Motivic homological stability of configuration spaces
aa r X i v : . [ m a t h . A T ] J u l MOTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES
GEOFFROY HOREL AND MARTIN PALMER
Abstract.
We prove that some of the classical homological stability results for config-uration spaces of points in manifolds can be lifted to motivic cohomology. Introduction
A classical result of McDuff and Segal states that the unordered configuration spaces of aconnected, open manifold M are homologically stable. More precisely, let M be a connectedmanifold homeomorphic to the interior of a manifold with non-empty boundary and write C n ( M ) = { ( x , . . . , x n ) ∈ M n | x i = x j for i = j } / Σ n for the unordered configuration space of n points in M . The theorem of McDuff and Segal(reproven more recently by Randal-Williams) is the following. Theorem 1.1 ([Seg73, McD75, Seg79, RW13]) . There are stabilization maps of the form C n ( M ) → C n +1 ( M ) such that the induced maps on integral homology H i ( C n ( M ); Z ) −→ H i ( C n +1 ( M ); Z ) are isomorphisms when n ≥ i . This is part of a more general phenomenon of homological stability that holds for manyother families of spaces or groups, including general linear groups [vdK80], mapping classgroups of surfaces [Har85], automorphism groups of free groups [Hat95, HV98] and modulispaces of high-dimensional manifolds [GRW17, GRW18].The goal of this paper is to lift Theorem 1.1 to a version for étale and motivic cohomol-ogy. If X is a smooth scheme over a number field, we denote by Conf n ( X ) its associatedunordered configuration scheme (described in §2). When X is not complete (and under aminor additional assumption), we construct in §4 stabilization maps(1.1) M (Conf n ( X )) −→ M (Conf n +1 ( X ))in the category of motives, whose Betti realizations agree with the classical stabilizationmaps for the unordered configuration spaces of the complex manifold M = X an consistingof the complex points of X . Our main result is then the following. Theorem 1.2 (Theorems 5.1 and 6.7) . Let X be as above and assume that its étale motive M et ( X ) is mixed Tate and that the complex manifold X an is connected. Then the maps ofétale motivic cohomology groups H p,qet (Conf n +1 ( X ); Λ) −→ H p,qet (Conf n ( X ); Λ) induced by (1.1) are isomorphisms for p ≤ n/ and any coefficient ring Λ . In the case when X = A d is affine space, the maps of motivic cohomology groups H p,q ( ] Conf n +1 ( A d ); Λ) −→ H p,q ( ] Conf n ( A d ); Λ) induced by (1.1) are isomorphisms for p ≤ n/ and any coefficient ring Λ . The statement for étale motivic cohomology is proven in §5 as a direct consequence of thetopological case (Theorem 1.1), a detection result for Betti realization (Theorem 3.7) andthe existence of the stabilization maps at the étale motivic level. The proof of the statementfor motivic cohomology in the case X = A d is proven in §6, and is more indirect, since theanalogous detection result does not hold in this case. Instead, we use a detection result forthe associated graded of the weight filtration, and the key topological input (Lemma 6.8) isa certain twisted homological stability result for the symmetric groups.Let us emphasize that the first part of the theorem applies to X = A d and gives us anétale motivic cohomological stability result. This is different from the second part of thetheorem which is a cohomological stability result for motivic cohomology (as opposed toétale motivic cohomology). The price to pay for this finer result is that we have to workwith ] Conf n instead of Conf n . The configuration space ] Conf n is a stacky version of theunordered configuration space (see §2 for a precise definition). Remark 1.3.
When X is the affine line A , the first statement of Theorem 1.2 (for étalemotivic cohomology) has previously been proven in [Hor16], except that the isomorphisms(in a stable range) of [Hor16] are induced not by motivic lifts of stabilization maps , butrather by motivic lifts of scanning maps C n ( C ) → Ω n S and of maps between the differentpath-components of Ω S . The paper [Hor16] also uses a different model (denoted by C n in[Hor16, §5]) for the configuration scheme Conf n ( A ).Throughout this paper, we denote by K a number field equipped with an embedding K → C and we denote by S the spectrum of K . For a smooth scheme X over S , we denoteby X an the set X ( C ) with its complex manifold structure.2. Quotient and stacky quotient
Let X be a smooth scheme over S and G be a finite group acting on the right on X .We denote by X/G the quotient of X by G in the category of smooth schemes whenever itexists. We denote by [ X/G ] the “stacky quotient” of X by G . This is a simplicial object inthe category of smooth schemes given by[ n ] X × G n . This is the nerve of the translation groupoid of the action of G on X .By mapping a fixed smooth scheme into a simplicial scheme, we can turn a simplicialscheme into a simplicial presheaf on the category of smooth schemes. We will use the samenotation to denote a simplicial scheme and the associated simplicial presheaf. Observe thatthere is a canonical map of simplicial scheme [ X/G ] → X/G (where we view
X/G as aconstant simplicial scheme).
