A^1-invariance of non-stable K_1-functors in the equicharacteristic case
aa r X i v : . [ m a t h . K T ] D ec A -INVARIANCE OF NON-STABLE K -FUNCTORS IN THE GEOMETRICCASE A. STAVROVA
Abstract.
We explain how the techniques developed by I. Panin for the proof of the geometriccase of the Serre–Grothendieck conjecture for isotropic reductive groups (I. Panin, A. Stavrova, N.Vavilov, 2015; I. Panin, 2019) yield similar injectivity and A -invariance theorems for non-stable K -functors associated to isotropic reductive groups. Namely, let G be a reductive group over acommutative ring R . We say that G has isotropic rank ≥ n , if every normal semisimple reductive R -subgroup of G contains ( G m ,R ) n . We show that if G has isotropic rank ≥ and R is a regulardomain containing a field, then K G ( R [ x ]) = K G ( R ) for any n ≥ , where K G ( R ) = G ( R ) /E ( R ) isthe corresponding non-stable K -functor, also called the Whitehead group of G . If R is, moreover,local, then we show that K G ( R ) → K G ( K ) is injective, where K is the field of fractions of R . Introduction
Let R be a commutative ring with 1. Let G be a reductive group scheme over R in the senseof [SGA3]. We say that G has isotropic rank ≥ n , if every normal semisimple reductive R -subgroupof G contains ( G m ,R ) n .The assumption that G has isotropic rank ≥ implies that G contains a proper parabolic subgroup.For any reductive group G over R and a parabolic subgroup P of G , one defines the elementarysubgroup E P ( R ) of G ( R ) as the subgroup generated by the R -points of the unipotent radicals of P and of an opposite parabolic subgroup P − , and considers the corresponding non-stable K -functor K G,P ( R ) = G ( R ) /E P ( R ) [PSt1, St14]. In particular, if A = k is a field and P is minimal, E ( k ) isnothing but the group G ( k ) + introduced by J. Tits [T1], and K G ( k ) is the subject of the Kneser–Titsproblem [G]. If G = GL n and P is a Borel subgroup, then K G ( R ) = GL n ( R ) /E n ( R ) , n ≥ , are theusual non-stable K -functors of algebraic K -theory.If G has isotropic rank ≥ , then K G,P ( R ) is independent of P by the main result of [PSt1], andwe denote it by K G ( R ) . See § 2 for a formal definition and further properties of K G .In [St14], we hproved that if G be a reductive group over a field k , such that every semisimplenormal subgroup of G contains ( G m ) , then K G ( k ) = K G ( k [ X , . . . , X n ]) for any n ≥ . This implied the following two statements. First, provided k is perfect, one has K G ( A ) = K G ( A [ x ]) forany regular ring A containing k . Second, provided that k is infinite and perfect, one has ker( K G ( A ) → K G ( K )) = 1 for any regular local ring A containing k , where K is the field of fractions of A . Thoseresults generalized several earlier results on split, i.e. Chevalley–Demazure, reductive groups; see [St14]for a historical survey.In the present text we show how the techniques developed by I. Panin for the proof of the geometriccase of the Serre–Grothendieck conjecture [PaStV15, Pa19, Pa18] allow to extend these results to thecase where G is defined over A and k is an arbitrary field. The main results are the following. Theorem 1.1.
Let k be a field, let A be a regular ring containing k . Let G be a reductive groupscheme over A of isotropic rank ≥ . Then there is a natural isomorphism K G ( A ) ∼ = K G ( A [ x ]) . Theorem 1.2.
Let k be a field, let A be a semilocal regular domain containing k , and let K be thefield of fractions of A . Let G be a reductive group scheme over A of isotropic rank ≥ . Then thenatural homomorphism K G ( A ) → K G ( K ) is injective. The author is a winner of the contest “Young Russian Mathematics”. The work was supported by RFBR 19-01-00513. -INVARIANCE OF NON-STABLE K -FUNCTORS IN THE GEOMETRIC CASE 2 Corollary 1.3.
