I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell}).$$ In some sense, this is a representable functor, i.e. there exists an $\ell$-adic Eilenberg-Maclane space (see Representability of Weil Cohomology Theories in Stable Motivic Homotopy Theory). I would like to know the dimension of this space. The normal way to compute the dimension of a moduli space is to study deformations. However, since the étale site of a scheme is determined by the underlying reduced structure, we see that $$F(X[\epsilon])=H^i_{ét}(X[\epsilon],\mathbb{Q}_{\ell})\cong H^i_{ét}(X,\mathbb{Q}_{\ell})=F(X).$$ However, confusingly, this implies that this analogue of the Eilenberg-Maclane space is zero dimensional, which seems very wrong to me. My guess would be that square-zero extension do not compute the dimension of a motivic spectra. Therefore it seems natural to ask how we can compute this dimension.
1 Answer
I see this is one of your first question on MO -- welcome! The topic of the question is certainly interesting but I think you need to put a little more effort in future questions into being clear (first and foremost with yourself) about the nature of the objects you're asking about. You don't need to understand the details of all the definitions in order to ask a question, but you shouldn't conflate a topological space and a vector space, for example, and if you have a fuzzy understanding of the distinction, ask a separate question about that first.
In particular there does not exist an $\ell$-adic Eilenberg MacLane space. Rather, there exists an ($\ell$-adic etale) Eilenberg-MacLane motive, which is an object of a different category from spaces. If you want to think of it as a "space-like" object, i.e., get an analog of the algebraic topology functor "Connective Spectra" --> "Topological spaces" which takes a spectrum to the underlying infinite loop spaces while replacing "Connective Spectra" with "Motivic spectra" (with some connectivity condition), you can absolutely not replace "spaces" with "algebraic varieties" (so, you do not expect motivic cohomology theories to be representable by algebraic varieties: think for example of the motive $B\mathbb{G}_m$, which admits nontrivial maps from a curve which are trivial outside a point). The weakest modification you could hope to make is to replace "spaces" with "algebraic varieties up to $\mathbb{A}^1$ homotopy equivalences", which already destroys the notion of dimension (since $*\cong\mathbb{A}^1$). Alternatively, you could replace spaces by "$\infty$-stacks" (essentially, simplicial spaces up to a notion of homotopy equivalence), but again the notion of dimension would get lost.
Moreover, even in topology, the Eilenberg-MacLane spaces can be extremely infinite-dimensional. As you perhaps know, the second Eilenberg-MacLane space of the integers, $K(\mathbb{Z}, 2),$ is $\mathbb{C}P^\infty,$ which has infinitely many nonzero cohomology groups. Things only get worse when you replace $\mathbb{Z}$ by $\mathbb{Q}_\ell$ and pass to motives.
This is not to say that you can't assign a dimension to this motive. For example dualizable objects in symmetric monoidal categories have a notion of dimension, and in the symmetric monoidal category of etale $\ell$-adic motives, the object you are interested in is the unit object and has dimension one.