Proposition 2.1.
Assume that the quotient
X/G exists in the category of smooth schemesand that the map X → X/G is an étale map. Then the canonical map [ X/G ] → X/G is anétale weak equivalence of simplicial presheaves.Proof.
Let F be a fibrant object in simplicial presheaves with the Jardine model structure.We denote by f the map induced by precomposition with the canonical map f : Map( X/G, F ) → Map([
X/G ] , F ) . In fact, [Hor16] also claimed to prove the second statement of Theorem 1.2 (for motivic cohomology)for the affine line X = A , but the proof contained an error; see the erratum [Hor]. OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 3
We need to prove that f is a weak equivalence of simplicial sets. The domain of f is simply F ( X/G ) while the codomain of f is the homotopy limit of the cosimplicial diagram[ n ] F ( X × G n )Now we observe that there is an isomorphism X × ( X/G ) X × ( X/G ) . . . × ( X/G ) X ∼ = X × G n where the fiber product on the left has n + 1 terms and this isomorphism is compatible withthe cosimplicial structures on both sides. So the map f can be identified with the map F ( X/G ) → holim ∆ ([ n ] F ( X × ( X/G ) ( n +1) )) . The latter map is a weak equivalence since F is fibrant and X → X/G is an étale cover. (cid:3)
For X a smooth scheme over S , we denote by PConf n ( X ) the complement in X n of thefat diagonal. The Σ n -action on X n restricts to a Σ n -action on PConf n ( X ). We denoteby Conf n ( X ) the quotient of PConf n ( X ) by the action of Σ n in the category of smoothschemes. We denote by ] Conf n ( X ) the stacky quotient [PConf n ( X ) / Σ n ]. Observe that inthis case the Σ n -action is free which implies that the assumptions of the previous propositionare satisfied. 3. Motives
Generalities.
We recall the construction of the category DA ( S, Λ). We start fromthe category of complexes of presheaves of Λ-modules on the site of smooth schemes over S and we force descent for étale hypercovers, contractibility of the affine line and invertibilityof the Tate motive for the tensor product. We define DM ( S, Λ) in a similar fashion exceptthat we start from the category of complexes of presheaves with transfers and we use theNisnevich topology instead of the étale topology. There is a left adjoint functor DM ( S, Λ) → DA ( S, Λ)which is an equivalence of categories when Λ is a Q -algebra. The category DM ( S, Λ) containsa collection of objects indexed by Z called the Tate twists and denoted by Λ( n ) , n ∈ Z . Thereare analogously defined motives in DA ( S, Λ). There is a symmetric monoidal categorystructure on both DM ( S, Λ) and DA ( S, Λ) such that the formulaΛ( i ) ⊗ Λ( j ) ∼ = Λ( i + j )holds for all integers i and j .A smooth scheme over S yields an object in DM ( S, Λ) and in DA ( S, Λ) denoted M ( X )and M et ( X ) respectively. The étale motivic cohomology with coefficients in Λ of a smoothscheme X over S is the bigraded collection of Λ-modules :H p,qet ( X, Λ) := Hom DA ( S, Λ) ( M et ( X ) , Λ( q ))Likewise the motivic cohomology is given byH p,q ( X, Λ( q )) := Hom DM ( S, Λ) ( M ( X ) , Λ)We can also define the motivic cohomology of [
X/G ] when G is a finite group acting ona smooth scheme X . For this we first define M ([ X/G ]) as the homotopy colimit of thesimplicial object of DM ( S, Λ) given by[ n ] M ( X × G n )and then we set H p,q ([ X/G ] , Λ) := Hom DM ( S, Λ) ([ X/G ] , Λ( q )) . OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 4
Remark 3.1.
The object M ([ X/G ]) can equivalently be defined as the homotopy orbits ofthe G -action on M ( X ). Remark 3.2.
The careful reader will have noticed that there is an abuse of terminology inthe definition above as there is no definition of the homotopy colimit of a simplicial object ina triangulated category. What we really mean is that DM ( S, Λ) is the homotopy categoryof a model category (or an ∞ -category) and that the diagram [ n ] M ( X × G n ) lifts tothe level of model categories. We can thus take the homotopy colimit in the model categoryand then consider the result as an object in the homotopy category. We will allow ourselvesto make this abuse in a few other places in the paper.We could define in a smilar fashion the étale motive of [ X/G ]. We have an isomorphism M et ([ X/G ]) ∼ = M ( X/G )whenever
X/G exists by Proposition 2.1. Since étale motivic cohomology coincides withmotivic cohomology when the ring of coefficients is a Q -algebra, we can also deduce thefollowing theorem. Theorem 3.3.
