We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson-Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson-Thomas theory: absolute, relative, and equivariant.
Abstract. The purpose of this paper is to study the p-part of motivic cohomology and algebraic K-theory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, H i (k, Z(n)) is uniquely p-divisible for i = n. This implies that the natural map K M n (k) − → Kn(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that Kn(k) is p-torsion free.As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example Kn(X, Z/p r ) = 0 for n > dimX. Another consequence is Gersten's conjecture with mod p-coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch's cycle complexes localized at p satisfy all BeilinsonLichtenbaum-Milne axioms for motivic complexes, except the vanishing conjecture.
Abstract. 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, the homotopy coniveau tower agrees with Voevodsky's 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 Voevodsky's slice filtration for P 1 -spectra. We also show that the 0th 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 recent results of Voevodsky on the 0th layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence.
We show that the multivariate additive higher Chow groups of a smooth affine k-scheme Spec (R) essentially of finite type over a perfect field k of characteristic = 2 form a differential graded module over the big de Rham-Witt complex W m Ω • R . In the univariate case, we show that additive higher Chow groups of Spec (R) form a Witt-complex over R. We use these structures to prove anétale descent for multivariate additive higher Chow groups.2010 Mathematics Subject Classification. Primary 14C25; Secondary 13F35, 19E15.
Abstract. We give a new proof of the theorem of Suslin-Voevodsky which shows that the Bloch-Kato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the Suslin-Voevodsky to positive characteristic.
No abstract
Let k be an algebraically closed field of characteristic zero. Let c : SH → SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In consequence, c induces an isomorphism c * : πn(E) −→ Πn,0(c(E))(k) for all spectra E and all n ∈ Z.Fix an embedding σ : k → C and let ReB : SH(k) → SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum over k, S k , has Betti realization which is strongly convergent. This gives a spectral sequence 'of motivic origin' converging to the homotopy groups of the sphere spectrum S ∈ SH; this spectral sequence at E 2 agrees with the E2 terms in the Adams-Novikov spectral sequence after a reindexing. Finally, we show that, for E a torsion object in SH(k) eff , the Betti realization induces an isomorphism Πn,0(E)(k) → πn(ReBE) for all n, generalizing the Suslin-Voevodsky theorem comparing mod N Suslin homology and mod N singular homology.As a special case, Theorem 1 implies the following corollary:Corollary 2. Let k be an algebraically closed field of characteristic zero. Let S k be the motivic sphere spectrum in SH(k) and S the classical sphere spectrum in SH. Then the constant presheaf functor induces an isomorphismIn fact, the corollary implies the theorem, by a density argument (see Lemma 6.5). Remark.(1) As pointed out by the referee, the functor c is induced by a (left) Quillen functor between model categories (see the proof of Lemma 6.5), so we do achieve a comparison of 'homotopy theories', as stated in the title, rather than just the underlying homotopy categories.(2) The functor c is not full in general. In fact, for a perfect field k, Morel [22, Lemma 3.10, Corollary 6.43] has constructed an isomorphism of Π 0,0 S k (k) with the Grothendieck-Witt group GW(k) of symmetric bilinear forms over k. As long as not every element of k is a square, the augmentation ideal in GW(k) is non-zero, hence c : π 0 (S) → Π 0,0 S k (k) is not surjective. Of course, if k is algebraically closed, then GW(k) = Z by rank, and thus c : π 0 (S) → Π 0,0 S k (k) is an isomorphism. This observation can be viewed as the starting point for our main result.We also have a homotopy analog of the theorem of Suslin-Voevodsky [33, Theorem 8.3] comparing Suslin homology and singular homology with mod N coefficients:Theorem 3. Let k be an algebraically closed field of characteristic zero with an embedding σ : k → C. Then, for all X ∈ Sm/k, all N > 1 and n ∈ Z, the Betti realization associated to σ induces an isomorphism
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