Morava K-theory of orthogonal groups and motives of projective quadrics

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.


Introduction
One of the questions of algebraic geometry over fields that are not algebraically closed is the study of projective homogeneous varieties.An effective method is to investigate various generalized cohomology theories of these varieties, constructed, for example, via algebraic cycles, vector bundles or cobordism classes.If G is an algebraic group that acts transitively on a projective homogeneous variety X, then the value of the cohomology theory A * (G) of the group already contains substantial information about the value of the cohomology theory A * (X).
In this article we examine A * = K(n) * , the algebraic Morava K-theory for prime 2, and G = SO m , the split special orthogonal group, i.e., the corresponding projective homogeneous varieties are quadrics and (higher) orthogonal Grassmannians.The computation of the ring K(n) * (SO m ) (Theorem 1.1) is one of the main results of this article.Among other things this provides new tools for the study of Morava motives, and as an instance of this principle, we determine all indecomposable summands of the K(n)-motive of a generic quadric Q (Theorem 1.2).
1.1.Overview of Morava K-theory.Levine and Morel defined a universal oriented cohomology theory, called the algebraic cobordism (see [LeMo]).It allows one to consider algebraic analogues of well studied topological oriented cohomology theories such as Morava K-theories over fields of characteristic 0.
The algebraic cobordism theory of Levine-Morel and arbitrary oriented cohomology theories [LePa,Le07,Vi15,PaSm,ViYa,NeZa,CPZ,Vi19,Se17,GiVi,PS21,Me02,Smi] are extensively studied now.Any oriented cohomology theory is endowed with a formal group law, and any formal group law over any commutative ring comes from a certain oriented cohomology theory in this sense.Among various theories corresponding to the same formal group law there exists a universal one called free.The class of free theories contains algebraic cobordism, Chow groups, i.e., algebraic cycles modulo rational equivalence, Grothendieck's K 0 , and Morava K-theories.One of the major features of these theories for our purposes is the Rost nilpotence principle for projective homogeneous varieties recently proven in [GiVi].
Vishik gave a geometric description of free theories in [Vi19], which allowed him to construct operations on algebraic cobordism, and later the fourth author used Vishik's description to construct operations from Morava K-theories, see [Se17,Se18].Vishik's results imply, in particular, that the category of free theories and multiplicative operations is equivalent to the category of 1-dimensional commutative graded formal group laws [Vi19,Theorem 6.9].
Working with quadratic forms, it is natural to consider localized at 2 coefficients Z (2) instead of integral, and, therefore, consider only formal group laws over Z (2) -algebras.Then by a theorem of Cartier we can restrict ourselves to a narrower class of formal group laws, called 2-typical (every formal group law over a Z (2) -algebra is isomorphic to a 2-typical one), in particular, it is natural to consider the universal 2-typical formal group law.It admits a standard construction as a formal group law over the polynomial ring with infinite number of variables Z (2) [v 1 , v 2 , . ..] defined by recurrent identities, see, e.g., [Rav,Appendix A2].Then passing to F 2 , specifying v k = 0 for k = n, and inverting v n , we obtain a formal group law over F 2 [v ±1  n ], and the corresponding free theory is called the Morava K-theory K(n) * .
It is worth remarking that Morava K-theories are related to higher powers of the fundamental ideal I in the Witt ring and, more generally, to cohomological invariants of algebraic groups.The first and the fourth authors stated in [SeS] a "Guiding Principle", which dates back to Voevodsky's program [Vo95] on the proof of the Bloch-Kato conjecture, and specifies this relation.In particular, the n-th Morava motive of a quadric depends only on its dimension and on the class of the quadratic form in the Witt ring modulo I n+2 [SeS,Proof of Proposition 6.18].
Recently, it was shown by the first and the fourth authors [SeS] that motives for the Morava K-theory allow to obtain new results on Chow groups of quadrics.Moreover, they most naturally fill in the gap between K 0 -motives and Chow motives.In this article this work is continued and is brought to the next level.
1.2.Overview of motives.The theory of motives envisioned by Grothendieck is supposed to bridge the study of cohomology of algebraic varieties with rational coefficients with the study of various geometric and arithmetic properties of the variety itself that are expressed via algebraic cycles.It turned out, however, that this theory is much more profound and fundamental and also accounts for phenomena of cohomology with integral coefficients.For example, the Chow and K 0 -motives with integral and finite coefficients are nowadays a common tool for the study of projective homogeneous varieties over non-algebraically closed fields.
One of the most striking applications of motives was Voevodsky's proof [Vo03,OVV] of the Milnor conjecture which relies on Rost's computation [Ros] of the motive of a Pfister quadric.More generally, the structure of Chow motives of norm varieties plays a crucial role in the proof of the Bloch-Kato conjecture by Rost and Voevodsky (see [Vo11]).
Motivic decompositions of projective homogeneous varieties were used to construct a new cohomological invariant for groups of type E 8 and to solve a problem of Serre about their finite subgroups (see [GaS] and [S16]).Besides, Garibaldi, Geldhauser, and Petrov used decompositions of Chow motives to relate the rationality of some parabolic subgroups of groups of type E 7 with the Rost invariant, proving a conjecture of Rost and solving a question of Springer in [GPS].
A new tool for the study of motives of homogeneous varieties was not long ago developed by the first and by the third author [PS21].Let G be a split semisimple algebraic group over a field and let A * be an oriented cohomology theory in the sense of Levine-Morel.Then there exists a functor from the category of A * -motives of twisted forms of smooth projective cellular G-varieties to the category of graded comodules over a bi-algebra obtained as the quotient of the Hopf algebra A * (G) modulo certain bi-ideal.
In the case of twisted flag varieties this quotient is called the J-invariant.Note that in the case of a generic twisted flag variety the J-invariant is just the Hopf algebra A * (G) itself.One of the main properties of the J-invariant is that it carries full information on the motivic decomposition type of generically split twisted flag varieties (see [PS21,Theorem 5.7]).
We remark finally that the J-invariant was introduced previously in [PSZ] and [Vi05] for the Chow theory, and it was an essential ingredient in the solution of problems mentioned above and, in particular, in the solution of a problem of Serre about groups of type E 8 and its finite subgroups.1.3.Overview of results.
1.3.1.Morava K-theory of orthogonal groups.The main result of the present article is a computation of the ring of the Morava K-theory for the prime 2 of split special orthogonal groups.
The investigation of the structure of oriented cohomology theories of split algebraic groups has a long history (see, for example, [Le93] and [Me97] for K 0 or [Kac,Table 2] for the Chow theory).For example, the structure of the Chow ring CH * SO m ) is explicitly computed by Marlin [Ma74].One can write a closed formula for CH * (SO m )/2 as (1) with explicit expressions for k i and r depending on m (see [Kac]).
The following theorem extends the above result to the case of the Morava K-theory.
Theorem 1.1 (Theorem 6.13).As an F 2 [v ±1 n ]-algebra the Morava K-theory of split orthogonal groups can be described explicitly as follows: We remark that the topological Morava K-theory of the connected component of the real compact Lie group of orthogonal matrices SO(m) is known only additively [Nis,Ra89].
The proof of the above theorem is very different from [Ma74,Kac], although it relies on (1).It is obtained by combining the following three claims that might be of independent interest.
The canonical pullback map along the natural inclusion We remark that for the topological Morava K-theory the natural map is not an isomorphism [Nis,Corollary 2.9].Recall that for an arbitrary smooth variety X there exists a canonical surjective ring homomorphism where τ denotes the topological filtration on the Morava K-theory.In general, ρ is an isomorphism only up to degree 2 n [Se17].However, for X = SO m we can prove the following result.
(ii) Unstable coincidence of the associated graded ring with Chow.
If m ≤ 2 n+1 , then the canonical ring homomorphism Finally, we show that the Hopf algebra structure on K(n) * (SO m ) allows to reconstruct the algebra structure of K(n) * (SO m ) from that of gr * τ K(n) * (SO m ).Here we use (1) and the explicit formulae for k i .
(iii) Reconstruction of the ring structure.
For m ≤ 2 n+1 the ring K . The three claims described above are proved by different methods.The proof of the stabilization result relies on Demazure differential operators (see [BGG, Dem, CPZ]), the unstable coincidence is based on the structure of algebraic cobordism studied by Vishik [Vi15] and the last statement as already mentioned requires a subtle study of Hopf algebras.
