Prime-to-residue-characteristic Tate modules and inertia

Assume AC and DC, inherited from the multiplication and strict-henselization suppliers. Let $A$ be an abelian variety over a field $K$ (Abelian varieties over a field), fix a separable closure $K^{\mathrm{s}\mathrm{e}\mathrm{p}}$ of $K$, and let $\mathrm{\ell }$ be a prime different from $\mathrm{char}K$. For each $\nu \ge 1$ the multiplication-by-${\mathrm{\ell }}^{\nu }$ endomorphism of $A$ is finite and faithfully flat and $A[{\mathrm{\ell }}^{\nu }]=A{\times }_{[{\mathrm{\ell }}^{\nu }],A,e}\mathrm{Spec}K$ is finite locally free by Nonzero multiplication on an abelian variety is finite and faithfully flat. It is etale because the differential of multiplication by ${\mathrm{\ell }}^{\nu }$ at the identity is the unit ${\mathrm{\ell }}^{\nu }$ times the identity; translations give an invertible differential everywhere, and Relative Jacobian criterion with its presentation hypothesis makes multiplication etale, as is its pullback along the unit section. The $\mathrm{\ell }$-adic Tate module is the inverse limit along the transition maps given by multiplication by $\mathrm{\ell }$, with the underlying inverse system of finite discrete groups (Inverse systems and inverse limits of modules, The inverse limit of finite groups carries the subspace topology from the product of discrete factors). Thus an element is a sequence $(a_{\nu }{)}_{\nu \ge 1}$ with $a_{\nu }\in A[{\mathrm{\ell }}^{\nu }](K^{\mathrm{s}\mathrm{e}\mathrm{p}})$ and $\mathrm{\ell }{a}_{\nu +1}=a_{\nu }$; coordinatewise scalar multiplication by (A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups) makes $T_{\mathrm{\ell }}(A)$ a ${\mathbb{Z}}_{\mathrm{\ell }}$-module. It is given the subspace topology from the product of the finite discrete torsion groups, so it is a profinite group and scalar multiplication is continuous. The absolute Galois group $\mathrm{Gal}(K^{\mathrm{s}\mathrm{e}\mathrm{p}}\mathrm{/}K)$ acts coordinatewise on $T_{\mathrm{\ell }}(A)$, and this action is ${\mathbb{Z}}_{\mathrm{\ell }}$-linear.

Let now $R$ be a discrete valuation ring with fraction field $K$, and choose a strict henselization $R^{\mathrm{s}\mathrm{h}}$ with fraction field $K^{\mathrm{s}\mathrm{h}}$ embedded in $K^{\mathrm{s}\mathrm{e}\mathrm{p}}$ through the fixed separable closure (Strict henselization of a DVR and smooth sections). The inertia group is $I=\mathrm{Gal}(K^{\mathrm{s}\mathrm{e}\mathrm{p}}\mathrm{/}{K}^{\mathrm{s}\mathrm{h}})\subseteq \mathrm{Gal}(K^{\mathrm{s}\mathrm{e}\mathrm{p}}\mathrm{/}K)\mathrm{.}$ The Tate module $T_{\mathrm{\ell }}(A)$ is unramified at $R$ when $I$ acts trivially on it. This definition specifies the action and the test; no freeness, torsion or reduction criterion is assumed.