The first Suslin homology sheaf of a split simply connected semisimple algebraic group
Let \(G\) be a split simply connected semisimple algebraic group. The aim of this article is to compute the first Suslin homology sheaf of \(G\), \(H_1^S(G)\). The results are as follows: let \(\mathcal Sm_k\) be the category of smooth \(k\)-schemes and \(\underline{C}^{\bullet}(X)\) be the Suslin complex of \(X\in \mathcal Sm_k\). On the other hand, let \(\underline{h}_i^{\mathrm{Nis}}(X)\) be the \((-i)\)-th Nisnevich cohomology sheaf of \( \underline{C}^{\bullet}(X)\), then one has that \(H^S_1(X)=\underline{h}_1^{\mathrm{Nis}}(X)(k)\). Furthermore, let \(\mathbb{Z}_{tr}\) be the Nisnevich sheaf associated to the constant presheaf \(U\mapsto \mathbb{Z}\). The author proves the following two theorems: Theorem A. Let \(k\) be a perfect field and \(G\) be a \(k\)-split simply connected semisimple algebraic group. Denote by \(d\) the number of irreducible components of the root system of \(G\) and by \(\underline{K}^M_2\) the unramified Milnor \(K\)-theory sheaf. Then \[ \underline{h}_0^{\mathrm{Nis}}(G)\simeq\mathbb{Z}_{tr}\text{ and }\underline{h}_1^{\mathrm{Nis}}(G)\simeq (\underline{K}^M_2)^{\oplus d}. \] In particular \(H^S_0(G)\simeq \mathbb{Z}=K^M_0(k)\) and \(H^S_1(G)\simeq (\underline{K}^M_2)^{\oplus d}\). The above theorem is a consequence of the following: Theorem B. Let \(k\) be a field and \(G\) be a \(k\)-split simply connected semisimple algebraic group. Denote by \(d\) the number of irreducible components of the root system of \(G\) and let \(M_*\) be a cycle module over \(k\) in the sense of \textit{M. Rost} [``Chow groups with coefficients'', Doc. Math., J. DMV 1, 319--393 (1996; Zbl 0864.14002)]. Then {\parindent=6mm \begin{itemize}\item[1.] \(H^0(G,M_n)\simeq M_n(k)\), and \item[2.] \(H^1(G,M_n)\simeq M_{n-2}(k)^{\oplus d}\), \end{itemize}} for all integers \(n\), where \(H^i(G,M_n)\) denotes the \(i\)-th cohomology group of the cycle module \(M_n\).
scientific article; zbMATH DE number 5987854
scientific article