aa r X i v : . [ m a t h . AG ] M a r Naive vs. genuine A -connectedness Anand Sawant
Abstract
We show that the triviality of sections of the sheaf of A -chain connected compo-nents of a space over finitely generated separable field extensions of the base field isnot sufficient to ensure the triviality of the sheaf of its A -chain connected components,contrary to the situation with genuine A -connected components. As a consequence,we show that there exists an A -connected scheme for which the Morel-Voevodskysingular construction is not A -local. Let k be a field and let Sm { k denote the category of smooth, finite-type schemes over k . Inthe 1990’s, Morel and Voevodsky [14] constructed the A -homotopy category H p k q by takinga suitable localization of the category of simplicial sheaves of sets on Sm { k for the Nisnevichtopology. Objects of H p k q are often called spaces . Analgous to algebraic topology, one thenstudies the A -homotopy sheaves of a (pointed) space p X , x q - the sheaf of A -connectedcomponents π A p X q , which is a sheaf of sets and the higher homotopy sheaves π A n p X , x q , for n ě
1, which are sheaves of groups. We will use the notation and terminology of [14]. Any(not necessarily smooth) scheme X over k can be viewed as an object of the A -homotopycategory H p k q (see the conventions stated at the beginning of Section 2). Recent worksin A -homotopy theory have indicated that the A -homotopy sheaves of schemes are oftenrelated to some of their interesting classical invariants.The simplest of objects in classical topology are the discrete topological spaces. Theanalogous notion in A -homotopy theory is that of A -invariant sheaves (see Section 2 forprecise definitions). In topology, the set of connected components of a topological spaceand the homotopy groups of a (pointed) topological space are discrete as topological spaces.Analogously, one can ask if the A -homotopy sheaves of a (pointed) space X are A -invariant.It has been shown by Morel [13, Theorem 6.1, Corollary 6.2] that the higher homotopysheaves π A n p X , x q , for n ě
1, are A -invariant. In fact, Morel shows much more - thesehigher A -homotopy sheaves are strongly A -invariant in the sense of [13, Definition 1.7](this shows that the higher A -homotopy sheaves are very special; for instance, they arebirational invariants of smooth, proper schemes). However, A -invariance of the sheaf of A -connected components is not yet known; this has been conjectured by Morel [13, Conjecture1.12]. It is worthwhile to mention that π A fails to be a birational invariant of smooth, properschemes [3, Example 4.8].There are two notions of A -connectedness in unstable A -homotopy theory. The naive notion is that of A -chain connected components of a space (see Definition 2.2), which is ob-tained by taking the Nisnevich sheafification of the presheaf that associates with any smooth1cheme U the set of morphisms from U to the space in question modulo the equivalence re-lation generated by naive A -homotopies. On the other hand, the genuine notion is that of A -connected components (see Definition 2.3) introduced by Morel-Voevodsky. These twonotions do not coincide in general, not even for smooth projective varieties over C (see [3,Section 4] for the first examples). Given a scheme X over k , one can infinitely iterate the con-struction of A -chain connected components to obtain the so-called universal A -invariantquotient of X , which is isomorphic to π A p X q provided the latter sheaf is A -invariant (thatis, Morel’s conjecture holds for X ). We recall these notions and known results about themin Section 2.A natural question is to characterize genuine A -connectedness of a scheme X over k interms of triviality of sections of π A p X q over field extensions of k . A result of Morel statesthat A -connectedness of a scheme X over an infinite field k in the genuine sense (that is,triviality of π A p X q as a sheaf) is equivalent to the triviality of π A p X qp Spec F q , where F runs over all finitely generated separable field extensions of k . In this short note, we examinethe analogous property for the sheaf of A -chain connected components in Section 