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The quotient map on the equivariant Grothendieck ring of varieties

For a scheme S with a good action of a finite abelian group G having enough roots of unity we show that the quotient map on the G-equivariant Grothendieck ring of varieties over S is well defined with image in the Grothendieck ring of varieties over S/G in the tame case, and in the modified Grothendieck ring in the wild case. To prove this we use a result on the class of the quotient of a vector space by a quasi-linear action in the Grothendieck ring of varieties due to Esnault and Viehweg, which we also generalize to the case of wild actions. As an application we deduce that the quotient of the motivic nearby fiber is a well defined invariant.


Introduction
The Grothendieck ring K 0 (Var S ) of varieties over a separated scheme S is as group spanned by isomorphism classes [X] of separated schemes X of finite type over S with relations allowing to cut and paste. The ring structure is given by the fiber product. This ring is useful because additive invariants of varieties, for example the Euler characteristic and the number of points over a finite fields in positive characteristic, factor through this ring. Therefore the Grothendieck ring of varieties and localizations of it are used in motivic integration as universal value rings.
Let S now be a scheme with a good action of a finite group G, and consider the category (Sch S,G ) of separated S-schemes with good G-action. A group action on S is called good if every orbit lies in an affine subscheme of S. This condition insures that the quotient exists in the category of schemes. For technical reasons we also assume that the quotient map S → S/G is finite and S/G is separated and locally Notherian. The equivariant Grothendieck ring K G 0 (Var S ) of varieties over S is generated by isomorphism classes [X] of objects X in this category. Whenever Y is a G-invariant closed subscheme of X, one asks the class of X to be equal to the sum of the class of Y and the class of X \ Y . Moreover one asks the classes of two affine bundles with affine G-action to be equal if they have the same rank and the same base. Hence in particular the class of every affine bundle with affine action is equal to the class of a trivial affine bundle with G-action induced by the action on the base. The ring structure is again given by the fiber product.
Using the equivariant Grothendieck ring of varieties as value ring allows us to also encode some group action on a scheme X, as done for example with the monodromy action on the motivic Zeta function, see [DL01]. To get a well defined theory of motivic integration with group actions, one needs to make actions on affine bundles 'trivial'. This is where the last relation in the definition of the equivariant Grothendieck ring of varieties actually comes from.
One can ask now how to relate the equivariant Grothendieck ring with the usual one. A natural thing to do is to divide out the action on the class of a variety, i.e. to send the class [X] ∈ K G 0 (Var S ) of a scheme X with good G-action to the class of its quotient X/G in K 0 (Var S/G ). Bittner showed that such a quotient map is well defined if S is a variety over a field of characteristic zero and the action of G on S is free, see [Bit05,Lemma 3.2]. The proof uses that in this case the action on an affine bundle in the category (Sch S,G ) is free, and thus the quotient is again an affine bundle. For general G-action on S this is not the case.
In this paper, we show that for an abelian group G, the quotient map on the G-equivariant Grothendieck ring is well defined in general if we put some extra assumptions on the stabilizers of the points of S, see Theorem 8.1. For s ∈ S/G, denote by F s the residue field of s, and by G s ⊂ G the stabilizer of a point s ′ ∈ S in the inverse image of s under the quotient map S → S/G. With this notation, we show the following theorem assuming that the G-actions on S is tame, i.e. that the characteristic of the residue field of every point of S is prime to |G|.
Theorem (tame case). Let G be a finite abelian group acting tamely on S. Assume for all s ∈ S/G that F s contains all |G s |-th roots of unity. Then there is a well defined group homomorphism to [X/G] ∈ K 0 (Var S/G ) for every X ∈ (Sch S,G ). If the action of G on S is wild, i.e. not tame, we have to modify the theorem a bit. This is due to the fact that if G acts wildly on a scheme X the quotient of a closed invariant subscheme of X only has a universal homeomorphism onto its image in the quotient X/G, which is in general not a piecewise isomorphism on the underlying reduced schemes, see Example 5.8. It is not known whether two such schemes have the same class in the usual Grothendieck ring of varieties. Hence in the wild case already the first relation in the Grothendieck ring of varieties causes problems. Therefore we need to work in the modified Grothendieck ring K mod 0 (Var S/G ), in which classes of varieties connected by universal homeomorphisms are equal. Hence in the wild case we get the following theorem.
Theorem (wild case). Let G be a finite abelian group. Assume for all s ∈ S/G that F s contains all |G s |-th roots of unity. Then there is a well defined group homomorphism K G 0 (Var S ) → K mod 0 (Var S/G ) sending [X] ∈ K G 0 (Var S ) to [X/G] ∈ K mod 0 (Var S/G ) for every X ∈ (Sch S,G ).
In order to prove that the quotient map is well defined, we need to control in particular quotients of affine bundles by affine actions. Such quotients are not affine bundles in general, in fact they can even be singular. We can show that the class of the quotient V /G of an affine bundle ϕ : V → B in the Grothendieck ring of varieties over S/G only depends on the rank d of the bundle and its base B, see Lemma 7.1, and hence two affine bundles with same rank and same base are mapped to the same element under the quotient map. We proof the lemma by showing that all fibers of the induced map ϕ G : V /G → B/G between the quotients of V and B actually have the class of an affine space of dimension d in the Grothendieck ring of varieties, and conclude then by spreading out. In the case of wild actions we have to work again in the modified Grothendieck ring of varieties.
