Brown's representability theorem is very usefull to show that the functor $$X \rightarrow H^i(X,A)$$ is representable. I would be interested to see if there exists an analogue of this statement in the context of algebraic geometry, in particular to show that algebraic geometric analogues of Eilenberg-Maclain spaces exist. For example, does there exists an ``l-adic Eilenberg-Maclain space", i.e. a scheme or stack which represents the functor $$X\rightarrow H_{et}^i(X,\mathbb{Q}_\ell).$$

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    $\begingroup$ Mathoverflow should be more appropriate for this question. $\endgroup$ Feb 18, 2020 at 4:41
  • $\begingroup$ I think you need to get into motivic homotopy theory to get a well behaved enough category for this. Certainly triangulated categories of motives can satisfy Brown representability, but this doesn't exactly give you a scheme or a stack. $\endgroup$ Feb 18, 2020 at 5:28
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    $\begingroup$ You should read "Local homotopy theory" by Jardine to understand what are Elenberg-Maclane spaces in the context of (pre)sheaves of simplicial sets. After that you should read "proétale topology for schemes" of Bhatt and Scholze to get that the $l$-adic cohomology is the mapping space into some simplicial sheaf (in the étale site this is not true). Combining the two should give a definition $l$-adic EM spaces $K(\mathbb{Q}_l,n)$. $\endgroup$ Feb 18, 2020 at 12:55
  • $\begingroup$ @jeanmfischer, thank you very much for your answer. If you put your comment in to an answer, I'm happy to accept it. $\endgroup$ Feb 19, 2020 at 18:19


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