The Motivic Cohomology of Stiefel Varieties

The Motivic Cohomology of Stiefel V arieties Ben W illiams November 13, 2018 Abstract The main result of this paper is a computation of the motivic cohomol- ogy of varieties of n × m -matrices of of rank m , including both the ring structure and the action of the reduced power operations. The argument proceeds by a comparison of the general linear group-scheme with a T a te suspension of a space which is A 1 -equivalent to projective n − 1-space with a dis joint basepoint. Key W ords Motivic cohomolog y , higher Chow groups, reduced po wer operations, Stiefel varieties. Mathema tics Subj ect Clas sification 2000 Primary: 19E15. Secondary: 20G20, 57T10. 1 Introduction The results of this paper are theorem 1 9, which computes the motivic cohomol- ogy (or higher Chow groups) of varieties of n × m -matr ice s of of rank m , the Stiefel varieties of the title, including the ring structur e, and theorems 20 , 21 which compute the action of the motivic reduced power operations, the first for the Stee nrod squares, the second for odd primes. The ring structure on H ∗ , ∗ ( GL ( n ) ) has a lready be en computed, in [Pus04], where the argument is via c omparison with higher K - theory . W e a re able to offer a different computa- tion of the same, which is more geometric in character , relyin g on an analysis of comparison maps of varieties rather than of cohomology theories. The equivalent computation for ´ etale cohomo logy a ppears in [Ray68]; the argument there is by comparison with singular cohomology , and that suffices to determine even the a ction of the reduced power operations on the ´ etale co- homology of GL ( n , Z ) , and from there on the ´ etale cohomology of GL ( n , k ) where k is an arbitrar y field. The universa l r ings of loc. cit. a re not quite S tief el varieties, but they are a ffine torsors over them. In principle, by slavish imita- tion of loc. cit. the methods of the present pape r allow one to prove that with sporadic exceptions the universal stably-free module of rank at least 2 over a field is not free, a nd to do so without recourse to a non-algebra ic category . 1 In this pa p e r we ded uce, by elementar y means, the additive structure of the cohomology of Stiefe l va rieties, proposition 5, then we deduce the effect on cohomology of two c ompar ison ma ps between the different Stiefel mani- folds, these are propositions 7 a nd 8. By use of these, the ring structur e a nd the reduced power opera tions in a ll cases may b e d educed from the case of GL ( n ) . W e present a map in homotopy G m ∧ P n − 1 + → GL ( n ) that is well- known in the cla ssica l cases of R ∗ ∧ R P n − 1 + → O ( n ) and C ∗ ∧ C P n − 1 + → U ( n ) , [Jam76, Chapter 3], where it fits into a larger pattern of ma ps from suspen- sions of so-called “stunted quasiprojective spaces” into Stief el manifolds. In the classical case, with some care, one may show that the cohomo logy of a Stiefel manifold is gener a ted as a ring by classes detected by maps to stunted projective spaces. The comparison of Stiefel manifolds with the appropriate stunted projective-spaces underlies much of the classical homoto py-theory of Stiefel manifolds, the theory is presented thoroughly in [Jam76], we mention in add ition only the spec tacular resolution of the problem of ve c tor fields on spheres in [ A da62]. Care is required even in cohomology calculations because the products of classes in the cohomology of S tiefel manifolds muddy the wa- ter . In A 1 -homotopy , there are a lso analogues of stunted projective space s, but the bigrad ing on motivic cohomology means that the products of generating classes may be disregarded in articulating the range in which the cohomolog y H ∗ , ∗ ( G m ∧ P n − 1 + ) a nd H ∗ , ∗ ( GL ( n ) ) coincide, and we arrive at a much simpler statement, theorem 18 without hav ing to mention the stunted spaces at a ll. In proving theorem 18, we employ a ca lculation in higher Chow groups. This c alculation is on the one hand attractive in its geometric and explicit char- acter but on the other hand it is the chief obstacle to ex te nding the scope of the arguments presented here to other theories than motivic cohomology . 2 Preliminaries W e compute motivic cohomology as a represented cohomology theory in the motivic- or A 1 -homotopy category of Morel & V oevodsky , see [M V99] f or the construction of this ca tegory . The best reference for the theory of motivic co- homology is [MVW06], a nd the proof that the theory presented there is repre- sentable in the category we claim can be found in [Del], subject to the restric- tion that the field k is p e rfect. Motivic cohomology , being a cohomolog y theory (again at least when k is perfect) equipped with suspension isomorphi sms f or both suspensions, Σ s , Σ t , is represented by a motivic spectrum, of course, but we never de a l explicitly with such objects. W e therefore fix a pe rfect field k . W e shall let R denote a fixed c ommuta- tive ring of coefficients. W e denote the terminal object, Spec k , by pt. If X is a finite type smooth k -scheme or more ge ner ally an element in the ca tegory s Shv Nis ( Sm / k ) ), we write H ∗ , ∗ ( X ; R ) for the bigraded motivic cohomolo gy ring of X . This is graded-commutative in the first grading, and commuta- tive in the second. If R → R ′ is a ring ma p, then there is a map of algebras 2 H ∗ , ∗ ( X ; R ) → H ∗ , ∗ ( X ; R ′ ) . It will be of some importance to us that most of our constructions are f unctorial in R , when the coefficient ring is not specified, therefore, it is to be understood that arbitrary coefficients R are meant and that the result is functorial in R . W e write M R for the ring H ∗ , ∗ ( pt; R ) . S ince pt is a terminal object, the ring H ∗ , ∗ ( X ; R ) is in fac t a n M R algebra. W e assume the