To compute the fibers of ϕ G we use a result on the class of the quotient of a vector space by a quasi-linear action of a finite abelian group G in the Grothendieck ring of varieties proved in [EV10, Lemma 1.1]. As in this paper only tame actions are considered, we generalize the result to the case of wild actions, see Proposition 6.2. More precisely, we show the following proposition.
K lifts to a k-linear action of G on a finite dimensional K-vector space V . If k contains all q-th roots of unity, then Note that this result and the analog result in the tame case are actually necessary for our theorem to hold. As in general it is wrong without assuming that k contains enough roots of unity, and if G is not abelian, the assumptions that G is abelian and the fields F s contain all |G s |-th roots of unity are necessary assumptions in our theorem, see Remark 8.3. How do we prove the proposition? In [EV10, Lemma 1.1] the analogous statement in the tame case was shown in the usual Grothendieck ring of varieties by decomposing V into eigenspaces. This cannot be done in the case of wild actions. Instead we construct a G-equivariant map from V to a vector space W of dimension one over K, use an induction argument to compute the fibers of the induce map between the quotients, and use again spreading out to conclude. To be able to compute the fibers separately, we need to work again in the modified Grothendieck ring of varieties.
As an application of our main theorem, we get that the quotient of the motivic nearby fiber is a well defined invariant with values in M k , the localization of K 0 (Var k ) with respect to L := [A 1 k ], with k a field of characteristic zero containing all roots of unity. The motivic nearby fiber is an invariant of a non-constant morphism f : X → A 1 k , with X an irreducible smooth k-variety, and was constructed in [DL01] as a limit of the motivic Zeta function. It takes values in Mμ X0 , with X 0 := f −1 (0),μ the profinite group of roots of unity, and Mμ X0 the localization of Kμ 0 (Var X0 ) with respect to the class of the affine line over X 0 .
We show that modulo L, the quotient of the motivic nearby fiber is equal to the motivic reduction R(f ) of f in the image of K 0 (Var k ) in M k , see Proposition 9.5. The motivic reduction of f is defined as the class of h −1 (X 0 ) in K 0 (Var k ) modulo L, where h : Y → X is any smooth modification of f , i.e. Y is a smooth k-variety and h is a proper morphism inducing an isomorphism Y \ h −1 (X 0 ) → X \ X 0 . The definition of R(f ) does not depend on the choice of such an h due to weak factorization.
From this result we deduce that, if X is a smooth variety with a proper, nonconstant morphism f : X → A 1 k , and the generic fiber , then the same holds for the special fiber X 0 of f in the image of K 0 (Var k ) in M k . This can be seen as a motivic analog of the main theorem in [Esn06, Theorem 1.1], which says the following: if V is an absolutely irreducible smooth projective variety over a local field K with finite residue field F which has a certain cohomological property, namely that thé etale cohomology of V × KK has coniveau 1, then the amount of points of the special fiber of every projective regular model of V is equal to 1 modulo |F |. We will explain this analogy in more details in Section 9.

Preliminaries
Fix a finite group G. Let S be a separated scheme endowed with a left action of G. If not mentioned otherwise, all group actions will be left actions. We say the action of G on S is good if every orbit of this action is contained in an affine open subscheme of S. By requiring the action to be good, one makes sure that the quotient exists in the category of schemes, see [Gro67, Exposé V.1]. We call an action tame if the characteristic of the residue field of every point s ∈ S is zero or positive and prime to the order of G. We call an action wild if it is not tame.
If not mentioned otherwise, we assume for the rest of the text that S is a separated scheme with a good G-action, that the quotient S/G is locally Notherian and separated, and that the quotient map S → S/G is finite. This is for example true if S is a separated scheme of finite type over a field k, and G acts on S by a group of k-morphisms, see [Gro67, Exposé V.1, Corollaire 1.5], thus it is not such a strong assumption.
We denote by (Sch S,G ) the category whose objects are separated schemes of finite type over S with a good G-action such that the structure map is G-equivariant, and whose morphisms are G-equivariant morphisms of S-schemes. One can check that the fiber product exists in this category by constructing a good G-action on the fiber product in the category of separated schemes of finite type.
. , x d ], but the K-structure on the first is given by sending s ∈ K to α −1 (s). By the definition of an affine action, α ′ is K-linear. Hence we have a ij x j for some a ij ∈ K. Using that α is a ringhomomorphism, we get that , because α| k = id by definition. Note that K is a Galois extension of k, and we have a surjective map So K is a k-vector space of dimension r := |Γ|. Now we can view E b as a K-vector space of dimension d, and hence also as a k-vector space of dimension rd. We have seen that the G-action on E b defines a k-linear action on E b which lifts the Galois action of Γ on K. This follows from Equation (1) and Equation (2). We call an action with this property quasi-linear.
As b is fixed under the action of G, the G-action on E and V restrict to E b and V b . Moreover ϕ b is G-equivariant. Take any g ∈ G, and let g E ∈ Aut(E b ) and Note that Remark 3.3 implies that g E is quasi-linear. Moreover g V (0) does only depend on g and the choice of 0, but not on v.
Remark 3.5. Let again V → B be a G-equivariant affine bundle of rank d with affine G-action. Let E → B be the associated vector bundle. Let b ∈ B be a fixed point, and let K be its residue field. Let H ⊂ G be the subgroup consisting of all elements of order prime to the characteristic of K. Assume that H is abelian.
View V b as a vector space over k = K G , and consider the action of H on V b . By Remark 3.4 we know that for every h ∈ H the corresponding automorphism sends We are now going to show that the action of H on V b has a fixed point. Therefore we view V b as a scheme over k, hence V b ∼ = A rd k = Spec(k[x 1 , . . . , x rd ]), and show by induction on n := |H| that the fixed point locus V H b ⊂ V b is isomorphic to A N k for some N ≥ 0. For n = 1, the statement is trivial.