Let k be a field, let A be a semilocal regular domain containing k , and let K be thefield of fractions of A . Let G be a simply connected semisimple group scheme over A of isotropic rank ≥ . Then K G ( A ) = K G ( A [ x ± , . . . , x ± n ]) . Properties of K G over general commutative rings Let R be a commutative ring with 1. Let G be an isotropic reductive group scheme over R , and let P be a parabolic subgroup of G in the sense of [SGA3]. Since the base Spec R is affine, the group P has a Levi subgroup L P [SGA3, Exp. XXVI Cor. 2.3]. There is a unique parabolic subgroup P − in G which is opposite to P with respect to L P , that is P − ∩ P = L P , cf. [SGA3, Exp. XXVI Th. 4.3.2].We denote by U P and U P − the unipotent radicals of P and P − respectively. Definition 2.1. [PSt1]
The elementary subgroup E P ( R ) corresponding to P is the subgroup of G ( R ) generated as an abstract group by U P ( R ) and U P − ( R ) . We denote by K G,P ( R ) = G ( R ) /E P ( R ) thepointed set of cosets gE P ( R ) , g ∈ G ( R ) . Note that if L ′ P is another Levi subgroup of P , then L ′ P and L P are conjugate by an element u ∈ U P ( R ) [SGA3, Exp. XXVI Cor. 1.8], hence the group E P ( R ) and the set K G,P ( R ) do notdepend on the choice of a Levi subgroup or an opposite subgroup P − (and so we do not include P − in the notation).If P is a strictly proper parabolic subgroup and G has isotropic rank ≥ , then K G,P is group-valuedand independent of P . Definition 2.2.
A parabolic subgroup P in G is called strictly proper , if it intersects properly everynormal semisimple subgroup of G . Theorem 2.3. [PSt1, Lemma 12, Theorem 1]
Let G be a reductive group over a commutative ring R , and let A be a commutative R -algebra. If for any maximal ideal m of R the isotropic rank of G R m is ≥ , then the subgroup E P ( A ) of G ( A ) is the same for any strictly proper parabolic A -subgroup P of G A , and is normal in G ( A ) . Definition 2.4.
Let G be a reductive group of isotropic rank ≥ over a commutative ring R . Forany strictly proper parabolic subgroup P of G over R , and any R -algebra A , we call the subgroup E ( A ) = E P ( A ) , where P is a strictly proper parabolic subgroup of G , the elementary subgroup of G ( A ) . The functor K G on the category of commutative R -algebras A , given by K G ( A ) = G ( A ) /E ( A ) ,is called the non-stable K -functor associated to G . The normality of the elementary subgroup implies that K G is a group-valued functor.We will use the following two properties of K G established in [St14, St15]. The following lemmawas established in [Su, Corollary 5.7] for G = GL n . Lemma 2.5. [St15, Lemma 2.7]
Let A be a commutative ring, and let G be a reductive group schemeover A , such that every semisimple normal subgroup of G is isotropic. Assume moreover that for anymaximal ideal m ⊆ A , every semisimple normal subgroup of G A m contains ( G m ,A m ) . Then for anymonic polynomial f ∈ A [ x ] the natural homomorphism K G ( A [ x ]) → K G ( A [ x ] f ) is injective. The following statement was proved for G = GL n , n ≥ , in [Su, Th. 3.1]. For the case of splitsemisimple groups the same result was obtained by Abe in [A, Th. 1.15]. Lemma 2.6. [PSt1, Lemma 17]
Let A be a commutative ring, and let G be a reductive group schemeover A , such that every semisimple normal subgroup of G is isotropic. Assume moreover that for anymaximal ideal m ⊆ A , every semisimple normal subgroup of G A m contains ( G m ,A m ) . Then for any g ( X ) ∈ G ( A [ X ]) such that g (0) ∈ E ( A ) and F m ( g ( X )) ∈ E ( A m [ X ]) for all maximal ideals m of A ,one has g ( X ) ∈ E ( A [ X ]) . The following lemma is a straightforward extension of [V, Lemma 2.4] for G = GL n and [A, Lemma3.7] for split reductive groups. -INVARIANCE OF NON-STABLE K -FUNCTORS IN THE GEOMETRIC CASE 3 Lemma 2.7. [St14, Corollary 3.5]