The following table summarizes the results of the computation for a few small numbers n and m (we set v n = 1 to simplify the formulae; grey cells indicate the stable range): Observe that the cell "K(n) * (SO 2 n+1 −1 )" is the first stable cell both in the corresponding row and the column.Although stabilization "in columns" is a general phenomenon, for an arbitrary X it happens only for K(n) * (X) with 2 n > dim X.We underline again that the stabilization "in rows" is a special phenomenon for K(n) * (SO m ), it does not hold for K(n) * top (SO(m)).This miraculous complementarity of two stabilizations is what allowed us to prove the main theorem.1.3.2.Applications to motives.In view of the discussion in Section 1.2, the above theorem establishes the ring structure of the J-invariant of orthogonal groups, and, consequently, automatically gives new information about the structure of the Morava motives of maximal orthogonal Grassmannians (see Corollary 6.14).
For example, as an immediate corollary of the above theorem and [PS21, Theorem 5.7] we obtain that for m > 2 n+1 the K(n)-motive of a connected component of the maximal orthogonal Grassmannian for a generic m-dimensional quadratic form with trivial discriminant is always non-trivially decomposable (see Corollary 6.14).Note that contrary to this by a result of Vishik its Chow motive is always indecomposable.
We also provide a complete computation of decompositions of the Morava motives of generic quadrics.Note that one can abstractly specialize this decomposition to an arbitrary smooth projective quadric, and we explicitly write the corresponding projectors.
The results of Levine-Morel [LeMo] and Vishik-Yagita [ViYa] imply that the decomposition of the Chow motive is the "roughest" one in the sense that any decomposition of the Chow motive of a smooth projective variety X can be lifted to a decomposition of the A * -motive of X for every oriented cohomology theory A * .
However, for an indecomposable Chow summand the corresponding A * -motive can be decomposable.For example, the Chow motive of a generic quadric is indecomposable (see [Vi04,Ka12]), on the other hand, by results of Swan and Panin [Swa,Pa94] the K 0motive of every positive dimensional quadric is decomposable and encodes, for example, its Clifford invariant.
The application of our computations to the motivic study of quadrics is the following.
Theorem 1.2 (Theorem 7.8).Let Q be a generic quadric of dimension D and n > 1.
We denote N = 2 n for D = 2d even or {i} and an indecomposable summand of rank N.Moreover, if D is odd, then over an algebraic closure this indecomposable summand splits as a sum of different twists of the motive of the point 2 n −2 i=0 M(pt){i}, and if D is even, then one of these twists appears twice.
Note that the last part of this theorem is important due to the periodicity of Tate motives for the n-th Morava K-theory with period 2 n − 1. Namely, one has The above theorem is a nice illustration of a general principle that the Morava Ktheories provide a sequence of deformations from K 0 to the Chow theory.For example, in the odd-dimensional case the rank of the indecomposable summand in the statement of the above theorem is (when n is growing) 3, 7, 15, . . .and eventually stabilizes at 2 ⌊log 2 (D+1)⌋ − 1, D + 1, D + 1, . . .when the motive of the whole quadric becomes indecomposable as it should also be according to results of Karpenko and Vishik for Chow motives.
The proof of the above theorem uses techniques of [PS21] and, in particular, relies on the explicit computations of the A * (SO m )-comodule structure of A * (Q) where m = dim Q + 2, A * = K(n) * , and Q is a split quadric.
1.4.Acknowledgement.We would like to thank Alexey Ananyevskiy for his useful suggestions and the anonymous referee for a careful reading of the preliminary version of this article and for his valuable comments.
The first-named author was supported by the DFG project AN 1545 "Equivariant and weak orientations in the motivic homotopy theory".The first-named and the secondnamed authors of the article were supported by the SPP 1786 "Homotopy theory and algebraic geometry" (DFG).The third-named author was supported by the BASIS foundation grant "Young Russia Mathematics".A part of this work was written when the second-named author was in St. Petersburg University supported by the BASIS foundation grants "Young Russia Mathematics" and the Ministry of Science and Higher Education of the Russian Federation, agreement No. 075-15-2022-287.
The results of Section 4-5, 7 constitute the major part of the second author's Ph.D. Thesis [Lav, Chapter I, II], however, the proofs in this paper are simplified and the exposition is clarified.
2. Background on oriented cohomology theories 2.1.Oriented cohomology theories.In the present article we always work over a field k of characteristic 0 and Sm k denotes the category of smooth quasi-projective varieties over k.We usually denote Spec k by pt.
Consider an oriented cohomology theory A * in the sense of Levine-Morel [LeMo] over k.Each oriented cohomology theory is equipped with a (commutative and onedimensional) formal group law F A , e.g., for A * = CH * the corresponding formal group law is additive, and for Grothendieck's K 0 [β ±1 ] the corresponding formal group law F K 0 (x, y) = x + y − βxy is multiplicative.
Levine and Morel constructed a theory of algebraic cobordism Ω * , which is a universal oriented cohomology theory.They also proved that Ω * (pt) is isomorphic to the Lazard ring L, and the formal group law F Ω is the universal formal group law [LeMo,Theorem 1.2.7].As a consequence, for any commutative Z-graded ring R, and any formal group law F (x, y) homogeneous of degree 1 as an element of R [[x, y]], there exists an oriented cohomology theory A * , more precisely, such that A * (pt) = R, and the corresponding formal group law F A (x, y) is equal to F (x, y).Such a theory A * is called free.
Free theories keep many properties of the algebraic cobordism, for instance, they are generically constant in the sense of [LeMo,Definition 4.4.1],see [LeMo,Corollary 1.2.11], and satisfy the localization property [LeMo,Definition 4.4.6,Theorem 1.2.8].Free theory A * also satisfies the following identity, which we call the Normalization identity following [Pa02,Theorem 1.1.8]: for any smooth divisor ι : D ֒→ X, and L(D) the corresponding line bundle as in [Har,Chapter II,Proposition 6.13], see [LeMo,Proposition 5.1.11].Actually, we will use it only for D a hypersurface in P n with L(D) = O(d), and A * = Ω * the algebraic cobordism theory.
2.2.Topological filtration.Assume that an oriented cohomology theory A * satisfies the localization property [LeMo,Definition 4.4.6],i.e., is exact for a closed subvariety i : Z → X of a smooth variety X with the open complement j : U → X (see [LeMo,Remark 2.1.4]for the definition of A * (Z) for Z not smooth).Then the two possible definitions of the topological filtration coincide: In fact, we will only consider free theories A in the present article.
We denote gr * τ A * (X) the associated graded ring of A * (X), and we consider the canonical map ρ : A * (pt) ⊗ Z CH * (X) ։ gr * τ A * (X) obtained by the universality of the Chow theory in the class of theories with the additive formal group law [LeMo,Theorem 4.5.1].The map ρ sends the class a ⊗ [Z] for a ∈ A −s (pt), i : Z ֒→ X a closed integral subscheme of codimension p, to the class of a • (i • f ) A Z in gr p τ A p−s (X), where f : Z → Z is a projective birational morphism with Z smooth, cf.[LeMo,Lemma 4.5.3] and [LeMo,Corollary 4.5.8].The map ρ is surjective for any free theory A * (in fact, for much more general A * , see [LeMo,Theorem 4.4.7]).
If A * is a free theory such that A >0 (pt) = 0, then the topological filtration on A * (X) is defined purely in terms of the structure of an A * (pt)-graded module.Namely, [LeMo,Theorem 4.5.7] and [Vi15,Section 4] for the case of algebraic cobordism).In particular, τ s A * (X) coincides with the ideal of A * (X) generated by all homogeneous elements of codimension at least s.
Remark.Observe that our choice of gradings on the associated graded ring coincides with the one in [Vi15, Section 4], but Levine and Morel in [LeMo,Subsection 4.5.2]denote by Gr s A p−s (X) what we denote by gr p τ A p−s (X).2.3.Morava K-theory.In the article we use notions from the theory of formal group laws (see [Rav, Haz]).For a prime p consider the universal p-typical formal group law F BP defined over the ring V ∼ = Z (p) [v 1 , v 2 , . ..], see [Rav,Theorem A2.1.25](here [Rav,Theorem A2.2.3] (in this situation v k are often called Hazewinkel's generators).Then choosing any n, and sending v k to 0 for k = n we obtain a formal group law with the logarithm where v n has degree 1 − p n .We call the corresponding free theory The term "Morava K-theory" can denote a family of free theories, as in [Se17,Se18,SeS].In the present work we prefer to use it, in contrast, only for the theory chosen above as in [PS14,PS21].
It is sometimes also convenient to consider the corresponding connective version of the Morava K-theory defined as a free theory with the same formal group law and another coefficient ring, or a version with coefficients in a Z (p) -algebra R, n ] , and, therefore, use the notation K(n) * (−) for it for shortness.We remark that in the present article we usually consider Morava K-theories for the prime p = 2 because we are mainly interested in quadrics.
We remark finally that the algebraic Morava K-theory as conjectured by Voevodsky in [Vo95] or as constructed in [LeTr] is a bi-graded "big" theory, and in the present article we only consider oriented cohomology theories in the sense of [LeMo], called sometimes "small".Our (small) Morava K-theory is the (2 * , * )-diagonal of the "big" theory of [LeTr], as shown in [Le09].