following vanishing results for a d -dimensional smooth scheme X : H p , q ( X ; R ) = 0 when p > 2 q , p > q + d or q < 0. At one point we employ the comparison theorem relating motivic cohomol- ogy and the higher Chow groups. For all these, see [MVW06]. Let Sm / k denote the category of smooth k -schemes, and let Aff / k denote the category of a ffine regular finite-type k -schemes. W e shall frequently make use of the following version of the Y oneda lemma Lemma 1. The functor Sm / k → Pre ( Aff / k ) given by X 7 → h X , where h X ( Spe c R ) denotes the set of maps Spec R → X over Spec k is a full, fa it hful embedding. Proof. The standard version of Y oneda’s lemma is that there is a full, faithful embedding Sm / k → Pre ( Sm / k ) . The functor we a re considering is the compo- sition Sm / k → Pre ( Sm / k ) → Pre ( Aff / k ) obtained by restricting the domain of the functors in P re ( Sm / k ) . One can write any Y in Sm / k as a c olimit of spectra of finite-type k -a lgebras. W e have Sm / k ( Y , X ) = Sm / k ( colim Spec A i , X ) = lim Sm / k ( Spec A i , X ) = lim X ( A i ) from which it follows that the f unctor Sm / k → Aff / k inherits fullness a nd fidelity from the Y oneda embedding by abstract-nonsense arguments. W e shall generally write X ( R ) for h X ( Spe c R ) . In practice this result means that rather than spec if ying a map of schemes X → Y explicitly , we shall happily exhibit a set-map X ( R ) → Y ( R ) , where R is an arbitrary finite-type k -algebra, and then observe that this set map is natural in R . The result is a map in Pre ( Aff / k ) , which is therefore (by the fullness and fidelity of Y oneda) also under stood as a map of schemes X → Y . 3 The Additive Structur e Proposition 2. Let X be a smoot h scheme and suppose E is an A n -bundle over X , and F is a sub-bundle with fiber A ℓ , t hen there is an exac t triangle of graded H ∗ , ∗ ( X ) - modules H ∗ , ∗ ( X ) τ j / / H ∗ , ∗ ( X ) x x q q q q q q q q q q H ∗ , ∗ ( E \ F ) ∂ f f N N N N N N N N N N N where | τ | = ( 2 n − 2 ℓ , n − ℓ ) 3 Proof. This is the localization exa ct triangle for the closed sub-bundle F ⊂ E , where H ∗ , ∗ ( E ) = H ∗ , ∗ ( F ) = H ∗ , ∗ ( X ) arising from the cofiber sequence (see [ MV99] for this and all other unrefer- enced assertions concerning A 1 -homotopy) E \ F / / E / / Th ( N ) where N is the normal-bundle of F in E . There are in two possible choices for τ , since τ and − τ serve equally well. W e make the c onvention that in any localization exact sequence a ssociated with a closed immersion of smooth schemes Z → X , viz. / / H ∗ , ∗ ( Z ) τ / / H ∗ , ∗ ( X ) / / H ∗ , ∗ ( X \ Z ) / / the class τ should be the class which, under the natural isomorphism of the above with the localization sequence in higher Chow groups, corresponds to the class represented by Z in C H ∗ ( Z , ∗ ) . In the case where F = X is the ze ro-bundle, then j takes τ to e ( E ) , the Euler class, a s proved in [V oe03]. In general, by identifying H ∗ , ∗ ( X ) τ with C H ∗ − n + ℓ ( F , ∗ ) , the higher Chow groups of F as a closed subscheme of E , and employing covar iant functoriality of higher Chow groups for the closed im- mersions X ֌ F ֌ E , we see that j ( τ ) e ( F ) = e ( E ) . As shall be the ca se throughout, H ∗ , ∗ ( X ) denotes cohomology with unspec- ified coefficients, R , and the result is understood to be natural in R . For the naturality of the localization sequence in R , one simply follows through the argument in [MVW06], which reduces it to the computation of H ∗ , ∗ ( P d ) = M R [ θ ] / ( θ d + 1 ) which is natural in R by elementary means, c.f . [Ful84]. W e note that | j ( τ ) | = ( 2 c , c ) , so if , as often happens, H 2 c , c ( X ) = 0, this triangle is a short exact sequence of H ∗ , ∗ ( X ) -modules: 0 / / H ∗ , ∗ ( X ) / / H ∗ , ∗ ( E \ F ) / / H ∗ , ∗ ( X ) ρ / / 0 Here | ρ | = ( 2 n − 2 ℓ − 1, n − ℓ ) Since H ∗ , ∗ ( X ) ρ is a free graded H ∗ , ∗ ( X ) -module, this short e x act sequence of graded modules splits, there is an isomorphism of H ∗ , ∗ ( X ) -modules H ∗ , ∗ ( E \ F ) ∼ = H ∗ , ∗ ( X ) ⊕ H ∗ , ∗ ( X ) ρ W e remark that | ρ | = ( 2 c − 1, c ) , so that 2 ρ 2 = 0 by anti-commutativity , we now see that ( a + b ρ ) ( c + d ρ ) = a c + ( ad + ( − 1 ) deg c bc ) ρ + ( − 1 ) deg d bd ρ 2 , so in many ca ses (e.g. when 1 /2 ∈ R ) the multiplicative structur e is f ully determined, and H ∗ , ∗ ( E \ F ) = H ∗ , ∗ ( X ) [ ρ ] / ( ρ 2 ) Observe that if H 2 n , n ( X ) = 0 for n > 0, a s often happens, then the same applies to H ∗ , ∗ ( E \ F ) . W e will ha ve occa sion late r to refer to the following two results, which ap- pear here for want of anywhere better to state them 4 Proposition 3. Let X be a scheme and suppose E is a Zariski-trivializeable fiber bun- dle with fiber F ≃ pt . Then E ≃ X . Proof. This is standard, see [DHI04]. When we use the term ‘bundle’, we shall mean a Zar iski-trivializable bun- dle over a scheme. The following two propositions allow us to identify affine bundles which are not necessarily vector-bundles. Proposition 4. Suppose X is a scheme, P is a projective bundle of rank n over X and Q is a projective subbundle of rank n − 1 , Th en P \ Q is a fiber bundle with fiber A n − 1 . Proof. This f ollows immediately by considering a Zariski open cover trivializ- ing both bundles. Let k be a field . Let W ( n , m ) denote the variety of full-rank n × m matrices over k , that is to say it is the open subscheme of A nm determined by the non- vanishing of at le a st one m × m -minor . W ithout loss of generality , m ≤ n . By a Stiefel V ariety we mean such a variety W ( n , m ) . Proposition 5. The cohomolo g y o f W ( n , m ) has the fo llowing p res entation as an M R - algebra: H ∗ , ∗ ( W ( n , m ) ; R ) = M R [ ρ n , . . . , ρ n − m + 1 ] I | ρ i | = ( 2 i − 1, i ) The ideal I is generated by