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A spectral sequence for de Rham cohomology. (English) Zbl 1230.14027

Let \(R\) be a complete discrete valuation ring of mixed characeristic, let \(K\) be its fraction field and let \(k\) be its residue field. Let \(X\) be a proper strictly semistable scheme over \(R\). Then the generic fibre \(X_K\) is smooth over \(K\), whereas the special fibre \(X_s\) is a strictly normal crossing divisor in \(X\), in particular, it is the union of smooth proper \(k\)-schemes \(Y_1,\ldots, Y_n\) intersecting transversally. Let \({\mathcal X}\) denote the formal completion of \(X\); for any subscheme \(E\) of \(Y\), the preimage \(]E[_{\mathcal X}\) of \(E\) under the specialization map \({\mathcal X}_K\to Y\) is an admissible open subset in the rigid analytic \(K\)-space \({\mathcal X}_K\) associated with \({\mathcal X}\).
For a non empty subset \(I\) of \(\{1,\ldots,n\}\) put \(Y_I=\cap_{i\in I}Y_i\) and \(U_I=Y_I-\cup_{I'\supsetneq I}Y_{I'}\). Let \(\Omega^{\bullet}_{c,I;{\mathcal X}}\) be the total complex of the bicomplex \[ \Omega^{\bullet}_{]Y_I[_{{\mathcal X}}/K}\longrightarrow (\text{ push forward of }\Omega^{\bullet}_{]Y_I-U_I[_{{\mathcal X}}/K}). \] Theorem. If \(|I|\geq2\) then there is a spectral sequence converging to \(H^*(]Y_I[_{{\mathcal X}},\Omega^{\bullet}_{c,I;{\mathcal X}})\) with \[ E_2^{pq}=H_{c,\text{rig}}^p(U_I/K)\otimes_K\bigwedge(V'_I) \] for a certain \(K\)-vector space \(V'_I\) of dimension \(|I|-1\).
As an application one obtains two formulae comparing the Euler Poincaré characteristic \(\chi_{\text{rig}}(X_s)\) of the rigid cohomology of \(X_s\) with the Euler Poincaré characteristic \(\chi_{dR}(X_K)\) of the de Rham cohomology of \(X_K\):
Proposition. \[ \chi_{\text{rig}}(X_s)-\chi_{dR}(X_K)=\sum_{|I|\geq2}\chi_{c}(U_I), \]
\[ \chi_{\text{rig}}(X_s)-\chi_{dR}(X_K)=\sum_{|I|\geq2}(-1)^I(|I|-1)(\Delta Y_I.\Delta Y_I) \] where \(\chi_{c}(U_I)\) is the Euler Poincaré characteristic of the rigid cohomology with proper support of \(U_I\), and \(\Delta Y_I.\Delta Y_I\) is the self intersection number of \(Y_I\).
One important ingredient in the proof is to elaborate further certain local cohomology calculations in a semistable reduction situation occuring in [E. Grosse-Klönne, Duke Math. J. 113, No. 1, 57–91 (2002; Zbl 1057.14023)].

MSC:

14F40 de Rham cohomology and algebraic geometry
14A15 Schemes and morphisms

Citations:

Zbl 1057.14023
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