2.4.Cellular varieties.For an oriented cohomology theory A * we consider the category of (effective) A-motives defined as in [Ma68].In particular, the morphisms between two smooth projective irreducible varieties in this category are given by A dim Y (X × Y ).The motive of a smooth projective variety X is denoted by M A (X) = M(X), and the Tate twist of M is denoted by M{n}.We call twisted motives of the point the Tate motives, and we say that the motive M(X) of a smooth projective variety X is split, if it is isomorphic to a direct sum of Tate motives.
We call a smooth projective variety X cellular if there exists a filtration of X by closed subschemes such that X i \ X i+1 is a disjoint union of affine spaces for each i.By [ViYa,Corollary 2.9] (and by the universality of the algebraic cobordism) the A-motive M A (X) of a cellular variety X is split.
For X cellular and for any smooth projective Z, we can obtain from (5) a similar filtration for X ×Z, and deduce by the same kind of arguments that the motive of X ×Z decomposes to a sum of twisted motives of Z, cf.[NeZa].This implies, in particular, that the map given by x⊗z → x×z is an isomorphism.We refer to this fact as the Künneth formula.
Natural examples of cellular varieties are split projective homogeneous varieties.
2.5.The Rost nilpotence principle.Let L/k be a field extension, and consider the extension of scalars map res A L/k for a free theory A * , see [LeMo,Example 1.2.10] and [GiVi,Example 2.7].The map res A L/k can be extended to the category of A-motives, and for M ∈ Mot A we write M L for res A L/k M. We say that the Rost nilpotence principle holds for M if for every field extension L/k the kernel of the map consists of nilpotent elements.
The Rost nilpotence principle is proven for the Chow motives of projective quadrics by Rost in [Ros].Alternative proofs of this result are given in [Vi98] and [Br03].The Rost nilpotence principle for Chow motives of projective homogeneous varieties is proven in [CGM] and [Br05], for generically split Chow motives of smooth projective varieties in [ViZa], and for Chow motives of surfaces in [Gi10,Gi14].By [ViYa,Corollary 2.8] the Rost nilpotence principle holds in these cases for algebraic cobordism or connective K-theory motives as well.
A recent result of Gille-Vishik [GiVi,Corollary 4.5] asserts that the Rost nilpotence principle holds for the motive M(X) of every projective homogeneous variety X for a semisimple algebraic group (e.g., for the motive M(Q) of any smooth projective quadric Q) and for every free theory A * .
2.6.The Krull-Schmidt Theorem.We say that the Krull-Schmidt Theorem holds for a motive M if for every two decompositions of M into finite direct sums of indecomposable motives Recall that a (non-commutative) ring S is local if for all a, b ∈ S with a + b = 1 it follows that at least one of a, b is invertible.The following proposition is a useful tool to prove the Krull-Schmidt Theorem [Bas,Chapter I,Theorem 3.6].
Proposition 2.1.Assume that an additive category C is idempotent complete.Let A i , B j be objects of C with local endomorphism rings and In particular, if any indecomposable direct summand of a motive M has a local endomorphism ring, we conclude that the Krull-Schmidt Theorem holds for M.This is the case, e.g., when the theory A * is free and has the property that A * (pt) is a K-algebra over some field K, all A k (pt) are finite dimensional vector spaces over K (e.g., for A * = K(n) * ), and M = M(X) for a smooth projective homogeneous variety X.Indeed, since M(X k ) is split, for an indecomposable (non-zero) summand N of M(X) the ring S = res k/k End(N ) is a finite-dimensional algebra over K. Since the Rost nilpotence principle holds for M(X), we conclude that S = 0, and S is indecomposable by [Bas,Chapter III,Proposition 2.10].This implies that S is local [Lam,Corollary 19.19].
Finally, apply the Rost nilpotence principle again to conclude that End(N ) is local itself.
Consider a twisted form X of a cellular variety, i.e., such that X L is cellular for a field extension L/k, and consider a free theory A * such that the Rost nilpotence principle and the Krull-Schmidt Theorem hold for the A-motive of X. Assume additionally that the A-motive of a point is indecomposable.Then for every summand N of M A (X) its image under the restriction map res L/k is a sum of Tate motives.The number of these Tate motives is called the rank of N .For N = 0 the Rost nilpotence principle implies that the rank of N is greater than 0. Clearly, using induction on the rank, one can show that the motive of X admits a decomposition into a finite direct sum of indecomposable summands.
We remark that the Krull-Schmidt Theorem for A * = CH * with integral coefficients is proven for motives of quadrics in [Vi04], cf. also [Hau].However, the counterexamples [ChMe,Example 9.4] and [CPSZ,Corollary 2.7] provide (twisted) projective homogeneous varieties for which the Krull-Schmidt Theorem fails (see also [SZh]).In [ChMe] Chernousov and Merkurjev proved the Krull-Schmidt Theorem for motives of projective varieties homogeneous under a simple group or an inner form of a split reductive group and A * = CH * (−; Z (p) ).

Background on algebraic groups
3.1.Schubert calculus.Let G denote a split semisimple group of rank l over k, T ∼ = (G m ) × l a fixed split maximal torus of G, and B a Borel subgroup containing T .Let Π = {α 1 , . . ., α l } be the respective set of simple roots of the root system Φ of G.For a subset Θ ⊆ Π consider the corresponding parabolic subgroup P Θ in G, generated by B and by the root subgroups U −α for α ∈ Θ.In particular, B = P ∅ , and the maximal parabolic subgroups are P i = P Π\{α i } .
For a parabolic subgroup P in G we denote its Levi part by L, its unipotent radical by U, and the opposite unipotent radical by U − .We also denote the commutator subgroup of L by C = [L, L].In the particular case G = SO m , P = P 1 , and The Weyl group W of G is generated by simple reflections s i = s α i corresponding to simple roots α i ∈ Π.We denote by l(v) the length of v ∈ W in simple reflections.The longest word of W is denoted by w 0 .
For P = P Θ consider W P = s i | i ∈ Θ and let us denote Then the map W P × W P → W sending a pair (u, v) to the product uv is a bijection and l(uv) = l(u) + l(v) [BjBr,Proposition 2.2.4].This immediately implies that W P is a set of the minimal representatives for the elements of W/W P .
Consider now the case A * = CH * , and let X w denote the classes [BwB/B] of Schubert varieties in CH * (G/B).Observe that these varieties are not necessarily smooth.It is well-known that {X w | w ∈ W } is a free basis of CH * (G/B).It can be also convenient to replace B by the corresponding opposite parabolic subgroup B − in the Bruhat decomposition, and introduce notation Z w for the classes of B − wB/B in CH(G/B).Observe that G acts on G/B with left translations in such a way that w 0 (BwB/B) = B − w 0 wB/B as subvarieties, and taking closures we obtain By [Gro,Lemma 1] the induced action of G on CH(G/B) is trivial, and, therefore, X w = Z w 0 w (see also [Dem,Proposition 1]).
More generally, the classes X w = [BwP/P ] for w ∈ W P form a free basis of CH * (G/P ).In the dual notation we set Z w = X w 0 ww P ∈ CH * (G/P ), where w P denotes the longest element of W P .Then the pullback map along the natural projection π : G/B → G/P sends Z w ∈ CH * (G/P ) to Z w ∈ CH * (G/B) for all w ∈ W P identifying CH * (G/P ) with the free abelian subgroup of CH * (G/B) generated by {Z w | w ∈ W P }, see also [GPS,Section 5.1].
The next theorem is proven in [CPZ,Theorem 13.13].
Theorem (Calmès-Petrov-Zainoulline).For any free theory A * there exists a free A * (pt)-basis of A * (G/B) consisting of homogeneous elements w coincide with X w defined above.More precisely, ζ w are given by resolutions of singularities of Schubert varieties BwB/B.

The characteristic map.
Recall that A * (BT ) for a classifying space BT of a split torus T is isomorphic to [CPZ,Definition 2.4], see [CZZ,Theorem 3.3], where F A is the formal group law of the theory A * .We fix a basis χ 1 , . . ., χ l of M and write x χ i for x i ∈ A * (BT ).For an arbitrary λ ∈ M we have an element x λ ∈ A * (BT ) defined according to the rule x λ+µ = F A (x λ , x µ ).
Following [LeMo,Remark 1.2.12] for any free theory A * and any smooth irreducible variety X with the function field K we consider a map deg A defined as the restriction to the generic K-point of X K We remark that the map deg A canonically splits by For an augmented algebra A we denote A + its augmentation ideal and we say that the sequence of augmented algebras is a right exact sequence of augmented L-algebras, where the first arrow is the characteristic map c = c Ω as defined in [CPZ,Definition 10.2] and the second one is the pullback along G → G/B.This, obviously, gives a similar sequence for any free theory A * .Observe that the above sequence can be continued to the left as in [CZZ,Theorem 10.2].