So let n > 1. Then there exists a nontrivial cyclic q-sub group H ′ of H for some prime q, prime to the characteristic of k. Consider the induced action of H ′ on V b . As q = p, we can use [EN11, Corollary 5.5], which follows from a theorem of Serre in [Ser09], to get that Herek is the algebraic closure of k. In is not the empty scheme. Let h ∈ H be a generator of H ′ . Then the corresponding automorphism of k[x 1 , . . . , is a nonempty linear subspace of V b , so in particular isomorphic to A N k for some N ≥ 0.
As H is abelian, it maps every point fixed by H ′ to a point fixed by H ′ . Hence This action is still given by some k-linear maps composed with some translation. As the order of H/H ′ is smaller than n, we can now use the induction assumption to get that has a point v 0 over k.
Let g ∈ H ⊂ G, and let g V be the corresponding automorphism of V b . Then g V (v 0 ) = v 0 . If we now chose 0 in Remark 3.4 to be v 0 , we get from Equation (3) that for all g ∈ H ⊂ G we have Note that we can also find a fixed point in Remark 3.5 using elementary calculations instead of [EN11, Corollary 5.5]. In both cases we need to assume that H ⊂ G is an abelian subgroup. Moreover it is crucial that the order of H is prime to the characteristic of K. In the case of wild actions there exist G-equivariant affine bundles with affine G-action such that there is no change of coordinates making the action quasi-linear. Here is an example of such an affine bundle: Example 3.6. Let k be a field of characteristic p > 0, and let G = Z/pZ. Let B = Spec(k), and consider V = Spec(k[x]) with the G-action given by sending x to x + 1. As this action has no fixed point, there is no way of changing coordinates to achieve that this action is linear. Let E = Spec(k[y]) be the trivial vector bundle of dimension 1 over B with trivial action of G. Consider the map given by sending (e, v) ∈ E × B V to e + v ∈ V . As (e, v + 1) is mapped to e + v + 1 this map is clearly G-equivariant. One can check that it induces an isomorphism

The equivariant Grothendieck ring of varieties
Definition 4.1. The equivariant Grothendieck ring of S-varieties K G 0 (Var S ) is defined as follows: as an abelian group, it is generated by isomorphism classes [X] of elements X ∈ (Sch S,G ). These generators are subject to the following relations: ( , whenever Y is a closed G-equivariant sub scheme of X (scissors relation). We denote by L S the class of the affine line A 1 S with G-action induced by the action on S as above. If the base scheme S is clear from the context, we write L instead of L S . We define M G S as the localization K G 0 (Var S )[L −1 S ]. Notation. If G is the trivial group {e}, we write K 0 (Var S ) and M S instead of K G 0 (Var S ) and M G S , receptively. Note that in this case Relation (2) becomes trivial.

The modified Grothendieck ring of varieties
Due to the nature of wild actions, we are not able to compute quotients of such actions in the usual Grothendieck ring by decomposing a scheme into G-invariant subschemes and computing the quotient separately on these subschemes. The quotient of a closed subscheme has in general a purely inseparable map to the image of this subscheme under the quotient map. But in the wild case this map might not be a piecewise isomorphism, as we will see in Example 5.8. We do not know whether the classes of two schemes connected with such a morphism have the same class in the Grothendieck ring of varieties. Therefore we now introduce the modified Grothendieck ring of varieties, in which their classes are the same.
Definition 5.1. A morphism of schemes f : Y → X is called a universal homeomorphism, if for every morphism of schemes X ′ → X the morphism of schemes f ′ : Y × X X ′ → X ′ induced by base change is a homeomorphism.
The modified Grothendieck ring of S-varieties is defined as the quotient Denote by L S the class of the affine line A 1 S . If the base scheme S is clear from the context, we write L instead of L S . We define M mod S as the localization Remark 5.3. If S is a Notherian Q-scheme, then the quotient map is an isomorphism, see [NS11, Corollary 3.8.3]. In particular this holds if S is a scheme of finite type over any field of characteristic 0.
It is not known whether it is an isomorphism in positive characteristic, see [CLNS14, Chapter 1, Remark 3.3.6]. The problem is that standard specializing morphisms used to distinguish elements in the Grothendieck ring factor through the modified Grothendieck ring, see [CLNS14, Chapter 1, Corollary 3.3.4].
We will now prove some technical lemmas which will be used later to compute quotients in the (modified) Grothendieck ring of varieties.
Lemma 5.4. (Spreading out for the modified Grothendick ring) Take a directed system of Noetherian commutative rings (A i , ϕ ij : A i → A j ), and denote by A the direct limit of this system in the category of rings. Then there exists an isomorphism of rings ϕ mod : lim Proof. Consider the ring morphism induced by the ringmorphism ϕ i : K 0 (Var Ai ) → K 0 (Var A ) given by sending the class of an A i -scheme U to the class of U × Spec(Ai) Spec(A). By [CLNS14, Chapter 1, Proposition 1.5.6], ϕ is an isomorphism. As a universal homeomorphism is stable under base change, for every universal homeomorphism f : X → Y between two A i -schemes, the base change of f to Spec(A) is also a universal homeomorphism. Hence we get well defined maps ϕ mod , which induce a well defined surjective map ϕ mod as in the claim.