Let G be a reductive group of isotropic rank ≥ over a commutativering B . Let φ : B → A be a homomorphism of commutative rings, and h ∈ B , f ∈ A non-nilpotentelements such that φ ( h ) ∈ f A × and φ : B/Bh → A/Af is an isomorphism. Assume moreover thatthe commutative square
Spec A f F f / / φ (cid:15) (cid:15) Spec A φ (cid:15) (cid:15) Spec B h F h / / Spec B is a distinguished Nisnevich square in the sense of [MoV, Def. 3.1.3] . Then the sequence of pointedsets K G ( B ) ( F h ,φ ) −−−−→ K G ( B h ) × K G ( A ) ( g ,g ) φ ( g ) F f ( g ) − −−−−−−−−−−−−−−−→ K G ( A f ) is exact. Proof of the main results
The following theorem proved in [Pa19] extends [PaStV15, Theorem 7.1] to arbitrary reductivegroups G and arbitrary fields k . Theorem 3.1. [Pa19, Theorem 2.5]
Let X be an affine k -smooth irreducible k -variety, and let x , x , . . . , x n be closed points in X . Let U = Spec ( O X, { x ,x ,...,x n } ) and f ∈ k [ X ] be a non-zerofunction vanishing at each point x i . Let G be a reductive group scheme over X , G U be its restrictionto U . Then there is a monic polynomial h ∈ O X, { x ,x ,...,x n } [ t ] , a commutative diagram of schemeswith the irreducible affine U -smooth Y (1) ( A × U ) hinc (cid:15) (cid:15) Y h := Y τ ∗ ( h ) τ h o o inc (cid:15) (cid:15) ( p X ) | Yh / / X finc (cid:15) (cid:15) ( A × U ) Y τ o o p X / / X and a morphism δ : U → Y subjecting to the following conditions: (i) the left hand side square is an elementary distinguished Nisnevich square in the category ofaffine U -smooth schemes in the sense of [MoV, Def. 3.1.3] ; (ii) p X ◦ δ = can : U → X , where can is the canonical morphism; (iii) τ ◦ δ = i : U → A × U is the zero section of the projection pr U : A × U → U ; (iv) h (1) ∈ O X, { x ,x ,...,x n } is a unit; (v) for p U := pr U ◦ τ there is a Y -group scheme isomorphism Φ : p ∗ U ( G U ) → p ∗ X ( G ) with δ ∗ (Φ) = id G U . Theorem 3.2.
Let k be a field, let A be a semilocal ring of several closed points on a smooth irreducible k -variety, and let K be the field of fractions of A . Let G be a reductive group over A such that everysemisimple normal subgroup of G contains ( G m ) . Then for any Noetherian k -algebra B , the naturalmap K G ( A ⊗ k B ) → K G ( K ⊗ k B ) has trivial kernel.Proof. Let g ∈ ker( K G ( A ⊗ k B ) → K G ( K ⊗ k B )) . Then there are a smooth irreducible affine k -variety X = Spec( C ) and f ∈ C = k [ X ] such that A is a semilocal ring of several closed points on X , G is defined over X and every semisimple normal subgroup of G contains ( G m ) over X , and g isdefined over C ⊗ k B and g ∈ ker( K G ( C ⊗ k B ) → K G ( C f ⊗ k B )) . Apply Theorem 3.1. Multiply thediagram (1) of U -schemes by U × k Spec B . By Lemma 2.7 there is an element ˜ g ( x ) ∈ ker (cid:0) K G ( A ⊗ k B [ x ]) → K G ( A ⊗ k B [ x ] h ) (cid:1) , such that ˜ g (0) = g . Since h is monic, by Lemma 2.5 we have ˜ g ( x ) = 1 . Hence g = 1 . (cid:3) -INVARIANCE OF NON-STABLE K -FUNCTORS IN THE GEOMETRIC CASE 4 Proof of Theorem 1.1.
Clearly, we can assume that k is a finite field or Q without loss of generality.The embedding k → A is geometrically regular, since k is perfect [Ma, (28.M), (28.N)]. Then byPopescu’s theorem [Po, Sw] A is a filtered direct limit of smooth k -algebras R . Since the groupscheme G and the unipotent radicals of its parabolic subgroups are finitely presented over A , thefunctors G ( − ) and E ( − ) = E P ( − ) commute with filtered direct limits. Hence we can replace A by asmooth k -algebra R . By the local-global principle Lemma 2.6, to show that K G ( R ) = K G ( R [ x ]) , it isenough to show that for every maximal localization R m of R . Let K be the field of fractions of R m .By Theorem 3.2 the maps K G ( R m ) → K G ( K ) and K G ( R m [ x ]) → K G ( K [ x ]) are injective. By [St14,Theorem 1.2] K G ( K ) = K G ( K [ x ]) . Hence K G ( R m ) = K G ( R m [ x ]) , and we are done. (cid:3) Proof of Theorem 1.2.
