Motivic Homotopy Theory — Effective and very effective motivic spectra

6 Effective and very effective motivic spectra

The category of (non-motivic) spectra admits a tt-structure, where the connective objects Sp0\operatorname{Sp}_{\geq 0} is the subcategory generated by SnS^ n for n0n \geq 0 under (sifted) colimits. In the motivic world, we have two kinds of spheres — S1S^1 and Gm\mathbb {G}_ m, which makes everything bigraded. Thus, we can have different notions of connectivity depending of which spheres we use.

Definition 6.1

Let SHeff(S)\mathcal{SH}^{\mathrm{eff}}(S) be the smallest stable subcategory (closed under direct sums) of SH(S)\mathcal{SH}(S) containing all ΣP1X+\Sigma ^\infty _{\mathbb {P}^1} X_+ for XSmSX \in \mathrm{Sm}_ S and closed under colimits. This is the category of effective spectra.

The inclusion GmnSHeff(S)SH(S)\mathbb {G}_ m^ n \wedge \mathcal{SH}^{\mathrm{eff}}(S) \hookrightarrow \mathcal{SH}(S) admits a right adjoint fnf_ n. This defines the slice filtration

fn+1EfnEfn1E. \cdots \to f_{n + 1} E \to f_ n E \to f_{n - 1} E \to \cdots .

Definition 6.2

The nn-slice of EE, denoted snEs_ n E, is defined by the cofiber sequence

fn+1EfnEsnE. f_{n + 1} E \to f_ n E \to s_ n E.

On the other hand, we can only allow for non-negative powers of SiS^ i but include negative powers of Gm\mathbb {G}_ m. This in fact defines a tt-structure on SH(S)\mathcal{SH}(S).

Definition 6.3

Let SH(S)0\mathcal{SH}(S)_{\geq 0} be the subcategory of SH(S)\mathcal{SH}(S) generated by ΣP1X+Gmq\Sigma ^\infty _{\mathbb {P}^1} X_+ \wedge \mathbb {G}_ m^{-q} under extensions and colimits.

Theorem 6.4

This forms part of a tt-structure on SH(S)\mathcal{SH}(S), called the homotopy tt-structure.

In the case of a perfect field, but not in general, we can characterize the homotopy tt-structure by the homotopy sheaves.

Theorem 6.5

If S=SpeckS = \operatorname{Spec}k and kk is a perfect field, then

SH(S)0={ESH(S)πp,q(E)=0 whenever pq<0}SH(S)0={ESH(S)πp,q(E)=0 whenever pq>0} \begin{aligned} \mathcal{SH}(S)_{\geq 0} & = \{ E \in \mathcal{SH}(S) \mid \pi _{p, q}(E) = 0\text{ whenever }p - q < 0\} \\ \mathcal{SH}(S)_{\leq 0} & = \{ E \in \mathcal{SH}(S) \mid \pi _{p, q}(E) = 0\text{ whenever }p - q> 0\} \end{aligned}

Finally, we can intersect these two.

Definition 6.6

We define

SHveff(S)=SHeff(S)SH(S)0. \mathcal{SH}^{\mathrm{veff}}(S) = \mathcal{SH}^{\mathrm{eff}}(S) \cap \mathcal{SH}(S)_{\geq 0}.

Equivalently, it is the full subcategory of SH(S)\mathcal{SH}(S) that generated by ΣP1X+\Sigma ^\infty _{\mathbb {P}^1} X_+ and under colimits. This is the category of very effective spectra.

Since the smash product preserves colimits and X+Y+=(X×)+X_+ \wedge Y_+ = (X \times )_+, we see that

Proposition 6.7

SH(S)veff\mathcal{SH}(S)^{\mathrm{veff}} is closed under the smash product.

Example 6.8

MGLMGL is very effective. To show this, we have to show that for γnBGLn\gamma _ n \to BGL_ n the universal vector bundle, the Thom spectrum Σ2n,nΣP1Th(γn)\Sigma ^{-2n, -n} \Sigma ^\infty _{\mathbb {P}^1}\mathrm{Th}(\gamma _ n) is in very effective. This is true for rank nn vector bundles in general, since it is true for trivial vector bundles, and vector bundles are locally trivial.