3 (seeTheorem 3.2). As a consequence, we obtain an example of an A -connected singular properscheme X for which the Morel-Voevodsky singular construction Sing A ˚ X is not A -local (seeExample 3.6). A -homotopy theory We begin this section by setting up the notation and conventions that will be used throughoutthe paper.Fix a base field k . We will henceforth denote by Sm { k the big Nisnevich site of smooth,finite-type schemes over k . We begin with the category of simplicial sheaves over Sm { k .Any scheme X over k can be seen as an object of this category as follows: consider the functor of points h X of X , which is the sheaf that associates with every U P Sm { k the set ofmorphisms of schemes over k from U to X . Any Nisnevich sheaf F on Sm { k can be viewedas a simplicially constant simplicial sheaf. More precisely, one considers the simplicial sheafin which the sheaf at every level is F and all the face and degeneracy maps are given by theidentity map. We will always denote the simplicial sheaf corresponding to h X for a scheme X over k by the same letter X .A morphism X Ñ Y of simplicial sheaves of sets on Sm { k is a local weak equivalence ifit induces an isomorphism on every stalk. The Nisnevich local injective model structure onthis category is the one in which the morphism of simplicial sheaves is a cofibration (resp. aweak equivalence) if and only if it is a monomorphism (resp. a local weak equivalence). Thecorresponding homotopy category is called the simplicial homotopy category and is denotedby H s p k q . The left Bousfield localization of the Nisnevich local injective model structurewith respect to the collection of all projection morphisms X ˆ A Ñ X , as X runs over allsimplicial sheaves, is called the A -model structure . The associated homotopy category iscalled the A -homotopy category and is denoted by H p k q . We will denote by ˚ the trivialone-point sheaf on Sm { k . We will abuse the notation and use ˚ to also denote a set withone element, whenever there is no confusion. Definition 2.1
A space (that is, a simplicial Nisnevich sheaf of sets on Sm { k ) X is said to2e A -local if the projection map U ˆ A Ñ U induces a bijectionHom H s p k q p U, X q Ñ Hom H s p k q p U ˆ A , X q . for every U P Sm { k . Note that a Nisnevich sheaf F on Sm { k is A -local if and only if it is A -invariant , that is, if the projection map U ˆ A Ñ U induces a bijection F p U q Ñ F p U ˆ A q ,for every U P Sm { k . Following standard convention, we say that a scheme is A -rigid if itis A -local as a space.Let X be a space. We now recall the singular construction on X defined by Morel-Voevodsky [14, p. 87-88]. Define Sing A ˚ X to be the simplicial sheaf given by p Sing A ˚ X q n “ Hom p ∆ n , X n q , where ∆ ‚ denotes the cosimplicial scheme∆ n “ Spec ˆ k r x , ..., x n sp ř i x i “ q ˙ with natural face and degeneracy maps analogous to the ones on topological simplices. Thereexists a natural transformation Id Ñ Sing A ˚ such that for any simplicial sheaf X , the mor-phism X Ñ Sing A ˚ p X q is an A -weak equivalence. Observe that the singular constructionSing A ˚ takes naive A -homotopies to simplicial homotopies.Given a simplicial sheaf of sets X on Sm { k , we will denote by π p X q the presheaf on Sm { k that associates with U P Sm { k the coequalizer of the diagram X p U q Ñ X p U q , wherethe maps are the face maps coming from the simplicial data of X . We will denote by π s p X q the Nisnevich sheafification of the presheaf π p X q . Definition 2.2
The sheaf of A -chain connected components of a space X is defined to be S p X q : “ π s p Sing A ˚ X q . Thus, S p X q is the Nisnevich sheafification of the presheaf that associates with any smoothscheme U the set X p U q{ „ , where „ is the equivalence relation generated by the image of X Ñ X ˆ X , where the maps are the face maps coming from the simplicial data of X (inother words, „ is the equivalence relation generated by naive A -homotopies). Definition 2.3