We still need to show that ϕ mod is injective. So let f : X → Y be a universal homeomorphism between A-schemes. By [Gro66, Theorem 8.8.2] there exist an i and a morphism of A i -schemes f i : X i → Y i such that the base change of f i to Spec(A) is f . By [Gro66,Theorem 8.10.5] f is a universal homeomorphism if and only if there is a j ≥ i such that the base change of f i induced by Spec(A j ) → Spec(A i ) is a universal homeomorphism. Hence ϕ mod is injective.
Recall that we assume that S is a separated scheme with good action of a finite group G, such that the quotient map S → S/G is finite and S/G is separated and locally Notherian. The next lemmas will enable us to decompose the quotient of schemes in the category (Sch G,S ) in the (modified) Grothendieck ring of varieties.
Lemma 5.5. Let X ∈ (Sch G,S ), and denote by π : X → X/G the quotient. Let Y ⊂ X be a closed G-invariant subscheme, and let Z be the image of Y under π. Then the G-action on X restricts to a good G-action on Y , and there exists a universal homeomorphism f : Y /G → Z. Hence in particular Proof. Let i : Y ֒→ X be the inclusion map. As Y ⊂ X is a G-invariant closed subscheme, the G-action on X restricts to Y . As every affine subscheme of X will restrict to an affine subscheme of Y , this action is good. Denote by π Y : Y → Y /G the quotient map. As i is G-equivariant, we get an induced map i G : We are going to show that i G : Y /G → Z is a universal homeomorphism. By [Gro65, 2.4.5.] it suffices to show that i G is finite, surjective and purely inseparable.
As both the points of X/G and Y /G are just orbits of the action of G, the map i G : Y /G → Z is a bijection on points.
As X is of finite type over S and hence over S/G using that S → S/G is finite, π is finite by [Gro67, Exposé V, Corollaire 1.5]. As i is proper, the same holds for π • i. As moreover π Y is surjective, i G is proper by [GW10, Proposition 12.59]. We have already seen that i G is quasi-finite, hence it is finite. It remains to show that i G is purely inseparable, i.e. that for all y ∈ Y /G the residue field L of y is purely inseparable over the residue field K of z := i G (y).
Using [Bou85, Capitre V.2, Théorème 2] we get the following: let G x be the stabilizer of a point x ∈ Y ⊂ X of the orbit of G over y and z, respectively, and let M be the residue field of x. Then M is normal over L and over K, and G x surjects on Gal(M, L) and Gal(K, L). Hence we get the following inclusions of fields As M is normal over K, M Gx is normal over K, too. We now can split this extension in a separable extension K ′ over K, and a purely inseparable extension M Gx over K ′ . Observe that K ′ is normal over K, and therefore K = K ′ Gal(K ′ ,K) . But Gal(K ′ , K) is a quotient of Gal(M Gx , K), and the latter is trivial. Hence K = K ′ . Therefore M Gx is purely inseparable over K, and hence the same holds for L. This finishes the proof.
Lemma 5.6. Assumptions and notation as in Lemma 5.5. Assume moreover that the action of G on X is tame. Then there exist a map f : Proof. Use the same notation as in the proof of Lemma 5.5. It follows from [Bou85, Capitre V.2, Proposition 5 and Corollaire] that M Gx is actually equal to L and to K, hence L = K. Hence we have a finite bijective map i G : Y /G → Z such that for every point y ∈ Y /G the residue field of y is isomorph with the residue field of i G (y). By [CLNS14, Chapter 1, Corollary 1.5.3] we can assume that Y /G and Z are reduced. Take a generic point η ∈ Y /G with function field κ η . The image i G (η) ∈ Z will be a generic point with function field isomorphic to κ η . Hence we can find open subschemes U ⊂ Y /G and V ⊂ Z, such that i G : U → V is an isomorphism. Now we can proceed with i G : Y /G \ U → Z \ V , and use Notherian induction to get that i G is a piecewise isomorphism. By [CLNS14, Chapter 1, Corollary 1.5.4] we get the claimed equation in K 0 (Var S/G ).
If the action of G on X is wild, we will really need to work in the modified Grothendieck ring, as the following examples show.
Example 5.7. Let p be a prime, G = Z/pZ and let k be a field of characteristic p.
, and consider the action of G on X given by sending x to x+y and y to y. We have that Nevertheless, Y and Z are isomorphic over k, but this isomorphism is not given by i G .
Example 5.8. Let p > 3 be a prime, let G = Z/pZ, and let k be a field of characteristic p. Let X = A 6 k = Spec(k[x, y, a, a ′ , b, b ′ ]), and consider the action of G on X given by sending One can check that Denote by π : X → X/G the quotient map. Consider Note that Y is G-invariant, and the induced action on Y is trivial. As ). Note that the residue field of the generic point of Z is isomorphic to the function field of the Note that the residue field of the generic point of Y = Y /G is isomorphic to the function field of E (p) . Now we compute the j-invariant of E and E (p) : j(E) = 1728 4a 3 4a 3 + 27b 2 and j(E (p) ) = 1728 4(a p ) 3 4(a p ) 3 + 27(b p ) 2 Hence E and E (p) will have the same j-invariant if and only if b 2 /a 3 = (b 2 /a 3 ) p . As this was only true if b 2 /a 3 was in F p , the j-invariants of E and E (p) are different. Therefore E and E (p) are not isomorphic, and hence Z and Y cannot be piecewise isomorphic. It is not known whether E and E (p) , and thus Z and Y , have the same class in the Grothendieck ring, see [CLNS14, Chapter 1, Remark 3.3.6].