relations ρ 2 i − a n , i ρ 2 i − 1 , wh ere the elements a n , i lie in M 1,1 R and satisfy 2 a n , i = 0 . W e shall later identify a n , i as {− 1 } , the image of the c la ss of − 1 in k ∗ = H 1,1 ( Spe c k ; Z ) under the map H ∗ , ∗ ( Spe c k ; Z ) → H ∗ , ∗ ( Spe c k ; R ) . Proof. If n = 1, there is only one possibility to consider , that of W ( 1, 1 ) = A 1 \ { 0 } , the cohomology of which is a lready known from [ V oe03], and is a s asserted in the propositio n. W e therefore assume n ≥ 2. The proof proceeds by induction on m , starting with m = 1 (we could start with W ( n , 0 ) = pt). In this c a se W ( n , 1 ) = A n \ { 0 } , and H ∗ , ∗ ( W ( n , 1 ) ) = M [ ρ n ] / ( ρ 2 n ) . W ( n , m − 1 ) is a dense open set of A nm , and a s such is a smooth scheme. If m < n , there is a trivial A n -bundle over W ( n , m − 1 ) , the fiber over a matrix A is the set of all n × m -matrices whose first m − 1 columns are the matrix A    v 1 A . . . v n    As a sub- bundle of this bundle, we find a trivial A m − 1 -bundle; the fiber of which over a k - point (i.e. a ma tr ix) A is the set of matrices where ( v 1 , . . . , v n ) 5 is in the row-space of A . Proposition 2 app lies in this setting, and we conclude that there e xists an exa ct triangle H ∗ , ∗ ( W ( n , m − 1 ) ) τ / / H ∗ , ∗ ( W ( n , m − 1 ) ) u u l l l l l l l l l l l l l l H ∗ , ∗ ( W ( n , m ) ) ∂ i i S S S S S S S S S S S S S S Since H 2 i , i ( W ( n , m − 1 ) ) = 0 by induction, so this triangle splits to give H ∗ , ∗ ( W ( m , n ) ) ∼ = H ∗ , ∗ ( W ( m − 1, n ) ) ⊕ H ∗ , ∗ ( W ( m − 1, n ) ) ρ n − m + 1 where | ρ n − m + 1 | = ( 2 ( n − m + 1 ) − 1 , n − m + 1 ) . B y graded-commutativity we have, 2 ρ 2 n − m + 1 = 0. By considering the bigrading on the motivic cohomology , and the vanishing results, we know that ρ 2 r ∈ H 4 r − 2 ,2 r ( W ( n , m ) ; R ) = H 4 r − 2 ,2 r ( W ( n , m − 1 ) ; R ) where r = n − m + 1 W e can describe H ∗ , ∗ ( W ( n , m − 1 ) ; R ) as an M -module as follows H ∗ , ∗ ( W ( n , m − 1 ) ; R ) ∼ = M ⊕ M i M ρ i ⊕ M i < j M ρ i ρ j ⊕ J where J is the submodule generated by multiples of at least three distinct classes of the form ρ i . Since wt ( ρ i ρ j ) = i + j > 2 ( n − m + 1 ) for a ll i , j ≥ n − m + 2 , it follows the higher product terms are irrelevant to the d etermina- tion of the cohomolo gy group H 4 n − 4 m + 2,2 n − 2 m + 2 ( W ( n , m − 1 ; R ) ) . There are two possibilities to consider . First, that n > 2 m − 1 , which by consideration of the grading f orces H 4 n − 4 m + 2,2 n − 2 m + 2 ( W ( n , m − 1 ) ; R ) = 0, and so ρ 2 n − m + 1 = 0. The other is n ≤ 2 m − 1, in which case H 4 n − 4 m + 2,2 n − 2 m + 2 ( W ( n , m − 1 ) ; R ) = M 1,1 R ρ 2 n − 2 m + 1 so that ρ 2 n − m + 1 = a n , m ρ 2 n − 2 m + 1 as required. The bigrading alluded to above forces 2 a n , m = 0. W e denote the cohomology ring H ∗ , ∗ ( W ( n , m ) ; R ) = M R [ ρ n , . . . , ρ n − m + 1 ] / I where the ideal I is understood to depend on n , m . W e shall need the f ollowin g technical lemma Lemma 6. Let Z → X be a closed immersion of irreducible smoot h sch emes, and let f : X ′ → X be a map o f smooth schemes such that f − 1 ( Z ) is again smooth and irreducible and so that is either f is flat or split by a flat m a p, in t he sense that there 6 exists a flat map s : X → X ′ such that s ◦ f = id X ′ . Th en th ere is a map of localizat ion sequences in m otivic cohomology / / H ∗ , ∗ ( Z ) τ   / / H ∗ , ∗ ( X ) / /   H ∗ , ∗ ( X \ Z )   / / / / H ∗ , ∗ ( f − 1 ( Z ) ) τ ′ / / H ∗ , ∗ ( X ′ ) / / H ∗ , ∗ ( X ′ \ f − 1 ( Z ) ) / / such that t h e la st two vertical arrows are the functorial maps on c o homology and such that τ 7 → τ ′ . The giving of references for results concerning higher Chow grou ps is de - ferred to the beginning of section 4. Proof. One begins by observing the e xistence in general of the following d ia- gram X ′ \ f − 1 ( Z ) / /   X ′ / /   Th N f − 1 ( Z ) → X ′      X \ Z / / X / / Th N Z → X where the dotted arrow exists for reasons of general nonsense. There is in general a map on cohomology a rising from the given diagr a m of cofiber sequences, but we ca nnot at this stage predict the behavior of the map induced by the dotted a rrow . When the map X ′ → X is flat, the pull- back f − 1 ( Z ) → Z is too. W e identify the motivic cohomology groups with the higher Chow groups, giving the localiz ation sequence / / C H ∗ ( Z , ∗ )      / / C H ∗ ( X , ∗ ) / /   C H ∗ ( X \ Z , ∗ )   / / / / C H ∗ ( f − 1 ( Z ) , ∗ ) / / C H ∗ ( X ′ , ∗ ) / / C H ∗ ( X ′ \ f − 1 ( Z ) , ∗ ) / / and in this case τ , τ ′ become the classes of the cycles [ Z ] , [ f − 1 Z ] . Since the map C H ∗ ( Z , ∗ ) → C H ∗ ( f − 1 ( Z ) , ∗ ) is the contravar iant ma p a ssociated with pull-back along a flat morphism, it follows immediately that τ 7→ τ ′ . The following results are analogues of classically known facts. Proposition 7. For m ′ ≤ m, th ere is a projection W ( n , m ) → W ( n , m ′ ) given by omission of the last m − m ′ -vectors. On cohom ology , this yields an inclusion M ( ρ n , . . . , ρ n − m ′ + 1 ) / I → M ( ρ n , . . . , ρ n − m ′ + 1 , . . . , ρ n − m + 1 ) / I 7 Proof. It suffices to prove the ca se m ′ = m − 1. In this case, the map W ( n , m ) → W ( n , m − 1 ) is the fiber bundle from which we computed the cohomology of W ( n , m ) , and the result on cohomology holds by inspection of the proof. Proposition 8. Giv en a nonzero rational po int, v ∈ ( A n \ 0 ) ( k ) , and a complemen- tary n − 1 -dimensional subspace U such t h at h v i ⊕ U = A n ( k ) , t h ere is a closed im- mersion φ v , U : W ( n − 1, m − 1 ) → W ( n , m ) given by identifying W ( n − 1, m − 1 ) with the space of ind ep endent m − 1 -frames in U , and t h en p re pending v. O n c o ho- mology, this yields th e surjection M [ ρ n , . . . , ρ n − m + 1 ] / I → M [ ρ n − 1 , . . . , ρ n − m + 1 ] / I with kernel ( ρ n ) . Proof. W e prove this by induction on the m , which is to sa y we deduce the case ( n , m ) from the case ( n , m − 1 ) . The base case of m = 1 is straightforward. Recall that we compute the cohomology of W ( n , m ) by forming a trivial bundle E n , m ≃ W ( n , m − 1 ) over W ( n , m − 1 ) , which on the level of R -points consists of matr ices    v 1 A . . . v n    and removing the trivial closed sub-bundle Z n , m where the vector ( v 1 , . . . , v n ) is in the span of the columns of A . There is then an open inclusion E n , m \ Z n , m ∼ = W ( n , m ) → E n , m ≃ W ( n , m − 1 ) The inclusion φ v , U : W ( n − 1, m − 1 ) → W ( n , m ) , without loss of generality can be a ssumed to act on field -valued points as as B  φ v , U / /  1 0 0 B  W e abbreviate this map of schemes to φ , and de note the ana logous maps W ( n − 1, m − 2 ) → W ( n , m − 1 ) , Z n − 1, m − 1 → Z n , m , E n − 1, m − 1 → E n , m etc. a lso by φ by abuse of notation. The following are pull-back diagra ms E n − 1, m − 1 / / E n , m Z n − 1, m − 1 O O / / Z n , m O O E n − 1, m − 1 / / E n , m W ( n − 1, m − 1 ) O O φ / / W ( n , m ) O O The second square above is homotopy equivalent to W ( n − 1, m − 2 ) φ / /   W ( n , m − 1 )   W ( n − 1, m − 1 ) φ / / W ( n , m ) 8 from which we deduce that the map φ ∗ : H ∗ , ∗ ( W ( n , m ) ) → H ∗ , ∗ ( W ( n − 1, m − 1 ) ) satisfies φ ∗ ( ρ j ) = ρ j for n − m + 2 ≤ j ≤ n − 1 and φ ∗ ( ρ n ) = 0, since this holds for W ( n − 1, m − 2 ) → W ( n , m − 1 ) by induction. The hard pa rt is the behavior of the element ρ n − m + 1 , which is in the kernel of H ∗ , ∗ ( W ( n , m ) ) → H ∗ , ∗ ( W ( n , m − 1 )) Recall that ρ n − m + 1 ∈ H ∗ , ∗ ( W ( n , m ) ) is the preimage of the Thom cla ss τ under the map ∂ : H ∗ , ∗ ( W ( n , m ) ) → H ∗ , ∗ ( W ( n , m − 1 ) ) τ = H ∗ , ∗ ( Z n , m ) τ W e should like to assert that the map of localization sequences / / H ∗ , ∗ ( W ( n , m ) ) φ ∗   ∂ / / H ∗ , ∗ ( Z n , m ) τ   / / H ∗ , ∗ ( E n , m )   / / / / H ∗ , ∗ ( W ( n − 1, m − 1 ) ) ∂ / / H ∗ , ∗ ( Z n − 1, m − 1 ) τ ′ / / H ∗ , ∗ ( E n − 1, m − 1 ) / / (1) one has τ 7 → τ ′ , beca use then chasing the commutative square of isomor- phisms Z ρ n − m − 1 = H 2 n − 2 m − 1, n − m ( W ( n , m ) ) / /   H 0,0 ( Z n , m τ = Z τ   Z ρ n − m − 1 = H 2 n − 2 m − 1, n − m ( W ( n − 1, m − 1 ) ) / / H 0,0 ( Z n − 1, m − 1 τ = Z τ ′ we have ρ n − m − 1 7→ ρ n − m − 1 as required. The difficulty is that the map g : E n − 1, m − 1 → E n , m is a closed immersion, rather than a flat or split map, for which we have ded uced this sort of natu- rality result in lemma 6. W e can however f actor g into such maps, which we denote only on the level of points, the obvious scheme-theoretic definitions a re suppressed. Let U n , m denote the variety of n , m -matrices which (on the level of k -points) ha v e a decomposition a s  u ∗ ∗ ∗ A ∗  where u ∈ k ∗ , and A ∈ W ( n − 1, m − 2 ) ( k ) . It goes without saying that this is a variety , since the conditions amount to the nonvanishing of cer ta in minors. It is also ea sily seen that U n , m is an open de nse subset of E n , m . W e have a map E n − 1, m − 1 → E n , m , given by B 7 →  1 ∗ ∗ B  9 and this map is obviously split by the projection onto the bottom-right n − 1 × m − 1- submatrix. The composition E n − 1, m − 1 → U n , m → E n , m is a fa ctorization of the map E n − 1, m − 1 → E n , m into a split map followed by an open immersion. The splitting of E n − 1, m − 1 is a projection onto a fa ctor , and since all schemes are flat over pt, the splitting is flat as well. W e ma y now use lemma 6 twice to conclude that in dia gra m (1) we have τ 7 → τ ′ , so that ρ n − m + 1 7→ ρ n − m + 1 , as asserted. 4 Higher I nt ersection Theory W e shall, as is standard, denote the algebraic d -simplex Spec k [ x 0 , . . . , x d ] / ( x 0 + · · · + x d − 1 ) ∼ = A d by ∆ d . The object ∆ • is cosimplicial in a n obvious way . It shall be convenient later to identify ∆ 1 in pa rticular with A 1 = Spec k [ t ] . Let X be a scheme of finite type over a field. The higher Chow groups of X , denoted C H i ( X , d ) are defined in [ B lo86] a s the homology of a certa in complex: C H i ( X , d ) = H d ( z i ( X , ∗ ) ) where z i ( X , d ) denotes the free abe lian group generated by cycles in X × ∆ d meeting all f aces of X × ∆ d properly . W e denote the d ifferential in this complex by δ . There is a comparison theorem, see [ MVW06, lecture 1 9], [V oe0 2], which states that f or any smooth scheme X over any field k , there is a n isomorphism between the motivic cohomolog y groups and the higher C how groups C H i ( X , d ) = H 2 i − d , i ( X , Z ) or the equivalent with Z replaced by a general coefficient ring R . The products on motivic cohomology and on higher Chow groups a re known to coincide, see[W ei99]. In the difficult paper [Blo94], the following result is proven in a n equivalent form ( the strong moving le mma) Theorem 9 (Bloch) . Let X be an equidimensional scheme, Y a closed equidimensional subscheme of cod imension c in X , U ∼ = X \ Y . Then for all i, there is an exact sequence of complexes 0 / / z i − c ( Y , ∗ ) / / z i ( X , ∗ ) / / z i a ( U , ∗ ) / / 0 where z i a ( U , ∗ ) is the subcomplex of z i ( U , ∗ ) generated by subvarieties γ whose closure γ ֌ X × ∆ ∗ meet all faces properly . Th e inclusion of complexes z i a ( U , ∗ ) ⊂ z i ( U , ∗ ) induces an isomo rp hism on homology groups. For a cycle α ∈ z i ( U , d ) , we c a n write α = ∑ N i = 1 n i A i for some subvar ieties A i of U × ∆ d , a nd n i ∈ Z \ { 0 } . W e can form the scheme-theoretic closure of A i in X × ∆ d , denoted A i . W e remark that A i × X × ∆ d ( U × ∆ d ) = A i [Har77, II.3]. W e define α N ∑ i = 1 n i A i 10 W e say that α meets a subvariety K ֌ X properly if every A i meets K properly . Suppose α is