Good reduction, coherent base change, and unramified torsion

Clause (a): the generic fibre $A_K$ is an abelian variety of dimension $g$ over the function field $K$ by [F1], and the abelian scheme $A$ itself is an abelian scheme model, so $A_K$ has good reduction over $S$ by definition; the fibre $A_s$ is an abelian variety of dimension $g$ for every closed $s\in S$ by [F1]. Any abelian scheme model of $A_K$ is a Neron model of $A_K$ by [F2], and such models are uniquely isomorphic compatibly with their specified generic-fibre identifications, so $A$ has exactly this compatible uniqueness and is the Neron model.

Clause (b): for every $S^{\mathrm{\prime }}\to S$ the base change $A_{S^{\mathrm{\prime }}}\to S^{\mathrm{\prime }}$ is an abelian scheme of relative dimension $g$ and represents the base change of the model by [F1]; it is therefore again a good-reduction model. If $S^{\mathrm{\prime }}$ is Dedekind, $S^{\mathrm{\prime }}\to S$ dominant and $K^{\mathrm{\prime }}$ is its function field, then $A_{K^{\mathrm{\prime }}}$ has good reduction over $S^{\mathrm{\prime }}$ with model $A_{S^{\mathrm{\prime }}}$ by [F2].

Clause (c): the map ${\phi }_s^q$ is the base-change map of Cohomology and base-change map, and [F3] gives, under surjectivity, an affine open $U\ni s$ with $h^{\ast }(R^q{f}_{\ast }\mathcal{F}{\mathrm{\mid }}_U)\to R^q{f}_{T\ast }^{\mathrm{\prime }}(g_T^{\ast }\mathcal{F})$ an isomorphism for every $T\to U$, and under the additional surjectivity of ${\phi }_s^{q-1}$ the finite local freeness of $R^q{f}_{\ast }\mathcal{F}$ on a neighbourhood. For $q=0$ the degree $-1$ surjectivity condition is automatic. For $q=0$ and $\mathcal{F}={\mathcal{O}}_A$, [F3] identifies the unit map ${\mathcal{O}}_S\to f_{\ast }{\mathcal{O}}_A$ with inverse given by evaluation along the identity section; for a line bundle flat over $S$, the successive-surjectivity clause yields finite locally free higher direct images where the base-change maps are isomorphisms.

Clause (e): locally near a closed point of $S$ the base is a discrete valuation ring $R$ with fraction field $K$ and residue field $k$, and the arbitrary-DVR existence theorem in [F4] first supplies a finite-type Neron model for the abelian variety in (e). Applying the criterion in [F4] to that model gives the equivalence between good reduction of $A_K$ over $R$, the Neron model being an abelian scheme, pointwise inertia-fixed $\mathrm{\ell }$-power torsion, and an unramified Tate module, for every $\mathrm{\ell }\mathrm{\ne }\mathrm{char}k$. For an abelian scheme model, [F5] shows that each $A[{\mathrm{\ell }}^{\nu }]$ is finite etale of rank ${\mathrm{\ell }}^{2g\nu }$ and commutes with every base change, specializing over a strict henselization to the special geometric torsion; this is the finite-torsion route to the criterion and claims no all-degree smooth-proper etale cohomology theorem.

Clause (d): the arguments in steps 1.1–1.3 are the scheme-theoretic identity, base-change and cohomology-and-base-change statements, none of which restricts the residue characteristic; only the surjectivity hypotheses of the cohomology-and-base-change theorem are used in (c), which is exactly why the finite-flatness conclusions are stated conditionally.

Combining steps 1.1–2.1 proves clauses (a)-(e); in particular both the coherent cohomology clauses (c)-(d) and the prime-to-residue-characteristic etale clause (e) are retained.