Zero cycles on homogeneous varieties

ZERO CYCLES ON HOMOGENEOUS V ARIETIES

DANIEL KRASHEN

Abstract.In this paper we study the group A0(X)of zero dimen-

sional cycles of degree0modulo rational equivalence on a projective

homogeneous algebraic variety X.To do this we translate rational

equivalence of0-cycles on a projective variety into R-equivalence

on symmetric powers of the variety.For certain homogeneous va-

rieties,we then relate these symmetric powers to moduli spaces

of′e tale subalgebras of central simple algebras which we construct.

This allows us to show A0(X)=0for certain classes of homoge-

neous varieties for groups of each of the classical types,extending

previous results of Swan/Karpenko,of Merkurjev,and of Panin.

1.Introduction

The study of algebraic cycles on quadric hypersurfaces has turned out to be unreasonably successful in its applications to quadratic forms. Karpenko,Izhboldhin,Rost,Merkurjev,Vishik and Voevodsky,to name a few,have used and developed the theory of algebraic cycles in order to solve a number of outstanding conjectures,most notably Voevodsky’s recent proof of the Milnor conjecture.

In part inspired by these great successes,there is much interest in studying algebraic cycles on and motives of general projective homo-geneous varieties,beyond the quadric hypersurfaces which arise in ap-plications to quadratic forms.Signi?cant progress has been made by various authors in this direction([Kar00,Bro,CGM05,SZ]). Despite the progress in understanding general projective homoge-neous varieties,the Chow groups of0-dimensional cycles for such va-rieties have remained somewhat mysterious.Whereas computations have been performed in various cases(see for example Swan[Swa89] and Merkurjev[Mer95]),the topic has so far resisted general statements or conjectures.

2DANIEL KRASHEN

In this paper,we compute the Chow group of zero cycles on various projective homogeneous varieties by showing that the group A0(X)of0 dimensional cycles of degree0modulo rational equivalence is trivial in many cases.We give examples of this for certain homogeneous varieties for groups of each of the classical types A n,B n,C n,D n.

More precisely,in the A n case(theorem7.3),we show that A0(X)= 0for X a Severi-Brauer variety(recovering a result of Panin),and for certain cases when X is a Severi-Brauer?ag variety.In all of these examples,we assume that either F is perfect or char(F)doesn’t divide the index of the underlying central simple algebra.

In the B n and D n cases(theorem8.8),we show that A0(X)=0 for any(orthogonal)involution variety X,assuming that char(F)=2. Involution varieties are twisted forms of quadric hypersurfaces intro-duced in[Tao94],and are de?ned in section8.This generalizes previous results of Swan([Swa89])and Karpenko who proved this when X is a quadric hypersurface,and Merkurjev([Mer95])who proved this when X has index2(see section2for the de?nition of index).

In the C n case(theorem8.13),we show that A0(X)=0for X= V2(A,σ)a2’nd generalized involution variety for a central simple alge-bra A with symplectic involutionσ(see section8)when ind(X)=1or2 and char(F)=2.This gives the?rst nontrivial computations of this group for such varieties.The case of higher index is still open.

To obtain our results we relate the Chow group of0-dimensional cycles to the more geometrically naive notion of R-equivalence(i.e. connecting points with rational curves)on symmetric powers of the original variety,along with the slightly weaker notion of H-equivalence which we introduce.This is explained in section3.Although in some sense,this idea is not new-various aspects of this idea over the complex ?eld appear in[Sam56],and similar ideas were used in Swan’s paper ([Swa89]),our formulation of this principle allows us to more fully exploit its uses.

From here,we show that the symmetric powers of certain homoge-neous varieties may be related to spaces which parametrize commuta-tive′e tale subalgebras in a central simple algebra.To make this con-nection precise,we de?ne moduli spaces of′e tale subalgebras in section 5.These spaces are very interesting in their own right,as many open questions in the area of central simple algebras concern the existence and structure of certain types of sub?elds in a division algebra.In sections6.1,6.2and6.3we determine show that in certain cases these moduli spaces are R-trivial,and in sections7and8we apply this to determining the Chow group of zero cycles for certain homogeneous varieties.

ZERO CYCLES ON HOMOGENEOUS V ARIETIES3 There are various known results concerning the group A0(X)for geo-metrically rationally connected varieties over certain?elds,particularly the?nite,local,and global cases(see[KS03],[Kol99],and[CT05]).For example,Colliot-Th′e l`e ne has conjectured that the torsion in CH0(X) is?nitely generated when F is p-adic,and has obtained positive results in certain cases([CT05]).

Over an arbitrary ground?eld,it is clear that the geometrically rationally connected varieties may have very complicated groups of zero cycles,and so it appears di?cult to know which classes of varieties have A0(X)=0.Even restricting to projective homogeneous varieties is not su?cient for this.For example,A.Vishik has pointed out the following example using a result of Karpenko and Merkurjev([KM90]): Proposition1.1.One may?nd a?eld F and a quadratic form q over a vector space V/F such that if we let X be the variety of2-dimensional totally isotropic subspaces of V,the group of CH0(X)is in?nitely generated(and therefore so is A0(X)).

