Rigidity for equivariant pseudo pretheories
aa r X i v : . [ m a t h . AG ] O c t RIGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES
JEREMIAH HELLER, CHARANYA RAVI, AND PAUL ARNE ØSTVÆR
Abstract.
We prove versions of the Suslin and Gabber rigidity theorems inthe setting of equivariant pseudo pretheories of smooth schemes over a fieldwith an action of a finite group. Examples include equivariant algebraic K -theory, presheaves with equivariant transfers, equivariant Suslin homology, andBredon motivic cohomology. Introduction
The classical rigidity theorems for algebraic K -theory are due to Suslin [Sus83]for extensions of algebraically closed fields, Gabber [Gab92] for Hensel local rings,and Gillet-Thomason [GT84] for strictly Hensel local rings. All known proofs rely on A -homotopy invariance and existence of transfer maps with certain nice properties.In his work on motives, Voevodsky introduced homotopy invariant pretheories ascontravariant functors on smooth schemes over a field enjoying certain transfer maps[Voe00a, Definition 3.1]. While algebraic K -theory admits transfer maps for relativesmooth curves, it is not an example of a pretheory [Voe00a, § K -theory is anexample, as well as equivariant Suslin homology, and Bredon motivic cohomologyin the sense of [HVØ15, Section 5].Our main results establish equivariant analogs of the Suslin-Voevodsky rigiditytheorems in [SV96, Section 4] (see Theorem 5.1, Theorem 5.4). Theorem 1.1.
Let k be a field, G be a finite group whose order is invertible in k , and let Sm Gk denote the category of smooth schemes over k equipped with anaction of G . Let F be a homotopy invariant equivariant pseudo pretheory on Sm Gk .Suppose that F is torsion of exponent coprime to char( k ) . (1) Let S = Spec( O hW,Gw ) be the Henselization of a smooth affine G -scheme W at the orbit Gw of a closed point. Let X → S be a smooth affine G-schemeof relative dimension one, admitting an equivariant good compactification.Then for all equivariant sections i , i : S → X which coincide on the closedorbit of S , we have i ∗ = i ∗ : F ( X ) → F ( S ) . (2) Let X be a smooth affine G -scheme and let x ∈ X be a closed point suchthat k ⊆ k ( x ) is separable. If every representation of G over k is a direct Mathematics Subject Classification. sum of one dimensional representations, then there is a naturally inducedisomorphism F ( Gx ) ∼ = −→ F (Spec( O hX,Gx )) . The condition in the second part of the theorem is satisfied whenever G is abelianand k contains a primitive d th root of unity, where d is the exponent of the group,by a theorem of Brauer, see e.g., [CR62, Theorem 41.1, Corollary 70.24].Rigidity theorems have been established for equivariant algebraic K -theory in[YØ09] and [Kri10, Theorem 1.4] at points with trivial stabilizers. The noveltyin Theorem 1.1 is that we allow points with nontrivial stabilizers. Note, however,that in [YØ09] the groups are more general, and [Kri10] deals with connectedsplit reductive groups. For works on rigidity results in related contexts, see e.g.,[AD], [Ayo14], [CD16], [HY07], [Jan], [Nes14], [PY02], [RØ06], [RØ08], [Tab], and[Yag11].A brief overview of the paper follows. Section 2 recalls notions in G -equivariantalgebraic geometry and shows an equivariant proper base change theorem for ´etalecohomology of Henselian pairs. After recalling equivariant divisors and equivariantcorrespondences, we define and give examples of equivariant pseudo pretheories inSection 3. Next in Section 4 we discuss the equivariant Nisnevich topology andequivariant good compactification for smooth affine relative curves. Our main re-sults are shown in Section 5. Finally, in Section 6 we show that exactness of theGersten complex for equivariant algebraic K -theory fails for the group G = Z / Z of order two acting on the affine line A k = Spec( k [ t ]) by t
7→ − t . This follows byapplying rigidity to the G -equivariant Grothendieck group K G of the Henselization O h A k ,Gx at the orbit of the closed point x = ( t ) ∈ A k . Acknowledgements.
Work on this paper took place at the Institut Mittag-Lefflerduring Spring 2017. We thank the institute for its hospitality and support. Theauthors gratefully acknowledge funding from the RCN Frontier Research GroupProject no. 250399 “Motivic Hopf equations.” Heller is supported by NSF GrantDMS-1710966. Østvær is supported by a Friedrich Wilhelm Bessel Research Awardfrom the Humboldt Foundation and a Nelder Visiting Fellowship from ImperialCollege London. The authors would like to thank the referee for a careful readingof this paper and for an insightful comment about the equivariant Gersten complex,which is included in the text as Remark 6.3.2.
Preliminaries
Throughout k is a field and G is a finite group whose order is coprime to char( k )(abusing the terminology we say that n is coprime to char( k ) if n is coprime to theexponential characteristic of k , i.e., n is invertible in k ). We view G as a groupscheme ` G Spec( k ) over Spec( k ). Let Sch Gk be the category of separated, finite typeschemes over Spec( k ) equipped with a left G -action, and equivariant morphisms.The smooth G -schemes over Spec( k ) form a full subcategory Sm Gk ⊆ Sch Gk . A G -scheme X is equivariantly irreducible if there exists an irreducible component X of X such that G · X = X . The fiber product X × Y of X, Y ∈ Sch Gk is a G -scheme with the diagonal G -action. For a finite dimensional k -vector space V , let A ( V ) := Spec(Sym( V ∨ )) and P ( V ) := Proj(Sym( V ∨ )). If V is a G -representationover k , we view A ( V ) and P ( V ) as G -schemes via the G -action on V . IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 3
For X ∈ Sch Gk we denote the categorical quotient of X by G (in the sense of[MFK94, Definition 0.5]) by X/G , provided it exists. Since G is a finite group, thecategorical quotient map π : X → X/G is in fact a uniform geometric quotient([MFK94, Definitions 0.6, 0.7]). If X is quasi-projective, then a quotient by a finitegroup π : X → X/G always exists.Let H ⊆ G be a subgroup and X ∈ Sch Hk . Then G × X is an H -scheme with theaction h ( g, x ) = ( gh − , hx ), and we define G × H X := ( G × X ) /H . The scheme G × H X has a left G -action through the action of G on itself. Since the H -action on G × X is free, π : G × X → G × H X is a principle H -bundle. In particular, π is ´etaleand surjective. It follows that if X is smooth, then so is G × H X . This defines a leftadjoint to the restriction functor Sm Gk → Sm Hk , given by G × H − : Sm Hk → Sm Gk .For X ∈ Sch Gk and x ∈ X a point, the set-theoretic stabilizer of x is the subgroup G x ⊆ G defined by G x = { g ∈ G | gx = x } . The orbit of x is Gx := G × G x { x } , withunderlying set { gx | g ∈ G } .2.1. G -sheaves. A G -sheaf on X is basically a sheaf with a G -action which iscompatible with the G -action on X . The precise definition goes as follows. Definition 2.1.
