GENERALIZED LÜROTH PROBLEMS HIERARCHIZED I SBNR - STABLY BIRATIONALIZED UNRAMIFIED SHEAVES AND LOWER RETRACT RATIONALITY

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GENERALIZED LÜROTH PROBLEMS, HIERARCHIZED I:

SBNR - STABLY BIRATIONALIZED UNRAMIFIED SHEAVES

AND LOWER RETRACT RATIONALITY

NORIHIKO MINAMI

Abstract. This is the ﬁrst of a series of papers, where we investigate hierarchies

of generalized Lüroth 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 a perfect base ﬁeld k

to be “retract (−i)-rational,.”

We achieve this goal by, for any Morel’s unramiﬁed sheaf S, constructing

SBNR - stably birationalized (Nisnevich) subsheaf Ssb of the unramiﬁed sheaf S,

in such a way that Ssb coincides with Sfor any proper smooth k-scheme of ﬁnite

type.

Such a stably birationalized Nisnevich subsheaf Ssb 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 spaces of ﬁnite group, presented as counterexamples to the

Noether problem of ﬁnite groups with the base complex number ﬁeld C.In fact,

for all of these examples, the game is not over from our hierarchical perspective!

An immediate consequence of our construction of Ssb is the stably birational

invariance of an arbitrary unramiﬁed sheaf Son 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) generalized

motivic cohomology theory is stably birational invariant on smooth proper k-

schemes of ﬁnite type (resp. arbitrary smooth k-schemes of ﬁnite type).

In the course of constructing Ssb,we have also shown a general local uni-

formization theorem of the ﬁrst kind for arbitrary geometric valuations.

1. Introduction

In 1875, Lüroth [L875] asked the converse of the obvious impliation

rational =⇒unirational

This is what is now called the Lüroth problem, and has drawn lots of attention,

Nowadays, as is surveyed e.g. in [K96][B16][AB17] [P18][V19] , the Lüroth problem

has been generalized to problems which ask the converse of various implications in

2020 Mathematics Subject Classiﬁcation. Primary 14E08,14M20,14F20,14F42,13A18,12F12,

19E15,55Q45; Secondary 14C35,13H05,14F22,20J06,55R35,19D45,14J70.

Key words and phrases. Lower Rationality = Higher Ruldeness, retract (−i)-rationality,

Lüroth’s problem, Noether’s problem, Morel’s unramiﬁed sheaf, stably birationalized Nisnevich

subsheaf, unramiﬁed cohomology group, unramiﬁed generalized motivic cohomology theory, stably

birationalized generalized motivic cohomology theory, First kind local uniformization of geometric

valuations .

2 NORIHIKO MINAMI

the following reﬁned implications:

rational =⇒stable rational =⇒retract rational

=⇒separably unirational =⇒separably rationally connected =⇒rationally connected

(1)

These problems are what we mean by generalized Lüroth problems.

On the other hand, there is an obvious, though rarely featured, hierarchy which

interpolates the rationality and the ruledness in the spirit of:

(2) lower rationality =higher ruledness

(For a precise deﬁnition, see Deﬁnition 1.1(i) below.)

In this series of papers, we embark on an organized study of the hierarchy of

hierarchies of rationality, obtained by introducing an appropriate hierarchy in the

spirit of (2) to each of (1). As a kick-oﬀ, we shall concentrate on the ﬁrst three of (1).

Fix the base ﬁeld k, which we shall soon (in fact, after the following Deﬁnition 1.1)

assume to be perfect. Then we have the following hierachies in the spirit of (2) for

the ﬁrst three of (1):

Deﬁnition 1.1. For a n-dimensional k-variety 1X, let us say:

(i) Xis (−i)-rational or (n−i)-ruled (0 ≤i≤n)

if there exist an i-dimensional smooth k-variety Zi,where Z0is taken to be

Spec kfor i= 0,and a birational map

An−i×Zi− − > X.

(ii) Xis stable (−i)-rational or stable (n−i)-ruled (0 ≤i≤n)

if there exist N∈Z≥n,an i-dimensional smooth k-variety Zi,where Z0is

taken to be Spec kfor i= 0,and a birational map

AN−n×An−i×Zi− − >AN−n×X.

(iii) Xis retract (−i)-rational or retract (n−i)-ruled (0 ≤i≤n)

if there exist N∈Z≥n,an i-dimensional smooth k-variety Zi,where Z0is

taken to be Spec kfor i= 0,and rational maps

f:X− − >AN−i×Zi, g :AN−i×Zi− − > X

such that the composition

g◦f:X− − > X

is deﬁned to be an identity on a dense open subset of X.