The sheaf of A -connected components of a space X is defined to be π A p X q : “ π s p L A X q , where L A denotes an A -fibrant replacement functor. A space X is said to be A -connected if π A p X q » ˚ .Morel-Voevodsky explicitly describe an A -fibrant replacement functor as follows: L A “ Ex ˝ p Ex ˝ Sing A ˚ q N ˝ Ex, where Ex denotes a simplicial fibrant replacement functor on the model category of simplicialNisnevich sheaves of sets over Sm { k [14, §
2, Lemma 2.6, p. 107]. There exists a naturaltransformation Id Ñ L A which factors through the natural transformation Id Ñ Sing A ˚ mentioned above. For any object X , the morphism X Ñ L A p X q is an A -weak equivalence.A result of Morel-Voevodsky [14, §
2, Corollary 3.22] describes what happens to the naturalmap Sing A ˚ X Ñ L A p X q after applying π s ; we record it below for the sake of convenience.3 emma 2.4 The canonical map S p X q Ñ π A p X q is an epimorphism, for every space X . If Sing A ˚ X is A -local, then the map S p X q Ñ π A p X q is an isomorphism. We will henceforth focus on a specific class of spaces, namely, sheaves of sets on the bigNisnevich site on Sm { k . We will eventually specialize to the case of schemes. Let F be aNisnevich sheaf on Sm { k . By Lemma 2.4, we have a sequence of epimorphisms F Ñ S p F q Ñ S p F q Ñ ¨ ¨ ¨ Ñ S n p F q Ñ ¨ ¨ ¨ , where S n ` p F q is defined inductively to be S p S n p F qq , for every n P N . We define L p F q : “ lim ÝÑ n S n p F q . (1)The following result was proved in [3] (see [3, Theorem 2.13, Remark 2.15, Corollary2.18]), which shows that L p F q is the universal A -invariant quotient of F . Theorem 2.5
Let F be a sheaf of sets on Sm { k . Then the sheaf L p F q is A -invariant.Moreover, if G is an A -invariant sheaf, then any map F Ñ G factors uniquely throughthe epimorphism F Ñ L p F q . Moreover, if π A p F q is A -invariant, then the canonical map L p F q Ñ π A p F q is an isomorphism. We will henceforth focus on schemes over a field. In view of Theorem 2.5, it is clear thata good understanding of L p X q is tantamount to understanding π A p X q . It is natural to askthe following question. Question 2.6
Let X be a smooth scheme over k . Does there exist n P N such that L p X q » S n p X q ?For every scheme X over a field k , we have the following commutative diagram in whichevery morphism is an epimorphism X / / ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ S p X q / / $ $ ■■■■■■■■■ S p X q / / ¨ ¨ ¨ / / L p X q π A p X q ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ where the existence of the map π A p X q Ñ L p X q making the diagram commute is a conse-quence of the A -invariance of L p X q (see [3, Lemma 2.8]). The morphism π A p X q Ñ L p X q is an isomorphism if π A p X q is A -invariant. An affirmative answer to Question 2.6 will givea conjectural but explicit geometric description of π A p X q . We end this section by enlistingthe known examples in which such an explicit description is available. Examples 2.7 ( A -rigid varieties) For an A -rigid variety X , one has isomorphisms ofNisnevich sheaves X » S p X q » π A p X q . Examples of A -rigid varieties include G m , algebraic tori, abelian varieties, curves of genus ě xamples 2.8 (Reductive algebraic groups) For any sheaf of groups G , it is knownthat π A p G q is A -invariant [7]. We therefore have π A p G q » L p G q “ lim ÝÑ n S n p G q . We willnow focus on the case where G is a reductive algebraic group over a field.(a) Isotropic groups.
Suppose that G satisfies the following isotropy condition : everyalmost k -simple component of the derived group G der of G contains a k -subgroupscheme isomorphic to G m . Under this hypothesis, Asok, Hoyois and Wendt haveshown that Sing A ˚ G is A -local [1, Theorem 2.3.2]. Therefore, for G satisfying theabove isotropy condition, one has S p G q » π A p G q » L p G q . The sections of this sheaf over fields can often be described explicitly. If G is a semisim-ple, simply connected group over an infinite field k satisfying the above isotropy hypoth-esis, then one has isomorphisms W p k, G q » S p G qp k q » G p k q{ R . This is a consequenceof [1, Theorem 2.3.2] and a classical result [9, Th´eor`eme 7.2]. Here W p k, G q denotesthe Whitehead group of G and G p k q{ R denotes the group of R-equivalence classes (see[9], for example, for precise definitions).(b)
Anisotropic groups.
