论文标题
普遍的吕伦问题，分层i：sbnr-稳定的生物化的未造成的滑轮和较低的缩回合理性
Generalized Lüroth problems, hierarchized I: SBNR -- stably birationalized unramified sheaves and lower retract rationality
论文作者
论文摘要
这是一系列论文中的第一篇，我们在其中调查了关于理性层次结构的广义{l}üroth问题的层次结构，从合理性和统治性之间的明显层次结构开始。我们的主要目标是在完美的基本$ k $上建立非常普通的有限类型的非常普遍的必要条件，以“缩回$（ - i）$ - 理性”。我们通过在任何莫雷尔（Morel）的未造成的捆绑$ s中构建“稳定的birational birational birational birational的nisnevich subsheaf” $ s_ {sb} $，其中$ s_ {sb} $与适当的平滑$ k $ k $ - $ k $ - $ k $ -schemes的有限型有限类型相吻合。这样一个稳定的生态化的Nisnevich Subsheaf $ s_ {sb} $为Artin-Mumford，Saltman，Colliot-Thélène-Ojanguren，Bogomolov，Peyre，Peyre，Peyre，colliot-thélène-théne-voisin和许多其他撤回型$ bg的不合理效果$ bg the $ bg bg的熟悉的不合理例子提供了新的启示。有限组$G。$的复杂数字基地案例实际上，对于所有这些示例，从我们的层次结构角度来看，游戏并没有结束！我们建造$ s_ {sb} $的后果是在适当的平滑$ k $ -schemes上的任意未遭受的捆绑$ s $的稳定birational birtination不变性。这特别意味着，对于任何一般的动机共同体学理论，其天真定义的未经（稳定生态化）的宝石动机共同体学理论在平滑的有限型$ k $ -schemes上是稳定的，稳定的有限型（分别$ k $ k $ k $ -s-schemes）。在构建$ s_ {sb}的过程中，$我们还展示了第一类的一般局部统一定理，用于任意几何估值。
This is the first of a series of papers, where we investigate hierarchies of generalized {L}üroth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal here was to construct very general necessary conditions for a smooth, not necessary proper, scheme of finite type over the perfect base field $k$ to be "retract $(-i)$-rational". We achieve this goal by constructing "stably birationalized Nisnevich subsheaf" $S_{sb}$ inside any Morel's unramified sheaf $S,$ where $S_{sb}$ coincides with $S$ on proper smooth $k$-schemes of finite type. Such a stably birationalized Nisnevich subsheaf $S_{sb}$ sheds a new light on the familiar irrational examples of Artin-Mumford, Saltman, Colliot-Thélène-Ojanguren, Bogomolov, Peyre, Colliot-Thélène-Voisin, and many other retract irrational classifying space $BG$ examples presented as counterexamples to the Noether problem of the complex number base field case for a finite group $G.$ In fact, for all of these examples, the game is not over from our hierarchical perspective! A consequence of our construction of $S_{sb}$ is the stably birational invariance of an arbitrary unramified sheaf $S$ on proper smooth $k$-schemes of finite type. This in particular implies that, for any generalized motivic cohomology theory, its naively defined unramified (resp. stably birationalized) gemeralized motivic cohomology theory is stably birational invariant on smooth proper $k$-schemes of finite type (resp. smooth $k$-schemes of finite type). In the course of constructing $S_{sb},$ we have also shown a general local uniformization theorem of the first kind for arbitrary geometric valuations.
