One of the main results on motivic Postnikov tower in this setting is Theorem 1. For E ∈SH(k), the slices snE have the natural structure of an HZ-module, and hence determine objects in the category of motives DM(k). The statement is a bit imprecise, as the following expansion will make clear: Ostvar-R¨ondigs [17, 18] have shown that the homotopy category of strict HZ- modules is equivalent to the category of motives DM(k). Additionally, Voevodsky Date: October 30, 2018. 2000 Mathematics Subject Classification. Primary 14C25, 19E15; Secondary 19E08 14F42, 55P42. Key words and phrases. Algebraic cycles, Morel-Voevodsky stable homotopy category, slice filtration. Research supported by the NSF grant DMS-0801220 and the Alexander von Humboldt Foundation. 1 2 MARC LEVINE [21] and the author [11] have shown that the 0th slice of the sphere spectrum S in SH(k) is isomorphic to HZ. Thus, for E ∈SH(k), the canonical S-module struc- ture on E induces an HZ-module structure on the slices snE, in SHS1(k). This has been refined to the model category level by Pelaez [19], showing that the slices of a P1-spectrum E have a natural structure of a strict HZ-module, hence are motives. Let SptS1(k) denote the category of S1-spectra. The motivic analog is the category of effective motives over k, DM eff(k). We consider the motivic Postnikov tower in the homotopy category of S1-spectra, SHS1(k), and ask the question: (1) is there a ring object in SptS1(k), HZeff, such that the homotopy cate- gory of HZeff modules is equivalent to the category of effective motives DM eff(k)? (2) What properties (if any) need an S1-spectrum E have so that the slices snE have a natural structure ofhave a natural structure as the Eilenberg- Maclane spectrum of a homotopy invariant complex of presheaves with transfer? Naturally, if HZeff exists as in (1), we are asking the slices in (2) to be (strict) HZeff modules. Of course, a natural candidate for HZeff would be the 0-S1- spectrum of HZ, Ω∞ P1HZ, but as far as I know, this property has not yet been investigated. As we shall see, the 0-S1-spectrum of a P1-spectrum does have the property that its (S1) slices are motives, while one can give examples of S1-spectra for which the 0th slice does not have this property. This suggests a relation of the question of the structure of the slices of an S1-spectrum with a motivic version of the recognition problem: (3) How can one tell if a given S1-spectrum is an n-fold P1-loop spectrum? In this paper, we prove two main results about the “motivic” structure on the slices of S1-spectra: Theorem 2. Suppose char k = 0. Let E be an S1-spectrum. Then for each n ≥1, there is a tower . . . →ρ≥p+1snE →ρ≥psnE →. . . →snE in SHS1(k) with the following properties: (1) the tower is natural in E. (2) Let ¯sp,nE be the cofiber of ρ≥p+1snE →ρ≥psnE. Then there is a homotopy invariant complex of presheaves with transfers ˆπp((snE)(n))∗∈DM eff −(k) and a natural isomorphism in SHS1(k), EM A1(ˆπp((snG)(n))∗) ∼= ¯sp,nE, where EM A1 : DMeff −(k) →SHS1(k) is the Eilenberg-Maclane spectrum functor. This result is proven in section 9. One can say a bit more about the tower appearing in theorem 2. For instance, holimp fib(ρ≥psnE →snE) is weakly equivalent to zero, so the spectral sequence associated to this tower is weakly convergent. If snE is globally N-connected (i.e., there is an N such that snE(X) is N-connected for all X ∈Sm/k) then the spectral sequence is strongly convergent. SLICES AND TRANSFERS 3 In other words, the higher slices of an arbitrary S1-spectrum have some sort of transfers “up to filtration”. The situation for the 0th slice appears to be more complicated, but for a P1-loop spectrum we have at least the following result: Theorem 3. Suppose char k = 0. Take E ∈SHP1(k). Then for all m, the homo- topy sheaf πm(s0ΩP1E) has a natural structure of a homotopy invariant sheaf with transfers. We actually prove a more precise result (corollary 8.5) which states that the 0th slice s0ΩP1E is itself a presheaf with transfers, with values in the stable homotopy category SH, i.e., s0ΩP1E has “transfers up to homotopy”. This raises the questio

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