Marc Levine

The homotopy coniveau tower

We examine the ‘homotopy coniveau tower’ for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, we show that the homotopy coniveau tower agrees with Voevodskys slice tower for S 1 -spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the Morel-Voevodsky stable homotopy category, and we identify this ℙ 1 -stable homotopy coniveau tower with Voevodskys slice tower for ℙ 1 -spectra. We also show that the zeroth layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general ℙ 1 -spectrum the structure of a module over motivic cohomology. This recovers and extends results of Voevodsky on the zeroth layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence.

The Arason invariant and mod 2 algebraic cycles

Introduction 73 1. Review of the Arason invariant 75 2. The special Clifford group 76 3. K-cohomology of split reductive algebraic groups 78 4. K-cohomology of BG 86 5. GL(N) and Cliff(n, n) 92 6. Two invariants for Clifford bundles 95 7. Snaking a Bloch-Ogus differential 100 8. Proof of Theorem 1 101 9. Application to quadratic forms 102 Appendix A. Toral descent 104 Appendix B. The Rost invariant 108 Appendix C. An amusing example 113 Acknowledgements 116 References 116

Chow groups of projective varieties of very small degree

(see [12], [5] and the references given there). These facts, together with various conjectures on the cohomology and Chow groups of algebraic varieties, suggest that the Chow groups of X might satisfy CHl(X)⊗Q = CHl(Pk)⊗Q = Q (∗) for l ≤ κ− 1 (compare with Remark 5.6 and Corollary 5.7). This is explicitly formulated by V. Srinivas and K. Paranjape in [16], Conjecture 1.8; the chain of reasoning goes roughly as follows. Suppose X is smooth. One expects a good filtration 0 = F j+1 ⊂ F j ⊂ . . . ⊂ F 0 = CH(X ×X)⊗Q, whose graded pieces F /F l+1 are controlled by H2j−l(X × X) (see [10]). According to Grothendieck’s generalized conjecture [8], the groups H (X) should be generated by the image under the Gysin morphism of the homology of a codimension κ subset, together with the classes coming from P. Applying this to the diagonal in X × X should then force the triviality of the Chow groups in the desired range. For zero-cycles, the conjecture (∗) follows from Roitman’s theorem (see [17] and [18]):

Tate Motives and the Vanishing Conjectures for Algebraic K-Theory

We give axioms for a triangulated ℚ-tensor category T, generated by “Tate objects” ℚ(a), which ensure the existence of a canonical weight filtration on T, and additional axioms which give rise to an abelian subcategory A generated by the ℚ(a). We show in addition that A is a Tannakian category, with fiber functor to graded ℚ-vectorspaces given by taking the associated graded with respect to the weight filtration. We then apply this to our construction of a triangulated motivic category over a field k, to show that, assuming the vanishing conjectures of Soule and Beilinson are true for k, there is a Tannakian category TM k which has many of the properties of the conjectural category of mixed Tate motives. In particular, the category TM k exists for k a number field.

Additive higher Chow groups of schemes

Abstract We show how to make the additive Chow groups of Bloch-Esnault, Rülling and Park into a graded module for Blochs higher Chow groups, in the case of a smooth projective variety over a field. This yields a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety. In case the base field k admits resolution of singularities, these properties allow us to apply the technique of Guillén and Navarro Aznar to define the additive Chow groups “with log poles at infinity” for an arbitrary finite-type k-scheme X. This theory has all the usual properties of a Borel-Moore theory on finite type k-schemes: it is covariantly functorial for projective morphisms, contravariantly functorial for morphisms of smooth schemes, and has a projective bundle formula, homotopy property, and Mayer-Vietoris and localization sequences. Finally, we show that the regulator map defined by Park from the additive Chow groups of 1-cycles to the modules of absolute Kähler differentials of an algebraically closed field of characteristic zero is surjective, giving evidence of a conjectured isomorphism between these two groups.

Cobordisme algébrique I

For any field k of characteristic 0 we prove for the algebraic cobordism the analogue of a theorem of Quillen on complex cobordism: the cobordism ring of the ground field is the Lazard ring L and for any smooth k-variety X, the algebraic cobordism ring Ω ∗ (X) is generated, as an L-module, by 1 ∈ Ω 0 (X) and the element of positive degrees. This implies Rosts conjectured degree formula. We also give a relation between the Chow ring, the K0 of a smooth k-variety X and Ω ∗ (X).  2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Witt sheaves and the η-inverted sphere spectrum

Ananyevskiy has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serres finiteness result. Theorem. Let k be a field of characteristic different from two. Then πnA1(Sk−)∗ is torsion for n>0. As an application, we define a category of Witt motives and show that rationally this category is equivalent to the minus part of SH(k)Q.

MOTIVES OF AZUMAYA ALGEBRAS

We study the slice filtration for the K -theory of a sheaf of Azumaya algebras A , and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing of SK 2 ( A ) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason’s theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H0(k,Z/n(1)) corresponding to a primitive nth root of unity.

Homology of algebraic varieties: An introduction to the works of Suslin and Voevodsky

We give an overview of the ideas Suslin and Voevodsky have introduced in their works on algebraic cycles and their relation to the mod-n homology of algebraic varieties.