A comparison of motivic and classical stable homotopy theories

By M. Levine. 2011

60 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/96bdb95aab113ece4ab25bb1dd87ed90241a9779

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Citations in the corpus

Synthetic spectra and the cellular motivic category (2018, background): “t k = 2b−a is good enough for Sa,b. Notice that the map πt,wS0,0 p →πtS 0 p is an isomorphism when t < 0, as then both groups vanish, the ﬁrst one by Morel’s connectivity. By Levine’s theorem, see [Lev14], it is also an isomorphism when w = 0, in fact both of these statements hold even before p-c”
Framed motives of algebraic varieties (after V. Voevodsky) (2014, methodology): “As a topological application, using the machinery of framed motives together with a theorem of Levine [16], we show in Theorem 11.9 that M fr ( pt )( pt ) , the framed motive of the point pt = Spec k evaluated at pt , is a quasi-ﬁbrant model (i.e. an Ω -spectrum in positive degrees) of the classical”
Motivic slices and coloured operads (2010, methodology): “Over fields of characteristic zero, the full slice s∗1 can be computed using the Hopkins-Morel isomorphism identifying MZ with a regular quotient of MGL: si1 is (up to a shift) the motivic Eilenberg-Mac Lane spectrum on the complex of abelian groups which computes the E2-term of the topological Adam”
The first stable homotopy groups of motivic spheres (2016, methodology): “An explicit answer to (1) is out of reach because the slices of 1 involve the E2-page of the topological Adams-Novikov spectral sequence as conjectured by Voevodsky [73] and verified by Levine [34] for fields.”