such that α meets the faces of X × ∆ d properly , then α = ( U → X ) ∗ ( α ) , so α ∈ z i a ( U , d ) . Proposition 10. As before , let X be a quasiprojective variety, let Y be a c losed sub- variety of pure cod imension c , let U = X − Y and let ι : U → X denote the op en embedding. Suppose α ∈ z i ( U , d ) is such t hat α meets th e faces of X × ∆ d prop- erly , th en t he connecting homomorphism ∂ : C H i ( U , d ) → C H i ( Y , d − 1 ) tak es the class of α to the class of δ ( α ) which h appens to lie in the subgrou p z i − c ( Y , d − 1 ) of z i − c ( X , d − 1 ) . Proof. First, since α is such that α meets the faces of X × ∆ d properly , it follows that α = ι ∗ ( α ) , so α ∈ z i a ( U , d ) . The localization sequence arises from the short exa ct sequence of complexes 0 / / z i − c ( Y , ∗ ) / / z i ( X , ∗ ) / / z i a ( U , ∗ ) / / 0 via the snake lemma. A diagra m chase now completes the argument. Proposition 1 1. Consider A n \ { 0 } as an open subscheme of A n in th e obvious way , so th er e is a localization sequence in higher Chow groups for pt, A n and A n \ { 0 } . The high er Ch ow groups C H ( pt ) = M are given a n explicit g enerator , ν . Wri te H 2 n − 1, n ( A n \ { 0 } , Z ) = C H n ( A n \ { 0 } , 1 ) = Z γ ⊕ Q , where Q = 0 for n ≥ 2 and Q = k ∗ for n = 1 , and wh ere γ is such that the boundary map ∂ : C H n ( A n \ { 0 } , 1 ) → C H 0 ( pt, 0 ) maps γ to ν . T he element γ may be repr esented by any curve in A n × ∆ 1 = Spec k [ x 1 , . . . , x n , t ] which fails to meet th e h yperplane t = 0 and meets t = 1 with multip licity one at x 1 = x 2 = · · · = x n = 0 only . Proof. The low-degree pa rt of the localization sequence is C H 0 ( pt, 1 ) = 0 / / C H n ( A n , 1 ) = Q / / C H n ( A n \ 0 , 1 ) ∂ / / C H 0 ( pt, 0 ) / / C H 0 ( A n , 0 ) = Z / / C H 0 ( A n \ 0, 0 ) = Z / / 0 Suppose C is a curve which does not meet t = 0 , a nd which meets t = 1 with multiplicity one at x 1 = · · · = x n = 0 only , then by proposition 10 the cycle [ C ] ∈ C H n ( A n − 0, 1 ) maps to the class of a point in C H 0 ( pt, 0 ) = Z , which is a generator , [ Ful8 4]. The assertion now follows from stra ightforward homologi cal algebra. Corollary 1 1.1. Suppose p ∈ A n \ { 0 } is a k -valued point. Wr ite p = ( p 1 , . . . , p n ) . The curve giv en by the equation γ p : ( x 1 − p 1 ) t + p 1 = ( x 2 − p 2 ) t + p 2 = · · · = ( x n − p n ) t + p n = 0 repr esents a canonical g enerator of C H n ( A n \ { 0 } , 1 ) . 11 Proof. One ve rifies easily that the propositio n a pplies. Corollary 11.2 . Consider t he map A n \ { 0 } → A n \ { 0 } giv en by multip lication by − 1 . This ma p induces the identity on cohom ology . Proof. The preimage of the curve γ p is the curve γ − p , but both represent the same generator of C H n ( A n \ { 0 } , 1 ) , so the result follows. W e ca n now prove two facts ab out the c ohomology of W ( n , m ) that should come as no surprise, the case of complex Stief el manifolds being our guide. Proposition 1 2. Let γ : W ( n , m ) → W ( n , m ) denote m ultiplication of t he first column by − 1 . Then γ ∗ is identity on c o homology Proof. By use of the comparison maps GL ( n ) → W ( n , m ) , se e proposition 7, we see that it suffices to prove this for GL ( n ) . By means of the sta ndard inclusion GL ( n − 1 ) → GL ( n ) and induction we see that it suffices to prove γ ∗ ( ρ n ) = ρ n , where ρ n is the highest-degree genera tor of H ∗ , ∗ ( GL ( n ) ; R ) . By the comparison map again, we see that it suffices to prove tha t τ : A n \ { 0 } → A n \ { 0 } has the required p roperty , but this is corollary 11.2 Proposition 13. Let σ ∈ Σ m , the sy mmetric group on m letters. Let f σ : W ( n , m ) → W ( n , m ) be the map t hat permutes th e columns of W ( n , m ) according to σ . Then f ∗ σ is the identity on cohom ology . Proof. W e can reduce immediately to the case where σ is a transposition, a nd from there we can assume without loss of generality that σ interchanges the first two columns. Let R be a finite-type k -algebr a. W e view W ( n , m ) as the space whose R -valued points are m -tuples of e le ments in R n satisfying certain conditions which we do not par ticularly need to know . In the ca se R = k , the condition is that the matrix is of full-rank in the usual way . W e can a ct on W ( n , m ) by the elementary matrix e i j ( λ ) e i j ( λ ) : ( v 1 , . . . , v m , v m + 1 ) 7→ ( v 1 , . . . , v i + λ v j , . . . , v m + 1 ) The two maps e i , j ( λ ) and e i , j ( 0 ) = id are homotopic, so e i , j ( λ ) induces the identity on cohomology . There is now a standard method to interchange two columns and change the sign of one by means of elementary oper a tion e i j ( λ ) , to wit e 12 ( 1 ) e 21 ( − 1 ) e 12 ( 1 ) . W e therefore know that the map ( v 1 , v 2 , v 3 , . . . , v m ) 7→ ( − v 2 , v 1 , v 3 , . . . , v m ) induces the identity on cohomology , but now proposition 12 allows us e v e n to undo the multiplication by − 1. 12 5 The Comparison Map: G m ∧ P n − 1 + → GL ( n ) It will be necessary in this section to pay attention to ba sepoints. The group schemes G m and GL n will be pointed by their identity e lements. When we deal with pointed spac es, we compute reduced motivic cohomology for p refe rence. W e establish a map in homotopy G m × P n − 1 → GL ( n ) , in fa ct we have a map from the half-smash product G m ∧ P n − 1 + → GL ( n ) (2) and we show this latter map induces isomorphism on a range of cohomology groups. W e view P n − 1 as be ing the space of lines in A n , and ˇ P n − 1 the space of hyperplanes in A n . Define a space F n − 1 as being the subbundle of P n − 1 × ˇ P n − 1 consisting of pairs ( L , U ) where L ∩ U = 0, or equivalently , where L + U = A n . More precisely , we take P n − 1 and construct P n − 1 × ˇ P n − 1 = P roj S ( O S [ y 0 , . . . , y n − 1 ]) where P roj denotes