3.3.BGG-Demazure operators.Divided difference operators ∆ i were defined independently by Bernstein-Gelfand-Gelfand [BGG] and Demazure [Dem] to describe the characteristic map c CH .These operators were generalized to arbitrary oriented cohomology theories in [CPZ].
Observe that the action of the Weyl group W on the group M of characters of T induces an action of W on A * (BT ) = A * (pt)[[M]] F A according to the rule s α (x λ ) = x λ−α ∨ (λ)α .We now define divided difference operators for Ω * by the formula and for any other theory by the change of coefficients [CPZ,Definition 3.5].
We have the following description of c CH in terms of divided difference operators.For a homogeneous u ∈ CH * (BT ) of degree s one has by [Dem,4.5.Théorème 1 (b)] Further, for a minimal parabolic subgroup P {α i } consider the natural projection and define an operator [CPZ,Theorem 13.13].
It also follows, e.g., from [CPZ,Lemma 13.3], that the sequence ∆ i 1 •. ..• ∆ i k does not depend on a reduced decomposition of w = s i 1 . . .s i k and equals 0 if it is not reduced.We write ∆ i for ∆ i if it does not lead to a confusion.
We can also describe the action of ∆ i on the Schubert basis.Take w ∈ W and assume that l(ws i ) = l(w) + 1 for some i.Then by [CPZ,Lemma 13.3] and [CPZ,Lemma 12.4 This means in particular that every X w can be obtained from ±pt = ±X 1 by a sequence of ∆ i and, moreover, since w 0 is greater than w in the weak Bruhat order for every w ∈ W \{w 0 } [BjBr, Definition 3.1.1(i)], there always exists a sequence (i 1 , . . ., i k ) such that Moreover, if we apply the same operator ∆ w −1 w 0 to another . .s js is not a reduced expression.In this sense ∆ w and X w are dual to each other.
Remark.The proof of [CPZ,Theorem 13.13] contains a misprint, more precisely, A Iw (z 0 ) should be changed to A I rev w (z 0 ).
Similar results are obtained in [CPZ] for any theory A * .We introduce the notation and for any simple root α i ∈ Π ⊂ M we denote As above, for a minimal parabolic subgroup P {α i } consider the natural projection π i : G/B → G/P {α i } and define the Demazure operator and the action of simple reflections on A * (G/B) by By [PS21,Lemma 3.8] we have the following Leibniz rule for Demazure operators: 9) and s i are A * (pt)-algebra homomorphisms defining an action of the Weyl group W on A * (G/B).Moreover, c A respects the action of the Weyl group (see the proof of [PS21, Lemma 3.8]).
3.4.Torsors.Let G be a smooth affine algebraic group over the base field k of characteristic 0. We identify the Galois cohomology group with the set of isomorphism classes of G-torsors over k.The natural homomorphism G → Aut(G) induces a map on Galois cohomology and, therefore, each G-torsor E defines an element ξ ∈ H 1 k, Aut(G) and the corresponding twisted form E G = ξ G of G, see [Ser] or [KMRT, Chapter VII].
For a representation ρ : G ֒→ GL m of G we call GL m → GL m /ρ(G) a standard classifying G-torsor, and its fiber over the generic point of X = GL m /ρ(G) we call a standard generic G-torsor over k(X) or just a generic torsor (see [GMS,Chapter 1,§5]).
Consider a (split) group G = SO m over k.Let E G be a generic group, i.e., the twisted form of G corresponding to a generic G-torsor E, see [PS17, Section 3].Strictly speaking, E G is defined over a field extension L of k, but we will denote SO m over L or L by the same letter G.For each parabolic subgroup P of G there exists a unique E G-homogeneous variety X which is a twisted form of G/P , see [ChMe, Section 1.1], moreover, X = E/P .For a maximal parabolic subgroup P = P 1 corresponding to the first simple root in the Dynkin diagram (according to the enumeration of Bourbaki) we refer to E/P 1 as the generic quadric of dimension D = m − 2. We also remark that E/P 1 is generic with respect to SO m , and there exist slightly different notions of generic quadric, which are compared in [Ka18].
Until the end of the section we reserve the notation Q for the split quadric G/P 1 .
For any G-torsor E and any free theory A * there exists a bi-algebra H * of [PS21, Definition 4.6] defined as an augmented algebra quotient of A * (G) by the image of A * (E) and a co-action of H * on A * (G/P ) for any parabolic subgroup P of G [PS21, Definition 4.10].It is shown in [PS21,Remark 4.8] that for A * = CH(−; F p ) the bialgebra H * carries essentially the same information as the J-invariant [PSZ,Vi05].We will be interested only in the case of a generic torsor E. In this case H * = A * (G) for every free theory A * [PS21, Example 4.7] and the co-action is given by the pullback map along the left multiplication by elements of G on G/P composed with the Künneth isomorphism [PS21,Lemma 4.3].
Consider smooth projective homogeneous varieties G/P , G/P ′ ∈ Sm k and let X and X ′ be the respective E-twisted forms.For an element α ∈ A * (X × X ′ ) define the realization map , where m α stands for the multiplication by α, and pr X , pr X ′ are the natural projections from X × X ′ to X or X ′ respectively.By [PS21, Theorem 4.14], is a homomorphism of A * (G)-comodules.This fact allows to consider an (additive) functor from the full subcategory of Mot A generated by the motives of smooth projective homogeneous E G-varieties to the category of graded A * (G)-comodules [PS21,Remark 4.15].In particular, the motivic decomposition of E/P gives us a decomposition of A * (G/P ) into a sum of A * (G)-comodules (as above, with some abuse of notation, we usually write G/P for X k ).

Cobordism ring of a split quadric
4.1.Multiplication in Ω * (Q) of a split quadric Q.For any smooth projective cellular variety X and an oriented cohomology theory A * it is easy to describe the structure of A * (X) as an abelian group.For a split quadric Q we will describe the multiplication in terms of the formal group law.For certain theories A * , e.g., for the Morava K-theory K(n) * with F 2 coefficients, the multiplication table has a very simple description.We also describe an explicit formula for the pushforward map along the structure morphism from Q to the point.
Let us denote H = H P n = c Ω 1 O P n (1) and recall that the powers of H k for 0 k n form a free basis of Ω * P n by the projective bundle formula.Moreover, H k coincide with the classes of projective subspaces of smaller dimensions in P n .
We follow the notation of [EKM,Chapter XIII,§ 68].Let Q be a smooth projective quadric of dimension D over k defined by the non-degenerate quadratic form ϕ on a vector space V of dimension D + 2. We write D = 2d for D even or D = 2d + 1 for D odd.We assume that the quadric Q is split, i.e., ϕ has the maximal possible Witt index d + 1.Let us denote by W a maximal totally isotropic subspace of V and by W ⊥ its orthogonal complement in V (for D even W ⊥ = W and for D odd W is a hyperplane in W ⊥ ).We denote by P(V ) the projective space of V which contains Q as a hypersurface.Then P(W ) is contained in Q and Q \ P(W ) → P(V /W ⊥ ) is a vector bundle We denote ).The elements l i for 0 i d and the powers h k of h for 0 k d form a free basis of Ω * (Q) over L, cf., e.g., [ViYa,Corollary 2.9] or [NeZa,Theorem 6.5].
Proposition 4.1.In the above notation the multiplication table of Ω * (Q) is determined by the identities where ] is the multiplication by 2 in the sense of the universal formal group law.
Proof.Observe that the basis elements l i and h k are homogeneous elements of Ω * (Q) of degree D − i and k, respectively, and L is graded by non-positive numbers.Therefore, in the decomposition of h d+1 as a sum of basis elements we necessarily have c k = 0 for all k by the degree reasons, i.e., h d+1 lies in the image of i Ω .Since Ω * P(W )) injects into Ω * P(V ) we can pushforward the above equality to Ω * P(V ) to determine a i .Let I : Q ֒→ P(V ) denote the inclusion, then The projection formula implies that the left hand side is equal to H d+1 I Ω (1) and putting D = Q in (2) we have Then, clearly, a D−d−i = b i .The same argument proves (11).
Finally, for the degree reasons we only need to consider (12) for D even and i = j = d, where l 2 d = al 0 for some a ∈ Z. Since Ω D (Q) ∼ = CH D (Q) by [LeMo,Lemma 4.5.10],we can determine a modulo L <0 .But for the Chow theory the result is well-known, see, e.g., [EKM,§68].
