5 Stable motivic homotopy theory
The stable motivic homotopy category is obtained by inverting in :
More generally, suppose we have a presentably symmetric monoidal -category (i.e. ) and . We can then define
where these (co)limits are taken in the category of large categories (or equivalently /).
If is symmetric, i.e. the cyclic permutation on is homotopic to the identity, then is symmetric monoidal, and the natural map is universal among maps in that send to an invertible object.
Note that there is always a map in satisfying this universal property, and always exists. The condition in the theorem ensures these two agree.
It is easy to check that is indeed symmetric by elementary row and column operations, so we can define
As in the topological world, we have an adjunction
Note that 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 .
For and , we define
Of course, these also lead to homotopy groups given by the global sections of the homotopy sheaves.
For and , and , we define
We shall briefly introduce three examples of motivic spectra.
Motivic cohomology is represented by a spectrum . If is smooth, motivic cohomology is equivalent to Bloch's higher Chow groups:
There are multiple ways one can construct , and I shall describe three. These mimic how one constructs the classical . These work over any perfect field, except for the second which only produces the right spectrum when the characteristic is .
Classically, we have the -category of chain complexes of abelian groups, and singular chains defines a natural map . This admits a right adjoint , and we can define .
In the motivic world, we can define the triangulated category of motives , which receives a map and we can repeat the previous paragraph.
Classically, we can construct as an infinite loop space directly using the Eilenberg–Maclane spaces, which one can in turn construct using the Dold–Thom theorem as . This works motivically as well.
Finally, we can define . Motivically, we can define as the zero slice of , which we will discuss later.
We can construct a motivic spectrum that represents (homotopy) -theory (which agrees with algebraic -theory for sufficiently nice schemes). This is obtained by constructing (as the colimit of Grassmannians), and then showing that . We then have an explicit infinite loop space.
We can construct the algebraic cobordism spectrum in the same way as we produce or . There is a universal vector bundle . We then define