Generalized L\"uroth problems, hierarchized I: SBNR -- stably birationalized unramified sheaves and lower retract rationality

Abstract

This is the first of a series of papers, where we investigate hierarchies of generalized {L}\"{u}roth 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-Th\'{e}l\`ene-Ojanguren, Bogomolov, Peyre, Colliot-Th\'{e}l\`ene-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.