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A general homological Kleiman-Bertini theorem

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A GENERAL HOMOLOGICAL KLEIMAN-BERTINI THEOREM SUSAN J.SIERRA Abstract.Let G be a smooth algebraic group acting on a variety X .Let F and E be coherent sheaves on X .We show that if all the higher T or sheaves of F against G -orbits vanish,then for generic g ∈G ,the sheaf T or X j (g F ,E )van-ishes for all j ≥1.This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser,itself generalizing the classical Kleiman-Bertini theorem,on generic transversality,under a general group action,of smooth subvarieties over an algebraically closed ?eld of characteristic 0.1.Introduction All schemes that we consider in this paper are of ?nite type over a ?xed ?eld k ;we make no assumptions on the characteristic of k .Our starting point is the following result of Miller and Speyer:Theorem 1.1.[MS]Let X be a variety with a transitive left action of a smooth algebraic group G .Let F and E be coherent sheaves on X ,and for all k -points g ∈G ,let g F denote the pushforward of F along multiplication by g .Then there is a dense Zariski open subset U of G such that,for all k -rational points g ∈U and for all j ≥1,the sheaf T or X j (g F ,E )is zero.As Miller and Speyer remark,their result is a homological generalization of the Kleiman-Bertini theorem:in characteristic 0,if F =O Z and E =O Y are structure sheaves of smooth subvarieties of X and G acts transitively on X ,then gZ and Y meet transversally for generic g ,implying that O gZ =g O Z and O Y have no higher T or .Motivated by this,if F and E are quasicoherent sheaves on X with T or X j (F ,E )=0for j ≥1,we will say that F and E are homologically transverse ;if E =O Y for some closed subscheme Y of X ,we will simply say that F and Y are

homologically transverse.

Homological transversality has a geometric meaning if F =O Z and E =O Y are structure sheaves of closed subschemes of X .If P is a component of Y ∩Z ,then Serre’s formula for the multiplicity of the intersection of Y and Z at P [Ha,p.427]is:

i (Y,Z ;P )= j ≥0

(?1)j len P (T or X j (F ,E )),

2SUSAN J.SIERRA

where the length is taken over the local ring at P.Thus if Y and Z are homologically transverse,their intersection multiplicity at P is simply the length of their scheme-theoretic intersection over the local ring at P.

It is natural to ask what conditions on the action of G are necessary to conclude that homological transversality is generic in the sense of Theorem1.1.In particular, the restriction to transitive actions is unfortunately strong,as it excludes important situations such as the torus action on P n.On the other hand,suppose that F is the structure sheaf of the closure of a non-dense orbit.Then for all k-points g∈G, we have T or X1(g F,F)=T or X1(F,F)=0,and so the conclusion of Theorem1.1 fails(as long as G(k)is dense in G).Thus for non-transitive group actions some additional hypothesis is necessary.

The main result of this paper is that there is a simple condition for homological transversality to be generic.This is:

Theorem1.2.Let X be a variety with a left action of a smooth algebraic group G, and let F be a coherent sheaf on X.Let

k,the pullback of F to X×

k)-orbit of x;

(2)For all coherent sheaves E on X,there is a Zariski open and dense subset U

of G such that for all k-rational points g∈U,the sheaf g F is homologically transverse to E.

Then(1)?(2).If k is algebraically closed,then(1)and(2)are equivalent.

If g is not k-rational,the sheaf g F can still be de?ned;in Section2we give this de?nition and a generalization of(2)that is equivalent to(1)in any setting(see Theorem2.1).

If G acts transitively on X in the sense of[MS],then the action is geometri-cally transitive,and so(1)is trivially satis?ed.Thus Theorem1.1follows from Theorem1.2.Since transversality of smooth subvarieties in characteristic0im-plies homological transversality,Theorem1.2also generalizes the following result of Robert Speiser:

Theorem1.3.[Sp,Theorem1.3]Suppose that k is algebraically closed of charac-teristic0.Let X be a smooth variety,and let G be a(necessarily smooth)algebraic group acting on X.Let Z be a smooth closed subvariety of X.If Z is transverse to every G-orbit in X,then for any smooth closed subvariety Y?X,there is a dense open subset U of G such that if g∈U,then gZ and Y are transverse.