Proof.For a given quadratic form q on V/F we may construct the variety X as above.Let Q be the quadric hypersurface in P(V)de?ned by the vanishing of q.Thinking of points in Q as isotropic lines in V,we may construct a Chow correspondence from X to Q by setting Z∈X×Q to be the subvariety described as

{(x,q)∈X×Q|q?x}.

This de?nes a homomorphism CH0(X)→CH1(Q).

In[KM90],the authors exhibit a quadratic form q on a7dimensional vector space such that the associated5-dimensional quadric Q has an in?nite family of independent nontrivial torsion cycles z i∈A4(Q)= CH1(Q).One may check by inspection that these cycles are in the image of the Chow correspondence above,and therefore give in?nitely many independent nontrivial elements in CH0(X).

I am grateful to A.Merkurjev who suggested this problem to me while I was a VIGRE assistant professor at UCLA,and whose helpful comments on various drafts of this paper were extremely useful.I would also like to thank D.Saltman who suggested to me the idea of using Pfa?ans to prove theorem6.7,and I.Panin who explained to me how to concretely think of the varieties associated to symplectic involutions.I am also grateful for the comments of an anonymous referee who reccomended the use of Hilbert schemes after reading a previous version of this paper.The use of Hilbert schemes of points has considerably cleaned up and shortened the exposition of the paper,as

4DANIEL KRASHEN

well as done away with almost all assumptions about the characteristic of the ground?eld.

After the appearance of this paper in preprint form,Viktor Petrov, Nikita Semenov and Kirill Zainoulline have subsequently applied these methods to compute groups of0cycles on homogeneous varieties for various exceptional groups([PSZ]).

2.Preliminaries and notation

Let F be a?eld.All schemes will be assumed to be separated and of?nite type over a?eld(generally F unless speci?ed otherwise).By a variety,we mean an integral scheme.If Z is a closed subscheme of a scheme X,we let[Z]denote the corresponding cycle.Suppose X and Y are schemes over F.For an extension?eld L/F we denote by X L the?ber product X×Spec(F)Spec(L).For a morphism f:X→Y, we write f(L):X(L)→Y(L)for the induced map on the L-points. We denote by F(X)the function?eld of X.We de?ne the index of a scheme X,as

ind(X)=GCD{[L:F]|L/F?nite?eld extension and X(L)=?}. If A is a central simple F algebra,we recall that its dimension is a square,and we de?ne the degree of A,deg(A)=

ZERO CYCLES ON HOMOGENEOUS V ARIETIES5 For a scheme X,we de?ne Z(X)to be the set of0dimensional cycles on X and Z n eff(X)to the the subset of degree n e?ective cycles in Z(X).We have a set map X[n](F)→Z n eff(X)de?ned by taking a subscheme z?X of degree n to its fundamental class[z].This gives a bijection between the cycles which are a disjoint union of spectrums of separable?eld extensions of F,and points in X(n)(F)?X[n](F).We will occasionally have to make use of cycles of other dimensions,and we will use the notation C i(X)to represent the group of i-dimensional cycles on X.

We say that a?eld L is prime to p closed if every?nite algebraic extension E/L has degree a power of p.An algebraic extension L/F is called a prime to p closure if for every?nite subextension F?L0?L, [L:F]is prime to p,and L is prime to p-closed.

Lemma2.1.Suppose X is a scheme over F with ind(X)=n,where either char(F)doesn’t divide n or F is perfect.Then X(n)(F)= X[n](F).

Proof.Given a point x∈X[n](F),x corresponds to a?nite subscheme Spec(R)?X,where R is a commutative F-algebra of dimension n. By taking a quotient by a maximal ideal of R,we obtain subscheme Spec(L)?X,L a?eld of degree at most n.Since ind(X)=n,we immediately conclude Spec(R)=Spec(L)and so R is a?eld.By our hypothesis,R is a separable?eld extension,and so we see that x corresponds to a point in X(n)(F)as claimed. Lemma2.2.Let X be a proper variety such that for any extension?eld

L/F,X(L)=?implies A0(X L)=0.If A0(X F

p )=0for each prime p

dividing ind(X)and every prime to p closure F p/F then A0(X)=0. Proof.Suppose?rst that p does not divide ind(X).It then follows that

X(F p)=?,and hence by the hypotheses,A0(X F

p )=0.Therefore,the

conditions of the lemma imply A0(X F

p )=0for all p.

We will show that A0(X)=0by showing that the degree map

deg:CH0(X)→Z is injective.Let deg p be the degree map af-ter?bering with F p.Consider the natural mapπp:X F

p→X, which is a?at morphism.Letα∈ker(deg),and assume that deg p

is injective.In this case,π?p(α)∈ker(deg p)=0.This means that we may?nd irreducible curves Z i?X F p,and a rational function r i∈R(Z i),such that div r i=α.But since these subvarieties Z i, and functions r i involve only a?nite number of coe?cients,they are de?ned over an?nite degree intermediate?eld E,where F?E?F p. But now we have that ifπE:X E→X is the natural map,then π?Eα=0.ButπE?π?Eα=[E:F]αtells us that[E:F]α=0,

6DANIEL KRASHEN

and so[E:F]∈ann Z(α).Therefore,since[E:F]is prime to p, ann Z(α)∈[E:F].But because this holds for every prime p,we must have that ann Z(α)is not contained in any maximal ideal of Z and hence ann Z(α)=Z.But this implies thatα=0.

