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This is the first of a series of papers, where we investigate hierarchies ofgeneralized {L}\"{u}roth problems on the hierarchy of rationality, startingwith the obvious hierarchy between the rationality and the ruledness. Ourprimary goal here was to construct very general necessary conditions for asmooth, 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'sunramified 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 ofArtin-Mumford, Saltman, Colliot-Th\'{e}l\`ene-Ojanguren, Bogomolov, Peyre,Colliot-Th\'{e}l\`ene-Voisin, and many other retract irrational classifyingspace $BG$ examples presented as counterexamples to the Noether problem of thecomplex number base field case for a finite group $G.$ In fact, for all ofthese examples, the game is not over from our hierarchical perspective! Aconsequence of our construction of $S_{sb}$ is the stably birational invarianceof an arbitrary unramified sheaf $S$ on proper smooth $k$-schemes of finitetype. This in particular implies that, for any generalized motivic cohomologytheory, its naively defined unramified (resp. stably birationalized)gemeralized motivic cohomology theory is stably birational invariant on smoothproper $k$-schemes of finite type (resp. smooth $k$-schemes of finite type). Inthe course of constructing $S_{sb},$ we have also shown a general localuniformization theorem of the first kind for arbitrary geometric valuations.

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