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ag.algebraic geometry – Bigraded endomorphisms of the motivic sphere over a bailiwick retort

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ag.algebraic geometry – Bigraded endomorphisms of the motivic sphere over a bailiwick

In An introduction to $mathbb A^1$-homotopy concept ([1]) and On the motivic $pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $bigoplus_{nin mathbb Z} [S^0, mathbb G_m^{wedge n}]$, the place the brackets denote “homotopy classes of maps in $mathbb P^1$-spectra over a consummate bailiwick $k$“.

In [2] there is not mighty extra to that assertion than that, however in [1] he offers the marginally extra widespread motivation of attempting to compute $[S^i,mathbb G_m^{wedge n}]$ for capricious $i,n$. Of passage, he mentions that for $i<0$ that is merely $0$, and the concept he describes is the $i=0$ illustration. But there appears to breathe no point out of the $i>0$ illustration.

One “patent” instinct for that to breathe sophisticated is that for $i>0$ and $n=0$, that may presumably breathe associated to computing the steady homotopy teams of spheres (e.g. if $okay$ can breathe embedded in $mathbb R$, then I arbitrator that the classical steady homotopy teams of $S^0$ splinter off those in $mathbb P^1$-spectra) – which isn’t an issue within the $i=0$ illustration (or $ileq 0$ extra typically).

But can we think about a computation of those teams by way of the classical steady homotopy teams of spheres ? We maintain a morphism from the graded ring $pi_*(S^0)$ to the bigraded ring $pi_{*,*}^{mathbb P^1}(S^0)$ given by the (unique) symmetric monoidal colimit preserving functor $mathbf{Sp}to mathcal{SH}^{mathbb P^1}(okay)$, so we will undoubtedly credence the latter as an algebra or a module over the previous :

Is there a path to painting this algebra/module construction “explicitly” ?

By “explicitly” I denote an outline just like Morel’s description of the $i=0$ illustration, the place we maintain an expression by way of Grothendieck-Witt teams. Speaking of which, I befall to carry a reference query in regards to the similar factor, so if it is arrogate, I’ll simply query it right here too :

Are there different references than [1] and [2] the place the computation of $[S^0,mathbb G_m^{wedge n}]$ is defined ? Ideally, extra “expository” references.

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