3.Cycles and equivalence relations

Let X be a scheme.We say that two points p1,p2∈X(F)are el-ementarily linked if there exists a rational morphismφ:P1 X such that p1,p2∈im(φ(F)).We de?ne R-equivalence to be the equiv-alence relation generated by this relation.Let X(F)/R denote the set of equivalence classes of points in X(F)under R-equivalence.We say that X is R-trivial in case X(F)/R is a set of cardinality1.

If f:X→Y is a morphism,we obtain a map of sets X(F)/R→Y(F)/R which we denote by f R.Note that this is well de?ned,since if p,q∈X(F)are elementarily linked via a rational map P1 X, then the composition P1 X→Y shows that f(p)and f(q)are elementarily linked as well.

Given points x,y∈X[n](F),we say that x and y are elementar-ily H-linked if there is a morphismφ:P1→X[n]such that[φ(0)]= [x],[φ(1)]=[y].We de?ne H-equivalence,denoted～H,to be the equiv-alence relation generated by elementary H-linkage.We say that an open subscheme U?X[n]is H-trivial if the H-equivalence classes U(F)/H form a set with one element.Note that for x,y∈X(n),[x]=[y]if and only if x=y.

We remark that the notions of R and H equivalence carry over in relative versions for any base scheme S by replacing P1F with P1S.In particular,if S～=Spec(⊕E i)where each E i is a?eld,it is easy to check that two points are R or H equivalent if and only if the corresponding points are equivalent with respect to each E i.

The?rst lemma we prove gives some justi?cation for considering H-equivalence:

Lemma3.1.Suppose X is a projective variety,andα,β∈X[n].Ifαandβare H-equivalent,then[α]and[β]are rationally equivalent. Proof.Without loss of generality,we may assume thatαandβare elementarily linked,and choose a morphismφ:P1→X[n]connecting these points(we may assumeφis a morphism and not just a rational map since the Hilbert scheme is proper).Pulling back the universal family on X[n]alongφ,we obtain a?at family F?X×P1of0 dimensional subvarieties of X of degree n on P1.By[Ful98],section1.6, any two specializations of this to points in P1are rationally equivalent. In particular,αandβare rationally equivalent.

ZERO CYCLES ON HOMOGENEOUS V ARIETIES7 Lemma3.2.Suppose X is a projective variety over F with dim(X)≥1.Then the map X[n](F)→Z n eff(X)is surjective.

Proof.Let z?X be an irreducible e?ective0cycle,say z～=Spec(L) for L/F a?nite?eld extension.It su?ces to show that for any r>1, there is a subscheme z?X with[ z]=r[z].Without loss of general-ity,we may assume that X=Spec(R)is a?ne.Let m?R be the maximal ideal corresponding to z.Since dim(R)≥1,we know that length(R/m k)is unbounded as k increases.In particular,there exists k>0such that R/m k has length≥r and R/m k?1has length

Corollary3.3.There is a natural bijection X[n](F)/H=Z n eff(X)/H. Proof.This follows immediately from lemma3.2. It is useful to have a relative version of lemma3.2for?at cycles over a curve:

Lemma3.4.Suppose X/F is a projective variety,C/F a curve and α∈C1(X×C)is an e?ective cycle such that every component of the support ofαis?nite and?at over C.Thenα=[Z]for some subscheme Z?X×C with Z→C?at.

Note that by the universal property of the Hilbert scheme,this Z must come from a morphism C→X[n]by pulling back the universal family.

Proof.Consider the restrictionα of the cycleαto X F(C)(the generic ?ber of the family X×C).We haveα ∈Z n eff(X F(C))for some n,and so we may use lemma3.2to?nd a subscheme Z ∈X F(C)representing it.We may interpret Z as a point in X[n]F(C)(F(C))=X[n](F(C)),and therefore obtain a rational morphism Spec(F(C))→X[n].Since C is a curve and X[n]is proper,we may complete this to a morphism C→X[n],and hence obtain a family Z?X×C.

8DANIEL KRASHEN

By construction it is clear that[Z]andαboth have the same re-striction to X F(C).We may therefore?nd an open subset U?C such that the cyclesαand[Z]are equal.From the fundamental sequence

C1(X×(C\U))→C1(X×C)→C1(X×U)→0,

we see that the di?erence cycle[Z]?αis supported entirely on C1(X×(C\U)).But since the support of each cycle is?at over P1,there cannot be any components supported in over P1\U,and therefore[Z]?α=0 as claimed.

If X/F is a projective variety and L/F a?nite?eld extension of degree n.We de?ne a map of sets

H:X(L)→Z n eff(X)

(φ:Spec(L)→X)→φ?[Spec(L)]

Lemma3.5.Let X/F be a projective variety,L/F a?nite?eld ex-tension,and suppose we have x,y∈X[n](L)with x～H y.Then H(x)～H H(y).In the case x,y∈X(n)(L)are elementarily linked, so are H(x)and H(y).

