Morel and Voevodsky construct the motivic stable homotopy category, a category through which all cohomology theories factor and where they are representable, by starting with a category of schemes, Yoneda-embedding it into simplicial presheaves, endowing those with the $\mathbb{A}^1$-local model structure, and then passing to $S^1 \wedge \mathbb{G}_m$-spectra. The last step ensures that smashing with $S^1$ or with $\mathbb{G}_m$ induce functors with a quasi-inverse on the homotopy category.
Inverting $S^1$ leads to a triangulated structure on the homotopy category, which is very welcome, but I would like a motivation for inverting $\mathbb{G}_m$. Since $\mathbb{P}^1$ is $\mathbb{A}^1$-equivalent to $S^1 \wedge \mathbb{G}_m$ I would also be content with a motivation to invert $\mathbb{P}^1$.
I must admit I already know some answers which certainly are reason enough to invert $\mathbb{G}_m$, e.g. (from slides by Marc Levine, start at page 64):
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Inverting $\mathbb{G}_m$ is necessary to produce a Gysin sequence
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The algebraic K-theory spectrum appears naturally as a $\mathbb{P}^1$-spectrum
However, I am greedy and would like to hear a motivation like the one for inverting the Lefschetz motive in the construction of pure motives: There one could say that for all envisaged realization functors which should factor through the category of pure motives, the effect of tensoring with the Lefschetz motive can be undone (e.g. is just a change of Galois representation leaving the cohomology groups unchanged).
Or, related to this, as Emerton explained in his nice answer here one has to invert the Lefschetz motive in order to make the Pure Motives a rigid tensor category. Ideally one would like the triangulated category of motives to arise as derived category of some rigid tensor category – if this was true, would it be reflected in the fact that $\mathbb{P}^1$ or $\mathbb{G}_m$ are invertible? (in this case of course one should ensure iinvertibility when constructing a candidate for this derived category)
Best Answer
I guess there are "internal" and "external" motivations. External for instance -- most natural examples of functors we have from the stable motivic homotopy category to some other category invert G_m (e.g. any of the usual realizations, or K-theory). Internal for instance -- we want the suspension spectra of varieties to be dualizable under the smash product.