Remark 1.2. We may also consider a slightly restrictive variant of the above deﬁni-

tions by demanding Zi(i∈Z≥1)to be further projective (of course, this is the same

as simply demanding properness by Chow’s lemma [sp18, 0200]), When char k = 0,

deﬁnitions are unchanged by this extra requirement, thanks to Hironaka [H64]. How-

ever, it is not clear for the case char k > 0,and we shall mostly adapt this more

restrictive deﬁnitions for the hierarchies in the sequel, where we are more concerned

with smooth projective varieties.

1By a variety, we mean an integral k-scheme, which is separatred and of ﬁnite type as in [sp18,

Tag 020D].

SBNR AND LOWER RETRACT RATIONALITY 3

From now on, we assume that the base ﬁeld kis perfect. Then, the concepts

presented in the above Deﬁnition 1.1(ii)(iii) are invariant with respect to the fol-

lowing standard equivalence relation (see e.g. [CTS07, §1]), as we shall show in

Proposition 1.4(iv) below:

Deﬁnition 1.3. Two varieties of possibly diﬀerent dimensions Xand Yare said

to be stable birational equivalent if for some natural numbers r, s, X ×Arand

Y×Asare birationally equivalent.

In fact, these hierarchies for the ﬁrst three of (1) enojy satisfactory properties:

Proposition 1.4. (i) When i= 0,those concepts presented in Deﬁnition 1.1 reduce

to the usual classical concepts (mentioned in the ﬁrst line of (1)).

0-rational = rational; stable 0-rational = stable rational;

retract 0-rational = retract rational.

(ii) Each concept in Deﬁnition 1.1 is a hierarchy; i.e. for any 0≤i≤j≤n,

(−i)-rational =⇒(−j)-rational;

stable (−i)-rational =⇒stable (−j)-rational;

retract (−i)-rational =⇒retract (−j)-rational.

(iii) Concepts in Deﬁnition 1.1 deﬁne a hierarchy of hierarchies stated in above (ii);

i.e. for any 0≤i≤n,

(−i)-rational =⇒stable (−i)-rational =⇒retract (−i)-rational

(iv) For any 0≤i≤n, stable (−i)-rationality and retract (−i)-rationality are stable

birational invariants in the sense of Deﬁnition 1.3.

Remark 1.5. (i) By deﬁnition, Xis stable rational if and only if Xis stable bira-

tionally equivalent to the point Spec k.

(ii) By deﬁnition, the concept of the stable birational equivalence is canonically ex-

tended from the usual varieties to ind-k-varieties {Wm}mwith each structure mor-

phism im:Wm→Wm+1 equipped with a retraction rm+1 :Wm+1 →Wmso that

rm+1 ◦im=idWm,making k(Wm+1)purely transcendental over k(Wm).For instance,

for a linear algebraic group Gand a scheme Xwith G-action, enjoying one of the

conditions in [EG98, Proposition 23], the geometric classifying spaces BG of [T99,

§1] [MV99, §4] and the Borel construction XG=EG×GX(see e.g. [K18][CJ19, 2.1])

are such ind-k-varieties which possess respective canonical stable birational types, in-

dependent of particular ind-k-variety representations, thanks to Bogomolov’s double

ﬁbration argument. (This stable birational independence can be argued and stated

purely in terms of the corresponding pro-k-algebras. Bogomolov’s double ﬁbration

argument is a culmination of many works [S919][EM73][L74] [BK85][CGR06, Lemma

4.4] [CTS07, 3.2], and christened “no-name lemma” by [D87].) This fact, amongst of

all, enables us to consider the classical Noether’s problem 2of a ﬁnite group Gin

terms of its classifying space BG, either by its geometric ind-k-variety model or its

algebraic pro-k-algebra model. (e.g. [L05, Prop.9.4.4][M17, 4.2]).

Corollary 1.6. For BG, and the Borel construction XG=EG×GX, the concepts of

stable (−i)-rationality and retract (−i)-rationality are well-deﬁned stable birational

invariants. □

2In fact, the retract rationality, which is the case i= 0 of Deﬁnition 1.1 (iii), was introduced by

Saltman [S84a, S84b] in his study of Noether’s problem (for Noether’s problem, consult surveys of

[S83][CTS07][H14][H20], or books of [JLY02][GMS03][L05]).

4 NORIHIKO MINAMI

For our later proof of Proposition 1.4 and especially to understand retract (−i)-

rationality better, we now state some remarks:

Remark 1.7. (i) Generalizing the setting of the retract (−i)-rationality in Deﬁni-

tion 1.1 (iii), let us consider the setting of the rational retraction. In other words,

let us consider the situation when k-varieties X, Y, with rational maps

f:X− − > Y, g :Y− − > X,

are given so that the composition

g◦f:X− − > X

is deﬁned to be an identity on a dense open subset of X. This is the same as saying

that there are some dense opens U⊆U0⊆X, V ⊆Ysuch that f, g restrict to

morphisms

f:U→V, g :V→U0,

whose composition

g◦f:U→U0

is the canonical inclusion in X.

