Isotropic reductive groups over discrete Hodge algebras
aa r X i v : . [ m a t h . K T ] A ug ISOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGEALGEBRAS
ANASTASIA STAVROVA
Abstract.
Let G be a reductive group over a commutative ring R . We say that G has isotropicrank ≥ n , if every normal semisimple reductive R -subgroup of G contains ( G m ,R ) n . We provethat if G has isotropic rank ≥ and R is a regular domain containing an infinite field k , then forany discrete Hodge algebra A = R [ x , . . . , x n ] /I over R , the map H Nis ( A, G ) → H Nis ( R, G ) induced by evaluation at x = . . . = x n = 0 , is a bijection. If k has characteristic , then,moreover, the map H ´et ( A, G ) → H ´et ( R, G ) has trivial kernel. We also prove that if k is perfect, G is defined over k , the isotropic rank of G is ≥ , and A is square-free, then K G ( A ) = K G ( R ) , where K G ( R ) = G ( R ) /E ( R ) is the corresponding non-stable K -functor, also calledthe Whitehead group of G . The corresponging statements for G = GL n were previously provedby Ton Vorst. Introduction
Let R be a commutative ring with 1. A commutative R -algebra A is called a discrete Hodgealgebra over R if A = R [ x , . . . , x n ] /I , where I is an ideal generated by monomials. If I isgenerated by square-free monomials, A is called a square-free discrete Hodge algebra. Thesimplest example of such an algebra is R [ x, y ] /xy . Square-free discrete Hodge algebras over afield are also called Stanley–Reisner rings.Serre’s conjecture on modules over polynomial rings, proved by D. Quillen and A. Suslin,states that any finitely generated projective module over a polynomial ring over a field is free.More generally, the Bass–Quillen conjecture [Bas73, Qui76] states that for any regular ring R ,every finitely generated projective module over R [ x , . . . , x n ] is extended from R . In [Vo1] T.Vorst proved that, once the Bass–Quillen conjecture holds for R , then it also holds for anydiscrete Hodge algebra A over R , i.e. every finitely generated projective A -module is extendedfrom R . Later, S. Mandal [Man85, Man86] used the same technique to extend several earlierresults on cancellation and extendability for modules and quadratic spaces over polynomialrings to discrete Hodge algebras.We generalize the observation of Vorst as follows. Grothendieck topologies are understoodin the sense of [Stacks, Tag 00ZD]. Theorem 1.1.
Let G be a faithfully flat affine group scheme locally of finite presentation overa commutative ring R . Let τ be a Grothendieck topology on R -schemes, refined by the fppftopology. Mathematics Subject Classification.
Key words and phrases.
Bass-Quillen conjecture, reductive group, G -torsor, non-stable K -functor, White-head group, simple algebraic group, discrete Hodge algebra, Stanley-Reisner ring, Milnor square.The author is a grantee of the contest “Young Russian Mathematics”. The work was supported by the RussianScience Foundation grant 14-21-00035. (i) If H τ ( R [ x , . . . , x n ] , G ) → H τ ( R, G ) has trivial kernel (respectively, is bijective) for any n ≥ , then for any square-free discrete Hodge algebra A [ x , . . . , x n ] /I over R , the canonicalmap H τ ( A, G ) → H τ ( R, G ) has trivial kernel (respectively, is bijective).(ii) Assume in addition that G is smooth. Then the claim of (i) holds for any discrete Hodgealgebra A over R . Let G be a reductive group scheme over R in the sense of [SGA3]. We say that G has isotropicrank ≥ n , if every normal semisimple reductive R -subgroup of G contains ( G m ,R ) n . Analogs ofthe Bass–Quillen conjecture for reductive groups G of isotropic rank ≥ have been establishedin many cases, most notably, for tori over regular rings, and for reductive groups over regularrings containing an infinite field, see [CTS, CTO, PSV15, AHW18] and the references therein. Itis known that, at least for reductive groups over an infinite perfect field, the isotropy conditionis necessary for the Bass–Quillen conjecture to hold [BS17].Combining Theorem 1.1 with several of the above results and the infinite field case of theSerre–Grothendieck conjecture [FP15], we obtain the following analogs of the Bass–Quillen con-jecture over discrete Hodge algebras. Note that, formally, the Nisnevich cohomology H Nis ( − , G ) is larger than the Zariski cohomology, however, the Serre–Grothendieck conjecture implies thatthey coincide on regular domains. From this point on, we assume all rings to be Noetherian. Corollary 1.2.
