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For a prime number   $\ell $  , we introduce and study oriented right-angled Artin pro-  $\ell $   groups   $G_{\Gamma ,\lambda }$  (oriented pro-  $\ell $   RAAGs for short) associated to a finite oriented graph   $\Gamma $   and a continuous group homomorphism   $\lambda \colon{\mathbb{Z}}_{\ell }\to{\mathbb{Z}}_{\ell }^{\times }$  . We show that an oriented pro-  $\ell $   RAAG   $G_{\Gamma ,\lambda }$   is a Bloch–Kato pro-  $\ell $   group if, and only if,   $(G_{\Gamma ,\lambda },\theta _{\Gamma ,\lambda })$   is an oriented pro-  $\ell $   group of elementary type, generalizing a recent result of I. Snopce and P. Zalesskiĭ—here   $\theta _{\Gamma ,\lambda }\colon G_{\Gamma ,\lambda }\to{\mathbb{Z}}_{\ell}^{\times }$   denotes the canonical   $\ell $  -orientation on   $G_{\Gamma ,\lambda }$  . This yields a plethora of new examples of pro-  $\ell $   groups that are not maximal pro-  $\ell $   Galois groups. We invest some effort in order to show that oriented right-angled Artin pro-  $\ell $   groups share many properties with right-angled Artin pro-  $\ell $  -groups or even discrete RAAG’s, for example, if   $\Gamma $   is a specially oriented chordal graph, then   $G_{\Gamma ,\lambda }$   is coherent generalizing a result of C. Droms. Moreover, in this case,   $(G_{\Gamma ,\lambda },\theta _{\Gamma ,\lambda })$   has the Positselski–Bogomolov property generalizing a result of H. Servatius, C. Droms, and B. Servatius for discrete RAAG’s. If   $\Gamma $   is a specially oriented chordal graph and   $\operatorname{Im}(\lambda )\subseteq 1+4{\mathbb{Z}}_{2}$   in case that   $\ell =2$  , then   $H^{\bullet }(G_{\Gamma ,\lambda },{\mathbb{F}}_{\ell }) \simeq \Lambda ^{\bullet }(\ddot{\Gamma }^{\textrm{op}})$   generalizing a well-known result of M. Salvetti (cf. [  39 ]). Dedicated to the memory of Avinoam Mann.

Oriented Right-Angled Artin Pro-ℓ Groups and Maximal Pro-ℓ Galois Groups