THE ZEROTH -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE

Galois 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...

A Comparison of Motivic and Classical Stable Homotopy Theories

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 sph...

Open Problems in the Motivic Stable Homotopy Theory, I

A new family in the stable homotopy groups of spheres

Let $p$ be a prime number greater than three. In this paper, we prove the existence of a new family of homotopy elements in the stable homotopy groups of spheres $pi_{ast}(S)$ which is represented by $h_nh_mtilde{beta}_{s+2}in {rm Ext}_A^{s+4, q[p^n+p^m+(s+2)p+(s+1)]+s}(mathbb{Z}_p,mathbb{Z}_p)$ up to nonzero scalar in the Adams spectral sequence, where $ngeq m+2>5$, $0leq sExt}_A^{s+2,q[(s+2)p...

Homotopy invariance of the sheaf WNis and of its cohomology

A conjecture of F.Morel states that the motivic group π0,0(k) of a perfect field k coincides with the Grothendieck-Witt group GW (k) of quadratic forms over k provided that char(k) 6= 2. Morel’s proof of the conjecture requires the the following result: the Nisnevich sheaf WNis associated with the presheaf X 7→ W (X) is homotopy invariant and all its Nisnevich cohomology are homotopy invariant ...