Let G be a reductive group scheme over a regular domain R containing aninfinite field k . Assume that G has isotropic rank ≥ . Then for any discrete Hodge algebra A = R [ x , . . . , x n ] /I over R , the map H Nis ( A, G ) → H Nis ( R, G ) , induced by evaluation at x = . . . = x n = 0 , is a bijection. If k has characteristic , then,moreover, the map H ´et ( A, G ) → H ´et ( R, G ) has trivial kernel. Theorem 1.1 and Corollary 1.2 are proved in § 3 and § 4 respectively.In parallel with the analog of the Bass–Quillen conjecture for discrete Hodge algebras, T.Vorst [Vo1, Theorem 1.1 (ii)] also established a similar result for the non-stable SK -functors K SL n ( R ) = SL n ( R ) /E n ( R ) , n ≥ , where E n ( R ) is the subgroup of SL n ( R ) generated by theelementary transvections e + te ij , ≤ i = j ≤ n , t ∈ R . Namely, one concludes that if K SL n ( R [ x , . . . , x n ]) = K SL n ( R ) , then the same holds for any discrete Hodge algebra over R .The corresponding analog of Serre’s problem is supplied by [Su]. V.I. Kopeiko extended thisobservation to symplectic groups [Ko].More generally, for any reductive group G over R and a parabolic subgroup P of G , onedefines the elementary subgroup E P ( R ) of G ( R ) as the subgroup generated by the R -pointsof unipotent radicals of parabolic subgroups of G , 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, E ( k ) is nothing but the group G ( k ) + introduced by J. Tits [T1], and K G ( k ) is the subject of theKneser–Tits problem [G]. If G has isotropic rank ≥ , then K G,P ( R ) is independent of P andwe denote it by K G ( R ) , see § 5 for a formal definition.We generalize the results of Vorst and Kopeiko as follows. The proofs of Theorem 1.3 andCorollaries 1.4–1.6 are given in § 5. SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 3
Theorem 1.3.
Let G be a reductive group scheme over a commutative Noetherian ring R , andlet P be a parabolic R -subgroup of G . If K G,P ( R [ x , . . . , x n ]) = K G,P ( R ) for any n ≥ , then K G,P ( A ) = K G,P ( R ) for any square-free discrete Hodge algebra A over R . Corollary 1.4.
Let G be a reductive group scheme of isotropic rank ≥ over a field k . Assumethat either R = k , or k is perfect and R is a regular ring containing k . Then for any square-freediscrete Hodge algebra A over R one has K G ( A ) = K G ( R ) . For non-square-free discrete Hodge algebras the above result cannot be true, since, for ex-ample, G m ,k ( k [ x ] /x ) = G m ,k ( k ) . However, if G is a Chevalley–Demazure (i.e. split) reductivegroup scheme, we are able to show that central subtori are, essentially, the only problem. Corollary 1.5.
Let G be a split simply connected semisimple group scheme over Z and let B be a Borel subgroup of G . For any commutative Noetherian ring R , if K G,B ( R [ x , . . . , x n ]) = K G,B ( R ) for any n ≥ , then K G,B ( A ) = K G,B ( R ) for any discrete Hodge algebra A over R . Corollary 1.6.
Let G be a split reductive group scheme over Z , such that every semisimplenormal subgroup of G has semisimple rank ≥ . Let R be a regular ring containing a field k . If G is simply connected semisimple or k has characteristic , then for any discrete Hodge algebra A over R the natural sequence of group homomorphisms → ker (cid:0) rad( G )( A ) → rad( G )( R ) (cid:1) → K G ( A ) → K G ( R ) → is exact. Assume that G is defined over an infinite perfect field k , and let R be a smooth k -algebra.By [AHW18, Theorem 4.1.3] (see also [Mor12] for the GL n case) we know that H Nis ( R, G ) =Hom H ( k ) (Spec( R ) , BG ) , where H ( k ) is the Morel–Voevodsky A -homotopy category over k [MoV].Combining [AHW18, Theorem 4.3.1] with [St14, Theorem 1.3], one concludes that K G ( R ) =Hom H ( k ) (Spec( R ) , G ) . The results of the present paper suggest that this relationship maysomehow extend to non-smooth k -algebras. See also Remark 5.7 in § 5.2. Discrete Hodge algebras as pull-backs
Following Vorst [Vo2], for any square-free discrete Hodge algebra A over R denote by m ( A ) the minimal integer m such that A ∼ = ( R [ x , . . . , x m ] /I )[ x m +1 , . . . , x n ] , where I is generated bysquare free monomials. Note that there is a natural bijective correspondence between simplicialsubcomplexes Σ of a standard n-simplex ∆ n and square-free discrete Hodge algebras which arequotients of R [ x , . . . , x n ] by the ideal generated by all monomials that do not occur as facesof Σ [Vo1, 3.3]. This yields an easy geometric proof of the following statement. Lemma 2.1. [Vo1, 3.4]