Let us assume that the base field k is infinite and perfect.Suppose now that G does not satisfy the above isotropy hypothesis; that means, thederived group G der of G has at least one almost k -simple factor which is anisotropic.In this case, it is known that Sing A ˚ G fails to be A -local [5, Theorem 4.7]. We nowassume that G is a semisimple anisotropic group. Note that one has W p k, G q “ G p k q in this case. A result of Borel-Tits implies in this case that S p G qp k q » G p k q (see [4,Lemma 3.7] for details). However, one has the following result (see [4, Theorem 4.2],[5, Theorem 3.6]): if G is a semisimple, simply connected group over an infinite perfectfield k which does not satisfy the above isotropy hypothesis, then one has canonicalisomorphisms π A p G qp k q » S p G qp k q » G p k q{ R. It is worthwhile to mention here that we do not yet know whether π A p G q agrees with S p G q as a sheaf.(c) A -connected reductive algebraic groups. Recall that a space X is said to be A -connected if π A p X q » ˚ . In [5, Theorem 5.2], A -connected reductive algebraicgroups have been characterized: a reductive algebraic group G over an infinite perfectfield k is A -connected if and only if G is semisimple, simply connected and R -trivial(that is G p F q{ R is trivial for every finitely generated separable field extension F of k ). Examples 2.9 (Proper varieties) (a) If X is a proper variety over k and if F is afinitely generated field extension of k , then one has S p X qp F q » π A p X qp F q (see [2,Theorem 2.4.3]). One also has S p X qp F q » S n p X qp F q , for every n (see [3, Theorem3.9, Corollary 3.10]).(b) The case of proper schemes of dimension ď X over k of dimension ď
1, one always has S p X q » S p X q ([3,Proposition 3.13]). Consequently, S p X q » π A p X q .5c) If X is a proper, non-uniruled surface over k , then one has S p X q » π A p X q » S p X q ([3, Theorem 3.14]).(d) The case of smooth projective ruled surfaces is surprisingly very complicated. If X is asmooth proper rational surface, then one has S p X q “ ˚ (see Corollary 3.3). However,one has π A p X q “ ˚ as well. We do not yet know if the sheaf S p X q for a rationalsurface X is trivial.Let us now assume that the characteristic of k is 0. The case of ruled surfaces whoseminimal model is of the form P ˆ C , where C is a smooth projective curve of genus ě C is A -rigid) is the most complicated one. If E is a P -bundleover C , then one has S p E q » π A p E q » C . If X is the surface obtained by blowingup one closed point on E and when k is assumed to be algebraically closed, one has S p X q ‰ S p X q . However, in this case one has S p X q » S p X q . The details will appearin a forthcoming paper [6]. A -connectedness on field-valued points Let X be a scheme over a field k . It is often much simpler to determine sections of thesheaf S p X q on smooth schemes which are the spectrum of a finitely generated separablefield extension of the base field k . A result of Morel (see [12, Lemma 6.1.3]) states thata space X over an infinite field k is A -connected (that is, π A p X q is trivial) if and onlyif π A p X qp Spec F q “ ˚ , for every finitely generated separable field extension F of k . Theargument given by Morel also works when the base field is finite, thanks to Gabber’s presen-tation lemma over finite fields proved in [10]. We wish to study the analogue of this result inthe context of the sheaf of A -chain connected components. The method used here closelyfollows the one employed by Morel in [12, Section 6.1] and in [11, Section 3.3]. Lemma 3.1