Proof.It su?ces to consider the case where x and y are elementarily H-linked.Therefore,we may reduce either to the case that x～R y or [x]=[y].If[x]=[y],we may write[x]=[y]=n[z]for z～=Spec(E)an irreducible subscheme,and E?L a sub?eld,n=[L:E].Therefore x and y may only di?er by an element of Gal(L/F)and so H(x)=H(y). We may therefore assume that x and y are elementarily linked. Chooseφ:P1E→X withφ(0)=x,φ(∞)=y,and letρ:P1E→P1 be the natural covering.Since the cycle(φ×ρ)?[P1E]∈C1(X×P1) satis?es the conditions of lemma3.4,it follows from lemma3.4and the remark just following it that we can?nd a morphismψ:P1→X[n], where n=[E:F]such that if C?P1×X is the corresponding family, [C]=(φ×ρ)?[P1E].

If we denote by i p:Spec(F)→P1,p=0,∞the inclusion of points on P1,and consider the pullback diagram:

(1)Spec(E)P1E

φ×ρ

Spec(F)i0

ZERO CYCLES ON HOMOGENEOUS V ARIETIES9 We have H(x)=x?(Spec(E))=x?i!0[P1E]which may be rewritten using[Ful98],theorem6.2as i!0(φ×ρ)?[P1E]=i!0[C]which by[Ful98], section10.1is the same as[i?10(C)]=[ψ(0)],and similarly H(y)= [ψ(∞)],showing that these points are elementarily H-linked. Suppose X/F is a projective variety.Given a zero-dimensional sub-scheme i:z →X[n],we obtain a family F?z×X.We de?ne the cycle[n]?(z)∈Z(X)by the formula[n]?(z)=π2?[F].

Lemma3.6.Let X/F be a projective variety.Then the map X[n][m](F)→

Z nm eff (X)de?ned by mapping a degree m scheme z?X[n]to[n]?[z]

passes to H-equivalence.

Proof.To show this,it su?ces to show that if we haveφ:P1→X[n][m],φ(0)=z,φ(1)=z then[n]?[z]～H[n]?[z ].To see this,we will construct a morphismψ:P1→X[mn]such that[ψ(0)]=[n]?[z]and [ψ(∞)]=[n]?[z ].By the universal property of the Hilbert scheme,this means that we really need to construct a family W?X×P1whose specializations over0and∞are[n]?[z]and[n]?[z ]respectively. Consider the family corresponding to the mapφ.This is a subscheme Z?X[n]×P1with?bers z and z over the points0and∞respectively. Pulling back the universal family on X[n]via the morphism Z→X[n], we obtain a family W →X×P1×Z,which is degree mn over P1,and such that each component of W is?at over P1.By lemma3.4,we may ?nd W?X×P1such that[ W]=π?[W],whereπ:X×P1×Z→X×P1 is the projection.It is now routine to check that the?bers over0and ∞of W give subschemes whose cycles are equal to[n]?[z]and[n]?[z ] respectively. De?nition3.7.For a scheme X and positive integers n,m,we de?ne X(n,m)to be the?ber product:

X(n,m)S m(S n X)

S nm X

Lemma3.8.Suppose F is prime to p closed,and X/F a quasipro-jective variety.Then the natural morphismπ:X(n,m)→X(nm)is surjective on F-points whenever n,m are powers of p.

Proof.Since we may identify S m S n X with the quotient

(X nm)/ (S n)m S m ,

10DANIEL KRASHEN

it follows that the degree of the map πis nm !

p ?1where v p is the p -adic valuation).Since πfactors through the ′e tale map X m +n ?→X (nm )it is also ′e tale.In

particular,if x ∈X (nm )(F ),the ?ber π?1(x )is ′e tale over Spec (F )and hence the spectrum of a direct sum of separable ?eld extensions ⊕L i .Since the total degree of this extension is prime to p ,there must be at least one of the ?eld extensions L i whose degree is not a multiple of p .But since F is prime to p closed,this implies that L i =F ,and so the ?ber has an F -point as desired. Corollary 3.9.Let X/F be a projective variety.There is a natural map X [n ][m ](F )/H →X [nm ]/H .In the case ind (X )=mn ,we have a natural map X (n )(m )(F )/H →X (mn )/H .If we also have that F is prime to p closed and m,n are p -powers then the map X (n )(m )(F )/H →X (mn )/H is surjective.

Proof.This is immediate from lemmas 2.1,3.6,3.8and corollary 3.3. Lemma 3.10.Let F be prime to p closed,and suppose X/F is a

projective variety.Fix a p -power n .Suppose X (n )L is H-trivial for every

?nite ?eld extension L/F of p -power degree m .Then X (nm )is H-trivial.In particular,we show that if α∈X (n )(F ),then for all β∈X (mn )(F ),we have β～H m [α].