Assume that the quotient
X/G exists in the category of smooth schemes.Let Λ be a Q -algebra. Then the canonical map [ X/G ] → X/G induces an isomorphism H p,q ( X/G, Λ) → H p,q ([ X/G ] , Λ) . Another important feature of these categories that we now recall is the so-called puritytheorem.
Theorem 3.4.
Let X be a smooth scheme and D be a smooth closed subscheme of X ofcodimension c , then there exists a cofiber sequence in DM ( S, Λ) . M ( X − D ) M ( i ) −−−→ M ( X ) → M ( D )( d )[2 d ] where i denotes the open inclusion X − D → X . There is a similar sequence in DA ( S, Λ) . Betti realization.
The functor X X an from smooth S -schemes to topologicalspaces induces a functor called Betti realization B ∗ : DA ( S, Λ) → D (Λ)and similarly a functor B ∗ : DM ( S, Λ) → D (Λ)The only things we will need to know about these functors is that they are symmetricmonoidal left adjoints (in fact they come from left Quillen functors) and: Fact 3.5.
The composite B ∗ ◦ M : Sm S → D (Λ) is naturally isomorphic to the functor X C ∗ ( X an , Λ) , where Sm S denotes the category ofsmooth schemes over S . Mixed Tate motives.
We denote by
DAT ( S, Λ) the smallest triagulated subcategoryof DA ( S, Λ) containing all the Tate twists Λ( n ) and closed under arbitrary coproducts andretracts. We define DMT ( S, Λ) analogously. An object of
DAT ( S, Λ) or
DMT ( S, Λ) willbe called a mixed Tate motive. Observe that the tensor product on DM ( S, Λ) and DA ( S, Λ)induces a symmetric monoidal structure on
DMT ( S, Λ) and
DAT ( S, Λ) respectively.
Proposition 3.6.
Let X be a smooth scheme over S that is such that M ( X ) is a mixedTate motive, then for each n , M (PConf n ( X )) is also a mixed Tate motive. OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 5
Proof.
Let P ( n ) be the set of subsets of { , . . . , n } with 2 elements. For P a subset of P ( n ),we denote by PConf n,P ( X ) the complement in X n of the diagonals indexed by P . We havePConf n, ∅ ( X ) = X n and PConf n,P ( n ) ( X ) = PConf n ( X ). We shall prove more generallythat M (PConf n,P ( X )) is mixed Tate for all P . We do this by induction on the pair ( n, | P | )with the lexicographic ordering. The result is obvious for | P | = 0. Now, assume that ( i, j )is an element of P and let Q = P − { ( i, j ) } . Then we can decompose PConf n,Q ( X ) as theunion of the open subscheme PConf n,P ( X ) and its closed complement which is isomorphicto PConf n − ,P ′ ( X ) for a certain P ′ ∈ P ( n − (cid:3) Betti realization and connectivity.
Finally, we will need the following theoremwhich relates the connectivity of a mixed Tate motive to the connectivity of its Betti re-alization. We denote by
DAT ( S, Λ) gm the smallest thick subcategory of DAT ( S, Λ) thatcontains all the Tate twists Λ( i ) , i ∈ Z . Theorem 3.7.