Let τ be a Grothendieck topology on X and F a τ -sheaf of abeliangroups. Write pr : G × X → X for the projection and µ : G × X → X for theaction map.(1) A G -linearization of F is an isomorphism φ : µ ∗ F ∼ = −→ pr ∗ F of sheaveson G × X which satisfies the cocycle condition pr ∗ ( φ ) ◦ ( Id G × µ ) ∗ ( φ ) =( m × Id X ) ∗ ( φ ) on G × G × X . Here m : G × G → G is the multiplicationand pr : G × G × X → G × X is the projection to second and third factors.(2) A G -sheaf (in the τ -topology) on X is a pair consisting of a τ -sheaf F together with a G -linearization φ of F . We simply write F for a G -sheaf,leaving the G -linearization understood.(3) A G -module M on X is a G -sheaf where M is a quasi-coherent O X -moduleand the G -linearization φ : µ ∗ M ∼ = pr ∗ M is an O G × X -module isomor-phism. A G -vector bundle on X is a G -module V whose underlying quasi-coherent O X -module is locally free. Remark 2.2.
Since G is finite, the data of a G -linearization of F is equivalent togiving a sheaf isomorphism φ g : F ∼ = −→ g ∗ F for each g ∈ G subject to the conditions φ e = id and φ gh = h ∗ ( φ g ) ◦ φ h for all g, h ∈ G . Remark 2.3.
Recall that if G acts on a commutative ring R , the skew group ring R ≀ G is the free left R -module with basis { [ g ] | g ∈ G } and multiplication is definedby setting ( r [ g ])( s [ h ]) = r ( g · s )[ gh ] and extending linearly. If G acts trivially on R ,then R ≀ G is simply the usual group ring RG .If X = Spec( R ), then the category of G -modules on X is equivalent to thecategory of left R ≀ G -modules. Further, if the order of G is invertible in R , then thecategory of G -vector bundles on X is equivalent to the category of left R ≀ G -moduleswhich are projective as R -modules. See e.g., [LS08, Section 1.1] for details.A G -equivariant morphism f : ( E , φ E ) → ( F , φ F ) of G -sheaves is a morphism f : E → F of sheaves compatible with the G -linearizations in the sense that φ F ◦ µ ∗ f = pr ∗ f ◦ φ E , or equivalently φ g ◦ f = g ∗ ( f ) ◦ φ g for all g ∈ G . Write Ab τ ( G, X )for the category of G -sheaves on X in the τ -topology. We note that Ab τ ( G, X ) hasenough injectives.
JEREMIAH HELLER, CHARANYA RAVI, AND PAUL ARNE ØSTVÆR
Given a G -sheaf ( F , φ g ), the morphisms φ g induce an action of the group G onthe group of global sections Γ( X, F ). We write Γ GX ( F ) = Γ( X, F ) G for the setof G -invariants of Γ( X, F ). This defines a functor Γ GX : Ab τ ( G, X ) → Ab fromthe category of G -sheaves to the category of abelian groups. The τ - G -cohomologygroups H pτ ( G ; X, M ) are defined as right derived functors H pτ ( G ; X, F ) := R p Γ GX ( F ) . Here Γ GX = ( − ) G ◦ Γ( X, − ) is a composite of left exact functors. Since the globalsections functor Γ( X, − ) sends injective G -sheaves to injective Z [ G ]-modules, theGrothendieck spectral sequence for this composition yields the bounded, convergentspectral sequence(2.4) E p,q = H p ( G, H qτ ( X, F )) ⇒ H p + qτ ( G ; X, F ) , where H ∗ ( G, − ) denotes the group cohomology of G . Moreover, the spectral se-quence induces a finite filtration on each H nτ ( G ; X, F ). Definition 2.5.
The G -equivariant Picard group Pic G ( X ) of X is the group of G -line bundles on X modulo equivariant isomorphisms, with group operation givenby tensor product. For an invariant closed subscheme Y ⊆ X , let Pic G ( X, Y )denote the group consisting of pairs ( L , φ ), where L is a G -line bundle on X and φ : O Y ∼ = −→ L| Y is an isomorphism of G -line bundles on Y , modulo equivariantisomorphisms respecting the trivializations on Y . The group Pic G ( X, Y ) is calledthe relative equivariant Picard group of X relative to Y .The following cohomological interpretations of the equivariant and the relativeequivariant Picard groups are standard, see [HVØ15, Theorem 2.7, Lemma 6.7]. Theorem 2.6.
Let X be a G -scheme. (1) There is a natural isomorphism
Pic G ( X ) ∼ = −→ H et ( G ; X, O ∗ X ) . (2) Let i : Y ֒ → X be an invariant closed subscheme. Then there is a naturalisomorphism Pic G ( X, Y ) ∼ = −→ H et ( G ; X, G X,Y ) , where G X,Y is the ´etale G -sheaf defined to be the kernel of the equivariant homomorphism O ∗ X → i ∗ O ∗ Y . We end this section by recording an equivariant version of Gabber’s proper basechange theorem for the cohomology of torsion ´etale G -sheaves, which will be neededto establish the equivariant version of Suslin’s rigidity theorem in Section 5. Definition 2.7. ([Ray70, Chapter XI, Definition 3]) Let A be a commutative ringand I ⊆ A an ideal which is contained in the Jacobson radical of A . The pair( A, I ) is said to be a
Henselian pair provided Hom A ( B, A ) → Hom A ( B, A/I ) issurjective for any ´etale A -algebra B . A G -action on a Henselian pair ( A, I ) issimply a G -action on A such that the ideal I is invariant. Theorem 2.8 (Equivariant Proper Base Change) . Let ( A, I ) be a Henselian pairwith G -action. Let f : Y → Spec( A ) be a proper equivariant map and define Y bythe pull-back Y i / / f ′ (cid:15) (cid:15) Y f (cid:15) (cid:15) Spec(
A/I ) j / / Spec( A ) . IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 5
Let F be a torsion ´etale G -sheaf on Y and write F = i ∗ F . Then the restrictionmap induces an isomorphism H n ´ et ( G ; Y, F ) ∼ = H n ´ et ( G ; Y , F ) for each n .Proof. Restriction induces a G -equivariant map H p ´ et ( Y, F ) → H p ´ et ( Y , F ). Gab-ber’s base change theorem [Gab94, Corollary 1] shows this is an isomorphism, andtherefore it induces an isomorphism in group cohomology. Thus the induced com-parison maps of spectral sequences (2.4) for ( Y, F ) and ( Y , F ) is an isomorphismon the E -page. This implies the desired isomorphism. (cid:3) Equivariant divisors and pseudo pretheories
We begin by recalling the notion of equivariant Cartier divisors and their prop-erties.3.1.
Equivariant divisors.
