Motivic Homotopy Theory — Stable motivic homotopy theory

5 Stable motivic homotopy theory

The stable motivic homotopy category is obtained by inverting P1\mathbb {P}^1 in H(S)\mathcal{H}(S)_*:

SH(S)=H(S)[(P1)1]. \mathcal{SH}(S) = \mathcal{H}(S)_*[(\mathbb {P}^1)^{-1}].

More generally, suppose we have a presentably symmetric monoidal \infty -category C\mathcal{C} (i.e. CCAlg(PrL)\mathcal{C} \in \operatorname{CAlg}(\mathcal{P}r^ L)) and XCX \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 PrL\mathcal{P}r^ L/PrR\mathcal{P}r^ R).

Theorem 5.1 (Robalo)

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

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

It is easy to check that P1\mathbb {P}^1 is indeed symmetric by elementary row and column operations, so we can define

Definition 5.2

SH(S)=H(S)[(P1)1]\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 ΣP1\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 SH(S)\mathcal{SH}(S).

Definition 5.3

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

πp,q(E)=π0A1(ΩP1(ESp,q)). \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 ESH(S)E \in \mathcal{SH}(S) and X(SmS)X \in (\mathrm{Sm}_ S)_*, and p,qZp, q \in \mathbb {Z}, we define

Ep,q(X)=πp,q(ΣP1XE)Ep,q(X)=[ΣP1X,Σp,qE]SH(S). \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 HZH\mathbb {Z}. If UU is smooth, motivic cohomology is equivalent to Bloch's higher Chow groups:

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

There are multiple ways one can construct HZH\mathbb {Z}, and I shall describe three. These mimic how one constructs the classical HZH\mathbb {Z}. These work over any perfect field, except for the second which only produces the right spectrum when the characteristic is 00.

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

    In the motivic world, we can define the triangulated category of motives DM(k)\mathrm{DM}(k), which receives a map SH(S)DM(k)\mathcal{SH}(S) \to \mathrm{DM}(k) and we can repeat the previous paragraph.

  2. Classically, we can construct HZH\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 Sym(Sn)\mathrm{Sym}^\infty (S^ n). This works motivically as well.

  3. Finally, we can define HZ=τ0SH\mathbb {Z}= \tau _{\leq 0} \mathbb {S}. Motivically, we can define HZH\mathbb {Z} as the zero slice of S\mathbb {S}, which we will discuss later.

Example 5.6

We can construct a motivic spectrum KGLKGL that represents (homotopy) KK-theory (which agrees with algebraic KK-theory for sufficiently nice schemes). This is obtained by constructing BGLBGL (as the colimit of Grassmannians), and then showing that ΩP1(BGL×Z)BGL×Z\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 MGLMGL in the same way as we produce MOMO or MUMU. There is a universal vector bundle γnBGLn\gamma _ n \to BGL_ n. We then define

MGL=colimΣ2n,nTh(γn). MGL = \operatorname*{colim}\Sigma ^{-2n, -n} \mathrm{Th}(\gamma _ n).