We remark that for the set U we construct in Theorem1.2,for any extension k′of k and any k′-rational g∈U×k′,then g F will be homologically transverse to E on X×k′.Further,in many situations U will automatically contain a k-rational point of G.This holds,in particular,if if k is in?nite,G is connected and a?ne, and either k is perfect or G is reductive,by[B,Corollary18.3].

Theorem1.2was proved in the course of an investigation of certain rings,de-termined by geometric data,that arise in the study of noncommutative alge-braic geometry.Given a variety X,an automorphismσof X and an invert-ible sheaf L on X,then Artin and Van den Bergh[AV]construct a twisted ho-mogeneous coordinate ring B=B(X,L,σ).The graded ring B is de?ned via B n=H0(X,L?Xσ?L?X···?X(σn?1)?L),with multiplication of sections given

A GENERAL HOMOLOGICAL KLEIMAN-BERTINI THEOREM3 by the action ofσ.A closed subscheme Z of X determines a graded right ideal I of B,generated by sections vanishing on Z.In[Si],we study the idealizer of I; that is,the maximal subring R of

B such that I is a two-sided ideal of R.It turns out that quite subtle properties of Z and its motion underσcontrol many of the properties of R;in particular,for R to be left Noetherian one needs that for any closed subscheme Y,all but?nitely manyσn Z are homologically transverse to Y. (For details,we refer the reader to[Si].)Thus we were naturally led to ask how often homological transversality can be considered“generic”behaviour,and what conditions on Z ensure this.

We make some remarks on notation.If x is any point of a scheme X,we denote the skyscraper sheaf at x by k x.For schemes X and Y,we will write X×Y for the product X×k Y.If k′is a?eld containing k,then we write X×k′for X×Spec k′. Finally,if X is a scheme with a(left)action of an algebraic group G,we will always denote the multiplication map byμ:G×X→X.

2.Generalizations

We begin this section by de?ning homological transversality more generally.If W and Y are schemes over a scheme X,with(quasi)coherent sheaves F on W and E on Y respectively,then for all j≥0there is a(quasi)coherent sheaf T or X j(F,E) on W×X Y.This sheaf is de?ned locally.Suppose that X=Spec R,W= Spec S and Y=Spec T are a?ne.Let(

G×W1×f X.

If Y is a scheme over X and E is a(quasi)coherent sheaf on Y,we will write

T or X j(g F,E)for the(quasi)coherent sheaf T or{g}×W→X←Y

j (g F,E)on W×X Y×

k(g).Note that if W=X and g is k-rational,then g F is simply the pushforward of F along multiplication by g.

In this context,we prove the following relative version of Theorem1.2: Theorem2.1.Let X be a scheme with a left action of a smooth algebraic group G,let f:W→X be a morphism of schemes,and let F be a coherent sheaf on W.

4SUSAN J.SIERRA

We de?ne maps:

G×WρX

whereρis the mapρ(g,w)=gf(w)induced by the action of G and p is projection

onto the second factor.

Then the following are equivalent:

(1)For all closed points x∈X×k is homologically

transverse to the closure of the G(

W f X

Let F be a coherent sheaf on W that is f-?at over X,and let E be a coherent sheaf on Y.For all a∈A,let W a denote the?ber of W over a,and let F a=F?W O W

a be the?ber of F over a.

Then there is a dense open U?A such that if a∈U,then F a is homologically transverse to E.

We note that we have not assumed that X,Y,W,or A is smooth.

3.Proofs

In this section we prove Theorem1.2,Theorem2.1,and Theorem2.2.We begin

by establishing some preparatory lemmas.

Lemma3.1.Let

X1αX3

be morphisms of schemes,and assume thatγis?at.Let G be a quasicoherent sheaf

on X1that is?at over X3.Let H be any quasicoherent sheaf on X3.Then for all j≥1,we have T or X2j(G,γ?H)=0.

A GENERAL HOMOLOGICAL KLEIMAN-BERTINI THEOREM5 Proof.We may reduce to the local case.Thus let x∈X1and let y=α(x)and

z=γ(y).Let S=O X

2,y and let R=O X

3,z

.Then(γ?H)y～=S?R H z.Since S is

?at over R,we have

Tor R j(G x,H z)～=Tor S j(G x,S?R H z)=T or X2j(G,γ?H)x

by?at base change.The left-hand side is0for j≥1since G is?at over X3.Thus for j≥1we have T or X2j(G,γ?H)=0.

To prove Theorem2.2,we show that a suitable modi?cation of the spectral sequences used in[MS]will work in our situation.Our key computation is the following lemma;compare to[MS,Proposition2].

