With ι Sample Clauses

With ι. Fp[[t]] → R defined as above, we set for any n ∈ N: X′(ϵ) :=XFp[[t]](1) ×Fp[[t]],ι R X′Ig(pn)(ϵ) :=XFp[[t]],Ig(pn)(1) ×Fp[[t]],ι R. Explicitly, like in (12), the formal scheme X′(ϵ) is locally over U = Spf(S) ⊆ XR of the form X′(ϵ)|U = Spf(S⟨X⟩/(XHa − tє)). It therefore coincides with the “X′(ϵ)” of III.2 [50]. We can again define a limit of the Igusa tower in the category of formal schemes: n X′Ig(p∞)(ϵ) = l←im− X′Ig(pn)(ϵ) → X′(ϵ). It is clear from this definition that for n ∈ N, the rigid generic fibre of X′Ig(pn)(ϵ) → X′(ϵ) is XI′g(pn)(ϵ) → X′(ϵ). We will see in the next section that the formal schemes we have just defined are in fact again integrally closed in their generic fibre, so that we can a posteriori see them as being the “normalisations” of XR(ϵ) in XI′g(pn)(ϵ) → X′(ϵ), even though this notion is in general problematic in the absence of excellence assumptions. p In particular, the reason we use XFp[[t]](1) = X−1 instead of Xr for r ≥ 0 is that base- change along the latter would not produce an integrally closed formal scheme. Nevertheless, we still have a natural map X′(p−kϵ) → XF [[t]](p−k), even though this is not the base-change: p