Algebraic Geometric Analogue of Brown's Representability

Question

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 $$X\rightarrow H_{et}^i(X,\mathbb{Q}_\ell).$$