Given a compatible subsystem {ρℓ}ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\{\rho _\ell \}_\ell $$\end{document} of n-dimensional ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}-adic Galois representations arising from étale cohomology of any complete, non-singular variety over a number field K, we define Γℓ:=ρℓ(Gal(K¯/K))\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _\ell := \rho _\ell ({\text {Gal}}(\bar{K}/K))$$\end{document} and let Gℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}_\ell $$\end{document} denote the Zariski closure of Γℓ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _\ell $$\end{document} in GLn\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}_n$$\end{document}. If Gℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}_\ell $$\end{document} is of Type A in the sense that all simple composition factors are of type A in the Cartan-Killing classification, then Γℓ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _\ell $$\end{document} is, in a suitable sense, maximal in Gℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}_\ell $$\end{document} for all ℓ≫0\documentclass[12pt]{minimal}
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\begin{document}$$\ell \gg 0$$\end{document}. As a corollary, if ρℓ\documentclass[12pt]{minimal}
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\begin{document}$$\rho _\ell $$\end{document} is semisimple and ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} is sufficiently large, then Gℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}_\ell $$\end{document} is unramified.