Let V be an irreducible smooth scheme over k and let W ã Ñ V be the inclusionof a dense open subscheme. Then S p V { W q » ˚ . Proof
Since we have epimorphisms V Ñ V { W Ñ S p V { W q , triviality of S p V { W q followsfrom the following statement: any point x P V has an open neighbourhood U such that S p U {p W X U qq is trivial.Let Z ã Ñ V be the closed immersion of the complement of W , with the reduced inducedsubscheme structure. By Gabber’s presentation lemma (see [8, Theorem 3.1.1] for the casewhere k is infinite and [10, Theorem 1.1] for the case where k is finite), x admits an openneighbourhood U and an ´etale morphism π : U Ñ A V , for some open subscheme V of A d ´ ,where d is the dimension of V at x , such that π induces a closed immersion Z X U ã Ñ A V satisfying Z X U “ π ´ p π p Z X U qq and such that Z X U Ñ V is a finite morphism. Therefore,we have an isomorphism of Nisnevich sheaves U {p U ´ Z X U q „ ÝÑ A V {p A V ´ π p Z X U qq . Hence, it suffices to check that S p A V {p A V ´ π p Z X U qqq is trivial. Now, since Z X U Ñ V is a finite morphism, Z X U Ñ P V is proper. This closed immersion does not intersect the6ection at infinity s : V Ñ P V . By Mayer-Vietoris excision (see [14, §
3, Lemma 1.6]), wehave an isomorphism of Nisnevich sheaves A V {p A V ´ π p Z X U qq „ ÝÑ P V {p P V ´ π p Z X U qq . Also observe that A V Ñ P V {p P V ´ π p Z X U qq is onto and that Sing A ˚ p A V q » Sing A ˚ p V q (since Sing A ˚ preserves A -weak equivalences). Thus, the composition V Ñ A V Ñ π s p Sing A ˚ p A V {p A V ´ π p Z X U qqqq Ñ π s p Sing A ˚ p P V {p P V ´ π p Z X U qqqq is surjective for any section V Ñ A V ; in particular, for the zero section. But, in P V , thezero section is A -homotopic to the section at infinity s . Since s p V q Ď P V ´ π p Z X U q ,it follows that V Ñ π s p Sing A ˚ p P V {p P V ´ π p Z X U qqqq “ S p P V {p P V ´ π p Z X U qqq is the trivial morphism, as desired. Theorem 3.2
Let k be a field and let X be a simplicial sheaf of sets on Sm { k . Supposethat S p X qp Spec F q “ ˚ , for every finitely generated separable field extension F of k . Then S p Ex Sing A ˚ X q » ˚ . Consequently, S p X q » ˚ . Proof
We need to show that for every U P Sm { k , the pointed set S p Sing A ˚ X qp U q is trivial.It suffices to show that for every morphism U Ñ S p Sing A ˚ X q , there is a Nisnevich cover V “ š V i Ñ U such that the composite V Ñ U Ñ S p Sing A ˚ X q is trivial. Claim:
For any irreducible, smooth k -scheme V and a morphism φ : V Ñ X , the composition V φ Ñ X Ñ Ex Sing A ˚ X Ñ S p Ex Sing A ˚ X q is trivial. Proof of the claim:
Let k p V q denote the function field of V . Since S p X qp Spec k p V qq “ lim ÝÑ W ã Ñ V nonempty open S p X qp W q is trivial by hypothesis, there exists a dense open subset W ã Ñ V such that the composite W Ñ V φ Ñ X is A -chain homotopic to the trivial morphism. Therefore the composite ofthis morphism with the morphism X Ñ Sing A ˚ X is simplicially homotopic to the trivialmorphism. Choose a simplicial fibrant replacement Sing A ˚ X Ñ Ex Sing A ˚ X . The composite W ã Ñ V φ Ñ X Ñ Sing A ˚ X Ñ Ex Sing A ˚ X continues to be simplicially homotopic to thetrivial map. We denote this simplicial homotopy by H : W ˆ ∆ Ñ Ex Sing A ˚ X , where H | W ˆt u is the trivial map and H | W ˆt u is induced by φ | W (here ∆ denotes the simplicial1-simplex). Consider the acyclic cofibration V ˆ t u Y W ˆ ∆ Ñ V ˆ ∆ . The maps V ˆ t u „ Ñ V φ Ñ X Ñ Ex Sing A ˚ X and H : W ˆ ∆ Ñ Ex Sing A ˚ X clearly glue to give a mapΦ : V ˆ t u Y W ˆ ∆ Ñ Ex Sing A ˚ X , which fits in the following commutative diagram. V ˆ t u Y W ˆ ∆ / / (cid:15) (cid:15) Ex Sing A ˚ X (cid:15) (cid:15) V ˆ ∆ / / ❦❦❦❦❦❦❦❦ ˚