However, we sometimes wish to restrict our attention to a particular dense open

U⊆U(⊆U0⊆X).Fortunately, we immediately get the following retraction given

by morphisms, satisfying our desire:

Uf//

id e

))

g−1(e

U)g//e

(ii) For any k-variety Xand any a∈Ad,we have an obvious retraction:

Xia

→Ad×XpX

↠X

x7→ (a, x)7→ x

However, we sometimes wish to restrict our attention to a particular dense open

V⊆Ad×X. Fortunately, we can still choose appropriate a∈Adand a non empty

open (consequently dense, because Xis a k-variety) U⊆X, V 0⊆V⊆Ad×Xwith

a restricted retraction given by honest morphisms of the form:

Ui/

_



_



r////U_



X

x7→(a,x)

ia/Ad×XpX////X,

which we think as a particular instance of rational retraction.

To see this, just choose any a∈Arwith i−1

a(V)∼

=V∩({a} × X)6=∅.Then, just

proceed as in (i) by setting:

V0:= V∩p−1

Xi−1

a(V)=V∩Ar×i−1

a(V)

U:= i−1

a(V)

Those dense open subschemes we wish to restrict our attention in the above Re-

mark 1.7 are actually some open subschemes of the smooth locus Uof the given

variety X. For this purpose, the smooth locus Ushould be non empty and dense

SBNR AND LOWER RETRACT RATIONALITY 5

open, which is guaranteed if Xis geometrically reduced. But this geometrically re-

duced condition is automatically satisﬁed under our perfect base ﬁeld kassumption.

For this, see [BLR90, §2.2, Proposition 16] [G65, p.68, Proposition 4.6.1] [P17, Propo-

sition 3.5.64, Warning 3.5.18, Prop.2.2.20]. We record the situation below following

[sp18, 0B8X] for our later purpose:

Proposition 1.8. Let kbe a perfect ﬁeld, i.e. 3every ﬁeld extension of kis separable

over k. Let Xbe a locally algebraic, i.e. 4locally of ﬁnite type k-scheme as in (12),

which is furthermore reduced, for example a variety over k. Then we have

{x∈X|X→Spec(k)is smooth at x}={x∈X| OX,x is regular},

which we call the smooth locus of X, is a dense open subscheme of X. □

Motivated by Propositin 1.4 and Noether’s problem, where BG is approximated

by non proper smooth varieties and the fact that the retract rationality implies the

higher vanishings of the unramiﬁed cohomologies was so successfully exploited to

produce many counter examples of Noether’s problem for the case k=C[S84b]

[B89] [P93] [CTS07], to cite just a few (for more, consult Hoshi’s surveys [H14][H20]

and [HKY20]) we are naturally led to the following problem:

The original motivation of this paper

 

Extend the following implications of the hierarchies in Propositin 1.4 for

not necessarily proper varieties to some hierarchy ?of algebraic, i.e. “coho-

mological”, stably birational invariants:

{(−i)-rational}i∈Z≥0=⇒ {stable (−i)-rational}i∈Z≥0

=⇒ {retract (−i)-rational}i∈Z≥0=⇒?

(3)

 

To search after ?,let us suppose a variety Xover a perfect ﬁeld kis retract (−i)-

rational. Then, for some Zi∈Smft

k,we have the corresponding rational retraction

as in Remark 1.7(i) with Y=AN−i×Zi.and e

U⊆U(⊆U0⊆X)the dense smooth

locus of U, as is guaranteed by Proposition 1.8.

Thus, denoting e

Usimply by Ufor ease of notation, we have the following diagram

of k-varieties:

(4) Uf//

idU

))

_



g−1(U)g//

_



U_



Xf//___

idX

AN−i×Zig//___ X

where vertical arrows are smooth dense open inclusions.

Then the following concept of Asok-Morel [AM11] is clearly of fundamental im-

portance for our purpose:

3See [sp18, 030Y,030Z].

4See [sp18, 06LG].

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GENERALIZEDLÜROTHPROBLEMS,HIERARCHIZEDI:SBNR-STABLYBIRATIONALIZEDUNRAMIFIEDSHEAVESANDLOWERRETRACTRATIONALITYNORIHIKOMINAMIAbstract.Thisistheﬁrstofaseriesofpapers,whereweinvestigatehierarchiesofgeneralizedLürothproblemsonthehierarchyofrationality,startingwiththeobvioushierarchybetweentherationalityan...

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