Let A be a square-free discrete Hodge algebra over R with m ( A ) > ,then there exist square-free discrete Hodge algebras A and A over R and a Cartesian squareof rings (1) A i (cid:15) (cid:15) i / / A j (cid:15) (cid:15) A [ x ] j / / A such that all maps are surjective, j is the evaluation at x = 0 , and m ( A ) < m ( A ) , m ( A ) The following lemma is a slightly modified version of [And, Lemma 5.7, attributed to C.Weibel and R. G. Swan]. Lemma 2.2. Let R be a commutative ring with 1, and let F be a covariant functor on thecategory of commutative finitely generated R -algebras with values in pointed sets. Let A = L i ≥ A i be a graded R -algebra such that the map F ( A [ x ]) → F ( A ) induced by the evaluation A [ x ] x −−→ A has trivial kernel (respectively, is bijective). Then the map F ( A ) → F ( A/ L i ≥ A i ) induced by the quotient homomorphism has trivial kernel (respectively, is bijective).Proof. Define f : A = L i ≥ A i → A [ x ] by f ( P ni =0 a i ) = P ni =0 a i x i . Let t a : A [ x ] → A [ x ] denotethe automorphism induced by x x + a , a ∈ A , and let e a : A [ x ] → A denote the evaluationat x = a . Since e ◦ f = id A , we conclude that F ( f ) is injective. By assumption F ( e ) hastrivial kernel, hence F ( e ◦ f ) = F ( e ) ◦ F ( f ) also has trivial kernel. But the latter map factorsthrough the quotient homomorphism A → A/ L i ≥ A i .Similarly, if F ( e ) is bijective, then F ( e ) = F ( e ◦ t ) is also bijective, and hence F ( f ) isbijective. Then F ( e ◦ f ) : F ( A ) → F ( A/ L i ≥ A i ) ∼ = F ( A ) is bijective. (cid:3) Corollary 2.3. Let R be a commutative ring with 1, and let F be a covariant functor onthe category of commutative finitely generated R -algebras with values in pointed sets. For anydiscrete Hodge algebra A = R [ x , . . . , x n ] /I over R , if F ( A [ x ]) x −−→ F ( A ) has trivial kernel(respectively, is bijective), then the canonical projection F ( A ) x i −−−→ F ( R ) has the same property.Proof. Lemma 2.2 applies, since the algebra A inherits the total degree grading from R [ x , . . . , x n ] . (cid:3) Lemma 2.4. Let R be a commutative ring with 1. Let F be a covariant functor on the categoryof commutative finitely generated R -algebras with values in pointed sets.(i) Assume that for any Cartesian square of commutative finitely generated R -algebras (2) A i (cid:15) (cid:15) i / / A j (cid:15) (cid:15) A [ x ] j / / A where j is surjective and j is the evaluation at x = 0 , the map of sets (3) F ( A ) ( i ,i ) −−−→ F ( A [ x ]) × F ( A ) has trivial kernel. Assume also that the maps g m : F ( R [ x , . . . , x m ]) → F ( R ) induced byevaluation at x = x = . . . = x m = 0 have trivial kernel for any m ≥ . Then for anysquare-free discrete Hodge algebra A = R [ x , . . . , x n ] /I over R the map F ( A ) → F ( R ) inducedby evaluation at x = x = . . . = x n = 0 has trivial kernel.(ii) Assume instead that for any square (2) , whenever F ( j ) bijective, F ( i ) is also bijective,and that all maps g m , m ≥ , are bijective. Then for any square-free discrete Hodge algebra A over R the map F ( A ) → F ( R ) is bijective.Proof. We apply induction on m ( A ) . If m ( A ) = 0 , then A = R [ x , . . . , x n ] , and the claimshold. Assume that the claim holds for any square-free discrete Hodge algebra C over R with SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 5 m ( C ) < m ( A ) . By Lemma 2.1 there is a Cartesian square (4). Assume (i). The inducedsquare(4) A [ y ] i (cid:15) (cid:15) i / / A [ y ] j (cid:15) (cid:15) A [ x, y ] j / / A [ y ] is also Cartesian, and hence F ( A [ y ]) ( i ,i ) −−−→ F ( A [ y ]) × F ( A [ x, y ]) has trivial kernel. Bythe induction hypothesis the maps F ( A [ y ]) → F ( A ) and F ( A [ x, y ]) → F ( A [ x ]) inducedby evaluation at y = 0 have trivial kernel, hence F ( A [ y ]) → F ( A ) has trivial kernel. ByCorollary 2.3 the map F ( A ) → F ( R ) has trivial kernel.In (ii), similarly, the map F ( A [ x ]) → F ( A ) is bijective by induction assumption, hence F ( A ) → F ( A ) is bijective. Again by induction assumption, F ( R ) → F ( A ) is bijective, hence F ( R ) → F ( A ) is bijective. (cid:3) Remark 2.5. Let A = R [ x , . . . , x n ] /I , where I is any ideal generated by monomials, be anydiscrete Hodge algebra over R . Let I ⊆ R [ x , . . . , x n ] be the ideal generated by monomials x − δ i , x − δ i , . . . x − δ in, n for all monomials x i x i . . . x i n n generating I ; here δ i, denotes Kroneckerdelta. Then R [ x , . . . , x n ] /I is a square-free discrete Hodge algebra, and the kernel of ρ : A → R [ x , . . . , x n ] /I is the nilpotent ideal J = I /I . Thus, if the functor F of Lemma 2.4 is such that F ( A ) → F ( A/J ) has trivial kernel, or, respectively, is bijective, then the claim (i), or, respectively, (ii)of the lemma holds for any discrete Hodge algebra.3. Milnor squares and G -torsors Recall that a Cartesian square of rings(5) A i (cid:15) (cid:15) i / / A j (cid:15) (cid:15) A j / / B is called a Milnor square, if at least one of the maps j , j is surjective [Mil71, §2]. Note thatif, say, j is surjective (respectively, split surjective), then i has the same property. J. Milnorshowed that these squares have patching property for finitely generated projective modules.We need the following extension of this result. Lemma 3.1. Let G be a faithfully flat affine group scheme locally of finite presentation over acommutative ring R . Consider a Milnor square of R -algebras (5) , where j is surjective.(i) For any fppf G -torsors E and E over A and A respectively, and any G -equivariantisomorphism φ : j ∗ ( E ) → j ∗ ( E ) over B , there is a G -torsor E = E ∪ φ E over A and G -equivariant isomorphisms ψ : i ∗ ( E ) → E , ψ : i ∗ ( E ) → E compatible with φ .(ii) For any fppf G -torsor E over A , there is a natural isomorphism i ∗ ( E ) ∪ id i ∗ ( E ) ∼ = −→ E of G -torsors over A . SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 6 Proof. Since G is affine, E and E are affine over the respective bases, see e.g [BLR90, §6.4].Let G = Spec( T ) , E = Spec( S ) , E = Spec( S ) . Then j ∗ ( E ) = Spec( S ⊗ A B ) and j ∗ ( E ) = Spec( S ⊗ A B ) are isomorphic B -algebras. We define E to be the spectrum of thefibered product S of rings S and S over S ⊗ A B , where the homomorphism S → S ⊗ A B factors through φ : S ⊗ A B → S ⊗ A B . In other words, E is the push-out of the diagram(6) j ∗ ( E ) ( j ) E (cid:15) (cid:15) φ / / j ∗ ( E ) ( j ) E / / E E in the category of ringed spaces, see e.g. [Fer03, Th. 5.1]. Note that the closed embedding ( j ) E base-changes to a closed embedding E → E , and j ∗ ( E ) ∼ = E × E E .By the universal property of the push-out E is naturally an A -scheme. Clearly, considered asan A -module, S is the Milnor-type patching of the flat A -module S and the flat A -module S in the sense of [Fer03, Th. 2.2]. In particular, S is flat over A and S ⊗ A A i ∼ = S i , i = 1 , .Since T is a faithfully flat R -algebra, G A × A E is faithfully flat over E , hence G A × A E ∼ = ( G A × A E ) ∪ id × φ ( G A × A E ) is the push-out of G A × A E and G A × A E . The universal property of push-out togetherwith the G -equivariance of φ then defines an action of G A on E , compatible with the actionsof G A on E and G A on E .As a topological space, E is isomorphic to the union of images of E and E [Fer03, Scolie4.3], hence E → Spec( A ) is surjective, and S is faithfully flat over A . Then tensoring with S also preserves fibered products of R -algebras, hence E × A E ∼ = ( E A × A E ) ∪ id × φ ( E A × A E ) ∼ = ( E × A E ) ∪ φ × φ ( E × A E ) . Since G A i × A i E i is isomorphic to E i × A i E i , i = 1 , , by means of the map ( g, x ) ( gx, x ) , weconclude that E × A E ∼ = ( G A × A E ) ∪ id × φ ( G A × A E ) ∼ = G A × A E. Since E → Spec( A ) is a faithfully flat and quasi-compact morphism, and G is locally of finitepresentation, we conclude that E → Spec( A ) is also locally of finite presentation by [Gro65,Proposition 2.7.1]. Hence E is an fppf G -torsor over A .To prove the last claim of the lemma, let E be any torsor over A . Note that both E and i ∗ ( E ) ∪ id i ∗ ( E ) are affine and flat over A , and there is a morphism of A -schemes i ∗ ( E ) ∪ id i ∗ ( E ) → E . By [Fer03, Th. 2.2 (iv)] patching of flat modules is an equivalence of categories, hence it isan isomorphism. Its G -equivariance is clear, since G A × A ( i ∗ ( E ) ∪ id i ∗ ( E )) is the patching of G A × A i ∗ ( E ) and G A × A i ∗ ( E ) . (cid:3) Lemma 3.2. In the setting of Lemma 3.1, assume