Lemma3.2.Given the notation of Theorem2.2,there is an open dense U?A such that for all a∈U and for all j≥0we have

T or W j(F?X E,q?k a)～=T or X j(F a,E)

as sheaves on W×X Y.

Note that F?X E is a sheaf on W×X Y and thus T or W j(F?X E,q?k a)is a sheaf on W×X Y×W W=W×X Y as required.

Proof.Since A is generically reduced,we may apply generic?atness to the mor-phism q:W→A.Thus there is an open dense subset U of A such that both W and F are?at over U.Let a∈U.Away from q?1(U),both sides of the equality we seek to establish are zero,and so the result is trivial.Since F|q?1(U)is still?at over X,without loss of generality we may replace W by q?1(U);that is,we may assume that both W and F are?at over A.

The question is local,so assume that X=Spec R,Y=Spec T,and W=Spec S are a?ne.Let E=Γ(Y,E)and let F=Γ(W,F).Let Q=Γ(W,q?k a);then Γ(W,F a)=F?S Q.We seek to show that

Tor S j(F?R E,Q)～=Tor R j(F?S Q,E)

as S?R T-modules.

We will work on W×X.For clarity,we lay out the various morphisms and corresponding ring maps in our situation.We have morphisms of schemes

W×X

p W φ

where p#(s)=s?1andφ#(s?r)=s·f#(r).We make the trivial observation that

B?R E=(S?k R)?R E～=S?k E.

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Let K?→F be a projective resolution of F,considered as a B-module via the mapφ#:B→S.As E is an R-module via the map r#:R→T,there is a B-action on S?k E;let L?→S?k E be a projective resolution over B.

Let P?,?be the double complex K??B L?.We claim the total complex of P?,?resolves F?B(S?k E).To see this,note that the rows of P?,?,which are of the form K??B L j,are acyclic,except in degree0,where the homology is F?B L j.The degree0horizontal homology forms a vertical complex whose homology computes Tor B j(F,S?k E).But S?k E～=B?R E,and B is a?at R-module.Therefore Tor B j(F,S?k E)～=Tor B j(F,B?R E)～=Tor R j(F,E)by the formula for?at base change for Tor.Since F is?at over R,this is zero for all j≥1.Thus,via the spectral sequence

H v j(H h i P?,?)?H i+j Tot P?,?

we see that the total complex of P?,?is acyclic,except in degree0,where the homology is F?B S?k E～=F?R E.

Consider the double complex P?,??S Q.Since Tot P?,?is a B-projective and therefore S-projective resolution of F?R E,the homology of the total complex of this double complex computes Tor S j(F?R E,Q).

Now consider the row K??B L j?S Q.As L j is B-projective and therefore B-?at,the i’th homology of this row is isomorphic to Tor S i(F,Q)?B L j.Since W and F are?at over A,by Lemma3.1we have Tor S i(F,Q)=0for all i≥1.Thus this row is acyclic except in degree0,where the homology is F?B L j?S Q.The vertical di?erentials on the degree0homology give a complex whose j’th homology is isomorphic to Tor B j(F?S Q,S?k E).As before,this is simply Tor R j(F?S Q,E).

Thus(via a spectral sequence)we see that the homology of the total complex of P?,??S Q computes Tor R j(F?S Q,E).But we have already seen that the homology of this total complex is isomorphic to Tor S j(F?R E,Q).Thus the two are isomorphic. Proof of Theorem2.2.By generic?atness,we may reduce without loss of generality to the case where W is?at over A.Since F and E are coherent sheaves on W and Y respectively,F?X E is a coherent sheaf on W×X Y.Applying generic?atness to the composition W×X Y→W→A,we obtain a dense open V?A such that F?X E is?at over V.Therefore,by Lemma3.1,if a∈V and j≥1,we have T or W j(F?X E,q?k a)=0.

We apply Lemma3.2to choose a dense open U?A such that for all j≥1,if a∈U,then T or W j(F?X E,q?k a)～=T or X j(F a,E).Thus if a is in the dense open set U∩V,then for all j≥1we have

T or X j(F a,E)～=T or W j(F?X E,q?k a)=0,

as required.

We now turn to the proof of Theorem2.1;for the remainder of this paper,we will adopt the hypotheses and notation given there.