Remark 5.9. Take X ∈ (Sch S,G ), let Y ⊂ X be a closed G-invariant subscheme and let U = X \ Y be the compliment. Let π : X → X/G be the quotient map. By [Gro67, Exposé V, Corollaire 1.5] we have that U/G ∼ = π(U ). Hence We do not know whether this is true or not, even if S = Spec(k) with k a finite field, see Example 5.8.
Lemma 5.10. Take two schemes V, B ∈ (Sch G,S ), and let ϕ : V → B be a Gequivariant morphism of finite type. Denote the induced map between the quotients with ϕ G : V /G → B/G. Let x ∈ B/G be a point with residue field k, let b be a point in B mapped to x under the quotient map, and let G b be the stabilizer of b. Assume that G b is a normal subgroup of G. Then G b acts on ϕ −1 (b), this action is good, and Proof. Let π : B → B/G and π V : V → V /G be the quotient maps. Note that Thus by Lemma 5.5 there is a universal homeomorphism . Hence we get a universal homeomorphism Thus in the modified Grothendieck ring we get Now consider the stabilizer G b of b ∈ π −1 (x). As G b is a subgroup of G, it acts on V and B. By construction, π −1 (x) is G-invariant and ϕ is G-equivariant, hence we get induced actions of G b on ϕ −1 (π −1 (x)) and π −1 (x), and an induced map As G b ⊂ G is a normal subgroup, we may consider H := G/G b . H acts on ϕ −1 (π −1 (x))/G b and π −1 (x)/G b , and ψ is H-equivariant. As π −1 (x) is the inverse image of the point x of the finite quotient map π, π −1 (x) is a finite union of points. Thus also π −1 (x)/G b is a finite union of points P 1 , . . . , P n , and hence ϕ −1 (π −1 (x))/G b ∼ = ψ −1 (P i ). As G b is the stabilizer of b and π −1 (x) is the orbit of b, the action of H on π −1 (x)/G b is free and transitive, i.e. for every pair i, j there is a unique h ij ∈ H with h ij (P i ) = (P j ). Hence h ij also maps ψ −1 (P i ) isomorphically to ψ −1 (P j ). Let W be the disjoint union of n copies of ψ −1 (P 1 ). Let H act on W as follows: for h ∈ H with h(P i ) = P j , let the corresponding automorphism of W map the i-th copy of ψ −1 (P 1 ) identically to the j-th copy. It is obvious that W/H ∼ = ψ −1 (P 1 ). Consider the map ϕ : W → ϕ −1 (π −1 (x))/G b given on the i-th copy of ψ −1 (P 1 ) by h 1i | ψ −1 (P1) . One can check that ϕ is a H-equivariant isomorphism with Hequivariant inverse, hence we get that Without loss of generality we may assume that P 1 is the image of b under the quotient map. As b is a component of All together we have Remark 5.11. Assumption and notation as in Lemma 5.10. If the characteristic of k is zero or prime to the order of G, we get that This holds, because in this case f is a finite, bijective map such that the map between the residue fields of the points are isomorphic, see the proof of Lemma 5.6. Hence the same holds for f k , and we can show, as done in Lemma 5.6, that ϕ −1 G (x) and ϕ −1 (π −1 (X))/G × Spec(k) ∼ = ϕ −1 (b)/G b have the same class in K 0 (Var k ).

Quotients of vector spaces by quasi-linear actions
The aim of this section is to show a version of the following proposition in the case of wild group actions. This proposition was proved in [EV10, Lemma 1.1] as a generalization of [Loo02, Lemma 5.1].
Proposition 6.1. [EV10, Lemma 1.1] Let G be a finite abelian group with quotient G → Γ. Let k be a field of characteristic zero, or positive characteristic prime to |G|, and let K/k be a Galois extension with Galois group Γ. Assume that the Galois action of Γ on K lifts to a k-linear action of G on a finite dimensional K-vector space V . If all |G|-th roots of unity lie in k, then In [EV10, Lemma 1.1] this proposition was only stated for characteristic zero. Going through the proof, one recognizes that the only assumptions on k which are used are that the characters of G are k-rational and that |G| is prime to the characteristic of k, and hence for every representation of G on a k-vector space V , there is a decomposition of V into eigenspaces over k. This is not true if the characteristic of k divides |G|, even if k is algebraically closed. Note furthermore that if the action of G is tame, we can decompose in the usual Grothendieck ring of varieties, see Lemma 5.6. Hence the proposition really holds in K 0 (Var k ), also if the characteristic of k is positive but prime to the order of G.
Working with wild actions, we cannot decompose V into eigenspaces, so we will not be able to use the stratification of V from [Loo02, Lemma 5.1] as done in [EV10, Lemma 1.1]. Instead we will first show the claim for the case dim K V = 1 with elementary methods, and then use a G-equivariant fibration ϕ : V → W to a vector space W of dimension 1 over K to conclude by induction. More precisely, we compute the classes of the fibers of the induced map ϕ G : V /G → W/G separately using the induction assumption and Lemma 5.10, and then we use spreading out, see Lemma 5.4. To be able to use Lemma 5.10 we need to work in the modified Grothendieck ring of varieties.