Let f : X → Y be a map in DAT ( S, Z ) gm . Assume that B ( f ) is an isomor-phism in negative homological degrees. Then, for all commutative rings Λ , the map Hom DA ( S, Λ) ( Y, Λ( q )[ p ]) → Hom DA ( S, Λ) ( X, Λ( q )[ p ]) induced by f is an isomorphism for all q and for all p < .Proof. Using the extension of scalars adjunction DA ( S, Z ) ⇆ DA ( S, Λ)we see that it suffices to prove that the map induced by f Hom DA ( S, Z ) ( Y, Λ( q )[ p ]) → Hom DA ( S, Z ) ( X, Λ( q )[ p ])is an isomorphism for all q and for all p <
0. We will in fact prove the more general factthat the map Hom DA ( S, Z ) ( Y, A ( q )[ p ]) → Hom DA ( S, Z ) ( X, A ( q )[ p ])is an isomorphism for any abelian group A . The case A = Q is classical and follows from theexistence of the motivic t -structure on DAT ( S, Q ) gm [Lev93] and the argument of [Hor16,Corollary 4.6] (beware that, contrary to what is stated in [Hor16], this corollary is incorrectwith integral coefficients. However, it is correct with rational coefficients).We will prove the result for A = Z /n in the paragraph below. Assuming this for themoment, then we can conclude as follows. Since the functor A Hom DA ( S, Z ) ( U, A ( q )[ p ])preserves filtered colimits, it is enough to prove the statement for A finitely generated. Sincethis functor also preserves direct sums, we can reduce to A = Z /n or A = Z . By assumption,we have the result for A = Z /n . Now, for any U ∈ DAT ( S, Z ) gm , we have a long exactsequence . . . → Hom( U, Q / Z ( q )[ p − → Hom( U, Z ( q )[ p ]) → Hom( U, Q ( q )[ p ]) → . . . (where all Homs are in DA ( S, Z )) induced by the short exact sequence0 → Z → Q → Q / Z → Z will follow from the result for Q and the result for Q / Z .To conclude in the latter case, it suffices to observe that Q / Z is a filtered colimit of cyclicgroups.Now, we treat the case A = Z /n . Suslin rigidity gives us an equivalence DA ( S, Z /n ) ≃ D ( S et , Z /n ). Moreover, viewed through the equivalence, the Betti realization functor canbe identified with i ∗ : D ( S et , Z /n ) → D ( T et , Z /n ) ≃ D ( Z /n ) OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 6 where T et = Spec( K ) is the small étale site of the algebraic closure of K and i ∗ is themap induced by the inclusion i : K → K (this is the Suslin rigidity theorem together withArtin’s theorem on the isomorphism between étale and singular cohomology). Moreover, for U ∈ D ( S et , Z /n ), we have an étale descent spectral sequence of the form H s (Γ , Hom D ( T et , Z /n ) ( i ∗ U, Z /n ( q )[ t ])) = ⇒ Hom D ( S et , Z /n ) ( U, Z /n ( q )[ s + t ])where H ∗ (Γ , − ) denotes Galois cohomology with respect to Γ = Gal( K/K ) (see for instance[AGV72, VIII, Corollaire 2.3]). Coming back to our situation, the map f : X → Y willinduce an isomorphism on the E -page of the étale descent spectral sequence in the range t <
0. Since moreover, this spectral sequence is zero for negative s , we see that the map f : X → Y will induce an isomorphism on the E ∞ -page for s + t < DA ( S, Z ) ( Y, Z /n ( q )[ p ]) → Hom DA ( S, Z ) ( Y, Z /n ( q )[ p ])for p < (cid:3) Construction of the stabilization map
In this section X is a smooth scheme over S . Assumption 4.1.
We assume that there exists a pair (
Y, D ) consisting of Y a smoothscheme over S , D a non-empty closed smooth subscheme such that X ∼ = Y − D . We makethe additional assumption that D has a K -point.The first part of the assumption is not very restrictive. Such a pair can be found as soonas X is not complete. Indeed, in that case, by a theorem of Nagata, X can be written asthe complement of a non-empty closed subscheme in a complete scheme X . Then, usingHironaka’s resolution of singularities, we can assume that X is smooth and that ∆ = X − X is a normal crossing divisor. If we write ∆ = ∪ ni =0 D i the decomposition of ∆ into irreduciblecomponents, we can then take Y = X − ∪ ni =1 D i and D = D ∩ Y .Under Assumption 4.1, we construct (see Construction 4.5 below) a stabilization map(4.1) M (PConf n ( X )) → M (PConf n +1 ( X ))that is Σ n -equivariant (see Lemma 4.6) in the category DM ( S, Λ). We can thus take ho-motopy orbits with respect to the symmetric group and get a map M ( ] Conf n ( X )) → M ([PConf n +1 ( X ) / Σ n ])Finally we can compose this with the obvious map M ([PConf n +1 ( X ) / Σ n ]) → M ([PConf n +1 ( X ) / Σ n +1 ]) ∼ = M ( ] Conf n +1 ( X ))and we get the stabilization map(4.2) M ( ] Conf n ( X )) → M ( ] Conf n +1 ( X )) . This is the map that will induce the isomorphisms in Theorems 5.1 and 6.7. The maps(4.1) and (4.2) have the correct Betti realizations by Lemma 4.7. Observe that in the étalesetting, by Proposition 2.1, the stabilization map above can equally be seen as a map ofétale motives: M (Conf n ( X )) → M (Conf n +1 ( X )) OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 7
Proposition 4.2.
Let X be a smooth scheme and D be a smooth closed subscheme ofcodimension c , let N ( j ) be the normal bundle of the inclusion j : D → X and let N ( j ) be the complement of the zero section of N ( j ) . Then there exists a map called the motivicexponential map exp : M ( N ( j )) → M ( X − D ) whose Betti realization is homotopic to the map induced by the tubular neighborhood inclusion C ∗ ( N ( j ) an ) ⊂ C ∗ (( X − D ) an ) Proof.