Let X be a G -scheme and Y ⊆ X an invariant closedsubscheme. Definition 3.1. (1) An equivariant Cartier divisor on X is an element ofΓ GX ( K ∗ X / O ∗ X ). The group of equivariant Cartier divisors on X is denoted byDiv G ( X ). An effective Cartier divisor D on X such that D ∈ Γ GX ( K ∗ X / O ∗ X )is called an equivariant effective Cartier divisor .(2) A relative equivariant Cartier divisor on X relative to Y is an equivariantCartier divisor D on X such that Supp( D ) ∩ Y = ∅ . Write Div G ( X, Y ) forthe subgroup of Div G ( X ) consisting of relative equivariant Cartier divisors.(3) A principal equivariant Cartier divisor is an invariant rational functionon X , i.e., an element in the image of Γ GX ( K ∗ X ) in Γ GX ( K ∗ X / O ∗ X ). In therelative setting, a principal equivariant Cartier divisor f on X is said to bea principal relative equivariant Cartier divisor if f is defined and equal to1 at points of Y .(4) Let Div Grat ( X ) denote the group of equivariant Cartier divisors on X modulothe principal equivariant Cartier divisors, and likewise write Div Grat ( X, Y )in the relative setting.Given a Cartier divisor D = { ( U i , f i ) } on X , we have an associated line bundle L D defined by L D | U i = O U i f − i . When D is an equivariant Cartier divisor it is easyto verify that the line bundle L D has a canonical G -linearization; write L D for the G -line bundle defined by this choice of linearization. If D is a relative equivariantCartier divisor relative to Y it is straightforward that L D | Y is trivial.Let Z d ( X ) (respectively Z d ( X )) denote the free group on dimension d (respec-tively codimension d ) cycles on X . The homomorphism cyc : Div ( X ) → Z ( X )is defined by cyc ( D ) = P Z ∈ X ord Z ( D ) Z , where X is the set of closed integralcodimension one subschemes. For a G -scheme X , the groups Z d ( X ) and Z d ( X )have natural G -actions and cyc is an equivariant homomorphism. Therefore weconclude the following. Lemma 3.2. ( [HVØ15, Lemma 2.11] ) For a smooth G -scheme X , cyc : Div( X ) →Z ( X ) is an equivariant isomorphism. Equivariant pseudo pretheories.
An equivariant pseudo pretheory is de-fined as a presheaf on Sm Gk with transfer maps associated to certain equivariantcorrespondences subject to some natural axioms. JEREMIAH HELLER, CHARANYA RAVI, AND PAUL ARNE ØSTVÆR
Definition 3.3. An equivariant pseudo pretheory on Sm Gk is an additive presheaf F : (Sm Gk ) op → Ab (i.e., F ( X ` Y ) = F ( X ) ⊕ F ( Y )) with transfer maps Tr D : F ( X ) → F ( S ) for any equivariant relative smooth affine curve X/S and effectiveequivariant Cartier divisor D on X which is finite and surjective over a componentof S , such that the following holds.(1) The transfer maps are compatible with pullbacks.(2) If D ( i ) is the divisor associated to an equivariant section i : S → X , thenTr D ( i ) = F ( i ) . (3) Let L D be the G -line bundle associated to D . If the restriction of L D to D ′ is trivial, then Tr D + Tr D ′ = Tr D + D ′ . As usual we extend all functors defined on the category Sm Gk to limits of smooth G -schemes with G -action (including semilocalizations of all smooth affine G -schemesat closed G -orbits) by taking direct limits. The above properties obviously remaintrue after such an extension as well. Definition 3.4.
A presheaf F on Sm Gk (or Sch Gk ) is said to be homotopy invariant if for any X ∈ Sm Gk (respectively in Sch Gk ) the projection map p : X × A → X induces an isomorphism p ∗ : F ( X ) ∼ = −→ F ( X × A ), where the G -action on X × A is induced by the given G -action on X and the trivial G -action on A .3.3. Examples of equivariant pseudo pretheories.
In the following we discussexamples of equivariant pseudo pretheories such as equivariant algebraic K -theory,equivariant Suslin homology, K G -presheaves with transfers, presheaves with equi-variant transfers, and equivariant motivic representable theories. Example 3.5. Presheaves with equivariant transfers.
For smooth schemes X , Y , the group of correspondences Cor k ( X, Y ) ⊆ Z dim( X ) ( X × Y ) is the subgroupof Z dim( X ) ( X × Y ) of cycles on X × Y which are finite over X and surjective oversome component of X . The category Cor k has the same objects as Sm /k andCor k ( X, Y ) are the morphisms between X and Y in this category. The equivariantcorrespondences Cor Gk ( X, Y ) between smooth G -schemes are correspondences Z : X → Y such that the square G × X Z × id / / µ (cid:15) (cid:15) G × Y µ (cid:15) (cid:15) X Z / / Y commutes in Cor k [HVØ15, Section 4]. Unravelling definitions we haveCor Gk ( X, Y ) = Cor k ( X, Y ) ∩ Z dim X ( X × Y ) G . Let Cor Gk denote the category whose objects are smooth G -schemes and morphismsare equivariant correspondences. There is a canonical inclusion Sm Gk ⊆ Cor Gk whichsends f : X → Y to its graph Γ f ⊆ X × Y . Definition 3.6. [HVØ15, Definition 4.1] A presheaf with equivariant transfers isa presheaf of abelian groups on the category Cor Gk . IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 7
Given an equivariant relative smooth affine curve
X/S and an effective equi-variant Cartier divisor D on X which is finite and surjective over S , note that D ∈ Cor Gk ( S, X ). Moreover, if D ( i ) is the divisor associated to an equivariant sec-tion i : S → X , then D ( i ) = Γ i in Cor Gk ( S, X ). Therefore if F is a presheaf withequivariant transfers, then F defines an additive presheaf on Sm Gk ⊆ Cor Gk suchthat for a divisor D as above, Tr D := F ( D ) : F ( X ) → F ( S ) satisfies conditions(1), (2) and (3) of Definition 3.3. Example 3.7. Equivariant K -theory. The G -equivariant algebraic K -theorygroup K Gi ( X ) of a scheme X with G -action is the i th homotopy group of thealgebraic K -theory spectrum K G ( X ) of the exact category of G -vector bundles on X . For n ≥
2, the equivariant K -groups with mod- n coefficients are defined as K Gi ( X ; n ) := π i ( K G ( X ) ∧ S /n ), for the mod- n Moore spectrum S /n .The equivariant algebraic K -theory groups K Gi define functors on Sch Gk (andSm GK ) by considering the category of “big G -vector bundles” ([FS02, AppendixC.4, C.5]). Let p : X → S be an equivariant relative smooth affine curve inSm Gk and let i D : D ֒ → X be an effective equivariant Cartier divisor on X suchthat p D := p | D : D → S is finite and surjective. Then p D : D → S is alsoflat. Let Tr D : K Gi ( X ) → K Gi ( S ) denote the map induced by the functor F D :Vect G ( X ) → Vect G ( S ) between the categories of G -vector bundles on X and S defined by P p D ∗ ◦ i ∗ D ( P ). By [Tho87, Theorem 4.1, Corollary 5.8(2)], K Gi is ahomotopy invariant functor on Sm Gk . We show that K Gi is an equivariant pseudopretheory on Sm Gk , so that K Gi ( − ; n ) is a homotopy invariant equivariant pseudo-pretheory on Sm Gk with n -torsion values. Lemma 3.8. If D and D ′ are effective equivariant Cartier divisors on X suchthat the restriction of the G -line bundle L D to the G -scheme D ′ is a trivial G -linebundle, then Tr D + D ′ = Tr D + Tr D ′ .Proof. We write i : D ֒ → D + D ′ and i ′ : D ′ ֒ → D + D ′ for the corresponding G -equivariant closed immersions. Let f ∈ Γ GD ′ ( L D | D ′ ) define the trivialization of L D on D ′ . Since L D defines the ideal sheaf of D , we have an exact sequence of G -equivariant coherent sheaves on D + D ′ :(3.9) 0 → i ′∗ ( O D ′ ) f −→ O D + D ′ → i ∗ ( O D ) → , where the maps are G -equivariant. Given P ∈ Vect G ( X ), the above exact sequencegives the following exact sequence:0 → i ′∗ ◦ i ∗ D ′ ( P ) → i ∗ D + D ′ ( P ) → i ∗ ◦ i ∗ D ( P ) → . Pushforward by the equivariant, finite, and flat map p D + D ′ gives an exact sequenceof G -vector bundles on S :0 → p D ′∗ ◦ i ∗ D ′ ( P ) → p D + D ′∗ ◦ i ∗ D + D ′ ( P ) → p D ∗ ◦ i ∗ D ( P ) → , which by definition of the transfer maps is the exact sequence of functors:0 → Tr D ′ ( P ) → Tr D + D ′ ( P ) → Tr D ( P ) → . Therefore by Waldhausen’s additivity theorem, [Wal85, Proposition 1.3.2(4)], weconclude that Tr D + D ′ = Tr D + Tr D ′ . (cid:3) JEREMIAH HELLER, CHARANYA RAVI, AND PAUL ARNE ØSTVÆR
Example 3.10. Equivariant Suslin Homology.
