Variants of formal nearby cycles (Q2921049)

This article introduces a variant of Berkovich's formal nearby cycle functor in the context of a (locally Noetherian, quasi-excellent) formal scheme \(\mathcal{X}\) over Spf \(R\), where \(R\) is a complete discrete valuation ring with separably closed residue field. This functor is constructed purely scheme-theoretically from \(\mathcal{X}\). The main result is rather technical, and we reproduce it here (in the notation below, \(\Lambda\) is the constant sheaf \(\mathbb{Z}/\ell^n\mathbb{Z},\) where \(\ell\) is invertible in \(R\)):NEWLINENEWLINETheorem. For a locally noetherian quasi-excellent formal scheme \(\mathcal{X}\) over Spf \(R\) such that \(\mathcal{X}_{\mathrm{red}}\) is separated, and a pair \(\mathcal{Z} = (\mathcal{Z}_1, \mathcal{Z}_2)\) of closed formal subschemes of the special fiber \(\mathcal{X}_s\) of \(\mathcal{X}\) with \(\mathcal{Z}_1 \subset \mathcal{Z}_2\), we can define an object \(R\Psi_{\mathcal{X}, \mathcal{Z}}\Lambda\) of \(D^+(\mathcal{X}_{\mathrm{red}}, \Lambda)\). It enjoys the following properties.NEWLINENEWLINE(i) \(R\Psi_{\mathcal{X}, \mathcal{Z}}\Lambda\) is functorial in \((\mathcal{X}, \mathcal{Z})\). In particular, if a group \(G\) acts on \(\mathcal{X}\) and \(\mathcal{Z}\) is stable under the action, then \(R\Psi_{\mathcal{X}}, \mathcal{Z}\Lambda\) has a natural \(G\)-equivariant structure.NEWLINENEWLINE(ii) If \(\mathcal{X}\) is obtained by completing a scheme \(X\) locally of finite type over \(R\) along a closed subscheme \(Y\) of \(X_s\) and \(\mathcal{Z}\) is induced from a pair \(Z = (\mathcal{Z}_1, \mathcal{Z}_2)\) of closed subschemes of \(X\) with \(Z_1 \subset Z_2\) (which are not necessarily contained in \(Y\)), then we have the functorial isomorphism NEWLINE\[NEWLINER\Psi_{\mathcal{X}, \mathcal{Z}}\Lambda \cong \left. (Rj_*Rj^!R\psi_X\Lambda) \right|_{Y_{\mathrm{red}}},NEWLINE\]NEWLINE where \(j\) denotes the natural immersion \(Z_2 \backslash Z_1 \hookrightarrow X_s\).NEWLINENEWLINE(iii) If \(\mathcal{X}\) is locally algebraizable or adic over Spf \(R\), then \(R\Psi_{\mathcal{X}}\Lambda := R\Psi_{\mathcal{X}, (\emptyset, \mathcal{X}_s)}\Lambda\) is canonically isomorphic to Berkovich's formal nearby cycle for \(\mathcal{X}\).NEWLINENEWLINE(iv) If \(\mathcal{X}\) is locally algebraizable and pseudo-compactifiable, we have a natural isomorphism NEWLINE\[NEWLINEH^q_c(t(\mathcal{X})_{\bar \eta}, \Lambda) \cong H^q_c(\mathcal{X}_{\mathrm{red}}, R\Psi_{\mathcal{X}, c}\Lambda),NEWLINE\]NEWLINE where \(t(\mathcal{X})_{\bar \eta}\) denotes the geometric generic fiber of \(\mathcal{X}\) and \(R\Psi_{\mathcal{X}, c}\Lambda\) denotes \(R\Psi_{\mathcal{X}, (\emptyset, \mathcal{X}_{\mathrm{red}})}\Lambda\).