Our initial computation of the above multiplication table was more awkward and the above simplification is suggested by Alexey Ananyevskiy.4.2.Pushforwards along structure morphisms.For X ∈ Sm k let χ : X → pt be the structure morphism and let us denote by [X] ∈ L the pushforward of 1 ∈ Ω * (X) along χ, i.e., [X] = χ Ω (1 Ω * (X) ).We will describe χ for X = Q a split quadric.
Proposition 4.2.In the notation of Proposition 4.1 we have the following formulae: Proof.Identity ( 14) is obvious and (15) follows from (2) with the use of the projection formula Obviously, the results of Propositions 4.1 and 4.2 can be applied to any oriented cohomology theory by the universality of algebraic cobordism.4.3.Morava with finite coefficients.Let log Ω (t) denote the logarithm of the universal formal group law over L ⊗ Q.The identity is known as the Mishchenko formula [Shi,Theorem 1].

Stabilization of the Morava K-theory for special orthogonal groups
In the present section we prove that K(n n ] stabilizes for m large enough.We will use in the proof the multiplication table for the Morava K-theory of a split quadric from Proposition 4.3 (see Subsection 5.5).

5.1.
The plan of the proof.In the present section and in Sections 6-7 we only work with the Morava K-theory with coefficients in F 2 [v ±1 n ] and, therefore, write K(n n ] for shortness.We are interested in the structure of K(n) * (SO m ) for a group variety SO m .We will prove the following Theorem 5.1.For m 2 n+1 + 1 the pullback map along the natural closed embedding induces an isomorphism of rings Recall that for a split reductive group G with a split maximal torus T the sequence n ]-algebras (see Subsection 3.2), where the first arrow is the characteristic map and the second one is the pullback along G → G/B.
On the one hand, K(n -module with a basis given by resolutions of singularities ζ w of Schubert varieties BwB/B for w in the Weyl group W of G, see Subsection 3.1, and we have an explicit description of K(n) * (BT ) as a ring On the other hand, the characteristic map c K(n) : K(n) * (BT ) → K(n) * (G/B) has a rather complicated form after these identifications, cf.Subsection 3.3.Instead of working with c directly we will deduce Theorem 5.1 from Propositions 5.2 and 5.3 below.The proof of Proposition 5.2 will follow the strategy of [PS21, Lemma 6.2] for part a) and [PS12] for part b).The proof of Proposition 5.3 will be based on the divided difference operators ∆ i (see Subsection 3.3).Proposition 5.3.Let m 2 n+1 + 1, and let Q denote a split projective quadric of dimension m − 2. Then the natural map The rest of this section is devoted to the proofs of the above propositions and is organized as follows.We start with an analogue of the Leray-Hirsch Theorem for free theories.Next, we use it to prove Proposition 5.2, and after that we investigate the divided difference operators and prove Proposition 5.3 and, finally, deduce Theorem 5.1.

The Leray-Hirsch Theorem. The required isomorphism
A * (P ) ∼ = A * (G) ⊗ A * (G/P ) A * (pt) (24) of Proposition 5.2 a) can be deduced from the A * (G/P )-module isomorphism obtained in [DoZh], see also [EdGr, DrTy].However, this argument does not prove that ( 24) is an isomorphism of rings, not just of A * (pt)-modules.To fix this, we will show that ( 24) is actually induced by a pullback map.
In the notation of Proposition 5. Now we will prove an analogue of Lemma 5.4 for any free theory.As a matter of fact, we deduce it from the Chow case with the use of the graded Nakayama Lemma.
Lemma 5.5 (Graded Nakayama's Lemma).Let M and N be graded L-modules, let N be finitely generated, and let f : M → N be a homomorphism of L-modules which preserves grading.Then a) Notation (Subring R A and free submodule V A ).For an oriented cohomology theory A * we denote by R A the image of A * (G/P ) in A * (G/B) under the pullback map and by V A the (free graded) A * (pt)-submodule of A * (G/B) generated by ζ A w 0 w , w ∈ W P , cf.Subsection 3.1.
Lemma 5.6.a) The map f : R A ⊗ A * (pt) V A → A * (G/B) which sends p ⊗ q to pq is surjective.b) The pullback map along the closed embedding Proof.Since V A and A * (P/B) are free A * (pt)-modules of the same rank, it is sufficient to prove that ι A | V A is surjective to get b).Therefore, we can assume that A * = Ω * .Since the L-module M = R Ω ⊗ L V Ω is graded, f preserves grading, and f ⊗ L Z is surjective by Lemma 5.4, we can apply Lemma 5.5 to get a).Similarly, we can apply Lemma 5.5 to 5.3.Proof of Proposition 5.2.First, we use the argument of [PS12,Lemma 2.2] to prove Proposition 5.2 b), i.e., to show that the natural pullback maps define isomorphisms Proof of Proposition 5.2 b).Since P → L is an affine fibration, and the inclusion L ֒→ P is its zero section, we have A * (P ) ∼ = A * (L).Next, for T ′ = T ∩ C and and use that the rows are exact.
Next, we prove the part a) of Proposition 5.2, i.e., establish an isomorphism of rings The argument is essentially the same as in [PS21,Lemma 6.2].
Proof of Proposition 5.2 a).First, we consider the surjective map from Lemma 5.6 a) followed by the pullback map: and tensor this sequence with A * (pt) over A * (G/P ).We obviously have and since P/B → G/B → G/P factors through a point, Therefore, we obtain the sequence and the composite map is an isomorphism by Lemma 5.6 b).Now, we use A * (P/B) ∼ = A * (L/B) and tensor the above sequence (of isomorphisms) with A * (pt) over A * (BT ).We have and similarly A * (L/B) ⊗ A * (BT ) A * (pt) = A * (L).Since the characteristic map commutes with pullbacks, we can conclude that the isomorphism A * (L) ∼ = A * (G) ⊗ A * (G/P ) A * (pt) is actually induced by the pullback along the inclusion L → G and, therefore, preserves the ring structure.We finish the proof with the use of Theorem 5.2 b).τ Ω(G/B) * of Ω * (G/B) with respect to the topological filtration (see Subsection 2.2).Moreover, the action of these induced operators can be described in terms of the action of the usual operators ∆ CH i on CH(G/B) (see Lemma 5.7 below).This observation will be very helpful in the next subsection.
Recall that τ 1 Ω * (G/B) = Ker(deg Ω ) is an ideal in Ω * (G/B) generated by the homogeneous elements of positive codimension, cf.[LeMo,Theorem 1.2.14], and τ s Ω * (G/B) is an ideal generated by the homogeneous elements of codimension at least s, see Subsection 2.2.Since the topological filtration respects the multiplication, clearly, Below for a sequence of indices I = (i 1 , . . ., i k ) we denote by |I| = k its length, and for a homogeneous element x ∈ Ω p (G/B) we denote by |x| = p its codimension.Proof.We can assume that I = {i}.Then for a homogeneous x ∈ Ω * (G/B) we have ).In particular, ∆ gr i is well-defined.To obtain the second statement, take x ∈ CH * (G/B) and let x be its lift in Ω * (G/B).Recall that ρ is L-linear and commutes with pullbacks and pushforwards.Then to the class of this pre-image in gr * τ Ω * (G/B).However, since κ i from ( 8) in fact lies in Let us briefly describe the plan of the proof realized below.We want to show that the natural map Let us denote N = 2 n for m even and N = 2 n − 1 for m odd.In our assumption m 2 n+1 + 1, we have N ≤ d.Recall from Proposition 4.3 that the ring K(n) * (Q) is generated as an K(n) * (pt)-algebra by two elements: h and l = l d connected by the following equation: To obtain the claim it is sufficient to show that h and l are mapped to zero in K(n) * (G).
Consider the natural map and let I = I K(n) denote the kernel of the map Then the claim is equivalent to the fact that H := π K(n) (h) and L := π K(n) (l) lie in I.However, I coincides with the ideal generated by (non-trivial) "generically rational" elements, i.e., the elements that lie in the image of the natural map where E ∈ H 1 (k, G) denotes the generic torsor (see Lemma 5.10 below for the precise statement).In particular, H lies in I, and it remains to show that L also does.
With this end we apply π to (25) and we claim that there exists a BGG-Demazure operator ∆ More precisely, using the Leibniz rule (9) we will show that ∆ K(n) and we will choose I 0 in such a way that a = ∆ K(n) Now we proceed with the proof of Proposition 5.3.
For A * = CH * the pullback map along π : G/B → G/P can be described explicitly, in particular, Indeed, in the notations of Section 3.1 the generator h equals Z s 1 and the pullback π CH always maps Z w ∈ CH * (G/P ) to Z w ∈ CH * (G/B) for every w ∈ W P .Moreover, the Chow ring structure CH * (G/P ) is described in [EKM,Propositions 68.1 and 68.2] and the combinatorial structure of W P immediately follows from [CGM,Example 9.2].