Proposition 6.2. Let G be a finite abelian group with quotient G → Γ. Let k be a field of characteristic p, let q be the greatest divisor of |G| prime to p, and let K/k be a Galois extension with Galois group Γ. Assume that the Galois action on K lifts to a k-linear action of G on a finite dimensional K-vector space V . If k contains all q-th roots of unity, then Proof. As G is a finite abelian group, we have that with p the characteristic of k, and q i prime to p. Set r := r i and q := q i . Note that |G| = p r q, and q is prime to p. Set d := dim K V . Then we have V ∼ = Spec(K[x 1 , . . . , x d ]) as schemes. The G-action on V is given by α 1 , . . . , α s in Aut k (K[x 1 , . . . , x d ]) with α p r l l = id, and β 1 , . . . , β t in Aut k (K[x 1 , . . . , x d ]) with β q l l = id, such that α l | K and β l | K generate the Galois action on K. As G is abelian, the α l and β l commute.
View V as a vector space over k. The G-action is given by A l , B l ∈ GL k (V ) by assumption. Moreover, A p r l l = id and B q l l = id, and the A l and B l commute. Note that, as the order of the B l is finite of rang prime to p, the B l are diagonalizable over k. As B q l l = id, all eigenvalues are q l -th roots of unity, and as q l divides q, all those eigenvalues are already in k by assumption. Hence B l is already diagonalizable over k. As the B l commute, we find a basis of V of common eigenvectors of all the B l . Now consider the A l . Let E be any intersection of eigenspaces of the B l . As the A l commute with the B l , A l (E) = E for all l. Recall that A p r l l = id, hence all eigenvalues of A l are p r l -th roots of unity, and as char(k) = p, all the eigenvalues are 1, i.e. in particular in k. So we find a k-basis of E such that A l has upper triangle form with only 1 on the diagonal. As the A l commute, we can even find a k-basis of E such that all the A l have upper triangle form. We can do this for all intersections of eigenspaces of the B l , hence we get a k-basis B := {v 1 , . . . , v s } of V such that all A l have upper triangle form with only 1 on the diagonal, and B consist only of eigenvectors of the B l .
Consider the subset of B containing those v i which do not lie in the sub-K-vector space of V spanned by the v j with j < i. This way we get a basis B ′ = {w 1 , . . . , w d } of V as K-vector space such that A l (w i ) = w i + j<i a lij w i for some a lij ∈ K, and B l (w i ) = µ li w i for some q l -th roots of unity µ li . Hence we may assume -after a change of coordinates -that for some a lij ∈ K, and some q l -th roots of unity µ li . We need to show that We will show a slightly more general statement by induction on d, we namely only ask that α l | K and β l | K generate the action of Γ = Gal(K, k), and that (5) α l (x i ) = x i + j<i a lij x j + a li and β l (x i ) = µ li x i for some a lij ∈ K and a li ∈ K, and some q l -th roots of unity.
Start with d = 1. Hence V ∼ = Spec(K[x 1 ]), α l (x 1 ) = x 1 + a l for some a l ∈ K, and β l (x 1 ) = µ l x 1 for some q l -th root of unity µ l . We will show the claim for d = 1 by jet another induction, this time on r. If r = 0, |G| is prime to p, so by [EV10, Lemma 1.1] the claim is true. Assume hence that the claim holds for r − 1.
Let G ′ := Z/pZ × {0} × · · · × {0} ⊂ G. Note that G ′ ⊂ G is a normal subgroup of order p and acts on V . The action is generated by γ = α p r 1 −1 1 , and hence γ(x 1 ) = x 1 + b for some b ∈ K. Note that K is a Galois extension of K ′ := K G ′ with Galois group generated by γ| K .
. We now may now assume that K/K ′ is a field extension of degree p. As the characteristic of K is equal to p, K/K ′ is an Arthin Schreier extension, thus K = K ′ (ω) for some ω in K, and γ(ω) = ω + 1. Hence ω 2 is mapped to ω 2 + 2ω + 1 and similarly for higher powers of ω. As γ is a linear map on the K ′ -vector space K of degree p, we find a basis {1, ω, v 3 , . . . , v p } of K over K ′ such that γ(v i ) = v i + v i−1 for i > 3, and γ(v 3 ) = v 3 + ω. We can write b in this basis, i.e. we have We have γ(y) = y + b p v p , and K[x 1 ] ∼ = K[y]. Using that the characteristic of K is p, we get that y = γ p (y) = y + b p , hence b p = 0. Therefore for some y as we have seen, and (K[ . For simplicity we write α l and β l also for α ′ l and β ′ l . Note that K ′ is a Galois extension of k and the Galois action is generated by α l | K ′ and β l | K ′ . In order to use the induction assumption, we still have to show that, maybe after some coordinate change, the α l and β l are given as in Equation (5).
