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Let $F$ be a CM field. In this paper, we prove the local-global compatibilityfor cohomological cuspidal automorphic representations of$\mathrm{GL}_n(\mathbb{A}_F)$ at $p \neq l$ by using certain potentialautomorphy theorems in many cases including higher dimensional cases. Moreover,we also prove the Ramanujan conjecture for cohomological cuspidal automorphicrepresentations of $\mathrm{GL}_2(\mathbb{A}_F )$.

On the potential automorphy and the local-global compatibility for the monodromy operators at $p \neq l$ over CM fields