The closed inclusions j and D ⊂ N ( j ) induce distinguished (Gysin) triangles M ( X − D ) → M ( X ) → M ( D )( c )[2 c ] and M ( N ( j )) → M ( N ( j )) → M ( D )( c )[2 c ] . In [Lev07, Section 5.2] Levine shows that one can construct an exponential mapexp : M ( N ( j )) → M ( X − D )such that the following square in DM ( S, Λ) commutes and is homotopy cocartesian: M ( N ( j )) M ( N ( j )) M ( X − D ) M ( X )(4.3)where the right-hand vertical map is induced by the composition N ( j ) → D j −→ X . Note thatLevine construct exp in the stable motivic homotopy category SH ( S ) instead of DM ( S, Λ)but the latter is simply the category of modules over the motivic Eilenberg-MacLane spec-trum in SH ( S ) so we can simply tensor Levine’s map with the motivic Eilenberg-MacLanespectrum to get the desired map.The map exp has the correct Betti realization. Indeed, since the Betti realization func-tor can be modelled by a left adjoint ∞ -functor between stable ∞ -categories, it preserveshomotopy cartesian squares. Therefore, applying Betti realization to the square 4.3, we geta homotopy cartesian square in D (Λ): C ∗ ( N ( j ) an ) C ∗ ( N ( j ) an ) C ∗ (( X − D ) an ) C ∗ ( X an )(4.4)On the other hand, we know from the excision theorem in classical algebraic topology andthe tubular neighborhood theorem that there is a homotopy cocartesian (and hence alsocartesian) square in D (Λ) which is the same as the one above but where the left-handvertical map is induced by the tubular neighborhood inclusion. Since homotopy pullbacksare unique up to weak equivalences, the Betti realization of exp must the homotopic to themap induced by the tubular neighborhood inclusion. (cid:3) Proposition 4.3.
In the setting of Proposition 4.2, suppose that X is equipped with anaction of a discrete group G that sends D to itself. Equip the normal bundle N ( j ) with thenatural induced G -action, which restricts to a G -action on the subscheme N ( j ) . Then themotivic exponential map map of Proposition 4.2 is G -equivariant.Proof. An examination of the construction of Levine (see [Lev07, Section 5.2]) shows thatthe map exp is natural in the data ( X, D ). (cid:3) OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 8
Definition 4.4.
Denote by PConf n +1 , ( Y, D ) ⊂ PConf n +1 ( Y ) the smooth subscheme ofordered ( n + 1)-point configurations in Y where the first n points lie in X , and denote byPConf n +1 , ( Y, D ) its closed smooth subscheme given by those configurations where, in addi-tion, the last point lies in D . Note that PConf n +1 , ( Y, D ) is isomorphic to PConf n ( X ) × D and its complement PConf n +1 , ( Y, D ) − PConf n +1 , ( Y, D ) is isomorphic to PConf n +1 ( X ). Construction 4.5.
The stabilization map (4.1) is constructed as follows. Denote the inclu-sion D → Y by i and the inclusion PConf n +1 , ( Y, D ) → PConf n +1 , ( Y, D ) by j . Choose a K -point ∗ ∈ N ( i ) (such a point exists by our assumption that D has a K -point). By con-struction, we have an identification N ( j ) = PConf n ( X ) × N ( i ), so the choice of ∗ inducesa map(4.5) M (PConf n ( X )) → M ( N ( j )) . By Proposition 4.2 and the identification of PConf n +1 , ( Y, D ) − PConf n +1 , ( Y, D ) withPConf n +1 ( X ), there is a map(4.6) M ( N ( j )) → M (PConf n +1 ( X )) , and (4.1) is defined to be the composition of these two maps. Lemma 4.6.
The stabilization map (4.1) is Σ n -equivariant.Proof. There is an action of Σ n on PConf n +1 , ( Y, D ) given by permuting the first n points,and this action preserves its subscheme PConf n +1 , ( Y, D ). This induces an action on N ( j ),which corresponds, under the identification N ( j ) = PConf n ( X ) × N ( i ), to the permutationaction on PConf n ( X ) and the trivial action on N ( i ). Hence the map (4.5) is Σ n -equivariant.By Proposition 4.3, the map (4.6) is also Σ n -equivariant. (cid:3) Lemma 4.7.
The Betti realizations of (4.1) and of (4.2) are the maps of chain complexesinduced by the classical stabilization maps for ordered and unordered configuration spacesrespectively.Proof.