If y = x 1 , there is nothing to show. Let y = x 1 +bx p 1 withb := b 1−p (p − 1) ∈ K. One can show that for every δ ∈ Aut(K ′ [y]) of finite order with δ(K ′ ) = K ′ , we have that δ(y) = c 0 + c 1 y for some c i ∈ K ′ , hence this holds in particular for the α l and the β l . As Consider now the case y = x 1 − b ′ with b ′ as above. View β l and γ = α p r 1 −1 1 again as morphism of K[x 1 ]. Recall that β l (x 1 ) = µ l x 1 and γ(x 1 ) = x 1 + b. By assumption all the β l and γ commute pairwise. Hence µ l x 1 + β l (b) = β l (x 1 + b) = β l (γ(x 1 )) = γ(β l (x 1 )) = γ(µ l x 1 ) = µ l x 1 + µ l b, so β l (b) = µ l b. Note that the β l | K are k-linear maps of the k-vector space K. As β q l l = id and all the q l -th roots of unity lie in k by assumption, there is a basis of eigenvectors of β l of K over k. As the β l commute with each other, we can even find a common basis of eigenvectors of all the β l . Hence, using that b ′ ∈ K, we can for some q l -th root of unity µ li . Moreover we assume that if for all l we have that µ li = µ lj , then i = j. Without loss of generality we may assume that µ l1 = µ l for all l. As γ(K) ⊂ K, and b ′ i ∈ K, γ(b ′ i ) = b ′ i +b i for someb i ∈ K. Using again that the β l and γ commute, we get that µ li b ′ i + µ libi = µ lib ′ i + β l (b i ), i.e. β l (b i ) = µ libi for all l. In particular the b i which are not zero are linear independent. As γ(b ′ ) = b ′ + b, we get that b = b i . Hence for all l we have Henceb i = 0 if µ li = µ l for at least one l. In particular this implies thatb : So all together we may assume that K[x 1 ] G ′ = K ′ [y] and α l (y) = y + a ′ l for some a ′ l ∈ K ′ , and β l (y) = µ l y. Hence we can use the induction assumption for the H-action on K ′ [y]. This proves Equation (4) for d = 1. Now assume that the claim holds for d−1. Look at V = Spec (K[x 1 , . . . , x d ]) with a G-action as in Equation (5). Note that the inclusion map K[x 1 ] ֒→ K[x 1 , . . . , x d ] is G-equivariant, if G acts on K[x 1 ] generated by α ′ l and β ′ l such that the α ′ l | K and β ′ l | K generate the Galois action on K, and α ′ l (x 1 ) = x 1 +a l1 and β ′ l (x 1 ) = µ l1 x 1 . To simplify notation we will use α l and β l also for α ′ l and β ′ l . Set W := Spec(K[x 1 ]), denote by ϕ the G-equivariant map from V to W , and let ϕ G : V /G → W/G be the induced map between the quotients. Let x ∈ W/G be any point with residue field κ x , and let w ∈ W be a point with residue field κ w in the inverse image of x under the quotient map π : W → W/G. Let G w ⊂ G be the stabilizer of w. We get an induced action of G w on V and W . Note that the action of G w on V is generated byα l = α s l l andβ l = β t l l for some s l > 0 and some t l > 0, hence we havẽ for someã lij ∈ K andã li ∈ K, and withμ li = µ ti li . By construction, w is fixed under this action of G w , and therefore we have Herex 1 denotes the image of x 1 in κ w . Note thatã li1x1 +ã li ∈ κ w . Moreover κ w is a Galois extension ofκ x := κ Gw w , and γ l | κw and δ l | κw generate the Galois action. All together we can use the induction assumption and get that Asκ x is purely inseparable over κ x by [Bou85, Capitre V.2, Théorème 2], and hence there is a universal homeomorphism f : As G is abelian and thus G w ⊂ G is a normal subgroup, we can use Lemma 5.10 to get that [U ] ∈ K mod 0 (Var k ). Note that W/G \ U consist of finitely many points P i . We already know that [ϕ −1 (6). Using the scissors relation in K mod 0 (Var S/G ), we get that Here we used the induction assumption again to get that [W/G] = L k ∈ K mod 0 (Var k ). Claim (4) now follows by induction. is purely inseparable, but ϕ −1 (0)/G and ϕ −1 G (0) are isomorphic over k, hence they have the same class in K 0 (Var k ). This suggests that Proposition 6.2 maybe also holds in the usual Grothendieck ring, at least if we assume that k is a finite field. But as the isomorphism between the fibers is not given in a canonical way, I do not know how to do this in general, and whether it is possible at all. In the case that dim K V = 1, it follows from the proof of Proposition 6.2.
Remark 6.4. In the proof of Proposition 6.2 we used the fact that G is abelian to get a G-invariant sub-vector space W of V . In fact this assumption is necessary. As was already observed in [EV10], it follows from [Eke09, Proposition 3.1, ii] and [Eke09, Corollary 5.2] that there exists an n-dimensional C-vector space V and a finite group G ⊂ GL(V ) such that whereM C is the completion of M C by the dimension fibration. So in particular there exists an m large enough such that L mn = [V m /G] ∈ K 0 (Var C ).
In fact there are explicit V and G for which this holds, which can be found in [Sal84]. The group in Saltman's example is a p-group, hence in particular solvable. Hence Proposition 6.1 will not hold in the case of solvable groups.
Remark 6.5. If one does not assume that k has enough roots of unity, Proposition 6.1 does not hold. There is namely a concrete counterexample, see [EV10,

Quotients of equivariant affine bundles
Now we use the result from the previous section to compute the class of the quotient of an equivariant affine bundle by an affine action in a (modified) Grothendieck ring of varieties. Take S with an action of G as before. For a point s ∈ S/G, denote by F s its residue field, and by G s ⊂ G the stabilizer of a point s ′ ∈ S lying in the inverse image of s under the quotient map S → S/G. A priori, G s depends on the choice of s ′ . But as all stabilizers of an orbit are conjugated, |G s | does not depend on s ′ .