7e now use the right lifting property of the projection map Ex Sing A ˚ X Ñ ˚ with respectto acyclic cofibrations to see that dotted arrow in the above diagram exists. It follows that φ : V Ñ Ex Sing A ˚ X is simplicially homotopic to a morphism φ : V Ñ Ex Sing A ˚ X whoserestriction to W is trivial. Thus, we get an induced morphism (of spaces) ¯ φ : V { W Ñ Ex Sing A ˚ X . Now, applying Lemma 3.1 and commutativity of the diagram V (cid:15) (cid:15) ( ( PPPPPPPPPPPPP V { W / / (cid:15) (cid:15) Ex Sing A ˚ X (cid:15) (cid:15) S p V { W q / / S p Ex Sing A ˚ X q proves the claim.We now complete the proof of the theorem using the claim. Since the natural map Ex Sing A ˚ X Ñ S p Ex Sing A ˚ X q is an epimorphism, there is a Nisnevich covering š V i Ñ U , where V i are irreducible smooth k -schemes such that every composite V i Ñ U Ñ S p Ex Sing A ˚ X q lifts to a morphism V i Ñ Ex Sing A ˚ X . Since Ex Sing A ˚ X is a simplicialfibrant replacement of X , each map V i Ñ Ex Sing A ˚ X is represented by a map V i Ñ Sing A ˚ X in the simplical homotopy category H s p k q . X (cid:15) (cid:15) Sing A ˚ X (cid:15) (cid:15) (cid:15) (cid:15) š V i ; ; ✈✈✈✈✈✈✈✈✈✈✈✈✈ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / U / / S p Ex Sing A ˚ X q Since the sheaf at simplicial level 0 of Sing A ˚ X is X , and since any map from a space ofsimplicial dimension 0 to another space is determined by a map at the 0th simplicial level,this map factors through the monomorphism X Ñ Sing A ˚ X . Thus, the theorem now followsfrom the claim, applied to each of the maps V i Ñ X . Corollary 3.3
Let X be a scheme over a field k such that S p X qp Spec F q “ ˚ , for everyfinitely generated separable field extension F of k . Then S p X q » ˚ . The condition in Corollary 3.3 can be seen to hold when X is a smooth proper rationalvariety over a field of characteristic 0. In general, it holds when X is a smooth proper R -trivial variety over a field of characteristic 0 (see [2, Theorem 2.4.3, Corollary 2.4.9]). Corollary 3.4
Let G be an A -connected reductive algebraic group over an infinite perfectfield k . Then S p G q » ˚ . Proof
This is a straightforward consequence of Theorem 3.2 and Examples 2.8 (a), (b).8 emark 3.5
Note that if G in Corollary 3.4 is such that every almost k -simple factor of G contains a copy of G m , then one has S p G q “ ˚ by [1, Theorem 2.3.2].We end this note with an example of a singular, projective scheme X for which S p X q ‰ ˚ ,but S p X qp Spec F q “ ˚ , for every finitely generated field extension F of k . This example waspointed out to the author by Chetan Balwe. Example 3.6
Let k be a field and let C ã Ñ P be an elliptic curve over k . Let X denote theblow up of P ˆ C ã Ñ P ˆ P at the closed point P “ pp q , Q q , where Q is a closed pointon C . Let E denote the exceptional divisor. We have an obvious morphism π : X Ñ C ,which is the composition of the blow-up morphism φ : X Ñ P ˆ C with the projection map P ˆ C Ñ C . Let X denote the plane tp qu ˆ P ; so X and X intersect in tp qu ˆ C .Let X : “ X Y X . Since S p X q “ ˚ , it is easy to see that S p X qp Spec F q “ ˚ , for every fieldextension F of k . By [2, Theorem 2.4.3], we have π A p X qp Spec F q “ ˚ , for every finitelygenerated field extension F of k . By [12, Lemma 6.1.3], we have π A p X q » ˚ .Let U be a smooth henselian local scheme of dimension 1 over