Abelian scheme torsion specialization is unramified

By [F1] the morphism $[{\mathrm{\ell }}^{\nu }]:A\to A$ is finite, flat and of degree ${\mathrm{\ell }}^{2g\nu }$, and since $\mathrm{\ell }$ is invertible on $R$ it is etale; its kernel $A[{\mathrm{\ell }}^{\nu }]$, the pullback along the identity section, is finite etale of rank ${\mathrm{\ell }}^{2g\nu }$ by [F2] (the rank is locally constant and equals ${\mathrm{\ell }}^{2g\nu }$ on the generic fibre of the connected base $\mathrm{Spec}R$). This proves (a).

Base change to $R^{\mathrm{s}\mathrm{h}}$: the scheme $A[{\mathrm{\ell }}^{\nu }{]}_{R^{\mathrm{s}\mathrm{h}}}$ is finite etale over the strictly henselian local ring $R^{\mathrm{s}\mathrm{h}}$ with separably closed residue field $k^s$, so by [F3] it is a disjoint union of copies of $\mathrm{Spec}{R}^{\mathrm{s}\mathrm{h}}$; in particular every geometric generic point is already rational over $K^{\mathrm{s}\mathrm{h}}$, and reduction is a bijection between the generic geometric points and the special separable points. Consequently $\mathrm{Gal}(K^{\mathrm{s}\mathrm{e}\mathrm{p}}\mathrm{/}{K}^{\mathrm{s}\mathrm{h}})$ fixes every point of $A[{\mathrm{\ell }}^{\nu }](K^{\mathrm{s}\mathrm{e}\mathrm{p}})$, so the inertia group $I=\mathrm{Gal}(K^{\mathrm{s}\mathrm{e}\mathrm{p}}\mathrm{/}{K}^{\mathrm{s}\mathrm{h}})$ acts trivially; the same holds on the inverse limit $T_{\mathrm{\ell }}(A)$, because inertia acts coordinatewise on the inverse limit and fixes every torsion coordinate. This is the precise finite-etale smooth-proper specialization statement used here; it is not general smooth proper base change for higher cohomology.