that j has a section s : B → A which is ahomomorphism of R -algebras, and let r : A → A be the induced section of i . If E is extendedfrom B , then E = E ∪ φ E is extended from A , i.e. E ∼ = s ∗ ( j ∗ ( E )) implies E ∼ = r ∗ ( E ) . Inparticular, E is a trivial G -torsor if and only if E and E are trivial G -torsors.Proof. Since E is faithfully flat over A , r ∗ ( E ) = E × A A is faithfully flat over A . ByLemma 3.1 (ii) the G -torsor r ∗ ( E ) over A is isomorphic to the push-out of the G -torsors SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 7 i ∗ ( r ∗ ( E )) over A and i ∗ ( r ∗ ( E )) = ( i ◦ r ) ∗ ( E ) = id ∗ A ( E ) = E over A , by means of thetrivial isomorphism of their restrictions to B . One has i ∗ ( r ∗ ( E )) = ( i ◦ r ) ∗ ( E ) = ( s ◦ j ) ∗ ( E ) = s ∗ ( j ∗ ( E )) . Then the isomorphism φ : j ∗ ( E ) → j ∗ ( E ) extends by means of s ∗ to the isomorphism of A -torsors E ∼ = s ∗ ( j ∗ ( E )) → i ∗ ( r ∗ ( E )) . Hence r ∗ ( E ) ∼ = E ∪ φ E by the unicity of thepush-out. (cid:3) Proof of Theorem 1.1. (i) We check the conditions of Lemma 2.4 for the functor H τ ( − , G ) . Forany square (2), Lemma 3.2 readily implies that that a G -torsor E over A is trivial, once i ∗ ( E ) and i ∗ ( E ) are trivial. In the bijective case, for any E over A , we know that i ∗ ( E ) is extendedfrom A . Then E is extended from A by Lemma 3.2.(ii) If G is smooth, then for any commutative R -algebra A and for any nilpotent ideal J of A , the map H fppf ( A, G ) → H fppf ( A/J, G ) is bijective [Gro68b, Str83]. Then by Re-mark 2.5 for any discrete Hodge algebra A = R [ x , . . . , x n ] /I over R one has H fppf ( A, G ) ∼ = H fppf ( R [ x , . . . , x n ] /I , G ) , where A ′ = R [ x , . . . , x n ] /I is a square-free discrete Hodge algebra.Then, clearly, H τ ( A, G ) → H τ ( A ′ , G ) is injective. If H τ ( A ′ , G ) → H τ ( R, G ) has trivial kernel,this implies that H τ ( A, G ) → H τ ( R, G ) has trivial kernel. In the bijective case, we concludethat H τ ( A, G ) → H τ ( R, G ) is injective, and its surjectivity is automatic since A → R has asection. (cid:3) Torsors under reductive group schemes In the present section we apply the results of § 3 to isotropic reductive groups and proveCorollary 1.2. Lemma 4.1. Let R be a regular semilocal domain, K be the fraction field of R . Let G, G ′ bereductive R -groups, and T be an R -group of multiplicative type such that there is a short exactsequence ( a ) 1 → G ′ → G → T → or ( b ) 1 → T → G ′ → G → of R -group schemes. For any n ≥ , if the natural maps H ´et ( R [ x , . . . , x n ] , G ′ ) → H ´et ( K [ x , . . . , x n ] , G ′ ) and H ´et ( R, G ) → H ´et ( K, G ) have trivial kernels, then H ´et ( R [ x , . . . , x n ] , G ) → H ´et ( K [ x , . . . , x n ] , G ) has trivial kernel.Proof. Since G and G ′ are smooth, we can replace their ´etale cohomology by fppf. For shortness,write x instead of x , . . . , x n , and x = 0 instead of x = . . . = x n = 0 .Consider first the case ( a ) . Let S be any of R , R [ x ] , K , K [ x ] , then we have an exact sequenceof pointed sets T ( S ) → H fppf ( S, G ′ ) → H fppf ( S, G ) → H fppf ( S, T ) . By [CTS, Lemma 2.4] H fppf ( S, T ) ∼ = H fppf ( S [ x ] , T ) , and by [CTS, Theorem 4.1] H fppf ( R, T ) → H fppf ( K, T ) is injective. Hence H fppf ( R [ x ] , T ) → H fppf ( K [ x ] , T ) is also injective. Hence any ξ ∈ ker (cid:0) H fppf ( R [ x ] , G ) → H fppf ( K [ x ] , G ) (cid:1) lifts to an element η ∈ H fppf ( R [ x ] , G ′ ) . Let ¯ η be theimage of η in H fppf ( G ′ , K [ x ]) . Then there is θ ∈ T ( K [ x ]) such that ¯ η is the image of θ . Since T is a group of multiplicative type, we have T ( K [ x ]) = T ( K ) . Hence ¯ η is extended from K . By SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 8 the assumptions on ξ and G , the torsor ξ | x =0 is trivial, hence η | x =0 has a preimage σ ∈ T ( R ) .Clearly, the image of σ in T ( K ) maps to ¯ η | x =0 . Note that the group T ( R [ x ]) = T ( R ) actson H fppf ( R [ x ] , G ′ ) by right shifts, see [Ser02, §5.5]. Since ¯ η is extended from K , the image of η · σ − in H fppf ( K [ x ] , G ′ ) is trivial. Hence η · σ − is trivial. Hence η is extended from R . Hence ξ is extended from R . Then ξ is trivial by the assumption on G .Consider the case ( b ) . For each S as above, we have an exact sequence H fppf ( S, T ) → H fppf ( S, G ′ ) → H fppf ( S, G ) → H fppf ( S, T ) . By [CTS, Theorem 4.3] H fppf ( R [ x ] , T ) → H fppf ( K ( x ) , T ) is injective, hence H fppf ( R [ x ] , T ) → H fppf ( K [ x ] , T ) is also injective. The rest of the proof is the same as in the previous case, with theonly difference that one uses the action of the commutative group H fppf ( R [ x ] , T ) ∼ = H fppf ( R, T ) on H fppf ( R [ x ] , G ′ ) , which is well-defined since T is central in G ′ ; see [Ser02, §5.7]. (cid:3) The following statement for simply connected semisimple reductive groups is a particularcase of [PSV15, Theorem 1.6]. We use this case, as well as the result of I. Panin and R. Fedorovon the Serre–Grothendieck conjecture [FP15], to obtain the case of general reductive groups. Theorem 4.2. Assume that R is a regular semilocal domain that contains an infinite field,and let K be its fraction field. Let G be a reductive group scheme over R of isotropic rank ≥ .Then for any n ≥ the natural map H ´et ( R [ x , . . . , x n ] , G ) → H ´et ( K [ x , . . . , x n ] , G ) has trivial kernel.Proof. In [PSV15, Theorem 1.6] the claim is proved under the assumption that G is simple andsimply connected. The case where G is an arbitrary simply connected reductive group followsimmediately by the Faddeev-Shapiro lemma, as in the proof of [PSV15, Theorem 11.1]. Now let G be an arbitrary reductive group, let der( G ) be its derived subgroup in the sense of [SGA3], andlet G sc be the simply connected cover of der( G ) . Then G sc and der( G ) are semisimple reductivegroups satisfying the same isotropy condition as G . There are two short exact sequences ofreductive R -groups → der( G ) → G → corad( G ) → , and → C → G sc → der( G ) → ,where corad( G ) and C are R -groups of multiplicative type [SGA3, Exp. XXII]. Note thatfor any reductive group H over R , the map H ´et ( R, H ) → H ´et ( K, H ) has trivial kernel by thecorresponding case of the Serre–Grothendieck conjecture [FP15, Theorem 1]. Since reductivegroups are smooth by definition, we can replace the ´etale topology by fppf. Hence these twoshort exact sequences are subject to Lemma 4.1. (cid:3) The following statement is a slight extension of [PSV15, Corollary 1.7]. Theorem 4.3. Assume that R is a regular domain that contains a field of characteristic . Let G be a reductive group scheme over R of isotropic rank ≥ . Then for any n ≥ the map H ´et ( R [ x ] , G ) → H ´et ( R, G ) , induced by evaluation at x = 0 , has trivial kernel.Proof. In [PSV15, Corollary 1.7] the claim is established under the assumption that G is simpleand simply connected. The proof for any reductive group is exactly the same using Theorem 4.2instead of its simply connected case [PSV15, Theorem 1.6]. (cid:3) SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 9 Theorem 4.4. Let G be a reductive group scheme over a regular domain R containing aninfinite field k . Assume that G has isotropic rank ≥ . Then the map H Nis ( R [ x ] , G ) x −−→ H Nis ( R, G ) is a bijection.Proof. We need to show that any Nisnevich G -torsor E over R [ x ] is extended from R . By [Tho87,Corollary 3.2] G is linear, hence by the local-global principle for torsors [AHW18, Theorem 3.2.5](see also [Mos, Korollar 3.5.2]) it is enough to prove the same claim for every maximal local-ization of R . Thus, we can assume that R is regular local. By Theorem 4.2 H ´et ( R [ x ] , G ) → H ´et ( K [ x ] , G ) has trivial kernel. By [CTO, Proposition 2.2] H ´et ( K [ x ] , G ) → H ´et ( K ( x ) , G ) hastrivial kernel. Hence H Nis ( R [ x ] , G ) → H Nis ( K ( x ) , G ) has trivial kernel. Since every Nisnevichtorsor over K ( x ) is trivial, therefore, every Nisnevich torsor over R [ x ] is trivial, and henceextended from R . (cid:3) Proof of Corollary 1.2. The first statement follows from Theorem 4.4 and Theorem 1.1 (ii).The second statement follows from Theorem 4.3 and Theorem 1.1 (ii). (cid:3) Non-stable K -functors 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 isaffine, the group P has a Levi subgroup L P [SGA3, Exp. XXVI Cor. 2.3]. There is a uniqueparabolic 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 5.1. [PSt1] The elementary subgroup E P ( R ) corresponding to P is the subgroupof 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 ) the pointed 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).The