the global projective-spectrum functor . Let Z denote the closed subscheme of P n − 1 × ˇ P n − 1 determined by the bihomogeneous equation x 0 y 0 + x 1 y 1 + · · · + x n − 1 y n − 1 = 0. Then we define F n − 1 as the complement ( P n − 1 × ˇ P n − 1 ) \ Z . Proposition 1 4. The c omposite F n − 1 → P n − 1 × ˇ P n − 1 → P n − 1 , the second map being projection, is a Zariski-trivializable bundle with fiber A n − 1 . In particular , F n − 1 ∼ → P n − 1 Proof. T aking A n − 1 to be a coordinate open subscheme of P n − 1 determined by e.g. x 0 6 = 0, we obtain the following pull-b a ck diagram U = P roj A n − 1 ( O A n [ y 0 , . . . , y n − 1 ]) \ Z | A n − 1 / /   F n − 1   A n − 1 / / P n − 1 The scheme U is the complement of a hyperplane in P n − 1 × ˇ P n − 1 , a nd so takes the form U ∼ = S pec A n − 1 ( O A n [ t 1 , . . . , t n − 1 ]) ∼ = A n − 1 × A n − 1 The projection U → A n − 1 is a projection onto a factor . Since the coordinate open subschemes A n − 1 form an open cover of P n − 1 , it follows that F n − 1 ≃ P n − 1 . In order to prove results concerning F n − 1 , it shall be useful to have the following definition to hand. 13 Definition: Let R be a commutative k -algebra. B y an n- generated split line- bundle we mean the following data. First, an isomorphism class of projective R - modules of rank 1, denoted L by abuse of notation; second, a class of surjections [ f ] : R n → L , where two surjections a re equivalent if they d iffer by a multiple of R × ; third, a class of splitting maps [ g ] : L → R n , a gain the maps are considered up to a ction of R × , and where any f ′ ∈ [ f ] is split by some g ′ ∈ [ g ] . If R → S is a map of k -algebras, and if ( L , f , g ) is an n -generated split line- bundle over R , then applica tion of S ⊗ R · yields the same over S . In this way , the assignment to R of the set of n -generated split line-bundles is a functor from the category of k -algebra s to the category of sets. Proposition 15. If R is a finite-ty pe k-a lg ebra, then th e set of R- points F n − 1 ( R ) is exactly the set of n -generated split line-bundles. Proof. It is generally known that Spec R → P n − 1 classifies isomorphism classes of ra nk-1 vec tor bundles, L , over R , equipped with a n equivalence class of surjections R n → L . This is a modification of a theorem of [Gro61, Chapte r 4 ]. It f ollows that Spec R → P n − 1 × ˇ P n − 1 classifies pa irs of equivalence classes of rank-1 projective modules, equipped with surjective ma p s ( f : R n → L 1 , g : ˇ R n → L 2 ) considered up to scalar multiplication by R × × R × . For convenience, let { e 1 , . . . , e n } be a basis of R n and let { ˇ e 1 , . . . , ˇ e n } be the dual ba sis. Let h : R → R n ⊗ R ˇ R n be the map given by the element ∑ n i = 1 e i ⊗ ˇ e i of the latter module. Given f , g , one obtains a composite map φ = ( f ⊗ g ) ◦ h : R → L 1 ⊗ R L 2 (3) Let Z be the closed subscheme of P n − 1 × ˇ P n − 1 determined by the equation x 1 y 1 + · · · + x n y n = 0 . L et m be a maximal ideal of R . Suppose that there is a dashed arrow making the following diagram commute: Spec R / m   / / _ _ _ _ _ Z   Spec R / / P n − 1 × ˇ P n − 1 If we ta ke the composite, Spec ( R / m ) → Spec R → P n − 1 × ˇ P n − 1 , then this has the effect of reducing our represented maps modulo m , and the result is two surjective maps over a field f : ( R / m ) n → R / m a nd g : ( ˇ R / m ) n → ˇ R / m . These can be ide ntified with two n -tuples [ a 1 ; . . . ; a n − 1 ] and [ b 1 ; . . . ; b n − 1 ] one in ( R / m ) n , the other in its dual, taken up to multiplication by ( R / m ) × . The map Spec R / m → P n − 1 × ˇ P n − 1 factors through Z if and only if a 1 b 1 + · · · + a n b n = 0, but this latter equation is precisely the statement that the reduction of the map φ of e quation (3) above, ( f ⊗ g ) · h , is nonzero. Since R is a finite-type k -algebra , one ha s a f actorization S pec R → F n − 1 = ( P n − 1 × ˇ P n − 1 ) \ Z → P n − 1 × ˇ P n − 1 if a nd only if no closed point of S p e c R lies in the closed subset Z of P n − 1 × ˇ P n − 1 , but this is equivalent to the state- ment that no matter which maximal idea l m ⊂ R is chosen, Spec R / m → 14 P n − 1 × ˇ P n − 1 does not factor through Z , and therefore to the statement that the reduction of φ at any maximal ideal of R is never 0. It follows that a map Spec R → P n − 1 × ˇ P n − 1 is the data of two equivalence classes of line bundles, L 1 , L 2 , each equipped with surjective maps f : R n → L 1 , g : R n → L 2 , consid- ered only up to scalar multiple, and where there is a nowher e-vanishing map of modules φ : R 1 → L 1 ⊗ R L 2 . Such a nowhere-vanishing map must be a n isomorphism, R 1 ∼ = L 1 ⊗ R L 2 , and we therefore have an identifying isomorphism L 2 = ˇ L 1 . W rite ( a 1 , . . . , a n ) for the image of the map f : R n → L 1 and ( b 1 , . . . , b n ) f or that of g : ˇ R n → ˇ L 1 , then the identifying isomorphism has been constructed specifically so that the element a 1 b 1 + · · · + a n b n ∈ L 1 ⊗ ˇ L 1 corresponds exactly to a generator of R 1 , to wit. a unit u ∈ R . S ince we a re working only with e quivalence- classes of presentations of L 1 , L 2 , we may if need be replace L 1 by u − 1 L 1 , and so we find that the composite L 1 ˇ g / / R n f / / L 1 is the identity , a s required. Proposition 16. For all n ∈ N , let X n denote the m otivic space X n = G m ∧ ( F n − 1 + ) (4) Then there are maps f n : X n → GL ( n ) , and maps h : X n → X n + 1 , which make the following diagram co m mute G m ∧ P n − 1 + id ∧ h +   X n o o h      / / GL ( n ) φ   G m ∧ P n + X n + 1 o o / / GL ( n + 1 ) (5) Where the maps h : P n − 1 → P n are the standard inclusion of P n − 1 → P n as the first n − 1 coord inat es. Proof. First we c onstruct a map f n : G m × X n → GL ( n ) . The first scheme represents the functor taking a