Recall that for I = (i 1 , . . ., i k ) we denote by Remark 5.8.In fact, we do not use explicit formulae for π CH (h N ) and for I 0 in the subsequent arguments.We use only the existence of Lemma 5.9.In the above notation, for a free theory A * let us denote Let E be a generic G-torsor, see Section 3.4, and let us denote by A * (E/B) the image of the restriction map A * (E/B) → A * (G/B).We call elements of this set rational.Since ∆ i can be defined on A * (E/B) by the same formula (8) as for A * (G/B), we conclude that ∆ i applied to a rational element is again rational.
Lemma 5.10.In the notation above, the ideal I A coincides with the ideal generated by the set A * (E/B) ∩ Ker(deg A ).

Now we can prove the following
Lemma 5.11.In the above notation, for any free theory A * , any set of indices I, and any s > |I| the element we conclude that π Ω (h s ) is rational, and, therefore, ∆ Ω I π Ω (h s ) is rational as well.On the other hand, since s > |I|, we see that ∆ Ω I π Ω (h s ) lies in Ker(deg Ω ) = τ 1 Ω * (G/B) by Lemma 5.7.Finally, we obtain the following Lemma 5.12.Let A * be a free theory, and let us denote In the proof we write ∆ for ∆ A .We show by induction on |I| ≤ N that ∆ I (H N L) − ∆ I (H N )L is a sum of elements of the following types: (1) an A * (G/B)-multiple of c A (x λ )∆ I (H N ) for some λ ∈ M; (2) an A * (G/B)-multiple of ∆ I ′ (H N ) for |I ′ | < |I|.Since c A (x λ ) ∈ I A , with the use of Lemma 5.11 we obtain the claim.
For the induction step we apply ∆ i to each summand.First, for a ∈ A * (G/B) we have where ), and we obtain summands of two types. Next, , we obtain two summands of the second type.
Finally, we have to apply where , and the proof is finished.Now we can finish the proof of Proposition 5.3, i.e., we can show that the map Proof of Proposition 5.3.Let h, l = l d ∈ K(n) * (Q) be as in Proposition 4.3, and π : G/B → Q denote the natural projection.In the notation above, we will prove that H = π K(n) (h) and L = π K(n) (l) belong to I K(n) .Indeed, H ∈ I K(n) by Lemma 5.11.Next, applying π K(n) to ( 25) we obtain the equation On the one hand, ∆ K(n) I 0 (H d+1 ) lies in I K(n) by Lemma 5.11.On the other hand, ∆ K(n) for some invertible a ∈ K(n) * (G/B) by Lemmata 5.12 and 5.9.This implies that L lies in I K(n) .
Recall that I K(n) coincides with the kernel of the map by ( 6).Then h, l ∈ K(n) * (Q) go to 0 in K(n) * (G).However, K(n) * (Q) is generated by h and l as algebra (see Proposition 4.3), and, therefore, the map Proof of Theorem 5.1.Let n be a natural number, G = SO m for m ≥ 2 n+1 + 1, P = P 1 the maximal parabolic subgroup corresponding to the first root, L the corresponding Levi subgroup (see Section 3.1).Then Q = G/P is a split m − 2 dimensional quadric, and Using Proposition 5.2 we conclude that where the isomorphism is induced by the pullback along the natural inclusion.
On the other hand, by Proposition 5.3 we conclude that the natural map K

Computation of the Morava K-theory ring for special orthogonal groups
In this section as previously we denote by K(n) * (−) the Morava K-theory modulo 2, in particular, K(n n ].We will compute the ring K(n) * (SO m ).
6.1.Chow ring of the special orthogonal group.The structure of the Chow ring of the group variety SO m modulo 2 is well-known (see [Kac, Table II]): . ., e s ]/(e 2 i − e 2i ), where m = 2s + 1 or m = 2s + 2, e i has degree i, and e 2i stands for 0 if 2i > s.The above identity can be written in a slightly different form: Moreover, the multiplication of SO m induces a Hopf algebra structure on CH * (SO m ; F 2 ) and the elements e 2i−1 are primitive (or Lie-like) for the comultiplication ∆, i.e., ∆(e 2i−1 ) = 1 ⊗ e 2i−1 + e 2i−1 ⊗ 1.
These notations do not contradict each other, and it is sometimes convenient to write e 2 k (2i−1) instead of e 2 k 2i−1 .
6.2.Hopf theoretic lemmata.The results of this section are based on the general statements about Hopf algebras.Consider a graded Hopf algebra over F p : . ., x p kr r ) where x i are primitive of positive degree, k i > 0. By an index set in this section we mean an r-tuple I = (d 1 , . . ., d r ) with 0 We call it the standard basis.
The following technical lemma is used in the proof of Proposition 6.2.
Lemma 6.1.In the above notation a) All primitive elements of H are equal to h x p q h j h for some j h and q h < k j h .b) Assume that for P ∈ H, Q ∈ H ⊗ Fp H and k ∈ N the equality Proof.Claim "a)" follows from the description of primitive elements in the tensor product of Hopf algebras and the easy case r = 1.In fact, it also follows from the argument below.
To obtain "b)" decompose P in the standard basis x I , and let f be the minimal possible non-negative integer such that for some tuple I = (d 1 , . . ., d r ) with d i = p f q, p ∤ q, the element x I appears in the decomposition of P .We can assume that f < k, since otherwise P = S p k for some S and the claim follows.
The primitivity of x i implies that ∆(x I ) has the monomial x p f i ⊗ x J with a non-zero coefficient in its decomposition for some J.Moreover, this monomial does not appear in the decomposition of ∆(x I ′ ) for I = I ′ .
However, P ⊗1+1⊗P +Q p k cannot contain an element x p f i ⊗x J in its decomposition for x J = 1, since f < k.This allows us to conclude that x I = x p f i .Then we can change P by P − x p f i and continue by induction.
The following purely Hopf theoretic proposition is a key ingredient in our argument.
Proposition 6.2.Let H be a Z-graded Hopf algebra over F p [v n ] which is free and finitely generated as an F p [v n ]-module (as usual, v n has degree 1 − p n ).Assume that H/v n has homogeneous generators x 1 , . . ., x r of positive degrees such that they all are primitive for the Hopf algebra structure and, moreover, there exists an isomorphism for some positive numbers k i .Assume furthermore that for every pair of numbers i, j from 1 to r (in particular, these may be equal) the equation has no solutions for x > 0, 0 ≤ y < k j , x, y ∈ Z. Then there exist homogeneous lifts x i of the variables x i to H such that 1 , . . ., x p kr r ) as an algebra.
Moreover, assume that for some i there exists x i ∈ H which lifts x i and such that ∆( x i ) can be written as an First, let us choose arbitrary lifts of x i to H and denote them x i .We will modify these generators of H to obtain relations of the correct form.
By the assumptions for every i and for some a i > 0 and Q i ∈ H a relation of the following form holds in H: We will describe a procedure that at each step chooses a new lift of x i so that the value a i in the relation ( 27) increases.Since the values a i are bounded above by the fact that H k = 0 for k >> 0, the procedure will stop after a finite number of steps, i.e., x p k i i for every i will become equal to zero.When this is achieved, we claim that H ∼ = F p [v n ][x 1 , . . ., x r ]/(x p k 1 1 , . . .x p kr r ).Let R ⊂ F p [v n ][x 1 , . . ., x r ] be the ideal of the relations of H. Since H is a free F p [v n ]-module, it follows that R/v n = (x p k 1 1 , . . .xp kr r ) and we have shown that (x p k 1 1 , . . .x p kr r ) ⊂ R. Therefore, R/(x p k 1 1 , . . .x p kr r ) /v n is zero, which implies R = (x p k 1 1 , . . .x p kr r ).To choose a different lift of x i we make a "change of variables" for some S ∈ H, so that x new i satisfies the relation ( Our goal is to find S such that a new i > a i .After we have performed this change, we will denote x new i just as x i by abuse of notation.Let us choose a list of relations among (27), one for every x j .Furthermore, we can assume that for every j element Q j is non-zero, i.e., Q j is not divisible by v n .Let i be the index of a variable x i with minimal a i =: a.The relation in our list for this i can be written as follows: where a ≥ 1, R ∈ H and P is a linear combination of x I with F p -coefficients Let Q t be linear combinations of x I ⊗ x J with F p -coefficients such that We now rewrite the sum containing (Q t ) p k i using relations between the variables from the chosen list of relations.If for some t ≥ t 0 we have that (Q t ) p k i = 0, then we claim that (Q t ) H. Indeed, if we express Q t as a linear combination of x I ⊗ x J with F p -coefficients, then (Q t ) p k j goes to zero in H ⊗ Fp H only if for all non-trivial summands x I ⊗ x J of Q t either x p k j I or x p k j J is zero in H.However, this means precisely that x q l with q > p k l appears in either x p k j I or x p k j J .Therefore, we can apply one of the relations ( 27) from the chosen list and using the fact that a ≤ a l by our choice, we can replace (Q t ) p k j by v a n Q ′ for some Q ′ .Using this and plugging in relation ( 29) we can rewrite (30) for some t 1 ≥ t 0 and some R ′ as follows: On the other hand, from (29) we have needs to be divisible by v n contradicting our assumptions.