For n ∈ N , the algebraic n -simplex ∆ n is ∆ n := Spec (cid:18) k [ t , · · · , t n ]( P i t i − (cid:19) and ∆ • = { ∆ n } n ≥ is a cosimplicial scheme with face and degeneracy maps givenby: ∂ r ( t j ) = t j if j < r j = rt j − if j > r δ r ( t j ) = t j if j < rt j + t j +1 if j = rt j +1 if j > r .We view ∆ • as a cosimplicial G -scheme with trivial G -action.For a smooth morphism f : X → S , let C ( X/S ) ⊆ Cor k ( S, X ) denote the groupof cycles on X which are finite and surjective over a component of S . If X, S ∈ Sch Gk and f is G -equivariant, then C ( X/S ) is a G -invariant subset of Cor k ( S, X ). Welet C • ( X/S ) G denote the chain complex associated to the simplicial abelian group n C n ( X/S ) G , where C n ( X/S ) := C ( X × ∆ n /S × ∆ n ). Definition 3.11.
The n th equivariant Suslin homology of X/S is defined as the n th homology group of the complex of abelian groups C • ( X/S ) G :H Sus n ( G ; X/S ) := H n C • ( X/S ) G . For a smooth G -scheme X over k , let Z tr,G ( X ) denote the presheaf with equivari-ant transfers given by the representable functors Z tr,G ( X )( U ) := C ( X × U/U ) G =Cor Gk ( U, X ) for each U ∈ Sm Gk . When G is trivial, this is the same as the presheaf c equi ( X/ Spec( k ) ,
0) studied in [Voe00b, Section 5.3]. Similarly for each n , thepresheaf U H Sus n ( G ; X × U/U ) is a homotopy invariant presheaf with equivarianttransfers. Therefore this defines a family of homotopy invariant equivariant pseudopretheories.
Lemma 3.12.
Let F be a homotopy invariant equivariant pseudo pretheory on Sm Gk . Let S be an equivariantly irreducible smooth semilocal G -scheme and X/S be a relative smooth affine curve. Let D and D ′ be effective equivariant Cartierdivisors on X which are finite and surjective over S . If the image of ( D − D ′ ) in H Sus0 ( G ; X/S ) vanishes, then Tr D = Tr D ′ . Here Tr D and Tr D ′ denote the transfermaps associated to D and D ′ , respectively.Proof. The proof follows as in [HVØ15, Lemma 6.3]. (cid:3)
Example 3.13. K G -presheaves. The notion of K -presheaves was introducedand studied by Walker in [Wal96] (see also [Sus03, Section 1]). Homotopy invariant K -presheaves satisfy many properties enjoyed by presheaves with transfers. Anequivariant generalisation of this notion was developed in [HKØ15, Section 6.2].We briefly recall the definition here.For X, Y ∈ Sch Gk , let P G ( X, Y ) denote the category of coherent G -modules on X × Y which are flat over X and whose support is finite over X . This is anexact subcategory of the abelian category of coherent G -modules on X × Y . Define K G ( X, Y ) := K ( P G ( X, Y )). Given
X, Y, Z ∈ Sm Gk , we have a natural biexactbifunctor P G ( X, Y ) × P G ( Y, Z ) → P G ( X, Z ) given by (
P, Q ) ( p XZ ) ∗ ( p ∗ XY ( P ) ⊗ p ∗ Y Z ( Q )), where the tensor product is taken over O X × Y × Z . Thus we get a naturalcomposition pairing of exact categories ◦ : K G ( X, Y ) × K G ( Y, Z ) → K G ( X, Z )and all these composition laws are associative. This allows us to define an additive
IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 9 category K (Sm Gk ) by taking the objects of Sm Gk to be the objects and definingHom K (Sm Gk ) ( X, Y ) = K G ( X, Y ). A K G -presheaf is an additive presheaf of abeliangroups on the category K (Sm Gk ). Equivariant algebraic K -theory K Gi ( − ) is a K G -presheaf for all i ; therefore, Example 3.7 is a special case of this one.There is a functor Sm Gk → K (Sm Gk ) which is the identity on objects and sendsa morphism q : X → Y to the structure sheaf O Γ q of the graph Γ q ⊆ X × Y . Inparticular, a K G -presheaf is also a presheaf on Sm Gk and we discuss below that itis in fact an equivariant pseudo pretheory.Given an equivariant relative smooth affine curve p : X → S and an effectiveequivariant Cartier divisor i D : D ֒ → X which is finite and surjective over S , themap p D := p | D : D → S is a finite and flat equivariant map. Let Γ tp D ⊆ S × D denote the transpose of the graph of p D and let O Γ tpD denote its structure sheaf.Then F tD := ( Id S × i D ) ∗ ( O Γ tpD ) ∈ P G ( S, X ). Define Tr D : F ( X ) → F ( S ) to be F ( F tD ). Then the transfer maps Tr D are clearly compatible with pullbacks andsections. If D and D ′ are as in Lemma 3.8, then the exact sequence (3.9) gives anexact sequence of coherent sheaves in P G ( S, X ):0 → F tD ′ → F tD + D ′ → F tD → . Using the additivity in K G ( S, X ), it follows that Tr D + D ′ = Tr D + Tr D ′ . Example 3.14. Bredon motivic cohomology.