finite-type k -algebra, R , to the set of e le ments of the form ( λ , ( L , φ , ψ ) ) , that is to say , consisting of elements λ ∈ R × and a surjection onto a ra nk-1 proj ective bundle φ : R n → L along with a splitting ψ : L → R n . The second scheme, GL ( n ) , represents the functor taking R to GL n ( R ) . The map f n can be constructed therefore as a natural transformation between functors. W e set up such a transformation as follows: the d ata φ , ψ amount to an isomorphism Φ : ker φ ⊕ L ∼ = − → R n W e can define a map Φ λ : R n ∼ = ker φ ⊕ L ( id, λ ) / / ker φ ⊕ L ∼ = R n 15 it has inverse ( id, λ − 1 ) , so it is an automorphism, and consequently a n element of GL n ( R ) . The tra nsformation taking ( λ , ( L , φ , ψ ) ) to Φ λ is natural, and so, by Y oneda’s lemma, is a map of schemes. The motivic space G m ∧ F n − 1 + is the shea f-theoretic quotient of the inclu- sion of sheaves of simplicial sets 1 × F n − 1 → G m × F n − 1 , in particular , f n will descend to a map f n : G m ∧ F n − 1 + if and only if the c omposite 1 × F n − 1 → G m × F n − 1 → GL ( n ) is contraction to a point. On the level of f unctors, how- ever , the first scheme represents the functor taking R to pa irs ( 1 , ( L , φ , ψ ) ) , and it is immediate that Φ 1 = id, so tha t the c omposite is indeed the constant map at the identity of GL ( n ) . W e ha v e f urnished therefore the requisite map G m ∧ F n − 1 + . As for the commutativity of the diagram, we recall tha t the standard inclu- sion P n − 1 → P n represents the natural transformation taking a surjection such as R N → L to the trivial extension R n ⊕ R → R n → L . W e lift this idea to F n − 1 , given a triple ( L , φ , ψ ) ∈ F n − 1 ( R ) , one can extend the split maps φ , ψ to maps φ and ψ , where φ : R n + 1 → L and φ splits φ , simply by adding a trivial summand to R n . This furnishes a natural transformation of functors, or a map of schemes, F n − 1 → F n , that makes the diagram P n − 1 h   / / F n − 1   P n / / F n commute. One may define h : X n → X n + 1 in dia gram ( 5 ) as the map given by application of G m ∧ ( · ) + to the map F n − 1 → F n immediately constructed. The map φ : GL ( n ) → GL ( n + 1 ) is obtained as a natura l transformation by taking A ∈ GL n ( R ) and constructing A ⊕ id : R n ⊕ R → R n ⊕ R . It is now routine to verify that d iagram (5 ) commutes. The construction G m ∧ X is denoted by Σ 1 t X a nd is called the T ate suspen- sion, [V oe03]. W e have H ∗ , ∗ ( G m ; R ) ∼ = M [ σ ] σ 2 − { − 1 } σ the relation being derived in loc. cit. It is easily seen that as rings, we have H ∗ , ∗ ( G m × X ; R ) ∼ = H ∗ , ∗ ( X ; R ) ⊗ M M [ σ ] ( σ 2 − { − 1 } σ ) and that ˜ H ( Σ 1 t X ; R ) is the split submodule (ideal) generated by σ , leading to a peculiar feature of the T ate suspension Proposition 1 7. Suppose x , y ∈ ˜ H ∗ , ∗ ( X ; R ) , and that σ x , σ y are their isomorph ic images in ˜ H ∗ , ∗ ( Σ 1 t X ; R ) . Then ( σ x ) ( σ y ) = {− 1 } σ ( x y ) . 16 W e now come to the main theorem of this paper Theorem 18. The map f n induces an isomo rp hism on cohomolog y H 2 j − 1, j ( GL ( n ) ; R ) → H 2 j − 1, j ( Σ 1 t F n − 1 + ) = H 2 j − 1, j ( Σ 1 t P n − 1 + ) in dimensions ( 2 j − 1, j ) where j ≥ 1 . Proof. W e first remark that when j is large, j > n , H 2 j − 1, j ( GL ( n ) ; R ) = H 2 j − 1, j ( Σ 1 t P n − 1 ; R ) = 0 so the result holds trivia lly in this range. W e restrict to the case 1 ≤ j ≤ n . The following a re known: H ∗ , ∗ ( GL n ) = M [ ρ n , . . . , ρ 1 ] / I | ρ i | = ( 2 i − 1, i ) ˜ H ∗ , ∗ ( Σ 1 t F n − 1 + ) = ˜ H ∗ , ∗ ( Σ 1 t P n − 1 + ) = n − 1 M i = 0 σ η i M | σ | = ( 1, 1 ) , | η | = ( 2, 1 ) It suffices to show that f ∗ n ( ρ i ) = σ η i − 1 . Since f ∗ n ( ρ i ) = ˜ f ∗ n ( ρ i ) , we ma y prove this f or the map ˜ f n : G m × F n − 1 , which has the benefit of being more explicitly geometric. W e prove this by induction on n . In the case n = 1, the space F n − 1 is trivial, and X n = G m = GL ( n ) . The map f 1 is the id entity ma p , so the result holds in this case. There is a diagram of varieties G m ∧ P n − 1 + id ∧ h +   X n o o h   / / GL ( n ) φ   G m ∧ P n + X n + 1 o o / / GL ( n + 1 ) (6) as previously constructed. W e understand the vertica l map on the lef t since we can rely on the theory of ordinary Chow groups, [Ful84, chapter 1], we know that the induced map H 2 j , j ( P n − 1 ) → H 2 j , j ( P n − 2 ) is an isomorphism for j < n − 1 , and so i ∗ is an isomorphism H 2 j − 1, j ( G m × F n − 1 ) ∼ = H 2 j − 1, j ( G m × F n − 2 ) for j < n . The maps φ ∗ i are a lso isomorphism s in this range, by proposition 8 and its corollary , so the diagram implies that the result holds except possibly for f ∗ n ( ρ n ) . The a rgument we use to prove f ∗ n ( ρ n ) = σ η n − 1 is based on the composition G m × F n − 1 / / g , , GL ( n ) π / / A n \ { 0 } 17 where the map π is projection on the first column. W e write g for the composi- tion of the two maps. Since H ∗ , ∗ ( A n \ { 0 } ) ∼ = M [ ι ] / ( ι 2 ) , with | ι | = ( 2 n − 1 , n ) , and π ∗ ( ι ) = ρ n , it suffices to prove that g ∗ ( ι ) = σ η n − 1 . For the sake of carrying out computations, it is helpful to ide ntify mo- tivic cohomology and higher C how groups, e.g. ide ntify H 2 n − 1, n ( A n \ { 0 } ) and C H n ( A n \ { 0 } , 1 ) . W e can write down an explicit generator for C H n ( A n \ { 0 } , 1 ) , see corollary 11. 