k with closed point u .Let γ : U Ñ X be a morphism that maps u on E and the generic point of U outside E . We begin by considering naive A -homotopies of U inside X , starting at γ . Let h bea morphism A ˆ U Ñ X such that h | t uˆ A “ γ . Since C is A -rigid, the composition π ˝ h : A ˆ U Ñ C factors through the projection A ˆ U Ñ U . Thus, h : A ˆ U Ñ X factors through X ˆ C U . By assumption, h is such that the point Q on C is in the imageof π ˝ h , that is, the image of the closed fiber A ˆ t u u intersects the exceptional divisor E .Hence, h ´ p P q is a closed subscheme of A ˆ U with support contained in A ˆ t u u . Since φ ˝ h can be lifted to X , we see that h ´ p P q is a closed subscheme of A ˆ U of codimension1. Therefore, the support of h ´ p P q must be exactly A ˆ t u u . Thus, h maps A ˆ t u u intothe exceptional divisor E .Now, let h : A ˆ U Ñ X be a morphism such that h | t uˆ U “ γ . Since A ˆ U isirreducible, this implies that h factors through the inclusion of X into X . The discussionin the above paragraph shows that h maps the whole closed fiber of A ˆ U on E . Hence,we cannot have S p X qp U q “ ˚ and consequently, S p X q ‰ ˚ . However, Theorem 3.2 impliesthat S p X q “ ˚ . Since π A p X q » ˚ as observed above, Sing A ˚ X cannot be A -local in viewof Lemma 2.4. Acknowledgement
The author thanks Chetan Balwe for suggesting Example 3.6 as wellas for his comments on this article and Marc Hoyois for a helpful discussion during theInternational Colloquium on K -theory at TIFR. The author also thanks the referee for acareful reading of the note and for comments that helped him improve the exposition. References [1] A. Asok, M. Hoyois and M. Wendt:
Affine representability results in A -homotopytheory II: principal bundles and homogeneous spaces , Preprint, arXiv: 1507:08020v3[math.AG] (2015).[2] A. Asok and F. Morel: Smooth varieties up to A -homotopy and algebraic h -cobordisms , Adv. Math. 227 (2011), no. 5, 1990–2058.93] C. Balwe, A. Hogadi and A. Sawant: A -connected components of schemes , Adv.Math. 282 (2015), 335–361.[4] C. Balwe and A. Sawant: R -equivalence and A -connectedness in anisotropicgroups , Int. Math. Res. Not. IMRN 2015, No. 22, 11816–11827.[5] C. Balwe and A. Sawant: A -connectedness in reductive algebraic groups , Preprint,arXiv: 1605:04535 [math.AG] (2016).[6] C. Balwe and A. Sawant: A -connected components of ruled surfaces , in prepara-tion.[7] U. Choudhury: Connectivity of motivic H -spaces , Algebr. Geom. Topol. 14 (2014),no. 1, 37–55.[8] J.-L. Colliot-Th´el`ene, R. Hoobler, B. Kahn: The Bloch-Ogus-Gabber theorem . Al-gebraic K -theory (Toronto, ON, 1996), 31-94, Fields Inst. Commun., 16, Amer.Math. Soc., Providence, RI, 1997.[9] P. Gille: Le probl`eme de Kneser-Tits , S´eminaire Bourbaki. Vol. 2007/2008.Ast´erisque No. 326 (2009), Exp. No. 983, vii, 39–81 (2010).[10] A. Hogadi and G. Kulkarni:
Gabber’s presentation lemma for finite fields , Preprint,arXiv: 1612:09393v2 [math.AG] (2016).[11] F. Morel:
An introduction to A -homotopy theory , Contemporary Developments inAlgebraic K-Theory, ICTP Lect. Notes, vol. XV, Abdus Salam Int. Cent. Theoret.Phys., Trieste (2004), 357 – 441 (electronic).[12] F. Morel: The stable A -connectivity theorems , K-Theory 35 (2005), 1–68.[13] F. Morel: A -algebraic topology over a field , Lecture Notes in Mathematics, Vol.2052, Springer, Heidelberg, 2012.[14] F. Morel and V. Voevodsky: A -homotopy theory of schemes Inst. Hautes ´EtudesSci. Publ. Math. 90(1999) 45–143.
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