As in the proof of Theorem 1.1 we are reduced to the case where A is a filtereddirect limit of smooth k -algebras R , and G is defined and has isotropic rank ≥ over R . Also,since A is a semilocal domain, we can assume that R is a domain with a fraction field L , and anelement g ∈ ker( K G ( A ) → K G ( K )) comes from an element g ′ ∈ K G ( R ) that vanishes in K G ( L ) . ByTheorem 3.2 combined with g ′ vanishes in every semilocalization of R at a finite set of maximal ideals,and hence in every semilocalization of R at a finite set of prime ideals. Since the map K G ( R ) → K G ( A ) factors through such a semilocalization of R , it follows that g = 1 . (cid:3) Proof of Corollary 1.3.
Recall that for any field K and any simply connected semisimple G of isotropicrank ≥ over K one has K G ( K [ x ± , . . . , x ± n ]) ∼ = K G ( K ) by [St14]. Then the claim follows fromTheorem 3.2 exactly as Theorem 1.2, via the following diagram:(2) K G ( R [ x ± , . . . , x ± n ]) x = ... = x n =1 / / (cid:15) (cid:15) K G ( R ) (cid:15) (cid:15) K G ( K [ x ± , . . . , x ± n ]) x = ... = x n =1 / / K G ( K ) . (cid:3) Remark 3.3.
It is clear that Theorem 3.1 can be applied to deduce analogs of Theorems 1.1 and 1.2for any reasonably good functor defined in terms of a reductive group scheme and satisfying prop-erties similar to Lemmas 2.5, 2.6, 2.7. The paradigmal example is the functor H ´et ( − , G ) , in whichcase Theorem 1.2 corresponds to the Serre–Grothendieck conjecture (the non-isotropic cases involveadditional modification of the counterpart of Lemma 2.5). One can axiomatize this approach similarlyto the “constant” case [CTO, Th´eor`eme 1.1], however, the axioms are, naturally, more cumbersome. References [A] E. Abe,
Whitehead groups of Chevalley groups over polynomial rings , Comm. Algebra (1983), 1271–1307.[CTO] J.-L. Colliot-Th´el`ene, M. Ojanguren, Espaces Principaux Homog`enes Localement Triviaux , Publ. Math. IH´ES , no. 2 (1992), 97–122.[SGA3] M. Demazure, A. Grothendieck, Sch´emas en groupes , Lecture Notes in Mathematics, vol. 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970.[G] Ph. Gille,
Le probl`eme de Kneser-Tits , S´em. Bourbaki (2007), 983-01–983-39.[MoV] F. Morel, V. Voevodsky, A -homotopy theory of schemes , Publ. Math. I.H.´E.S. (1999), 45–143.[Ma] H. Matsumura, Commutative algebra , second ed., Math. Lect. Note Series , Benjamin/Cummings Pub-lishing Co., Inc., Reading, Massachusetts, 1980.[PaStV15] I. Panin, A. Stavrova, and N. Vavilov, On Grothendieck-Serre’s conjecture concerning principal G -bundlesover reductive group schemes: I , Compos. Math. (2015), no. 3, 535–567.[Pa18] I. Panin, On Grothendieck–Serre conjecture concerning principal bundles , Proc. Int. Cong. of Math. 2018,Rio de Janeiro, vol. 1, 201–222.[Pa19] I. Panin,
Nice triples and the Grothendieck-Serre conjecture concerning principal G -bundles over reductivegroup schemes , Duke Math. J. (2019), no. 2, 351–375. MR 3909899[PSt1] V. Petrov, A. Stavrova, Elementary subgroups of isotropic reductive groups , St. Petersburg Math. J. (2009), 625–644.[Po] D. Popescu, Letter to the Editor: General N´eron desingularization and approximation , Nagoya Math. J. (1990), 45–53.[St14] A. Stavrova,
Homotopy invariance of non-stable K -functors , J. K-Theory (2014), 199–248.[St15] A. Stavrova, Non-stable K -functors of multiloop groups , Canad. J. Math. (2016), 150–178. -INVARIANCE OF NON-STABLE K -FUNCTORS IN THE GEOMETRIC CASE 5 [Su] A. A. Suslin, On the structure of the special linear group over polynomial rings . Math. USSR Izv. (1977),221–238.[Sw] R. G. Swan, N´eron-Popescu desingularization , in Algebra and Geometry (Taipei, 1995), Lect. Alg. Geom. (1998), 135–198. Int. Press, Cambridge, MA.[T1] J. Tits, Algebraic and abstract simple groups , Ann. of Math. (1964), 313–329.[V] T. Vorst, The general linear group of polynomial rings over regular rings, Comm. Algebra (1981), 499–509. Chebyshev Laboratory, St. Petersburg State University, Russia
E-mail address ::