Abelian class groups of reductive group schemes
Let \(K\) be a number field or a function field in one variable over a finite field \(k\). Let \(S\) be a nonempty open subscheme of \(\mathrm{Spec}(\mathcal{O}_K)\) when \(K\) is a number field or a nonempty open affine subscheme of the smooth projective irreducible curve over \(k\) with function field \(K\) when \(K\) is a function field. For each open subscheme \(U\) of \(S\), let \(A_S(U)\) be the ring of \(U\)-integral adeles of \(S\) and let \(A_S = \varinjlim_U A_S(U)\). Suppose that \(G\) is an affine group scheme of finite type over \(S\) with smooth generic fiber, consider the class set of \(G\) \[ C(G) = G(A_S(S)) \backslash G(A_S) / G(K). \] In general, \(C(G)\) is just a set and carries no additioanl structure. In this paper, the author uses the idea of ``abelian group approximation'' as in the theory of quadratic forms to study \(C(G)\) when \(G\) is a reductive \(S\)-group scheme. Let \(S_{\mathrm{fl}}\) be the small fppf site over \(S\) and \(H^i_{\mathrm{ab}}(S_{\mathrm{fl}},G)\) be the abelian cohomology groups of \(G\), define \[ C_{\mathrm{ab}}(G) = \mathrm{Ker}(H^1_{\mathrm{ab}}(S_{\mathrm{fl}},G) \to H^1_{\mathrm{ab}}(K_{\mathrm{fl}},G)). \] \vskip 0.1in The first main result of the paper is Theorem 1: Assume that \(K\) has no primes. Then there exist a natural right action of \(H_{\mathrm{ab}}^0(S_{\mathrm{fl}},G)\) on \(H^1(S_{\mathrm{\'{e}t}},\tilde{G})\) where \(\tilde{G}\) is the simply connected central cover of the derived group \(G\), and a canonical exact sequence of pointed sets \[ 1 \to H^1(S_{\text{ét}},\tilde{G})/H_{\mathrm{ab}}^0(S_{\mathrm{fl}},G) \to C(G) \to C_{\mathrm{ab}}(G) \to 1, \] where the first nontrivial map is injective. When \(K\) has no real primes and the simply-connected central cover of \(G_K\) satisfies the strong approximation property with respect to the set of primes of \(K\) which do not correspond to a closed point of \(S\), there exists a canonical isomorphism between \(C(G)\) and \(C_{\mathrm{ab}}(G)\). The second main result of the paper is Theorem 2: Let \(G\) be a reductive group scheme over \(S\) and let \(1 \to F \to H \to G \to 1\) be a flasque resolution of \(G\). Set \(R = H^{\mathrm{tor}}\). Then the given resolution induces an exact sequence of finitely generated abelian groups \[ \mu(S) \hookrightarrow F(S) \to R(S) \to H^0_{\mathrm{ab}}(S_{\mathrm{fl}}, G) \to H^1(S_{\mathrm{\'{e}t}}, F) \to C(R) \to C_{\mathrm{ab}}(G) \to 1, \] where \(\mu\) is the fundamental group of the derived group of \(G\). \vskip 0.1in The third main result of the paper is Theorem 3: Assume that \(G_K = G \times_K \mathrm{Spec}(K)\) admits a smooth \(K\)-compactification. There exists a perfect pairing of finite groups \[ C_{\mathrm{ab}}(G) \times \mathrm{Br}^S_{\mathrm{a},\mathrm{nr}}(G_K)/B(G_K) \to \mathbb{Q}/\mathbb{Z} \] where \(B(G_K) = \mathrm{Ker}(\mathrm{Br}_a(G_K) \to \prod_{\mathrm{all} \; v}\mathrm{Br}_a(G_{K_v}))\).
scientific article; zbMATH DE number 6221771
scientific article