We show now that a < t 1 p k i cannot happen.Indeed, in this case we would obtain that P is primitive (and non-zero by the assumption).Moreover, relation (29) gives us that p k i deg(x i ) = deg P − (1 − p n )a.However, all primitive elements in H are of the form h x p q h j h for some j h and q h < k j h by Lemma 6.1 a), and, therefore, the equation on the degrees can be rewritten as p k i deg(x i ) = (p n − 1)a + p s h (deg x j h ) for each h.This equation has no solutions by our assumption (26).
Let us now consider the case a = t 1 p k i .Again comparing two expressions of ∆ x p k i i we see that ∆(P ) in the quotient Hopf algebra has the form P ⊗ 1 + 1 ⊗ P + Q p k i t 1 .By Lemma 6.1 b) it follows that P − S p k i is primitive for some S ∈ H.However, assumption (26) again implies that such a primitive element equals zero, i.e., P = S p k i .Now take any lift S ∈ H of S and define Then relation ( 29) can be rewritten as ( for some Q i new , so that we have performed the required modification of the variables.To finish the proof of the proposition we need to explain that if we choose x i such that ∆(x i ) is an F p [v n ]-linear combination of x b i ⊗ x c i , b, c ≥ 0, then the procedure described above does not modify x i .Indeed, in this case every Q t in (30) is divisible by x i ⊗ x i (i.e., in every monomial x b i ⊗ x c i we have b, c ≥ 1), and, therefore, rewriting (30) as described above we will get t 1 p k i ≥ 2a > a.In particular, the case a = t 1 p k i never happens, and the variable x i is not modified by our procedure.
Remark 6.3.The following example shows that there exists a graded Hopf algebra H that is free and of finite rank as an F 2 [v n ]-module for v n of negative degree and does not satisfy (26) and the claim of Proposition 6.2.Let n = 1, p = 2, and consider the following Hopf algebra: 3 ), where deg v 1 = −1, deg x i = i and the comultiplication is given by the following formulae It is easy to check that H/v 1 and H/(v 1 − 1) are not isomorphic as algebras.In other words, condition (26) in Proposition 6.2 cannot be omitted.6.3.Associated graded ring of the connective Morava K-theory.In this section we consider the connective Morava K-theory modulo p and denote it by CK(n) * (−).In particular, CK(n) * (pt) = F p [v n ] and deg v n = 1 − p n .We will work with the topological filtration τ * on the connected Morava K-theory, see Subsection 2.2.We remark that each associated graded piece of the topological filtration is itself graded, so that * τ CK(n) * (−) is a bi-graded theory, and after a change of grading it coincides with the graded ring associated with the filtration on CK cf. the analogous statement on algebraic cobordism in [LeMo,Theorem 4.5.7] and [Vi15,Section 4].
Recall that a torsion in a graded F p [v n ]-module M = (M i ) i∈Z can only be a v Z n -torsion.This simple fact implies the following result.Proposition 6.4.Let G be a split reductive group.i) The connective Morava K-theory of G has the following (non-canonical) form as a graded F p [v n ]-module: for some homogeneous x i , y j ∈ CK(n) * (G), d j ≥ 0 (and the direct sums are finite).If x i , y j denote the classes of x i , y j in the associated graded ring gr Then the canonical map from [LeMo,Corollary 4.5.8]sends x i to x i and y j to y j .
iii) The Hopf algebra structure on CK(n) * (G) induces a Hopf algebra structure on gr * τ CK(n) * (G), and the map (31) is a morphism of Hopf algebras.Proof.i) Since CK(n) * (G) is finitely generated over F p [v n ], e.g., by (6), the first claim of i) follows from the remark on torsion above.To get the second claim, recall that the topological filtration on CK(n) * (G) coincides with the filtration by the powers of and the map ρ is F p [v n ]linear and sends an element of CH * (G; F p ) to the class of its lift in CK(n) * (G).
iii) The comultiplication on gr * τ CK(n) * (G) induced from CK(n) * (G) coincides with the comultiplication induced from CH * (G; F p [v n ]) via ρ.Since the latter defines a Hopf algebra structure, iii) follows.
6.4.Computation of the associated graded ring.In the notation of the previous section, we assume additionally that p = 2 and G = SO m .
In this section we will compute gr * τ CK(n) * (SO m ) and show that CK(n) * (SO m ) is a finitely generated free F 2 [v n ]-module for m ≤ 2 n+1 .Luckily, this is precisely the range of m when the n-th Morava K-theory of SO m is not stabilised, see Theorem 5.1.
The key ingredient in the proof of this result is Vishik's theorem on the relations in algebraic cobordism [Vi15,Theorem 4.3], see Lemma 6.5 below.
Recall that Proposition 5.2 implies that for Q m−2 a split (m − 2)-dimensional quadric SO m /P 1 , and the isomorphism is induced by the pullback along the natural inclusion of SO m−2 into SO m .Moreover, Lemma 5.10 implies that the image of h ∈ CK(n) * (Q m−2 ) is trivial in CK(n) * (SO m ), and, therefore, the kernel of the pullback map is generated (as an ideal) by a single element y ∈ CK(n) * (SO m ), more precisely, by the image of l ∈ CK(n) * (Q m−2 ) where l is the class of an ⌊(m − 2)/2⌋-dimensional linear subspace.Since l 2 is divisible by h, we conclude that y 2 = 0.
Lemma 6.5.In the above notation let m ≤ 2 n+1 .Then the image y of y in gr has generator 1 in degree 0 and other generators in positive degrees with relations in positive degrees (as F 2 [v n ]-module).Moreover, the latter generators can be chosen to be the lifts of any homogeneous F 2 -basis of CH * (G; F 2 ), see [Vi15,Proof of Theorem 4.3].
In particular, an element x ∈ CK(n) i (G), i ≤ 2 n − 1, projecting to a non-trivial element in CH i (G; F 2 ), cannot be a v Z n -torsion.Next, recall that the topological filtration on CK(n) * (G) coincides with the filtration by the powers of v n .Thus, if y ∈ gr * τ CK * (n) is a v r n -torsion for some r, then v r n y = v r+1 n z for some z ∈ CK(n) * (G).However, this implies that v r n (y − v n z) = 0 contradicting the above observation (cf.Proposition 6.4 i).
As another corollary of Theorem 6.13 we also obtain a similar result about spin groups.
Corollary 6.15.For the group Spin m with m ≤ 2 n+1 , n ≥ 1, the ring K Obviously, the ideal generated by c τ 1 K(n)(BT Spin m ) coincides with the ideal generated by c(̟ l ) and , where ̟ l denotes the last fundamental weight.Our choice of e 1 in Theorem 6.10 guaranties that c(̟ l ) goes to e ⌋ for m ≥ 2 n+1 + 1.Now, the stabilization is clear, and the result for small m follows from the analogous result on CH * (Spin m ), see [Kac, Table II].
7. Application: Morava K-theory motives of projective quadrics 7.1.Rational projectors.As before we denote by K(n) * the Morava K-theory with the coefficient ring F 2 [v ±1 n ].In this section we always assume that n ≥ 2. Our objective is to describe a decomposition of the K(n)-motive M K(n) (Q) of a generic quadric Q into indecomposable summands.We start with the following proposition.
Proposition 7.1.Let Q be a smooth projective quadric (not necessarily generic) of dimension D ≥ 2 n − 1, n > 1.Then its K(n)-motive M(Q) decomposes as a direct sum of D − 2 n + 2 Tate motives and a motive N of rank 2 n for D even or 2 n − 1 for D odd.
Proof.Let us denote D ′ = D − 2 n + 1.Then using Proposition 4.3 we conclude that a system of D ′ + 1 rational orthogonal projectors corresponding to Tate summands.
We will show that the remaining summand N is in general indecomposable.For a split quadric Q we can give an explicit decomposition of N into Tate summands in terms of orthogonal projectors.Consider the basis h i , l i , 0 ≤ i ≤ d of the ring K(n) * (Q), where D = dim Q is equal to 2d or 2d + 1 defined in Proposition 4.1, and let us denote h i = 0 = l i for i < 0. (i) The diagonal where δ 0, D mod 4 = 0, if D ≡ 0 mod 4; 1, if D ≡ 0 mod 4. (ii) The projectors ) define a decomposition of ∆ into a sum of 2d+2 orthogonal Tate motives.Observe that π i = h i × l i and π D ′ −i = l i × h i for i < d ′ .
