Let G be a reductive group over a commutative ring R. We say that G has isotropic rank \(\ge n\), if every normal semisimple reductive R-subgroup of G contains \(({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n\). We prove that if G has isotropic rank \(\ge 1\) and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra \(A=R[x_1,\ldots ,x_n]/I\) over R, the map \(H^1_{\mathrm {Nis}}(A,G)\rightarrow H^1_{\mathrm {Nis}}(R,G)\) induced by evaluation at \(x_1=\cdots =x_n=0\), is a bijection. If k has characteristic 0, then, moreover, the map \(H^1_{\acute{\mathrm{e}}\mathrm {t}}(A,G)\rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(R,G)\) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is \(\ge 2\), and A is square-free, then \(K_1^G(A)=K_1^G(R)\), where \(K_1^G(R)=G(R)/E(R)\) is the corresponding non-stable \(K_1\)-functor, also called the Whitehead group of G. The corresponding statements for \(G={{\mathrm{GL}}}_n\) were previously proved by Ton Vorst.

中文翻译：

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离散Hodge代数上的各向同性还原群

令G为交换环R上的还原性基团。我们说G具有各向同性秩\（\ ge n \），如果G的每个普通半简单还原R-子群都包含\（（{{\ mathrm {{{\ mathbf {G}}} _ m}}}​​ __ {， R}）^ n \）。我们证明如果G具有各向同性的秩\（\ ge 1 \）并且R是包含无限字段k的正则域，那么对于任何离散的Hodge代数\（A = R [x_1，\ ldots，x_n] / I \）在R上，通过在处进行评估得出的映射\（H ^ 1 _ {\ mathrm {Nis}}（A，G）\ rightarrow H ^ 1 _ {\ mathrm {Nis}}（R，G）\）\（x_1 = \ cdots = x_n = 0 \）是双射。如果k具有特征0，则映射\（H ^ 1 _ {\ acute {\ mathrm {e}} \ mathrm {t}}（A，G）\ rightarrow H ^ 1 _ {\ acute {\ mathrm { e}} \ mathrm {t}}（R，G）\）的内核很小。我们还证明，如果ķ是完美的，ģ被过定义ķ，的各向同性秩ģ是\（\ GE 2 \） ，和甲是正方形-自由，然后\（K_1 ^ G（A）= K_1 ^ G（ R）\），其中\（K_1 ^ G（R）= G（R）/ E（R）\）是相应的非稳定\（K_1 \）- functor，也称为G的怀特黑德组。对应的语句\（G = {{\ mathrm {GL}}} _ n \）之前由Ton Vorst证明。