Proof.By corollary 3.9,it is su?cient to show that X (n )(m )is H-trivial.Choose α∈X (n )(F ),which is nonempty by the hypothesis.We will show that given β∈X (n )(m )(F ),we can write [β]～H m [α].We may write β=H ( β

)for some β∈X (n )(E )where E/F is a degree m ′e tale extension.Choose α∈X (n )(F ),and de?ne α∈X (n )(E )via composing αwith the structure morphism Spec (E )→Spec (F ).We

then have H ( α)=n [α].Since X (n )E is H-trivial, α～H β

and by lemma 3.5,m [α]～H [β]as desired. Corollary 3.11.Suppose X/F is a projective variety with F is prime

to p -closed and such that for every ?nite ?eld extension L/F ,X (indX L

)L is H-trivial.Then for every p -power n ≥ind (X ),X (n )is H-trivial.

Proof.By lemma 3.10,it su?ces to show that X ind (X )L is H-trivial,

where [L :F ]=n/ind (X ).We prove this by induction on ind (X ).If ind (X )=1,the hypothesis implies that X E is R-trivial for every extension E/F and the conclusion follows from lemma 3.10(setting n =1in the statement of the lemma).

ZERO CYCLES ON HOMOGENEOUS V ARIETIES11 For the general induction case,we either have ind(X L)=ind(X) or ind(X L)

implies X ind(X)

L =X ind(X L)

L

is H-trivial.In the latter case,we have

m(ind(X L))=ind(X)for some p-power m,and by lemma3.10,to

show that X ind(X)

L is H-trivial,it su?ces to show that X ind(X L)

E

is H-

trivial for E/L a degree m extension.Therefore,the result follows from the induction step. Theorem3.12.Suppose X/F is a projective variety with F is prime to p closed,p=char(F)or F perfect and such that(X L)(ind(X L))H-trivial for every?nite?eld extension L/F.Then A0(X)=0. Proof.Letα∈X(i),where i=ind(X).Since by assumption on the characteristic every prime cycleβis represented by a point in X(n)(F) for some p-power n,it follows from corollary3.11and lemma3.10that [β]～H n

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similarly that f (x )～=x ,since otherwise the image would have smaller degree,contradicting ind (X )=ind (Y ).Therefore,f induces a map X (m )→Y (m ).We note that this may also be seen as the rational map induced by the morphism S m X →S m Y .Since every x ∈X (m )is of the form Spec (L )for L/F a degree m ?eld extension,we have a commutative diagram such that the vertical arrows are surjective: [L :F ]X (L )H

[L :F ]

Y (L )

H Y (m )(F )

It is clear by tracing the diagram that the map on the bottom must be surjective,and it is also clear that it must preserve R-equivalence classes.

We need only show therefore that the map is injective on R-equivalence classes in the case X and Y are projective.Without loss of generality,we may assume that we have x,x ∈X (m )(F )such that y =f (x )and y =f (x )are elementarily linked.Choose P 1→Y (m )linking y and y .In the case that y =y ～=Spec (L ),we have that x,x may be lifted to elements of X (L )which both lie in the same ?ber over a point in Y (L ).Since by hypothesis,the ?bers of f are R-trivial,we therefore have x ～R x due to the fact that H preserves R-equivalence in this case.

We are therefore done if we may show that there is a morphism P 1→X [m ]connecting some point in the ?ber over y with a point over the ?ber of y .We begin by choosing a morphism φ:P 1→Y [m ]connecting y and y ,and consider the pullback of the universal family.This gives a curve C ?Y ×P 1such that the projection C →P 1is degree m and such that the ?bers over 0and ∞are equal to y and y respectively.If we consider the projection morphism C →Y restricted to the generic point Spec (F (C )),we obtain a point in Y (F (C )).Since f has R-trivial ?bers,the ?ber over this point in X is nonempty and hence there is a morphism Spec (F (C ))→X such that its composition with f gives the original map Spec (F (C ))→Y .Let C →C be the normalization of C .Since X is projective,we get a morphism C →X such that the digram

C

ZERO CYCLES ON HOMOGENEOUS V ARIETIES13 is commutative.In particular,if we let D?X×P1be the image of C,then D is birational to C and by the universal property of X[m], de?nes a morphism P1→X[m].If we let U?P1be the open set on which C→P1is′e tale,one may check that we have a commutative diagram

Uφ|U

14DANIEL KRASHEN

type (n 1,...,n k ),denoted V n 1,...,n k (A ),is the variety whose points cor-respond to ?ags of ideals I n 1?I n 2?···?I n k ,where I n i has reduced dimension n k .More precisely V n 1,...,n k (A )represents the following func-tor:V n 1,...,n k (A )(R )= (I 1,...,I k ) I i ∈Gr (n i n,A )(R )is a right ideal of A R and I i ?I i +1

In particular,in the case k =1,the variety V i (A )is the i ’th gen-eralized Severi-Brauer variety of A ([Bla91]),which parametrizes right ideals of A which are locally direct summands of reduced rank i .The same de?nition generalizes easily to sheaves of Azumaya algebras of constant degree over a base scheme S .

Theorem 4.2.Suppose A is a central simple F -algebra.Then the Severi-Brauer ?ag variety V n 1,...,n k (A )is stably R-isomorphic to V d (D ),where D is any central simple algebra Brauer equivalent to A and

d =GCD {n 1,...,n k ,ind (A )}.

