On Grothendieck-Serre's conjecture concerning principal \(G\)-bundles over reductive group schemes. II (Q2827311)

A conjecture of Grothendieck and Serre on torsors dating back to 1958 (see [Séminaire Claude Chevalley (2e année) Tome 3. (French) École Normale Supérieure. Paris: Secrétariat Mathématique (1958; Zbl 0098.13101)] predicts that for a regular local ring \(R\) with fraction field \(K\) and a reductive group \(R\)-scheme \(G\), the specialization map NEWLINE\[NEWLINE H^1_{\text{ét}}(R, G) \to H^1_{\text{ét}}(K, G) NEWLINE\]NEWLINE is always injective.When \(R\) contains an infinite field \(k\), this has been proved in [\textit{R. Fedorov} and the author, Publ. Math., Inst. Hautes Étud. Sci. 122, 169--193 (2015; Zbl 1330.14077)]. That proof hinges upon the Main Theorem 1.0.3 in this article stated as follows. Let \(\mathcal{O}\) be a semi-local ring with finitely many closed points on a smooth affine variety \(X\) over \(k\). Assume that for all semisimple simply connected \(\mathcal{O}\)-group scheme \(H\), the map \(H^1_{\text{ét}}(\mathcal{O}, H) \to H^1_{\text{ét}}(k(X), H)\) is injective, then the same holds for any reductive \(\mathcal{O}\)-group scheme \(G\), with \(K = \text{Frac}(\mathcal{O})\) replacing \(k(X)\).NEWLINENEWLINEThis is in turn built on two purity Theorems 1.0.1 and 1.0.2, which are of independent interest. The first one asserts that in the circumstance above, consider a smooth \(\mathcal{O}\)-morphism of reductive group schemes \(\mu: G \to C\) such that \(\text{Ker}(\mu)\) is a reductive \(\mathcal{O}\)-group scheme, then there is an exact sequence NEWLINE\[NEWLINE 0 \to C(\mathcal{O})/\mu(G(\mathcal{O})) \to C(K)/\mu(G(K)) \to \bigoplus_{ \text{ht}(\mathfrak{p})=1 } C(K) \big/ C(\mathcal{O}_p)\mu(G(K)) \to 0. NEWLINE\]NEWLINENEWLINENEWLINEFor the second purity result, let \(i: Z \hookrightarrow G\) be a central closed subgroup scheme of a semisimple \(\mathcal{O}\)-group scheme \(G\), with \(G' = G/i(Z)\). Then there is an exact sequence NEWLINE\[NEWLINE 0 \to \dfrac{ H^1_{\text{fppt}}(\mathcal{O}, Z) }{\text{im}(\delta_{\mathcal{O}}) } \to \dfrac{ H^1_{\text{fppt}}(K, Z) }{\text{im}(\delta_K) } \to \bigoplus_{ \text{ht}(\mathfrak{p})=1 } \dfrac{ H^1_{\text{fppt}}(K, Z) }{ H^1_{\text{fppt}}(\mathcal{O}_{\mathfrak{p}}, Z) + \text{im}(\delta_K) } \to 0 NEWLINE\]NEWLINE where \(\delta_\star\) stand for the coboundary maps \(G'(\star) \to H^1_{\text{fppt}}(\star, Z)\), and fppt indicates the topology of finitely presented flat morphisms.