A Comparison of Motivic and Classical Stable Homotopy Theories
نویسنده
چکیده
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 particular, c induces an isomorphism c∗ : πn(E)→ Πn,0(c(E)) for all spectra E. Fix an embedding σ : k → C and let ReB : SH(k)→ SH be the associated Betti realization. Let Sk be the motivic sphere spectrum. We show that the Tate-Postnikov tower for Sk . . .→ fn+1Sk → fnSk → . . .→ f0Sk = Sk has Betti realization which is strongly convergent, in fact Re(fnSk) is n − 1 connected. This gives a spectral sequence “of algebro-geometric origin” converging to the homotopy groups of S; this spectral sequence at E2 agrees with the E2 terms in the Adams-Novikov spectral sequence.
منابع مشابه
Cohomology operations and algebraic geometry
This manuscript is based on a ten hours series of seminars I delivered in August of 2003 at the Nagoya Institute of Technology as part of the workshop on homotopy theory organized by Norihiko Minami and following the Kinosaki conference in honor of Goro Nishida. One of the most striking applications of homotopy theory in “exotic” contexes is Voevodsky’s proof of the Milnor Conjecture. This conj...
متن کاملThe Homotopy Coniveau Filtration
We examine the “homotopy coniveau tower” for a general cohomology theory on smooth k-schemes, satisfying some natural axioms, and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. We show how these constructions lead to a tower of functors on the Morel-Voevodsky stable homotopy category, and identify this stable homotopy coniveau tower with Voevodsky’s s...
متن کاملGalois Equivariance and Stable Motivic Homotopy Theory
For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full a...
متن کاملClassical and Motivic Adams Charts
Abstract. This document contains large-format Adams charts that compute 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. The charts are essentially complete through the 59-stem and contain partial results to the 70-stem. In the classical context, we believe that these are the most accurate charts of their kind. We also include Adams charts for ...
متن کاملThe Motivic Adams Spectral Sequence
We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence, and use these tools to give new proofs of some results in classical algebraic topology. We also consider a m...
متن کاملBirational Motivic Homotopy Theories and the Slice Filtration
We show that there is an equivalence of categories between the orthogonal components for the slice filtration and the birational motivic stable homotopy categories which are constructed in this paper. Relying on this equivalence, we are able to describe the slices for projective spaces (including P), Thom spaces and blow ups. 2000 Mathematics Subject Classification: 14F42
متن کامل