Geodesic
The Motivic Cofiber of $\tau$
Documenta mathematica, Tome 23 (2018), pp. 1077-1127
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Consider the Tate twist τ∈H0,1(S0,0) in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map τ:S0,−1→S0,0, with cofiber Cτ. We show that this motivic 2-cell complex can be endowed with a unique E∞​ ring structure. Moreover, this promotes the known isomorphism π∗,∗​Cτ≅ExtBP∗​BP∗,∗​(BP∗​,BP∗​) to an isomorphism of rings which also preserves higher products. We then consider the closed symmetric monoidal category of Cτ-modules (Cτ​Mod,−∧Cτ​−) which lives in the kernel of Betti realization. Given a motivic spectrum X, the Cτ-induced spectrum X∧Cτ is usually better behaved and easier to understand than X itself. We specifically illustrate this concept in the examples of the mod 2 Eilenberg-Maclane spectrum HF2​, the mod 2 Moore spectrum S0,0/2 and the connective hermitian K-theory spectrum kq.
DOI : 10.4171/dm/642
Classification : 14F42, 55P43, 55S10
Mots-clés : motivic cohomology, motivic homotopy theory, cofibre of τ, E∞​-structure, hermitian K-theory 
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     author = {Bogdan Gheorghe},
     title = {The {Motivic} {Cofiber} of $\tau$},
     journal = {Documenta mathematica},
     pages = {1077--1127},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/642},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/642/}
}
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Bogdan Gheorghe. The Motivic Cofiber of $\tau$. Documenta mathematica, Tome 23 (2018), pp. 1077-1127. doi: 10.4171/dm/642

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