# 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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