Bredon motivic cohomologyintroduced in [HVØ15, Section 5] and further studied in [HVØ16] (for smoothvarieties equipped with Z / Z -action) is an equivariant generalization of motiviccohomology for finite group actions.For a smooth G -scheme X over k , recall that Z tr,G ( X ) denotes the presheaf withequivariant transfers given by Z tr,G ( X )( − ) := Cor Gk ( − , X ). If F is a presheaf ofabelian groups on Sm Gk , write C ∗ F ( X ) for the cochain complex associated to thesimplicial abelian group F ( X × ∆ • ). For a finite dimensional representation V of G , let Z G ( V ) denote the complex of presheaves with equivariant transfers given by: Z G ( V ) := C ∗ ( Z tr,G ( P ( V ⊕ / Z tr,G ( P ( V )))[ − V )] . The
Bredon motivic cohomology of a smooth G -variety X is defined to be theequivariant Nisnevich hypercohomology with coefficients in Z G ( V ): H nG ( X, Z ( V )) := H nGNis ( X, Z G ( V )) . (See Section 4.1 for the definition of the equivariant Nisnevich site.)The fact that Bredon motivic cohomology define presheaves with equivarianttransfers follows from [Voe00c, Proposition 3.1.9] in the case of a trivial group andis proved in [HVØ16, Corollary 3.8] for Z / Z . The case of finite groups followsverbatim from the fact that smooth G -schemes have finite equivariant Nisnevichcohomological dimension [HVØ15, Corollary 3.9] and [HVØ15, Theorem 4.15(3)].Therefore Bredon motivic cohomology define equivariant pseudo pretheories.4. Equivariant Nisnevich topology and compactifications
In this section we discuss the notions of equivariant Nisnevich topology and equi-variant good compactification of equivariant smooth relative curves. We establishsome of their properties which are needed in the proofs of our rigidity theorems.
Equivariant Nisnevich topology.
We recall briefly the equivariant Nis-nevich topology on Sm Gk for finite groups, first introduced by Voevodsky in [Del09,Section 3.1]. Definition 4.1. A distinguished square in Sch Gk is a cartesian square(4.2) B (cid:15) (cid:15) / / Y p (cid:15) (cid:15) A (cid:31) (cid:127) j / / X, where j is an equivariant open immersion, p an equivariant ´etale morphism, andthe induced map ( Y r B ) red → ( X r A ) red is an isomorphism. The collection ofdistinguished squares forms a cd -structure in the sense of [Voe10, Definition 2.1].The associated Grothendieck topology is called the equivariant Nisnevich topology.We write (Sm Gk ) GNis (resp. (Sch Gk ) GNis ) for the respective sites of smooth G -schemes and G -schemes equipped with the equivariant Nisnevich topology .Equivariant Nisnevich covers admit the following equivalent characterizations(see [HKØ15, Propositions 2.15, 2.17]). Proposition 4.3.
Let f : Y → X be an equivariant ´etale map between G -schemes.The following are equivalent. (1) The map f is an equivariant Nisnevich cover. (2) There exists a sequence of invariant closed subschemes ∅ = Z m +1 ⊆ Z m ⊆ · · · ⊆ Z ⊆ Z = X such that f | f − ( Z i − Z i +1 ) : f − ( Z i − Z i +1 ) → Z i − Z i +1 has an equivariantsection. (3) For every x ∈ X , there exists a point y ∈ Y such that f induces isomor-phisms of residue fields k ( x ) ∼ = k ( y ) and set-theoretic stabilizers G y ∼ = G x . Let X ∈ Sch Gk and suppose x ∈ X has an invariant open affine neighborhood.Then the semilocal ring O X,Gx has a natural G -action which induces a G -actionon the Henselian semilocal ring O hX,Gx with a single closed orbit. Any semilocalHenselian affine G -scheme over k with a single orbit is equivariantly isomorphic toSpec( O hY,Gy ) for some affine G -scheme Y and y ∈ Y .For X ∈ Sch Gk and any x ∈ X , let N G ( Gx ) denote the filtering category ofequivariant ´etale neighborhoods of Gx . Its objects are pairs ( p : U → X, s ), where U is an equivariantly irreducible G -scheme, p is an equivariant ´etale map, and s : Gx → U is an equivariant section of p over Gx . A morphism from ( U → X, s ) to( V → X, s ′ ) in N G ( Gx ) is a map f : U → V making the evident triangles commute.Although x ∈ X might not be contained in any G -invariant affine neighborhood, itmakes sense to consider G × G x Spec( O hX,x ) and according to [HVØ15, Proposition3.13] we have:(4.4) lim U ∈ N G ( Gx ) U ∼ = Spec( O hG × Gx X,Gx ) ∼ = G × G x Spec( O hX,x ) . Further if x ∈ X has an invariant affine neighborhood then there is a canonical G -isomorphism(4.5) G × G x Spec( O hX,x ) ∼ = −→ Spec( O hX,Gx ) . IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 11
For a Nisnevich sheaf F on Sm Gk , X ∈ Sm Gk , and x ∈ X , we set p ∗ x F := F (Spec( O hG × Gx X,Gx )) = colim U ∈ N G ( Gx ) F ( U ) . Then p ∗ x defines a fiber functor from the category of sheaves to sets, i.e., it commuteswith colimits and finite products and so determines a point of the G -equivariantNisnevich topos. It is known that the set of points { p ∗ x | x ∈ X, X ∈ Sm Gk } forms aconservative set of points for (Sm Gk ) GNis (see [HVØ15, Theorem 3.14]).4.2.
Suslin homology of equivariant curves.
An equivariant map p : X → S is an equivariant curve if all of its fibers have dimension one. Definition 4.6.
Say that a smooth equivariant curve p : X → S admits a goodcompactification if p factors as X (cid:31) (cid:127) j / / p (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ X p (cid:15) (cid:15) S, where X is normal, p is a proper equivariant curve, j is an equivariant open em-bedding, and X ∞ = ( X r X ) red has an invariant open affine neighborhood in X .The following lemma about base change is straightforward to verify. Lemma 4.7.
Let X → S be an equivariant smooth curve and S ′ → S be anequivariant map, where S, S ′ are affine G -schemes (smooth or a local or semilocal G -scheme which is a limit of smooth G -schemes). If X → S admits an equivariantgood compactification, then the smooth equivariant curve X ′ = X × S S ′ → S ′ alsoadmits an equivariant good compactification. If S is affine and X → S is an equivariant smooth quasi-affine curve with equi-variant good compactification X and X ∞ = ( X r X ) red , then the equivariant Suslinhomology of X/S can be interpreted in terms of relative equivariant Cartier divi-sors (see [SV96, Theorem 3.1] when G is trivial, and [HVØ15, Theorem 6.12] foran extension to the equivariant case):(4.8) H Sus n ( G ; X/S ) ∼ = ( Div
Grat ( X, X ∞ ) n = 00 n > . Lemma 4.9.
