TITLE: Why does one invert $G_m$ in the construction of the motivic stable homotopy category?
QUESTION [16 upvotes]: 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):

Inverting $\mathbb{G}_m$ is necessary to produce a Gysin sequence
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)

REPLY [12 votes]: $H^1({\mathbb G}_m)$ is the same motive as $H^2(\mathbb P^1)$, so I believe that inverting $\mathbb G_m$ is the same as inverting the Lefschetz motive.  (Topologically, fundamental
classes all ultimately arise from the $H^1$ of the circle, so we must  invert this
if fundamental classes are to be invertible.)  In the pure context, one is not allowed to talk about ${\mathbb G}_m$, and so talks of $H^2(\mathbb P^1)$ instead; but I do think it is the same process.
Note that this is compatible with Dustin Clausen's and Marty's answers: in all realizations (Galois representations, computing 
periods, ... ) of motives, we can and do invert the Lefschetz motive (since it just becomes
the inverse cyclotomic character, or $2 \pi i$, or ... ).    Incidentally, related to David Roberts's answer, smashing with
$\mathbb G_m$ is what number theorists call a Tate twist, I believe, and on the level of Galois representations it it is just tensoring with the cyclotomic character (here I am thinking of $H_1(\mathbb G_m)$).   So asking for $\mathbb G_m$ to be invertible is the same as asking that one can perform Tate twists (of arbitrary integer power) on the level of motives.
Now if $M$ is a motive over a number field, with $L$-function $L(M,s)$,
and $M(n)$ is its $n$th Tate twist, then the $L$-function of $M(n)$ is simply 
$L(M,s + n)$.  So the Tate twisting parameter is the same as the parameter $s$ in
the $L$-function (restricted to integer values, of course).  Thus the desire to have
this parameter really be an integer (and not just a natural number) also has deep roots
in the conjectured relations between special values of $L$-functions and the arithmetic geometry of motives.  (If one looks at the simplest $L$-function, namely the Riemann zeta function, it has special values at positive and negative integers, and one certainly wants to consider all of these and relate all of them to motivic considerations.)