The classical stabilization map for ordered configuration spaces is homotopic to thecomposition of(4.7) x ( x, ∗ ) : PConf n ( X ) an → PConf n ( X ) an × N ( i ) an ∼ = N ( j ) an with the tubular neighborhood inclusion(4.8) N ( j ) an → PConf n +1 ( X ) an . We therefore need to show that C ∗ ((4.7); Λ) ∼ = B ∗ ((4.5)) and C ∗ ((4.8); Λ) ∼ = B ∗ ((4.6)). Thefirst of these identifications follows directly from Fact 3.5 and the second follows from thelast part of Proposition 4.2. This shows that the Betti realization of (4.1) is the map inducedby the classical stabilization map for ordered configuration spaces.The classical stabilization map s un : Conf n ( X ) an → Conf n +1 ( X ) an for unordered con-figuration spaces is obtained from the classical stabilization map s ord : PConf n ( X ) an → PConf n +1 ( X ) an for ordered configuration spaces by taking quotients with respect to thenatural symmetric group actions. Since these actions are free and properly discontinuous,we may equivalently take homotopy orbits instead of quotients. At the beginning of thissection, we defined (4.2) by taking homotopy orbits (in the category of motives) of symmet-ric group actions on the map (4.1). The fact that C ∗ ( s un ; Λ) ∼ = B ∗ ((4.2)) thus follows fromthe fact that C ∗ ( s ord ; Λ) ∼ = B ∗ ((4.1)) – proved in the paragraph above – and the fact thatboth B ∗ and C ∗ ( − ; Λ) preserve homotopy colimits, since they are both left adjoints. (cid:3) OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 9 Homological stability for open schemes
In this section, we still assume that X is a smooth S -scheme that can be written as X = Y − D with Y a smooth scheme and D a non-empty smooth closed subscheme of Y .We further assume that we have chosen an S -point of the punctured normal bundle of theinclusion D → Y . Hence, we have a stabilization map M ( ] Conf n ( X )) → M ( ] Conf n +1 ( X ))as explained in the previous section. This stabilization map induces a map in étale motiviccohomologyH p,qet (Conf n +1 ( X ) , Λ) ∼ = H p,qet ( ] Conf n +1 ( X ) , Λ) → H p,qet ( ] Conf n ( X ) , Λ) ∼ = H p,qet (Conf n ( X ) , Λ) . Theorem 5.1.
Assum that X satisfies the condtions above and assume further that M ( X ) is a mixed Tate motive and that X an is a connected topological space. Then the stabilizationmap H p,qet (Conf n +1 ( X ) , Λ) → H p,qet (Conf n ( X ) , Λ) is an isomorphism for p ≤ n/ .Proof. This follows from the fact that the Betti realization detects connectivity (Theorem3.7) and the statement in the topological case (Theorem 1.1). (cid:3) Stability for X = A d In the case where X = A d we can prove stability in the category DM ( S, Λ) instead of DA ( S, Λ). Note that a similar result was claimed in [Hor16] but the proof is incorrect. Itwas based on the claim that the Betti realization functor on
DMT ( S, Λ) detects connectivitybut this is not the case. Here we replace the Betti realization by the associated graded forthe weight filtration which indeed detects connectivity and is still sufficiently close to theBetti realization in this case.6.1.
The weight filtration.
For X ∈ DMT ( S, Λ), we denote by . . . → w ≥ n ( X ) → w ≥ n − ( X ) → . . . → X the weight filtration of X . If we denote by w ≥ n DMT (Λ) the smallest localizing subcategoryof
DMT (Λ) containing the objects Λ( i ) with i ≥ n , then w ≥ n is by definition the rightadjoint to the inclusion w ≥ n DMT (Λ) → DMT (Λ)We denote by w n ( X ) the cofiber of the map w ≥ n +1 ( X ) → w ≥ n ( X ). We will need thefollowing fact about this functor. Proposition 6.1.
Let X be an object of DMT ( S, Λ) . Then the motive w n ( X ) is of theform A ( n ) for some object A of D (Λ) . Proposition 6.2.
Let M ∈ DMT (Λ) . Assume that for each n , the object B ◦ w n ( M ) isconnective, then for m < , we have H m,q ( M, Λ) = 0
Proof.
The filtration . . . → w ≥ n ( M ) → w ≥ n − ( M ) → . . . → M induces a spectral sequence E u,v = H u + v,q ( w u ( M ) , Λ) = ⇒ H u + v,q ( M, Λ) OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 10 therefore, it suffices to prove that H u + v,q ( w u M, Λ) = 0 for u + v <
0. By the previousProposition, the object w u ( M ) is of the form A ( u ) where A is in D (Λ). The fact that B ◦ w u ( M ) is connective implies that A is connective. It follows that A lies is in the fullsubcategory of D (Λ) generated under colimits by Λ[0]. Hence, without loss of generality, wemay assume that A = Λ[0]. But then the result is true since the group H n,q ( S, Λ) vanisheswhenever n < (cid:3) The weight filtration of
PConf n ( A d ) . We will need to compute the weight filtrationof the scheme PConf n ( A d ). For this purpose, we introduce a definition. We take α a positiverational number and we write α = p/q with p and q two positive coprime integers. Definition 6.3.