Let S = lim α ∈ A S α be a cofiltered limit where the S α are quasi-projective G -schemes over k and the transition maps are equivariant and affine. If f : X → S is a finite type equivariant map, then there is λ , a finite type G -scheme X λ over k , and an equivariant map f λ : X λ → S λ fitting into a Cartesian square X / / f (cid:15) (cid:15) X λf λ (cid:15) (cid:15) S / / S λ . Moreover if f is satisfies any of the properties: (i) affine, (ii) open, (iii) smooth,(iv) proper, then f λ can be chosen to have the same properties. Proof.
Let T α = S α /G and T = lim α T α . By [Gro66, Th´eor`eme 8.8.2] there is β and a map of finite type T β -schemes f β : X β → S β such that X ∼ = X β × S β S andunder this isomorphism f is the pullback of f β . Moreover if f satisfies some of theproperties (i)-(iv), then f β can be chosen to satisfy the same properties [Gro66,Th´eor`eme 8.10.5], [Gro67, Proposition 17.7.8]. For α ≥ β , set X α = X β × S β S α .We have that Aut T ( X ) ∼ = colim α Aut T α ( X α ). Since G is finite, the homomorphism G → Aut T ( X ) factors through some Aut T λ ( X λ ), i.e., we may choose X λ to have a G -action. Increasing λ we can further assume that f λ is equivariant. (cid:3) Lemma 4.10.
Let S = lim α ∈ A S α be a cofiltered limit where S α ∈ Sm Gk are affineand the transition maps are equivariant ´etale. Let X → S be a smooth equivariantaffine curve admitting good compactification. (1) H Sus n ( G ; X/S ) ∼ = colim β H Sus n ( G ; X β /S β ) where X β → S β are smooth equi-variant curves with good compactification. (2) H Sus0 ( G ; X/S ) ∼ = Div Grat ( X, X ∞ ) and H Sus i ( G ; X/S ) = 0 for i > .Proof. Let X ⊆ X be an equivariant good compactification. By the previouslemma, there is a smooth, affine, equivariant map X α → S α , with equivariantcompactification X α → S α with X α r X α has an affine neighborhood, such that X ∼ = X α × S α S and X ∼ = X α × S α S . For any generic point η ′ ∈ X α lying over ageneric point η ∈ S α , we have dim( O X α ,η ′ ) = dim( O S α ,η )+1. Thus there is an opensubset of U ⊆ S α over which the fibers of X α , X α are one dimensional. Since U contains the image of S in S α , there is λ ≥ α such that X λ and X λ are equivariantcurves over S λ , where X β = X α × S α S β for β ≥ α and similarly for X β . Replacing X λ by its normalization, we see that X λ → S λ admits good compactification. Wethus have that X → S is isomorphic to the cofiltered limit lim β ≥ λ ( X β → S β )of smooth affine equivariant curves admitting good compactification. Moreover,we have colim β C n ( X β /S β ) ∼ = C n ( X/S ) and taking fixed points and homologycommutes with filtered colimits, yielding (1).Write X → S as a filtered limit lim β ∈ B ( X β → S β ) of equivariant curves withgood compactification. Moreover we can assume B has a minimal element 0and X β = X × S S β is a good compactification of X β . Write Y β = X β r X β . Under the isomorphism (4.8), the map H Sus0 ( G ; X β /S β ) → H Sus0 ( G ; X α /S α )agrees with the map Div Grat ( X β , Y β ) → Div
Grat ( X α , Y α ) and so H Sus n ( G ; X/S ) ∼ =colim β Div
Grat ( X β , Y β ). Finally, note that colim β Div
Grat ( X β , Y β ) ∼ = Div Grat ( X, X ∞ ). (cid:3) Corollary 4.11.
Let F be a homotopy invariant equivariant pseudo pretheory on Sm Gk and X → S as in the statement of the previous lemma. Then there is a pairingof abelian groups H Sus0 ( G ; X/S ) ⊗ F ( X ) → F ( S ) . Proposition 4.12.
Let S = Spec( O hW,Gw ) be the Henselization of a smooth affine G -scheme W at an orbit Gw . Let p : X → S be a smooth equivariant affine curvewith an equivariant good compactification. Let X → S be the fiber over the closedorbit S in S . Then for any n coprime to char( k ) , restriction induces an injection H Sus0 ( G ; X/S ) /n ֒ → H Sus0 ( G ; X /S ) /n. Proof.
Let X be the equivariant good compactification of X over S such that Y = ( X r X ) red has an invariant open neighborhood in X . By Lemma 4.10(2) and IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 13 [HVØ15, Proposition 6.8] it suffices to show that the restriction Pic G ( X, Y ) /n → Pic G ( X , Y ) /n is injective. This follows as in the proof of [SV96, Theorem 4.3], byreplacing ´etale cohomology with H ∗ et ( G ; − ) and classical proper base change withTheorem 2.8. (cid:3) Rigidity for equivariant pseudo pretheories
In this section we establish versions of the rigidity theorems of Suslin [Sus83],Gabber [Gab92], and Gillet and Thomason [GT84] in the setting of equivariantpseudo pretheories.
Theorem 5.1 (Equivariant Suslin Rigidity) . Let F be a homotopy invariant equi-variant pseudo pretheory on Sm Gk which takes values in torsion abelian groups ofexponent coprime to char( k ) . Let S = Spec( O hW,Gw ) be the Henselization of asmooth affine G -scheme W at a closed orbit, and X → S a smooth affine equivari-ant curve admitting good compactification. If i , i : S → X are two equivariantsections which coincide on the closed orbit of S , then i ∗ = i ∗ : F ( X ) → F ( S ) .Proof. For any n , F n = ker( n : F → F ) is again a homotopy invariant equivari-ant pseudo pretheory and F = ∪ n F n . Thus it suffices to consider the case when nF = 0. We may assume that X is equivariantly irreducible. The images of thesections i j are closed subschemes W j ⊆ X which are elements of C ( X/S ) G . Bydefinition we have i ∗ j = Tr W j . By Lemma 3.12 it suffices to show that W − W becomes zero in H Sus0 ( G ; X/S ) /n . Proposition 4.12 shows that there is an injectionH Sus0 ( G ; X/S ) /n ֒ → H Sus0 ( G ; X /S ) /n , where X is the fiber over the closed orbit S of S . Since i and i coincide on the closed orbit, we conclude that W − W iszero in H Sus0 ( G ; X/S ) /n . (cid:3) Recall that we write R ≀ G for the skew group ring. Lemma 5.2.
Let X → Z be a map in Sm Gk , with X affine, Z = Spec( L ) where L is a field, and x ∈ X an invariant closed point such that k ( x ) ∼ = L . Then there isa commutative diagram in Sm Gk X φ / / (cid:28) (cid:28) ✿✿✿✿✿✿✿ V (cid:2) (cid:2) ✆✆✆✆✆✆✆ Z, where V is an equivariant vector bundle over Z , φ is ´etale at x , and φ ( x ) = 0 .Proof. Write X = Spec( A ) and m ⊆ A for the maximal ideal corresponding to x . Since | G | is invertible in L , the surjection of L ≀ G -modules m → m/m hasa splitting. The resulting map of L ≀ G -modules m/m → m ⊆ A induces theequivariant ring map Sym( m/m ) → A . Applying Spec yields the desired map. (cid:3) Lemma 5.3.