This result relies on a number of intermediate results:

Proposition 4.3.Suppose A is a central simple F -algebra,and we have positive integers n 1<···

Corollary 4.4.Suppose A is a central simple F -algebra,and we have positive integers n such that ind (A )|n .Then any two F -points in the generalized Severi-Brauer variety V n (A )are elementarily linked.In particular,V n (A )is R-trivial.

Lemma 4.5.Suppose A =End r.D (V )for some F -central division algebra D ,where V is a right D -space.Let i =deg (D ),and let I ?A be a right ideal of reduced dimension ri (note that every ideal has reduced dimension a multiple of i ).Then there exists a D -subspace W ?V of dimension r such that I =Hom r.D (V,W )?End r.D (V ).

Equivalently,writing A =M m (D ),we may consider I to be the set of matricies such that each column is a vector in W .

Proof.Choose a right ideal I ?A ,and let W =im (I ).It is enough to show that I =Hom r.D (V,W ).The claim concerning reduced di-mension will follow immediately from a dimension count.Since it is clear by de?nition that I ?Hom r.D (V,W ),it remains to show that the reverse inclusion holds.We do this by showing that I contains a basis for Hom r.D (V,W ).Let e 1,...,e m be a basis for V ,and f 1,...,f r be a basis for W .We must show that the transformation T i,j ∈I where

ZERO CYCLES ON HOMOGENEOUS V ARIETIES 15

T i,j (e k )=f j δi,k .Since f i ∈im (i )for some i ∈I ,we know that there exists v ∈V such that i (v )=f j .Now de?ne a ∈End r.D (V )to be given by a (e k )=vδi,k .Now,ia (e k )=f j δi,k and ia ∈I as desired. proof of 4.3.Let A =M m (D )for some division algebra D with deg (D )=i =ind (A ),mi =n =deg (A ).We may therefore write A =End r.D (V )for some right D -space V of dimension m .Choose ?ags of ideals

(I 1,...,I k ),(I 1,...,I k )∈V n 1,...,n k (A )(F ).

We will show that there is a rational morphism f :A 1 V n 1,...,n k (A ),such that f (0)=(I 1,...,I k )and f (1)=(I 1,...,I k ).By lemma 4.5,we may write I j =Hom (V,W j ),I j =Hom (V,W j ).

Choose bases w j,1,...,w j,l j for W j and w j,1,...,w j,l j for W j ,where

l j =n j /i .De?ne morphisms f j,l :A 1→V by f j,l (t )=w j,l t +w j,l (1?t ).

We may combine these to get rational morphisms A 1 Gr (n j n,A )by taking t to the n j n -dimensional space of matricies in M m (D )whose columns are right D -linear combinations of the vectors w j,1t +w j,1(1?

t ),...,w j,l j t +w j,l j (1?t ).By 4.5,this corresponds to a rational mor-phism f j :A 1→V n j (A ).One may check that f j (0)=I j and f j (1)=I j .

Further,for any t ,f j (t )?f j +1(t ).Therefore,we may put these together to yield a rational morphism f :A 1 V n 1,...,n k (A )with f (0)=(I 1,...,I k )and f (1)=(I 1,...,I k ). Remark 4.6.In fact the proof above shows that if we are given n

proof of theorem 4.2.Let X =V n 1,...,n k (A )and Y =V d (D ).Consider the product variety X ×Y together with its natural projections π1,π2onto X and Y respectively.I claim that both projections have R-trivial ?bers,which would prove the theorem.

Suppose we have x :Spec (L )→X or x :Spec (L )→Y .This would imply that X (L )=?or Y (L )=?,and in either case this in turn says that ind (A )|d .Since the scheme theoretic ?ber over x is isomorphic to either X L or Y L respectively,we know that since ind (A L )=ind (D L )|d that the ?bers are R-trivial by proposition 4.3. De?nition 4.7.Suppose A is a central simple algebra and I ?A is a right ideal of reduced dimension l .Given integers n 1,...,n k

16DANIEL KRASHEN

For these varieties,we have a theorem which generalizes a result from [Art82]on Severi-Brauer varieties:

Theorem4.8.Suppose A is a central simple algebra.Let I?A be a right ideal of reduced dimension l.Then there exists a degree l algebra D which is Brauer equivalent to A such that for any n1,...,n k

V n

1,...,n k (I)=V n

1,...,n k

(D)

In order to prove this theorem,we will use the following lemma: Lemma4.9.Let A be an Azumaya algebra with center R,a Noetherian commutative ring,and suppose that I is a right ideal of A such that A/I is a projective R-module.Then there exists an idempotent element e∈I such that I=eA.

Proof.Since A/I is projective as an R-module,by[DI71]it is also a projective(right)A-module.This implies that the short exact sequence

0→I→A→A/I→0

splits as a sequence of right A-modules,and therefore,there exists a right ideal J?A such that A=I⊕J.We may therefore uniquely write1=e+f,with e∈I and f∈J.Now,

e=(e+f)e=e2+fe.