following lemma generalizes [Vo1, Theorem 2.1 (ii)]. Lemma 5.2. Let G be a reductive group scheme over a commutative ring R , and let P be aproper parabolic subgroup of G . For any Milnor square of R -algebras (5) , where j is surjective,the induced map of sets (7) K G,P ( A ) ( i ,i ) −−−→ K G,P ( A ) × K G,P ( B ) K G,P ( A ) is surjective. If, moreover, j is split surjective with R -algebra section map s : B → A , and j is surjective, then the induced square (8) K G,P ( A ) i (cid:15) (cid:15) i / / K G,P ( A ) j (cid:15) (cid:15) K G,P ( A ) j / / K G,P ( B ) SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 10 is a Cartesian square of sets.Proof. To prove surjectivity of (7), let g ∈ G ( A ) , g ∈ G ( A ) be such that j ( g ) ∈ j ( g ) E P ( B ) .Since E P ( A ) surjects onto E P ( B ) , adjusting g we obtain j ( g ) = j ( g ) . Since G is left exact,there is g ∈ G ( A ) such that i ( g ) = g and i ( g ) = g .Next, assume that j is split surjective and j is surjective, then all four homomorphismsof (5) are surjective, and i is also split. Let g , g ∈ G ( A ) be such that i ( g ) ∈ i ( g ) E P ( A ) and i ( g ) ∈ i ( g ) E P ( A ) . Then i ( g − g ) ∈ E P ( A ) and i ( g − g ) ∈ E P ( A ) . Then g = g − g satisfies i ( g ) ∈ E P ( A ) and i ( g ) ∈ E P ( A ) . We are going to show that g ∈ E P ( A ) .Since i is surjective, adjusting g by an element of E P ( A ) , we can assume that i ( g ) = 1 . Let s : B → A be a splitting of j . By [St14, Lemma 4.1] one has G ( A , ker( j )) ∩ E P ( A ) = E P ( A , ker( j )) = E P (ker( j )) E P ( s ( B )) . Since j ( i ( g )) = j ( i ( g )) = 1 , one has i ( g ) ∈ E P (ker( j )) E P ( s ( B )) . Since the square is Carte-sian, ker( j ) ⊆ i (ker( i )) . Therefore, we can lift any element of E P (ker( j )) to an element of E P (ker( i )) . Since, moreover, j is surjective, i ( g ) has a preimage in E P ( A, ker( i )) E P ( A ) = E P ( A, ker( i )) . Since the square is Cartesian and G is left exact, we conclude that g ∈ E P ( A, ker( i )) . This finishes the proof. (cid:3) Proof of Theorem 1.3. The claim follows immediately from Lemma 2.4 and Lemma 5.2 (ii). (cid:3) If P is a strictly proper parabolic subgroup and G has isotropic rank ≥ , then K G,P isgroup-valued and independent of P . Definition 5.3. A parabolic subgroup P in G is called strictly proper , if it intersects properlyevery normal semisimple subgroup of G . Theorem 5.4. [PSt1, Lemma 12, Theorem 1] Let G be a reductive group over a commutativering R , and let A be a commutative R -algebra. If for any maximal ideal m of R the isotropicrank of G R m is ≥ , then the subgroup E P ( A ) of G ( A ) is the same for any strictly properparabolic A -subgroup P of G A , and is normal in G ( A ) . Definition 5.5. Let G be a reductive group of isotropic rank ≥ over a commutative ring R .For any strictly proper parabolic subgroup P of G over R , and any R -algebra A , we call thesubgroup E ( A ) = E P ( A ) , where P is a strictly proper parabolic subgroup of G , the elementarysubgroup 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. Proof of Corollary 1.4. Under the given assumptions K G ( R [ x , . . . , x n ]) = K G ( R ) for any n ≥ by [St14, Theorem 1.2] and [St14, Theorem 1.3] respectively. Hence the claim follows fromTheorem 1.3. (cid:3) Proof of Corollary 1.5. Let A = R [ x , . . . , x n ] /I , where I is any ideal generated by monomials.By Remark 2.5 there is a square-free discrete Hodge algebra A ′ = A/J over R , where J isan ideal of A generated by a finite set of nilpotent elements. Let B − be a Borel R -subgroupof G opposite to B , let U B and U B − be their unipotent radicals, and let T = B ∩ B − be amaximal torus of G . By Theorem 1.3 we know that K G,B ( A ′ ) = K G,B ( R ) . We need to showthat the natural homomorphism p : K G,B ( A ) → K G,B ( A ′ ) is injective. Let g ∈ G ( A ) be such SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 11 that p ( g ) ∈ E B ( A ′ ) . Adjusting g by an element of E B ( A ) , we can assume that p ( g ) = 1 . Then U B T U − B ⊆ G is a principal open subscheme of G corresponding to a function f ∈ R [ G ] andisomorphic to the direct product U B × T × U B − , see [SGA3, Exp. XXVI, Remarque 4.3.6]and [Ma, p. 9]. Since p ( g ) = 1 , we have f ( p ( g )) ∈ ( A/J ) × . Since J is nilpotent, it followsthat f ( g ) ∈ A × , and hence g ∈ U B ( A ) T ( A ) U B − ( A ) . Since T is a split maximal torus and G issimply connected semisimple, we have T ( A ) ≤ E ( A ) . Hence g ∈ E ( A ) , as required. (cid:3) Lemma 5.6. Let R be a Noetherian ring, and let J ⊆ R be an ideal such that ( R, J ) is aHenselian pair. Let → T → G ′ → G → be a short exact sequence of R -group schemes,where T is a smooth R -group of multiplicative type, and G, G ′ be two reductive R -groups. Let P ′ be a parabolic R -subgroup of G ′ , and let P be its image in G . There is a short exact sequenceof groups (9) → ker (cid:0) T ( R ) → T ( R/J ) (cid:1) → ker (cid:0) K G ′ ,P ′ ( R ) → K G ′ ,P ′ ( R/J ) (cid:1) → ker (cid:0) K G,P ( R ) → K G,P ( R/J ) (cid:1) → . Proof. Let S be one of R , R/J . We have a short exact sequence of groups → T ( S ) → G ′ ( S ) → G ( S ) → H fppf ( S, T ) . Since the image of E P ′ ( S ) in G ( S ) coincides with E P ( S ) , there is an induced sequence → T ( S ) → K G ′ ,P ′ ( S ) → K G,P ( S ) . Hence the maps in (9) are well-defined, and it is a complex.To prove the exactness of (9) at the third term, let g ∈ G ′ ( R ) be such that its image in G ( R ) belongs to E P ( R ) . Adjusting g by an element of E P ′ ( R ) , we can assume that g is in the kernelof G ′ ( R ) → G ( R ) . Then g belongs to the image of T ( R ) .To prove the exactness of (9) at the fourth term, assume that g ∈ G ( R ) maps to an ele-ment of E P ( R/J ) . Since E P ( R ) → E P ( R/J ) is surjective, adjusting g we can assume that g maps to ∈ G ( R/J ) . Since T is smooth, we can replace its fppf cohomology with ´etale, and H ´et ( R, T ) ∼ = H ´et ( R/J, T ) by [Gro68b] (see also [Str83]). Hence g lifts to an element h ∈ G ′ ( R ) .By assumption the image of h in G ′ ( R/J ) lifts to T ( R/J ) . Since ( R, J ) is a Henselian pair,and T is smooth, the map T ( R ) → T ( R/J ) is surjective. Adjusting h by an element of T ( R ) ,we obtain a new preimage h ′ ∈ G ′ ( R ) of g such that its image in G ( R/J ) is trivial. Hence h ′ ∈ E P ′ ( R ) and g ∈ E P ( R ) , as required. (cid:3) Proof of Corollary 1.6. Since R contains a field, it contains a perfect field k . Since G is split,it is defined over k . Then K G ( R [ x , . . . , x n ]) = K G ( R ) for any n ≥ by [St14, Theorem 1.3].Let G sc be the simply connected cover of the semisimple group G ss = G/ rad( G ) . Then byCorollary 1.5 K G sc ( A ) = K G sc ( R ) . If G = G sc , this finishes the proof.To treat the general case, recall that by Remark 2.5 there is a square-free discrete Hodgealgebra A ′ = A/J over R , where J is an ideal of A generated by a finite set of nilpotent elements.Clearly, ( A, J ) is a Henselian pair. There is a short exact sequence → C → G sc → G ad → , where C is an R -group of multiplicative type. By Theorem 1.3 K G sc ( A/J ) = K G sc ( R ) ,and K G ss ( A/J ) = K G ss ( R ) . In particular, K G sc ( A ) = K G sc ( A/J ) . Hence by Lemma 5.6 K G ss ( A ) → K G ss ( A/J ) has trivial kernel. Since K G ss ( A ) → K G ss ( R ) is obviously surjective, SOTROPIC REDUCTIVE GROUPS OVER DISCRETE HODGE ALGEBRAS 12 we conclude that K G ss ( A ) = K G ss ( A/J ) = K G ss ( R ) . Applying Lemma 5.6 again, we obtain ker (cid:0) rad( G )( A ) → rad( G )( A/J ) (cid:1) = ker (cid:0) K G ( A ) → K G ( A/J ) (cid:1) . It remains to note that by Theorem 1.3 K G ( A/J ) = K G ( R ) and rad( G )( A/J ) = rad( G )( R ) . (cid:3) Remark 5.7. Let G be a smooth group scheme over a commutative Noetherian ring R . Let H be a normal subgroup subfunctor of the functor represented by G on the category of com-mutative finitely generated R -algebras, such that for any surjective homomorphism φ : A → B of such algebras one has H ( B ) ≤ φ ( G ( A )) . (If G is a reductive group of isotropic rank ≥ ,one can take H = E .) For any Milnor square of R -algebras (5) there is an exact sequence ofpointed sets G ( A ) /H ( A ) ( i ,i ) −−−→ G ( A ) /H ( A ) × G ( A ) /H ( A ) λ −→ G ( B ) /H ( B ) → δ −→ H ´et ( A, G ) ( i ∗ ,i ∗ ) −−−→ H ´et ( A , G ) × H ´et ( A , G ) . 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Vorst, The general linear group of discrete Hodge algebras , Ring theory (Antwerp, 1985), LectureNotes in Math., vol. 1197, Springer, Berlin, 1986, pp. 225–231. Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29B, 199178 SaintPetersburg, Russia E-mail address ::