The proof is straightforward.
The summands l −1 × l 1 and l 1 × l −1 denote zero according to our convention.
7.2.Computation of the co-action.We denote by G the split group SO m and by P 1 the maximal parabolic subgroup corresponding to the first simple root in the Dynkin diagram.Let Q = G/P 1 denote the split quadric of dimension m − 2 = 2d or m − 2 = 2d + 1.We will describe the co-action see Section 3.4.Observe that since K(n) * (Q) is generated as an algebra by the elements h and l = l d in the notation of Proposition 4.1 and since ρ(h) = 1 ⊗ h by [PS21,Lemma 4.12], the co-action is determined by ρ(l).
It is convenient here to work with the connective Morava K-theory CK(n) * .Sending v n to 0 we obtain a surjective map from the connective Morava K-theory onto the Chow theory.
Assume that m ≤ 2 n+1 , n > 1 and recall that for some q I ∈ K(n) * (Q), where the last sum is taken over e I = e i for 1 ≤ i ≤ s. for some homogeneous q I ∈ CK(n) * (Q) \ 0. Observe that the codimension of q I is at most m − 2. Since ρ CK(n) (l) is homogeneous of codimension s, none of such e I can be equal to e 2 x 2i−1 for some x ≤ k i − 1 by dimensional reasons.Now the result about K(n) * follows.
Lemma 7.4.Let Q = Q 2 n −1 be a split (2 n − 1)-dimensional quadric, n > 1.The co-action of K(n) * (SO 2 n +1 ) on K(n) * (Q 2 n −1 ) is given by for some q I ∈ K(n) * (Q), where the last sum is taken with e I = e i for all 1 ≤ i ≤ s.
Proof.As in Lemma 7.3 we have that for some q I ∈ CK(n)(Q) \ 0. Since ρ CK(n) (l) is homogeneous of codimension 2 n−1 , by dimensional reasons there exists only one possibility when e I in the sum above is equal to some e s , more precisely, the case e I = e 2 n−1 cannot be excluded.Moreover, we will show that such a summand indeed appears.So, the K(n)-co-action is determined by the formula for some c ∈ F 2 and q I ∈ K(n) * (Q).We claim that c = 1.Let E be a generic SO 2 n +1 -torsor.Recall that a motivic decomposition of E/P 1 provides a decomposition of A * (G/P 1 ) into a sum of A * (G)-comodules. Since , and the diagonal is equal to ∆ = defines a summand of M K(n) (E/P 1 ).Then using [PS21,Theorem 4.14] and [PS21,Remark 4.15] we conclude that the realization functor maps the motivic summands M 1 = (M(E/P ), π) and M 2 = (M(E/P ), ∆−π) to the K(n) * (SO 2 n +1 )-subcomodules • 1 and of K(n) * (Q), respectively (here denotes the restriction map res k/k , see Subsection 3.4).Since ρ(l) belongs to the submodule K(n) * (SO 2 n +1 ) ⊗ C 2 , it remains to collect the terms of ρ(l) lying in F 2 [v ±1  n ]e 2 n−1 ⊗ K(n) * (Q) to conclude that c = 1.
7.3.Motive of a generic quadric.We proved in Proposition 7.1 that the K(n)motive (n > 1) of any smooth projective quadric of dimension D ≥ 2 n −1 has D +2−2 n Tate summands.In the present subsection we show that the remaining summand of a generic quadric is indecomposable.This reproves, in particular, that the Chow motive of a generic quadric is indecomposable, see [Vi04,Ka12].As before G denotes SO m , P 1 denotes the first maximal parabolic subgroup, and E denotes a generic torsor.Recall that s = ⌊ m−1 2 ⌋ equals d for D = 2d even, and s = d + 1 for D = 2d + 1 odd.Since C i are graded and the dimension of the quadric is small, we can assume that either l ∈ C 1 or l + h d ∈ C 1 .Since C 1 is a sub-comodule, we conclude that ρ(l) or ρ(l + h d ) belongs to K(n) * (SO 2 n +1 ) ⊗ C 1 .Using the description of ρ(l) from Lemma 7.3 and collecting the terms lying in F 2 [v ±1  n ]e 1 ⊗ K(n) * (Q) we conclude that h s−1 ∈ C 1 .But similarly, each l k for 0 ≤ k ≤ d − 1 lies in one of C i 's, and since C i 's are subcomodules, ρ(l k ) belongs to the submodule K(n) * (SO 2 n +1 ) ⊗ C i .Then the description of ρ(l k ) from Lemma 7.3 implies that h s−1 ∈ C i .However, the sum is direct, so that all l k lie in C 1 .
Next, using the description of ρ(l) we conclude that all h k for 0 ≤ k ≤ s − 1 lie in C 1 .This finishes the proof for D odd, and for D even it remains to use the description of ρ(l d−1 ) to show that h d ∈ C 1 .Therefore, C 1 = K(n) * (M), C 2 = 0, and M is indecomposable.
Proposition 5.2.a) For any free theory A * and any parabolic subgroup P in a split semisimple group G the pullback map along the closed embedding induces an isomorphism of rings A * (P ) ∼ = A * (G) ⊗ A * (G/P ) A * (pt).b) For the Levi part L of a parabolic subgroup P and C = [L, L] the pullback maps along the natural closed embeddings induce isomorphisms of rings A * (P ) ∼ = A * (L) ∼ = A * (C).
2 let us denote by B ′ the Borel subgroup of C, i.e., B ′ = B ∩ C. Observe that P/B and C/B ′ coincide as varieties and the ring CH * (P/B) has a free basis consisting of the classes Z ′ w of closures of Bruhat cells (B ′ ) − wB ′ /B ′ in C/B ′ , or, equivalently, of closures of (B ′ ) − wB/B in P/B, w ∈ W P .We introduce the following notation.Notation (Subring R and subgroup V ).Let us denote by R a free abelian subgroup of CH * (G/B) with the basis Z w , w ∈ W P , and by V a free abelian subgroup with the basis Z w , w ∈ W P (cf.Subsection 3.1).We have the following isomorphisms of abelian groups: R ∼ = CH * (G/P ), V ∼ = CH * (P/B), moreover, the pullback map π CH : CH * (G/P ) → CH * (G/B) is injective, and R = Im(π CH ), i.e., R is a subring of CH * (G/B) isomorphic to CH * (G/P ).The next lemma is just [EdGr, Proposition 1] applied to our situation.We remark that U − P • P/B is an open subvariety of G/B, and the multiplication U − P × (P/B) → (U − P • P )/B defines an isomorphism of schemes.Lemma 5.4.a) The map R ⊗ Z V → CH * (G/B) sending p ⊗ q to pq is a bijection.b) The isomorphism of abelian groups CH * (P/B) ∼ = V identifying Z ′ w with Z w , w ∈ W P , is a section of the pullback map along the closed embedding ι : P/B → G/B.Proof.For a), see [PS21, Lemma 6.2].For b), decompose ι as a composition of a closed embedding i : P/B → (U − P • P )/B followed by an open embedding j : (U − P • P )/B ֒→ G/B.The pullback j * (Z w ) is given by Z w ∩ (U − P • P )/B, and the latter coincides with U − P × (B ′ ) − wB/B as a subvariety, since both schemes are reduced.Next, i is the zero section of the trivial fibration U − P • P/B → P/B and, therefore, sends U − P × (B ′ ) − wB/B to Z ′ w .
composition of divided difference operators, and ∆ CH I = ∆ CH w if w = s i 1 . . .s i k is a reduced decomposition (see Subsection 3.3).Using the duality of the Demazure operators and the Schubert basis (7) we conclude that For any free theory A * consider the characteristic map c A : A * (BT ) → A * (G/B), and let I A = I A (G/B) denote the ideal of A * (G/B) generated by the set Im(c A ) ∩ Ker(deg A ).

Proof.
It follows from [PS21, Lemma 7.2] that ρ CK(n) (l) = s i=1 e i ⊗ h s−i + 1 ⊗ l + v n e I ⊗ q I

Lemma 7. 5 .
For n > 1 the K(n)-motive M of a generic quadric E/P 1 of dimension D = m − 2 such that 0 < D ≤ 2 n − 2 is indecomposable.Proof.Assume that M decomposes as a direct sum M = M 1 ⊕ M 2 .Then by[PS21,  Theorem 4.14] and[PS21, Remark 4.15] the realization K(n) * (M) of the motive M over k is a direct sum of the realizations C 1 = K(n) * (M 1 ) and C 2 = K(n) * (M 2 ) of the motives M 1 and M 2 over k (see Subsection 3.4).
defined in Proposition 4.1.The multiplication table can be deduced from the identities h and if h is an element of H or of H ⊗ Fp[vn] H, then let h denote the image of h modulo v n in H or, respectively, in H ⊗ Fp H.