Lemma3.3.Let R,R′,S,and T be commutative rings,and let

R′

A GENERAL HOMOLOGICAL KLEIMAN-BERTINI THEOREM 7

be a commutative diagram of ring homomorphisms,such that R ′

R and T S are ?at.

Let N be an R

-module.Then for all j ≥0,we have that Tor R ′

j (N ?R R ′,T )～=Tor R j (N,S )?S T.

Proof.Let P ?→N be a projective resolution of N .Consider the complex (3.4)P ??R R ′?R ′T ～=P ??R T ～=P ??R S ?S T.

Since R ′R is ?at,P ??R R ′is a projective resolution of N ?R R ′.Thus the j’th homology of (3.4)computes Tor R ′

j (N ?R R ′,T ).Since T S is ?at,this homology is

isomorphic to H j (P ??R S )?S T .Thus Tor R ′

j (N ?R R ′,T )～=Tor R j (N,S )?S T .

Lemma 3.5.Let x be a closed point of X .Consider the multiplication map

μx :G ×{x }→X.

Then for all j ≥0we have

(3.6)T or X j (F ,O G ×{x })～=T or G ×X j (p ?F ,μ?k x )

If k is algebraically closed,then we also have

(3.7)T or G ×X j (p ?F ,μ?k x )～=T or X j (F ,O

1×f G ×{

x }

ψμx s s s s s s

s s

s s W f

where πis the induced map and p is projection onto the second factor.Since ψ2=Id G ×X and μ=p ?ψ,we have that μ?k x ～=ψ?p ?k x ～=ψ?O G ×{x },considered as sheaves on G ×X .Then the isomorphism (3.6)is a direct consequence of the ?atness of p and Lemma 3.3.If k is algebraically closed,then πis also ?at,and so the isomorphism (3.7)also follows from Lemma 3.3. Proof of Theorem 2.1.(3)?(2).Assume (3).Let E be a coherent sheaf on Y .Consider the maps:

8SUSAN J.SIERRA

Since G is smooth,it is generically reduced.Thus we may apply Theorem2.2 to theρ-?at sheaf p?F to obtain a dense open U?G such that if g∈U is a closed point,thenρmakes(p?F)g homologically transverse to E.Butρ|{g}×W is the map used to de?ne T or X j(g F,E);that is,considered as sheaves over X,(p?F)g～=g F. Thus(2)holds.

(2)?(3).The morphismρfactors as

G×W1×f X.

Since the multiplication mapμis the composition of an automorphism of G×X and projection,it is?at.Therefore for any quasicoherent N on X and M on G×W and for any closed point z∈G×W,we have

(3.9)T or G×X

j (M,μ?N)z～=T or O X,ρ(z)

(M z,Nρ(z)),

as in the proof of Lemma3.1.

If p?F fails to be?at over X,then?atness fails against the structure sheaf of some closed point x∈X,by the local criterion for?atness[E,Theorem6.8].Thus to check that p?F is?at over X,it is equivalent to test?atness against structure sheaves of closed points of X.By(3.9),we see that p?F isρ-?at over X if and only if

(3.10)T or G×X

(p?F,μ?k x)=0for all closed points x∈X and for all j≥1. Applying Lemma3.5,we see that the?atness of p?F is equivalent to the vanishing (3.11)T or X j(F,O G×{x})=0for all closed points x∈X and for all j≥1.

Assume(2).We will show that(3.11)holds for all x∈X.Fix a closed point x∈X and consider the morphismμx:G×{x}→X.By assumption,there is a closed point g∈G such that g F is homologically transverse to O G×{x}.Let k′=k(g)and let g′be the canonical k′-point of G×k′lying over g.Let G′=G×k′and let X′=X×k′.Let F′be the pullback of F to W′=W×k′.Consider the commutative diagram

G×{x}×k′μx×1X′

～=

X{g}×W.

X′W′

X W,

A GENERAL HOMOLOGICAL KLEIMAN-BERTINI THEOREM9

(1)?(3).Theρ-?atness of F is not a?ected by base extension,so without loss of generality we may assume that k is algebraically closed.Then(3)follows directly from Lemma3.5and the criterion(3.10)for?atness.

(3)?(1).As before,we may assume that k is algebraically closed.Let x be

a closed point of X.We have seen that(3)and(2)are equivalent;by(2)applied to E=O Gx are homolog-ically transverse.By G(k)-equivariance,F and g?1Gx are homologically transverse. Proof of Theorem1.2.If F is homologically transverse to orbit closures upon ex-tending to