1, for instance the curve γ in A n \ { 0 } × ∆ 1 given by t ( x 1 − 1 ) = − 1, x 2 = x 3 = · · · = x n = 0. W e ca n also write down a n ex- plicit generator , µ , for a class σ η n − 1 0 ∈ G m × P n − 1 , writing x and a 0 , . . . , a n − 1 for the coordin ates on each, and t for the coordinate f unction on ∆ 1 , σ η n − 1 is explicitly represented by the product of the subvariety of P n − 1 given by a 0 = 1, a 2 = 0 . . . , a n − 1 = 0, which represents η n − 1 , with the cycle given by t ( x − 1 ) = − 1 on G m × ∆ 1 . W e have therefore a closed subvariety , γ , in A n \ { 0 } × ∆ 1 , and another closed subvariety µ ∈ G m × P n − 1 , representing the cohomology classes we wish to relate to one another . W e shall show that the pull-bac ks of each to G m × F n − 1 × ∆ 1 coincide. For this, it shall a gain be a dvantageous to take the functorial point of view . The variety A n \ { 0 } × ∆ 1 represents, when applied to a k -algebra R , uni- modular columns ( x 1 , . . . , x n ) t ∈ R n of height n , along with a paramete r t ∈ R . The subvarie ty γ represents the unimodular columns and pa r ameters for which t ( x 1 − 1 ) = − 1 and x i = 0 for i > 1. Note that f or such pairs, we have t ∈ R × and t 6 = 1 (since otherwise x 1 = 0) . The pull-back of γ to GL ( n ) × ∆ 1 represents pairs ( A , t ) , where A ∈ GL n ( R ) and t ∈ R , satisfying ( A ( 1 , 0 , . . . , 0 ) t , t ) ∈ γ ( R ) . W e observe that, writing e 1 = ( 1, 0, . . . , 0 ) t , this implies A e 1 = ( 1 − t − 1 ) e 1 . Note further that 1 − t − 1 6 = 1. Pulling γ back a second time, to G m × F n − 1 × ∆ 1 , we obtain the set of triples ( λ , ( L , φ , ψ ) , t ) , where λ ∈ R × , ( L , φ , ψ ) is a split n - generated line bun- dle, and t is a parameter , and where the invertible linear tra nsformation Φ λ : ker φ ⊕ L → ker φ ⊕ L along with the pa rameter t lies in the pull-back of γ ( R ) to GL n ( R ) . Decomposing e 1 = v + w , where v ∈ ker φ and w ∈ L , we see that Φ L ( e 1 ) = v + λ w = ( 1 − t − 1 ) e 1 = ( 1 − t − 1 ) ( v + w ) , which by uniqueness of the decomposition, forces λ = ( 1 − t − 1 ) , and v = 0 . Consequently , L is the rank-1 split subbundle of R n generated by e 1 , and we have t ( λ − 1 ) = − 1 . On the other hand, the variety µ ⊂ G m × P n − 1 × ∆ 1 represents triples ( λ , L , t ) ∈ R × × P n − 1 ( R ) × R , where L is exac tly the ra nk-1 free module ge n- erated by e 1 , and where t ( λ − 1 ) = − 1. The pull-back of µ to G m × F n − 1 × ∆ 1 coincides with that of γ , as cla imed. W e are now in a position to pa y off at last the d e bt we owe regarding the product structure of H ∗ , ∗ ( W ( n , m ) ; R ) . Theorem 19. The cohomolog y of W ( n , m ) has t he following presentation as a graded- commutativ e M R -algebra: H ∗ , ∗ ( W ( n , m ) ; R ) = M R [ ρ n , . . . , ρ n − m + 1 ] I | ρ i | = ( 2 i − 1, i ) 18 The ideal I is generated by relations ρ 2 i − { − 1 } ρ 2 i − 1 , wh ere { − 1 } ∈ M 1,1 R is the image of − 1 ∈ k ∗ = M Z under th e m ap M Z → M R . Proof. It suffices to deal with the ca se R = Z . It suffices also to consider only the case m = n , since we can use the inclusion H ∗ , ∗ ( W ( n , m ) ) ⊂ H ∗ , ∗ ( GL ( n ) ) to deduce it for all n , m . W e have proved everything a lready in propositi on 5, except that in the re- lation ρ 2 i − a ρ 2 i − 1 , we were unable to show a was nontrivial. W e consider the map G m × F n − 1 + → GL ( n ) , which induces a map of r ings on cohomolog y . In the induced map , we have ρ i 7→ σ η i − 1 , and so ρ 2 i 7→ − σ 2 η 2 i − 2 = { − 1 } σ η 2 i − 2 . Since this is nontrivial if 2 i − 2 ≤ n − 1, it follows that ρ 2 i is similarly nontriv- ial. In the case n = m this result, although computed by a different method, appears in [Pus04]. W e can compute the action of the reduced power opera tions of [V oe03] on the cohomology H ∗ , ∗ ( W ( n , m ) ; Z / p ) by means of the comparison theorem. Theorem 20. Supp ose the ground-field k has characteristic different from 2 . Repre sent H ∗ , ∗ ( W ( n , m ) ; Z /2 ) as M 2 [ ρ n , . . . , ρ n − k + 1 ] / I . Th e even motivic Steenr od square s act as Sq 2 i ( ρ j ) = ( ( j − 1 i ) ρ j + i if i + j ≤ n 0 otherwise The odd squares vanish for dimensional reasons. Theorem 21. Let p be an odd p rime and suppose the ground-field h a s ch aracteristic different from p. Represent H ∗ , ∗ ( W ( n , m ) ; Z / p ) as M p [ ρ n , . . . , ρ n − k + 1 ] / I . The reduced p ower operations act as P i ( ρ j ) = ( ( j − 1 i ) ρ i p + j − i if i p + j − i ≤ n 0 otherwise The Bockstein vanishes o n these classes for dimensional reasons. Observe that in both ca ses, since the cohomology ring is multiplicatively generated by the ρ j , the given calculations suffice to deduce the reduced- power operations in full on the appropriate cohomology ring. Proof. W e prove only the case of p = 2, the other cases being much the same. W e observe that Sq 2 j is honest squaring on H 2 j , j ( P n ; Z /2 ) , on the classes η i ∈ H 2 i , i ( P n ; Z /2 ) the Bockstein va nishes, and as a consequence the expected Cartan formula obtains for c a lculating η i + i ′ , it is a simple matter of induction to show that S q 2 i ( θ j ) = ( j i ) θ j + i . There is an inclusion of H ∗ , ∗ ( W ( n , m ) ; Z /2 ) ⊂ H ∗ , ∗ ( GL ( n ) ; Z /2 ) arising from the projection map, see proposition 7 . It suffices therefore to compute 19 the ac tion of the squares on H ∗ , ∗ ( GL ( n ) ; Z /2 ) . Using the p revious propositio n and the dec omposition in equation (4 ), we have isomorphisms H 2 n − 1, n ( GL ( n ) ; Z /2 ) ∼ = H 2 n − 1, n ( Σ 1 t P n − 1 ; Z /2 ) The reduced power operations are stable not only with respect to the simplicial suspension, but are a lso stable with respect to the T ate suspension. This a llows a transfer of the calcula tion on P n to the ca lculation on GL n via the comparison of theorem 18 T o be precise, we have f ∗ n ( Sq 2 i ρ j ) = Sq 2 i f ∗ n ( ρ j ) = Sq 2 i σ η j − 1 = σ Sq 2 i η j − 1 = ( ( j − 1 i ) σ η j + i − 1 = ( j − 1 i ) f ∗ n ( ρ j + i ) if i + j ≤ n 0 otherwise Since f ∗ n is an isomorphism on these groups, the result follows. References [Ada62 ] J. F . Ad ams. V ector fields on spheres. Ann. of Math. (2) , 75:6 03–63 2, 1962. [Blo86] Spencer Bloch. Algebraic cycles and higher K - theory . 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