We say that an object X ∈ DM ( S, Λ) is α -pure if the following twoconditions are satisfied. • If n is not a multiple of q , then the map B ( w ≥ n +1 ( X )) → B ( w ≥ n ( X ))is an isomorphism. • If n is a multiple of q then the map B ( w ≥ n ( X )) → B ( X )exhibits B ( w ≥ n ( X )) as the αn -connective cover of B ( X ).In other words, an object is α -pure if, up to rescaling by α , the weight filtration inducesthe Postnikov filtration upon application of the Betti realization functor. Observe that if q = 1, then, B ( X ) has homology concentrated in degrees that are multiple of p . Example 6.4.
Take X = M ( P n ). Then it is a classical computation that X = Λ(0) ⊕ Λ(1)[2] ⊕ . . . ⊕ Λ( n )[2 n ] , and it follows that X is 2-pure.For an example where α is not an integer, take X = M ( A d − { } ). Then one has X = Λ(0) ⊕ Λ( d )[2 d − X is ( d − d )-pure. Proposition 6.5.
Write α = p/q with p and q two coprime positive integers. (1) If X is α -pure, then B ( w qm ( X )) ∼ = H pm ( X an )[ pm ] and B ( w n ( X )) = 0 if n is not amultiple of q . (2) If X is α -pure, then X ( p )[ q ] is also α -pure. (3) If X and Z are α -pure and Y ∈ DM (Λ) fits in a cofiber sequence X → Y → Z then Y is also α -pure.Proof. We prove (1). If n is not a multiple of q , then, by definition, the map B ( w ≥ n +1 ( X )) → B ( w ≥ n ( X ))is an isomorphism, this implies that w n ( X ) = 0. If n = qm , we apply B to the cofibersequence w ≥ qm +1 ( X ) → w ≥ qm ( X ) → w qm ( X ) . By definition, B ( w ≥ qm ( X )) is the pm -connective cover of B ( X ) and B ( w ≥ qm +1 ( X )) is the p ( m + 1)-connective cover of X , it follows that B ( w m ( X )) ∼ = H pm ( X an )[ pm ].The proof of (2) is elementary, once we observe that w ≥ n ( X ( p )) ∼ = w ≥ n − p ( X ). OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 11
In order to prove (3), we use an alternative characterization of α -pure objects. An object X of DM (Λ) is α -pure if the homology of B ( w ≥ n ( X )) concentrated in degrees ≥ ⌈ αn ⌉ andthe homology of B ( w The object w k (PConf n ( A d )) is trivial if d does not divide k and w kd (PConf n ( A d )) ∼ = H k (2 d − (PConf n ( C d ))[ k (2 d − . Proof. By (1) of the previous proposition, it suffices to prove that the motive of PConf n ( A d )is ( d − d )-pure. We will prove more generally that this result holds for the complement ofa good arrangement of codimension d subspaces of an affine space. So let us consider X = A n − S i ∈ I V i a complement of a good arrangement of codimension d subspaces. Weproceed by induction on the cardinality of I . The result is obvious if I is empty. Nowassume that I = J ⊔ { i } . Let us denote by Y = A n − S j ∈ J V j . Then, X is an open subsetof Y whose closed complement, denoted Z , is the complement of a good arrangement ofcodimension d subspaces. We thus have a cofiber sequence M ( Z )( d )[2 d − → M ( X ) → M ( Y )thus the result follows from the induction hypothesis and (2) and (3) of Proposition 6.5. (cid:3) The main theorem.Theorem 6.7. The map H p,q (Conf n +1 ( A d ) , Λ) → H p,q (Conf n ( A d ) , Λ) is an isomorphism for p ≤ n/ .Proof. By Proposition 6.2, it suffices to prove that, for each a , the stabilization map w a (Conf n ( A d )) → w a (Conf n +1 ( A d ))is n/ w a commutes with colimits, it is equivalent to prove that w a (PConf n ( A d )) Σ n → w a (PConf n +1 ( A d )) Σ n +1 is n/ d does notdivide a (so the map is ∞ -connected), so we may assume that a = kd .By Lemma 6.6 and the spectral sequence H ∗ ( G ; H ∗ ( C )) ⇒ H ∗ ( C G ) for a G -equivariantchain complex C , we haveH l ( w kd (PConf n ( A d )) Σ n ) ∼ = H l − k (2 d − (Σ n , H k (2 d − (PConf n ( C d ))) , so it suffices to prove that the mapH l − k (2 d − (Σ n , H k (2 d − (PConf n ( C d ))) → H l − k (2 d − (Σ n +1 , H k (2 d − (PConf n +1 ( C d )))is an isomorphism