Since e2∈I,fe∈J this gives fe∈I∩J=0and so e2=e.Finally, I=(e+f)I=eI+fI,and this gives fI∈J∩I=0.Consequently, we have eA?I=eI?eA so I=eA as desired. proof of theorem4.8.By4.9,we know that I=eA for some idempo-tent e∈A.Set D=eAe.

Let X I=V n

1,...,n k (I),and X D=V n

1,...,n k

(D).To prove the theo-

rem,we will construct mutually inverse maps(natural transformations of functors)φ:X I→X D andψ:X D→X I.For a commutative Noetherian F-algebra R,and for J=(J1,...,J k)∈X I(R),we de?ne φ(J)=(J1e,...,J k e)=(eJ1e,...,eJ k e).For K=(K1,...,K k)∈X D(R),de?neφ(K)=(K1A R,...,K k A R).To see that these are mu-tually inverse,we need to show that for each i=1,...,k,we have J i eA=J i and that K i A R e=K i.For the second we have

K i A R e=K i e=K i

since K i?eA R e.For the?rst,we note that by the lemma,we have J i=hA i for some idempotent h.But then

J i?J i eA R=J i I?J2i=hA R hA R=hA R=J i

and so J i=J i eA R and we are done.

ZERO CYCLES ON HOMOGENEOUS V ARIETIES17

5.Moduli spaces of′e tale subalgebras

Let S be a Noetherian scheme,and let A be a sheaf of Azumaya alge-bras over S.Our goal in this section is to study the functor′e t(A),which

associates to every S-scheme X,the set of sheaves of commutative′e tale

subalgebras of A X.We will show that this functor is representable by

a scheme which may be described in terms of the generalized Severi-Brauer variety of A.

Unless said otherwise,all products are?ber products over S.If X is

an S-scheme with structure morphism f:X→S,then we write A X

for the sheaf of O X-algebras f?(A).For a S-scheme Y,we occasionally write Y X for Y×X,thought of as an X-scheme.

Every sheaf of′e tale subalgebras may be assigned a discrete invariant,

which we call its type,and therefore our moduli scheme is actually a

disjoint union of other moduli spaces.

To begin,let us de?ne the notion of type.

De?nition5.1.Let R be a local ring,and B/R an Azumaya algebra.

If e∈B is an idempotent,we de?ne the rank of e,denoted r(e)to be the reduced rank of the right ideal eB.

Let E be a sheaf of′e tale subalgebras of A/S,and let p∈S.Let

R be the local ring of p in the′e tale topology(so that R is a strictly Henselian local ring).Then taking′e tale stalks,we see that E p is an ′e tale subalgebra of A p/R,and it follows that

E p=⊕k i=1Re i,

for a uniquely de?ned collection of idempotents e i,which are each minimal idempotents in S p.

De?nition5.2.The type of E at the point p is the unordered collec-

tion of positive integers[r(e1),...,r(e m)].

De?nition5.3.We say that E has type[n1,...,n m]if if has this type

for each point p∈S.

Remark5.4.Since1= e i,the ideals I i=e i A span A.Further it is easy to see that the ideals I i are linearly independent since e i a=e j b implies e i a=e i e i a=e i e j b=0.We therefore know that the numbers making up the type of E give a partition of deg(A p).

Some additional notation for partitions will be useful.Letρ=

[n1,...,n m].For a positive integer i,letρ(i)be the number of oc-currences of i inρ.Let S(ρ)be the set of distinct integers n i occurring

18DANIEL KRASHEN

inρ,and let N(ρ)=|S(ρ)|.Let

(ρ)= i∈S(ρ)ρ(i)=m

be the length of the partition.

Suppose A/S is an sheaf of Azumaya algebras,and suppose S is a connected,Noetherian scheme.Letρ=[n1,...,n m]be a partition of n=deg(A).Let′e tρ(A)be the functor which associates to every S scheme X the set of′e tale subalgebras of A X of typeρ.That is,if X has structure map f:X→S,

′e tρ(A)(X)= sub-O X-modules

subalgebras of f?A of typeρ

E?f?A E is a sheaf of commutative′e tale Our?rst goal will be to describe the scheme which represents this functor.We use the following notation:

V(A)ρ= i∈S(ρ)V i(A)ρ(i).

We de?ne V(A)ρ?to be the open subscheme parametrizing ideals which are linearly independent.That is to say,for a S-scheme X,if I1,...,I ρis a collection of sheaves of ideals in A X,representing a point in V(A)ρ(X),then by de?nition,this point lies in V(A)ρ?if and only if ⊕I i=A.

Let Sρbe the subgroup i∈S(ρ)Sρ(i)of the symmetric group S n.For each i,we have an action of Sρ(i)on V i(A)ρ(i)by permuting the factors. This induces an action of Sρon V(A)ρ,and on V(A)ρ?.Denote the quotients of these actions by SρV(A)and V(A)(ρ)?respectively.We note that since the action on V(A)ρ?is free,the quotient morphism V(A)ρ?→is a Galois covering with group Sρ.

Theorem5.5.Letρ=[n1,...,n m]be a partition of n.Then the functor′e t(A)ρis represented by the scheme V(A)(ρ)?.

