论文标题
对称群体的理性函数和半线性表示的不变字段
Invariant fields of rational functions and semilinear representations of symmetric groups over them
论文作者
论文摘要
让$ k $为一个领域，而$ g $是赋予紧凑型拓扑的一组自动形态。在许多情况下，很自然地研究$ k $ g $的$ k $ semilinear代表$ sm_k（g）$ sm_k（g）$。 $ sm_k（g）$的类别为semisimple（在这种情况下，$ k $是$ sm_k（g）$）的生成器，并且仅当$ g $是预发的。在本说明中，我们研究了无限套装$ s $的所有排列的非表述集团$ g $的情况。结果表明，类别$ sm_k（g）$是本地noetherian的；形态是“局部分裂”。给定一个字段$ f $和一个子字段$ k \ neq f $代数以$ f $关闭，其中一个主要结果描述了类别的Gabriel Spectra（及相关对象）$ SM_K（g）$ g $ g $ g $ invariant $ invariant $ k $ k $ k $ f _ $ f_ $ f _ $ a $ a $ a $ a $ a $ k a $ a $ k的$ k，s $ k的$ k n a $ k n a $ s $ k n $ $ f $。特别是，对象$ f_ {k，s} $结果是任何$ g $ invariant subfield $ k $ of $ f_ {k，s} $的类别$ sm_k（g）$的注入性cogogenerator。 As an application, when transcendence degree of $F|k$ is 1, a correspondence between the $G$-invariant subfields of $F_{k,S}$ algebraically closed in $F_{k,S}$ and certain systems of isogenies of `generically $F$-pointed' torsors over absolutely irreducible one-dimensional algebraic $k$-groups is constructed, so far only in特征0。$ g $的唯一不可约的有限维平滑表示是微不足道的。但是，有一个不变的“交叉比例”子场$ k $ k $ k（s）$，使得$ g $的不可约有的有限维平滑$ k $ -semilinear表示对应于$ pgl_ {2，k} $的$ pgl_的不可减至的代数表示。通常，任何光滑的$ g $ - field $ k $都承认平滑的$ g $ field扩展$ l | k $，因此$ l $是$ sm_l（g）$的cogenerator。
Let $K$ be a field and $G$ be a group of its automorphisms endowed with the compact-open topology. There are many situations, where it is natural to study the category $Sm_K(G)$ of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. The category $Sm_K(G)$ is semisimple (in which case $K$ is a generator of $Sm_K(G)$) if and only if $G$ is precompact. In this note we study the case of the non-precompact group $G$ of all permutations of an infinite set $S$. It is shown that the categories $Sm_K(G)$ are locally noetherian; the morphisms are `locally split'. Given a field $F$ and a subfield $k\neq F$ algebraically closed in $F$, one of principal results describes the Gabriel spectra (and related objects) of the categories $Sm_K(G)$ for some of $G$-invariant subfields $K$ of the fraction field $F_{k,S}$ of the tensor product over $k$ of the labeled by $S$ copies of $F$. In particular, the object $F_{k,S}$ turns out to be an injective cogenerator of the category $Sm_K(G)$ for any $G$-invariant subfield $K$ of $F_{k,S}$. As an application, when transcendence degree of $F|k$ is 1, a correspondence between the $G$-invariant subfields of $F_{k,S}$ algebraically closed in $F_{k,S}$ and certain systems of isogenies of `generically $F$-pointed' torsors over absolutely irreducible one-dimensional algebraic $k$-groups is constructed, so far only in characteristic 0. The only irreducible finite-dimensional smooth representation of $G$ is trivial. However, there is an invariant `cross-ratio' subfield $K$ of $k(S)$ such that the irreducible finite-dimensional smooth $K$-semilinear representations of $G$ correspond to the irreducible algebraic representations of $PGL_{2,k}$. In general, any smooth $G$-field $K$ admits a smooth $G$-field extension $L|K$ such that $L$ is a cogenerator of $Sm_L(G)$.
