6 Effective and very effective motivic spectra
The category of (non-motivic) spectra admits a -structure, where the connective objects is the subcategory generated by for under (sifted) colimits. In the motivic world, we have two kinds of spheres — and , which makes everything bigraded. Thus, we can have different notions of connectivity depending of which spheres we use.
Let be the smallest stable subcategory (closed under direct sums) of containing all for and closed under colimits. This is the category of effective spectra.
The inclusion admits a right adjoint . This defines the slice filtration
The -slice of , denoted , is defined by the cofiber sequence
On the other hand, we can only allow for non-negative powers of but include negative powers of . This in fact defines a -structure on .
Let be the subcategory of generated by under extensions and colimits.
This forms part of a -structure on , called the homotopy -structure.
In the case of a perfect field, but not in general, we can characterize the homotopy -structure by the homotopy sheaves.
If and is a perfect field, then
Finally, we can intersect these two.
Equivalently, it is the full subcategory of that generated by and under colimits. This is the category of very effective spectra.
Since the smash product preserves colimits and , we see that
is closed under the smash product.
is very effective. To show this, we have to show that for the universal vector bundle, the Thom spectrum is in very effective. This is true for rank vector bundles in general, since it is true for trivial vector bundles, and vector bundles are locally trivial.