Proof.To begin,we?rst note that both′e t(A)ρand the functor repre-sented by V(A)(ρ)?are sheaves in the′e tale topology.Therefore,to show that these functors are naturally isomorphic,it su?ces to construct a natural transformationψ:V(A)(ρ)?→′e t(A)ρ,and then show that this morphism induces isomorphisms on the level of stalks.

Let X be an S-scheme,and let p:X→V(A)(ρ)?.To de?neψ(X)(p), since both functors are′e tale sheaves,it su?ces to de?ne it on an′e tale

ZERO CYCLES ON HOMOGENEOUS V ARIETIES19 cover of X.Let X be the pullback in the diagram

(2) X V(A)ρ?

π

V(A)(ρ)?

Since the quotient morphismπis′e tale,so is the morphism X→X. Therefore we see that after passing to an′e tale cover,and replacing X by X,we may assume that p=π(q)for some q∈V(A)ρ?(X).Passing to another cover,we may also assume that X=Spec(R).

Since p=π(q),we may?nd right ideals I1,...,I (ρ)of A R such that ⊕I i=A R,which represent q.Writing

1= e i,e i∈I i,

we de?ne E p=⊕e i R.This is a split′e tale extension of R,which is a subalgebra of A,and we setψ(p)=E p.One may check that this de?nes a morphism of sheaves.Note that this de?nition with respect to an′e tale cover gives a general de?nition since the association (I1,...,I )→E p is Sρinvariant.

To see thatψis an isomorphism,it su?ces to check that it is an isomorphism on′e tale stalks.In other words,we may restrict to the case that X=Spec(R),where R is a strictly Henselian local ring. We?rst show thatψis injective.Suppose E is an′e tale subalgebra of A R of typeρ.Since R is strictly Henselian,we have

E=⊕

i∈S(ρ)ρ(i)

⊕j=1e i,j R.

By de?nition,since the type of E isρ,if we let I i,j=e i,j A,then we the tuple of ideals(I i,j)de?nes a point q∈V(A)ρ?(R).Further,since e i,j=1,we actually have q∈V(A)ρ?(R).If we let p=π(q),then tracing through the above map yieldsψ(R)(p)=E.Thereforeψis surjective.

To see that it is injective,we suppose that we have a pair of points p,p ∈V(A)(ρ)?(R).By forming the pullbacks as in equation2,since R is strictly Henselian,we immediately?nd that in each case,because X is an′e tale cover of X,it is a split′e tale extension,and hence we have sections.This means we may write

p=π(I1,...,I (ρ)),p =π(I 1,...,I (ρ)).

Note that in order to show that p=p is su?ces to prove that the ideals are equal after reordering.Now,if E p=E p ,then both rings have the

20DANIEL KRASHEN

same minimal idempotents.However,by remark5.4,the ideals are generated by these idempotents.Therefore,the ideals coincide after reordering,and we are done. Since we now know that the functor′e tρ(A)is representable,we will abuse notation slightly and refer to it and the representing variety by the same name.

De?nition5.6.′e t(A)is the disjoint union of the schemes′e tρ(A)asρranges over all the partitions of n=deg(A).

Corollary5.7.The functor which associates to any S-scheme X the set of′e tale subalgebras of A X is representable by′e t(A).

Remark5.8.By associating to an′e tale subalgebra E?A X its under-lying module,we obtain a natural transformation to the Grassmannian functor,′e tρ(A)→Gr( (ρ),A).

6.Subfields of central simple algebras

In this section and for the remainder of the paper,we specialize back to the case where S=Spec(F),and A is a central simple F-algebra. If E is an′e tale subalgebra of A,then taking the′e tale stalk at Spec(F) amounts to extending scalars to the separable closure F sep of F.Let G be the absolute Galois group of F sep over F.Writing

E?F sep～=⊕i∈S(ρ)ρ(i)⊕j=1e i,j F sep,

we have an action of G on the idempotents e i,j.One may check that the idempotents e i,j are permuted by G,and there is a correspondence be-tween the orbits of this action and the idempotents of E.In particular we have

Lemma 6.1.In the notation above,if E is a sub?eld of A,then |S(ρ)|=1.

Proof.E is a?eld if and only if G acts transitively on the set of idem-potents.On the other hand,this action must also preserve the rank of an idempotent,which implies that all the idempotents have the same rank. Therefore,if we are interested in studying the sub?elds of a central simple algebra,we may restrict attention to partitions of the above type.If m|n=deg(A),we write

′e t m(A)=′e t[n

,...,n

m

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石器时代的食人魔，有铁头功技能：造成20伤害并震晕一回合，附加当前对方蓝魔法数值的伤害。

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看起来暴乱已经激怒了堕落的精灵，精灵boss和它的两个手下以及一个巨大的钢铁侠一起出现在这里。一番嘘寒问暖之后站在最前面的精灵族战士手持双刀前来切磋，解决掉开胃菜之后，对上身后那位女性战法师。她有一个Mirror Shield技能：将对方的防御值全部吸到自己身上，持续10+1/8自身蓝魔法的回合，而她初始防御就有47，所以使用该技能后基本上防御值超过100，注意使用减防技能。另一个Dark Blast技能如果紫色魔法多的话伤害也会很高，而且只消耗3的红色魔法。

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