Motivic Homotopy TheoryEffective and very effective motivic spectra

# 6 Effective and very effective motivic spectra

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

Definition 6.1

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

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

$\cdots \to f_{n + 1} E \to f_n E \to f_{n - 1} E \to \cdots .$

Definition 6.2

The $n$-slice of $E$, denoted $s_n E$, is defined by the cofiber sequence

$f_{n + 1} E \to f_n E \to s_n E.$

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

Definition 6.3

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

Theorem 6.4

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

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

Theorem 6.5

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

\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

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

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

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

Proposition 6.7

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

Example 6.8

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