Let x ∈ X be an invariant closed point, X → Spec( L ) , and V be as inthe previous lemma. Assume that there is an equivariant vector bundle isomorphism V ∼ = W ⊕ V ′ , where W has rank dim( X ) − , and let p : X → W be the resultingmap. Then there are invariant open affine neighborhoods U ⊆ X and S ⊆ W of x and respectively, such that p induces a smooth equivariant curve U → S admittinggood compactification. Proof.
First consider the case where X ⊆ V is an invariant open subscheme withclosure X = W × P ( V ′ ⊕ O L ). For any a ∈ X , the fiber of X p ( a ) has dimension oneand so ( X \ X ) p ( a ) must be finite over p ( a ) (where X \ X is considered as a closedsubscheme with reduced structure). Since X is projective over an affine scheme,there is an invariant affine neighborhood A ⊆ X of the finite set of closed points( X \ X ) . Then Z = ( X r X ) r (( X r X ) ∩ A ) is closed in X and so has closedimage in W . Now let S ⊆ W be an invariant affine neighborhood of 0 which missesthe image of Z and is contained in p ( X ) (we can find an affine neighborhood withthese properties and the intersection over all the translates by g ∈ G is an invariantneighborhood). Now let U = X S and U ′ = X S . Then U ′ r U has an invariantaffine neighborhood. Let U be the normalization of U ′ . Then U inherits a G -actionfrom that on U ′ and contains U as an invariant open subscheme. Since U → U ′ is finite, U r U is contained in an invariant affine neighborhood. Now U → S is asmooth equivariant curve with good compactification U .In the general case, since φ : X → V is ´etale at x , there is an open invariantaffine neighborhood on which φ is ´etale, so shrinking X , we may assume φ is ´etale.By the previous paragraph, there are invariant affine neighborhoods M ⊆ φ ( X ) of0 and S ⊆ W such that M → S is an equivariant smooth affine curve with goodcompactification M . Then U := φ − ( M ) → M is equivariant and quasi-finite andso the equivariant version of Zariski’s main theorem (see [LMB00, Theorem 16.5])yields an equivariant factorization of U → M as the composition of an invariantopen immersion U ֒ → U and an equivariant finite map q : U → M . Replacing U by its normalization, we may assume U is normal. Since M is an equivariant goodcompactification of M over S and q is affine, it follows that U is an equivariantgood compactification of U over S . (cid:3) Theorem 5.4 (Equivariant Gabber Rigidity) . Assume that every G -representationover k is a direct sum of one dimensional representations. Let F be a homotopyinvariant equivariant pseudo pretheory on Sm Gk with torsion values of exponentcoprime to char( k ) . If X is a smooth affine G -scheme over k of pure dimension d and x ∈ X is a closed point such that k ⊆ k ( x ) is separable, then there is anisomorphism: F ( Gx ) ∼ = −→ F (Spec( O hX,Gx )) . Proof.
We proceed by induction on d = dim( X ), the case d = 0 being clear. By(4.5), there is an equivariant isomorphism G × G x Spec( O hX,x ) ∼ = → Spec( O hX,Gx ) . Thus we are reduced to showing there is an isomorphism ǫ ∗ F (Spec( k ( x ))) ∼ = → ǫ ∗ F (Spec( O hX,x )) , where ǫ ( − ) = G × G x ( − ) and ǫ ∗ F := F ◦ ǫ . Note that ǫ ∗ F is a homotopy invariantequivariant pseudo pretheory on Sm G x k which is torsion of exponent coprime tochar( k ). Replacing G by G x and F by ǫ ∗ F it suffices to consider the case where Gx consists of a single point.The projection X x → X sends equivariant ´etale neighborhoods of x ∈ X x toequivariant ´etale neighborhoods of x ∈ X . If U → X is an equivariant ´etaleneighborhood of x ∈ X , then U x → X is an equivariant ´etale neighborhood of IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 15 x ∈ X mapping to U . This implies that Spec( O hX x ,x ) ∼ = Spec( O hX,x ) and so we mayreplace X with X x and assume there is an equivariant map X → Spec( L ), where L = k ( x ) (equipped with the corresponding G -action). Furthermore, by Lemma 5.2there is an equivariant vector bundle V over Spec( L ) such that O hX,x ∼ = O h V , andso it suffices to assume X = V and x = 0 L ∈ V .The assumption on G implies that there is a representation V ′ over k and an equi-variant isomorphism V ∼ = A ( V ′ ) L , see e.g., the beginning of the proof of [HVØ15,Theorem 8.11]. In particular, V is a direct sum of equivariant line bundles. Let i : W ⊆ V be a rank d − i ∗ induces anisomorphism F (Spec( O h V , )) ∼ = F (Spec( O h W , )), since 0 L ∈ W and the inductionhypothesis implies that F ( W ) ∼ = F (0 L ). The inclusion i is split by the projection p : V → W , so it suffices to see that i ∗ is injective.Suppose that [ α ] ∈ F (Spec( O h V , )) is such that i ∗ ([ α ]) = 0. By definition F (Spec( O h V , )) = colim U →V F ( U ), where the colimit is over equivariant ´etale neigh-borhoods of 0 L ∈ V . Thus, there is a representative α ∈ F ( U ) of [ α ] where U → V is an affine equivariant ´etale neighborhood of 0 L . There is a canonical equivariantmap π : Spec( O h V , ) → U .After shrinking U , there is a smooth affine equivariant curve U → Y , admitting agood compactification, by Lemma 5.3, where Y ⊆ W is an invariant neighborhoodof 0. Consider the following commutative diagram of equivariant maps:Spec( O h V , ) s i * * j i & & ▼▼▼▼▼▼▼▼▼▼▼▼ id " " ˜ U q / / q (cid:15) (cid:15) U (cid:15) (cid:15) Spec( O h V , ) p / / Spec( O h W , ) / / Y, where the rectangle is a pullback. By Lemma 4.7, ˜ U → Spec( O h V , ) is a smoothaffine equivariant curve admitting good compactification. The maps s := π and s := π ◦ i ◦ p induce equivariant sections j , j : Spec( O h V , ) → ˜ U of q . Thesections j , j agree on the closed orbit by construction and therefore j ∗ = j ∗ byTheorem 5.1. Thus [ α ] = π ∗ α = p ∗ i ∗ π ∗ α = 0. (cid:3) On the equivariant Gersten resolution
For an affine G -scheme X ∈ Sch Gk , let M G ( X ) denote the abelian categoryof G -equivariant coherent O X -modules. For p ≥
