5 Stable motivic homotopy theory

The stable motivic homotopy category is obtained by inverting $\mathbb {P}^1$ in $\mathcal{H}(S)_*$ :

\mathcal{SH}(S) = \mathcal{H}(S)_*[(\mathbb {P}^1)^{-1}].

More generally, suppose we have a presentably symmetric monoidal $\infty$ -category $\mathcal{C}$ (i.e. $\mathcal{C} \in \operatorname{CAlg}(\mathcal{P}r^L)$ ) and $X \in \mathcal{C}$ . We can then define

$\begin{useimager} \[ \Stab_X(\mathcal{C}) = \colim \left(\begin{tikzcd}[column sep=large] \mathcal{C} \ar[r, "-\otimes X"] & \mathcal{C} \ar[r, "-\otimes X"] & \mathcal{C} \ar[r, "-\otimes X"] & \cdots \end{tikzcd}\right), \] \end{useimager}$

or equivalently

$\begin{useimager} \[ \Stab_X(\mathcal{C}) = \lim \left(\begin{tikzcd}[column sep=large] \mathcal{C} & \mathcal{C} \ar[l, "(-)^X"'] & \mathcal{C} \ar[l, "(-)^X"'] & \cdots \ar[l, "(-)^X"'] \end{tikzcd} \right), \] \end{useimager}$

where these (co)limits are taken in the category of large categories (or equivalently $\mathcal{P}r^L$ / $\mathcal{P}r^R$ ).

Theorem 5.1 (Robalo)

If $X$ is symmetric, i.e. the cyclic permutation on $X \otimes X \otimes X$ is homotopic to the identity, then $\operatorname{Stab}_X(\mathcal{C}) \in \operatorname{CAlg}(\mathcal{P}r^L)$ is symmetric monoidal, and the natural map $\mathcal{C} \to \operatorname{Stab}_X(\mathcal{C})$ is universal among maps in $\operatorname{CAlg}(\mathcal{P}r^L)$ that send $X$ to an invertible object.

Note that there is always a map in $\operatorname{CAlg}(\mathcal{P}r^L)$ satisfying this universal property, and $\operatorname{Stab}_X(\mathcal{C})$ always exists. The condition in the theorem ensures these two agree.

It is easy to check that $\mathbb {P}^1$ is indeed symmetric by elementary row and column operations, so we can define

Definition 5.2

$\mathcal{SH}(S) = \mathcal{H}(S)_*[(\mathbb {P}^1)^{-1}]$ .

As in the topological world, we have an adjunction

$\begin{useimager} \[ \begin{tikzcd} \H(S)_* \ar[r, yshift = 2, "\Sigma^\infty_{\P^1}"] & \SH(S) \ar[l, yshift=-2, "\Omega^\infty_{\P^1}"] \end{tikzcd} \] \end{useimager}$

Note that $\Sigma ^\infty _{\mathbb {P}^1}$ is by definition symmetric monoidal, and the fact that the tensor product preserves colimits in both variables tells us how to compute the tensor product in $\mathcal{SH}(S)$ .

Definition 5.3

For $E \in \mathcal{SH}(S)$ and $i, j \in \mathbb {Z}$ , we define

\pi _{p, q}(E) = \pi _0^{\mathbb {A}^1}(\Omega ^\infty _{\mathbb {P}^1}(E \wedge S^{-p, -q})).

Of course, these also lead to homotopy groups given by the global sections of the homotopy sheaves.

Definition 5.4

For $E \in \mathcal{SH}(S)$ and $X \in (\mathrm{Sm}_S)_*$ , and $p, q \in \mathbb {Z}$ , we define

\begin{aligned} E_{p, q}(X) & = \pi _{p, q}(\Sigma ^\infty _{\mathbb {P}^1} X \wedge E)\\ E^{p, q}(X) & = [\Sigma ^\infty _{\mathbb {P}^1} X, \Sigma ^{p, q} E]_{\mathcal{SH}(S)}. \end{aligned}

We shall briefly introduce three examples of motivic spectra.

Example 5.5

Motivic cohomology is represented by a spectrum $H\mathbb {Z}$ . If $U$ is smooth, motivic cohomology is equivalent to Bloch's higher Chow groups:

H\mathbb {Z}^{p, q}(U_+) \cong CH^q(U, 2q - p).

There are multiple ways one can construct $H\mathbb {Z}$ , and I shall describe three. These mimic how one constructs the classical $H\mathbb {Z}$ . These work over any perfect field, except for the second which only produces the right spectrum when the characteristic is $0$ .

Classically, we have the $\infty$ -category $\mathrm{Ch}(\mathrm{Ab})$ of chain complexes of abelian groups, and singular chains defines a natural map $i^*: \operatorname{Sp}\to \mathrm{Ch}(\mathrm{Ab})$ . This admits a right adjoint $i_*$ , and we can define $H\mathbb {Z}= i_* i^* \mathbb {S}$ .

In the motivic world, we can define the triangulated category of motives $\mathrm{DM}(k)$ , which receives a map $\mathcal{SH}(S) \to \mathrm{DM}(k)$ and we can repeat the previous paragraph.
Classically, we can construct $H\mathbb {Z}$ as an infinite loop space directly using the Eilenberg–Maclane spaces, which one can in turn construct using the Dold–Thom theorem as $\mathrm{Sym}^\infty (S^n)$ . This works motivically as well.
Finally, we can define $H\mathbb {Z}= \tau _{\leq 0} \mathbb {S}$ . Motivically, we can define $H\mathbb {Z}$ as the zero slice of $\mathbb {S}$ , which we will discuss later.

Example 5.6

We can construct a motivic spectrum $KGL$ that represents (homotopy) $K$ -theory (which agrees with algebraic $K$ -theory for sufficiently nice schemes). This is obtained by constructing $BGL$ (as the colimit of Grassmannians), and then showing that $\Omega _{\mathbb {P}^1} (BGL \times \mathbb {Z}) \cong BGL \times \mathbb {Z}$ . We then have an explicit infinite loop space.

Example 5.7

We can construct the algebraic cobordism spectrum $MGL$ in the same way as we produce $MO$ or $MU$ . There is a universal vector bundle $\gamma _n \to BGL_n$ . We then define

MGL = \operatorname*{colim}\Sigma ^{-2n, -n} \mathrm{Th}(\gamma _n).