in the range l ≤ n/ 2. By Lemma 6.8 below, this map is an isomorphismin the range l ≤ n/ k (2 d − − k. Note that k (2 d − − k ≥ 0, since k ≥ d ≥ 1, so we are done. (cid:3) Lemma 6.8. For any dimension m ≥ and coefficient ring Λ , the stabilization map H q (Σ n , H r (PConf n ( R m ); Λ)) → H q (Σ n +1 , H r (PConf n +1 ( R m ); Λ)) is an isomorphism for q ≤ n − rm − . OTIVIC HOMOLOGICAL STABILITY OF CONFIGURATION SPACES 12 Proof. For fixed m ≥ 2, the assignment S PConf S ( R m ), where S is a finite set, naturallyextends to a functor FI ♯ → hTop, by [CEF15, Proposition 6.4.2], and hence the assignment S H r (PConf S ( R m ); Λ)extends to a functor FI ♯ → Λ-Mod. We will show that this functor is polynomial of degree ≤ r/ ( m − M = R ∞ and X = ∗ . (Theorem A of [Pal18] is stated only in the case Λ = Z , so that Λ-Mod is thecategory of abelian groups, but the results of [Pal18] generalise immediately to any abeliancategory, including Λ-Mod.)As an aside, we note that applying [RWW17, Theorem A] with C = FI, A = ∅ , X = {∗} (where we may take k = 2) and N = 0 also gives the stated result, although only in theworse range of degrees q ≤ n − rm − − 1. The stated result would also follow from [Bet02,Theorem 4.3] if the functor H r (PConf • ( R m ); Λ) : FI ♯ → Λ-Mod could be extended furtherto the category of finite sets and all partially-defined functions (not just partially-definedinjections). However, there does not seem to be a natural extension to a functor on thislarger category.We now show that the functor T = H r (PConf • ( R m ); Λ) : FI ♯ → Λ-Mod is polynomialof degree at most d = ⌊ r/ ( m − ⌋ . Recall that this means that ∆ d +1 T = 0, where ∆ isthe operation on functors from FI ♯ to an abelian category defined in [Pal18, §3.1]. In theterminology of [Dja16] this is equivalent to saying that T is strongly polynomial of strongdegree at most d , when considered as a functor on the subcategory Θ = FI ⊂ FI ♯ . (Notethat the operation ∆ of [Pal18] corresponds to the operation δ of [Dja16] and, in the case M = Θ of [Dja16], it suffices to consider only δ , rather than δ a for all objects a of M ,since Θ is generated as a monoidal category by the object 1.) Thus [Dja16, Proposition 4.4]implies that it is equivalent to prove that T | FI is generated in degrees at most d . This, inturn, is equivalent, by [CEF15, Remark 2.3.8], to the condition that H ( T | FI ) i = 0 for all i > d , where H is the left adjoint of the inclusion of Fun(FB , Λ-Mod) into Fun(FI , Λ-Mod),where FB is the category of finite sets and bijections. The proof of Theorem 4.1.7 of [CEF15]shows that this condition will hold as long as, for all n ≥ 0, the Λ-module T ( n ) = H r (PConf n ( R m ); Λ)is generated by at most O ( n d ) elements. It therefore remains to verify this last condition.We first consider the case Λ = Z . By [FH01, Theorem V.4.1], the N -graded ringH ∗ (PConf n ( R m ); Z )is generated by (cid:0) n (cid:1) elements, all in degree m − 1, subject to certain relations (the cohomolog-ical Yang-Baxter relations). It follows that the abelian group H r (PConf n ( R m ); Z ) is trivialunless r = i ( m − i in (cid:0) n (cid:1) variables. The number of such monomials is at most (cid:0) n (cid:1) i ∼ O ( n i ) = O ( n d ) , and so H r (PConf n ( R m ); Z ) is generated by at most O ( n d ) elements. Since the cohomologygroups H ∗ (PConf n ( R m ); Z ) are free in all degrees, by [FH01, Theorem V.1.1], and the spacePConf n ( R m ) has the homotopy type of a finite CW-complex (see [FH01, §VI.8–10]), theuniversal coefficient theorem implies that the homology groups H ∗ (PConf n ( R m ); Z ) are alsofree in all degrees and H r (PConf n ( R m ); Z ) has the same rank as H r (PConf n ( R m ); Z ), so itis also generated by at most O ( n d ) elements. 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