0, let M G,p ( X ) ⊂ M G ( X ) de-note the Serre subcategory of coherent sheaves F whose support is a subscheme ofcodimension ≥ p in X . Since F is equivariant, the support is an invariant closedsubscheme of X . Let S G,p ( X ) denote the set of all distinct set-theoretic G -orbits[ x ] in X of codimension p points x of X . Consider the filtration of M G ( X ) by Serresubcategories M G ( X ) = M G, ( X ) ⊃ M G, ( X ) ⊃ M G, ( X ) ⊃ · · · ⊃ M G,p ( X ) · · · . Since the natural exact functor M G,p ( X ) → ` [ x ] ∈ S G,p ( X ) S n M G (Spec( O X,Gx /J nGx ))has kernel M G,p +1 ( X ) and admits a section functor, by [Gab62, Proposition III.2.5]we have an equivalence of categories: M G,p ( X ) M G,p +1 ( X ) ≃ −→ a [ x ] ∈ S G,p ( X ) [ n M G (Spec( O X,Gx /J nGx )) , where J Gx denotes the Jacobson radical of the semilocal ring O X,Gx . The Devissagetheorem [Qui73, Theorem 4], the Chinese remainder theorem and the equivalenceof equivariant K -theory and G -theory for regular G -schemes [Tho87, Theorem 5.7]imply that K Gq ( a y ∈ [ x ] Spec( k ( y ))) ≃ G Gq ( a y ∈ [ x ] Spec( k ( y ))) ≃ K q ( M G (Spec( O X,Gx /J nGx ))) , for every n . This yields an isomorphism for the union along all n . Further for any x ∈ X , we have the Morita isomorphism [Tho87, Proposition 6.3] K Gq ( a y ∈ [ x ] Spec( k ( y ))) ≃ K G x q (Spec( k ( x ))) . Combining the above and by [Qui73, Theorem 5], for each p ≥ · · · → K i ( M G,p +1 ( X )) → K i ( M G,p ( X )) → ` [ x ] ∈ S G,p ( X ) K G x i (Spec( k ( x ))) → K i − ( M G,p +1 ( X )) → · · · . The above gives rise to a strongly convergent spectral sequence E p,q = a [ x ] ∈ S G,p ( X ) K G x − p − q (Spec( k ( x ))) ⇒ G G − p − q ( X ) . For X ∈ Sm Gk , the spectral sequence yields a sequence of abelian groups(6.1)0 −→ K Gn ( X ) −→ ` [ x ] ∈ S G, ( X ) K G x n (Spec( k ( x ))) d −→ ` [ x ] ∈ S G, ( X ) K G x n − (Spec( k ( x ))) d −→ ` [ x ] ∈ S G, ( X ) K G x n − (Spec( k ( x ))) d −→ · · · , where d is the differential on the E -terms of the spectral sequence.The Gersten conjecture states that (6.1) is exact if G is trivial and X = Spec( R ),where R is a regular local ring. This is known for regular local rings containinga field, the geometric case was proved by Quillen [Qui73, Theorem 5.11] and thegeneral equicharacteristic case was proved by Sherman [She78] in the 1-dimensionalcase and Panin [Pan03] for higher dimensions. If X is a regular local ring containinga field with a trivial G -action, where G is a finite diagonalizable group, then theGersten sequence (6.1) is simply the tensor product of the non-equivariant Gerstensequence with the group ring Z [ G ] (by [Ser68, Section 3.4]), and is therefore exact.If the action of G is non-trivial, we discuss in Example 6.2 below that the sequence(6.1) need not be exact even for n = 0. Example 6.2.
Let G = Z / Z act on X = A k = Spec( k [ t ]) via the map t
7→ − t .For the closed point x = ( t ) ∈ A k the Henselization O hX,x is the ring of algebraicformal power series in t over k . We compute the G -equivariant K with mod- l IGIDITY FOR EQUIVARIANT PSEUDO PRETHEORIES 17 coefficients of A x ) := Spec( O X,Gx ), Spec( O hX,Gx ), the orbit Gx , and the genericpoint η ∈ X .By [Tho87, Proposition 6.2] there is an isomorphism K G ( Gx ) ∼ = −→ K G x (Spec( k )) , where the set-theoretic stabilizer G x of x is equal to G = Z / Z . We have K G (Spec( k ); l ) ∼ = K G (Spec( k )) ⊗ Z /l Z ∼ = Z /l Z ⊕ Z /l Z . Thus for a field k of characteristic coprime to 2, l , Theorem 5.4 implies K G (Spec( O hX,Gx ); l ) ∼ = K G ( Gx ; l ) ∼ = Z /l Z ⊕ Z /l Z . The natural map π : A x ) → Spec( k ) affords a G -equivariant factorization: A x ) π / / j ' ' PPPPPP
Spec( k ) A k . π ❧❧❧❧❧❧ Here j ∗ : K G ( A k ) → K G ( A x ) ) is surjective by the localization exact sequence, and π ∗ : K G (Spec( k )) → K G ( A k ) is an isomorphism [Tho87, Theorems 2.7, 5.7, 4.1]. Itfollows that π ∗ : K G (Spec( k )) → K G ( A x ) ) is surjective. Since π : A x → Spec( k )has an equivariant section given by t π ∗ : K G (Spec( k )) → K G ( A x ) ) is alsoinjective. Therefore K G ( A x ) ; l ) ∼ = K G (Spec( k ); l ) ∼ = Z /l Z ⊕ Z /l Z .For the generic point η = Spec( k ( t )), note that the G -action on k ( t ) is freeand k ( t ) G = k ( t ). Therefore, K G ( η ; l ) ∼ = K ( k ( t )) ⊗ Z /l Z ∼ = Z /l Z so that K G ( A x ) ; l ) ≇ K G ( η ; l ). Remark 6.3.
As pointed out by the referee, the Gersten complex for A x ) withaction of the group G = Z / Z given by t
7→ − t as in the above example can beanalyzed using the localization sequence as follows. Under the notations of example6.2, we get an exact sequence: · · · → K G (Spec( k ( t )) ∂ −→ K G (Spec( k )) x ∗ −→ K G ( A x ) ) η ∗ −→ K G (Spec( k ( t ))) . Now the closed point x ∈ A x ) can be seen as the zero set of the diagonal sectionof the line bundle L = A x ) × A k → A x ) , where A k has the above non-trivial G -action. By a variant of the excess intersection formula for equivariant K -theory[K¨oc98, Theorem 3.8], x ∗ (1) = 1 − [ L ], and this class is non-zero in K G ( A x ) ).Thus η ∗ is not injective. The above considerations give the geometric reason forthis: as soon as the top Chern class (in equivariant K -theory of the point) of thenormal bundle is non-trivial, then x ∗ is non-zero and η ∗ is not injective. In thecases considered in other articles, the normal bundle has trivial action, so the topChern class is zero and the map η ∗ is injective.The rigidity property and the exactness of the Gersten sequence (6.1) are twoimportant properties of algebraic K -theory of semilocal rings. In Example 3.7 andTheorem 5.4, we prove the rigidity theorem for equivariant K -theory of schemeswith finite group actions. Example 6.2 (see also [Ngu16, Section 5.3]) shows thatthe Gersten sequence is not exact for equivariant K -theory of semilocal rings withnon-trivial Z / Z -actions. In this respect the cases of trivial and non-trivial actionsare very different. References [AD] A. Ananyevskiy and A. Druzhinin,
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