Lemma 7.1. Let G be a finite abelian group, and let ϕ : V → B be a G-equivariant affine bundle of rank d with affine G-action in the category (Sch S,G ). Assume that the residue field F s of every point s ∈ S/G contains all |G s |-th roots of unity. Then If the action of G on S is tame, we get Proof. Let ϕ : V → B be as in the claim, and let ϕ G : V /G → B/G be the induced map between the quotients. Let x ∈ B/G be a point with residue field k. As B/G is an S/G-scheme, x lies over a point s ∈ S/G, and k contains the residue field F s of s. Let b ∈ B be a point mapped to x under the quotient map, let K be the residue field of b and let G b be the stabilizer of b under the action of G. Without loss of generality we may assume that b is mapped to s ′ under the structure map B → S. Hence as this map is G-equivariant, G s is a subgroup of G b . As G is abelian, G b ⊂ G is a normal subgroup, and hence Lemma 5.10 implies that If the characteristic of F s is zero or prime to the order of G, we get this equation also in K 0 (Var k ), see Remark 5.11. As ϕ : V → B is a G-equivariant affine bundle of rank d with affine G-action, and G b is a subgroup of G, the induced action of G b makes it into a G b -equivariant affine bundle of rank d with affine G b -action. As G b is the stabilizer of b, b is fixed under the action of G b . Hence by Remark 3.4, V b := ϕ −1 (b) is a K-vector space of rank d, and the G b -action on V restricts to V b . If the characteristic of k and hence of F s is p, this action is generated by automorphisms A l , B l ∈ Aut(V b ) such that the order of the A l is a power of p and the order of the B l divides |G b | and is prime to p. As G b is a subgroup of G s , this means in particular that the order of B l divides |G s |. If the characteristic of k is zero, we may assume that the action is only generated by B l s as above. Also from Remark 3.4 we get that for all As G b ⊂ G is abelian, we may assume by Remark 3.5 that w l = 0.
Note thatk := K G b is a field extension of k and hence of F s . By Remark 3.3, the A ′ l and the B ′ l define ak-linear action on a K-vector space V b lifting the Galois action of Gal(K,k) on K. By assumption F s contains all |G s |-th roots of unity, hence the same holds fork. So using this and that G b is abelian, we can find as in the proof of Proposition 6.2 a K-basis such that the A ′ l have upper triangle form with only 1 on the diagonal, and the B ′ l are diagonal with q s -th roots of unity as eigenvalues. All together we may assume that V b ∼ = Spec(K[x 1 , . . . , x d ]) as schemes, and that the G-action on V b is given by α l , β l ∈ Aut(K[x 1 , . . . , x d ]) such that the α l | K and β l | K generate the Gal(K,k)-action on K, and for some a lij , a li ∈ K and some q-th roots of unity µ li . But this is exactly the setting for which we showed Proposition 6.2, see Equation (5). Hence we get that As G b is the stabilizer of b ∈ B under the action of G, and x is the image of b under π, by [Bou85, Capitre V.2, Théorème 2]k is a purely inseparable extension of k. Hence Lk = L k ∈ K mod 0 (Var k ). Putting everything together we get for every point x ∈ B/G with residue field k that If the characteristic of F s and hence the characteristic of k is zero or prime to the order of G, by [Bou85, Capitre V.2, Proposition 5 and Corollaire]k = k. As the A l are trivial in this case, the G-action on V b is actually quasi-linear, and hence by Proposition 6.1 we get that Note that, without loss of generality, we may assume that B/G is integral in the claim. This is true, because we can decompose B/G in irreducible schemes using the scissors relation, and by [CLNS14, Chapter 1, Corollary 1.5.3] we only need to consider the underlying reduced scheme structure. Let η ∈ B/G be the generic point with residue field κ η . By Lemma 5.4 there is an isomorphism This implies using Equation (7) for η that we can find an open U ⊂ B/G such that As the separated map U → S/G induces a forgetful map from K mod 0 (Var U ) to K mod 0 (Var S/G ), we get that [ϕ −1 G (U )] = L d S/G [U ] ∈ K mod 0 (Var S/G ). Proceed with a generic point of B/G \ U . By Noetherian induction we get, using the scissors relation in the modified Grothendieck ring, that If the action of G on S is tame, and hence the characteristic of the residue field of every point s ∈ S/G is zero or prime to the order of G, we can use that by [CLNS14, Chapter 1, Proposition 3.2.3] is an isomorphism, and Equation (8) to conclude with an analog argument as above that Example 7.2. Take a B ∈ (Sch S,G ), and let A d B → B be the trivial vector bundle with G-action induced by the action on B.
As A d S/G is flat over S/G, it also follows directly from [Gro67, Exposé V, Propo-

The quotient map on the equivariant Grothendieck ring of varieties
In [Bit05, Lemma 3.2] it was shown that, if G acts freely on a variety S of characteristic zero, taking the quotient defines a map from K G 0 (Var S ) to K 0 (Var S/G ). We will now show that such a map exists for more general S and G. To make sure that the quotient X/G of a separated S-scheme of finite type X with good G-action is a separated S/G-scheme of finite type, we assume, as before, that S/G is locally Notherian and separated, and that the quotient map π S : S → S/G is finite. For a point s ∈ S/G, we denote again by F s the residue field of s and by G s ⊂ G the stabilizer of a point s ′ ∈ π −1 (s) ⊂ S.
Proving that the quotient map is well defined, the main problem is to show that the second relation in Definition 4.1 does not cause troubles, hence we have to control quotients of G-equivariant affine bundles by affine G-actions. In the proof of [Bit05, Lemma 3.2] it is shown that, if the action of G on S is free, the quotient of such a bundle is again an affine bundle. Without the freeness assumption the quotient can be even singular, hence in particular it is no an affine bundle in general. But we have seen in Lemma 7.1 that in the modified Grothendieck ring, or in the usual Grothendieck ring in the case of tame actions, the class of the quotient of a G-equivariant affine bundle only depends on its rank and its base. This enables us to show the following theorem.
Theorem 8.1. Let G be a finite abelian group. Assume that the residue field F s of any point s ∈ S/G contains all |G s |-th roots of unity. Then there is a well defined group homomorphism (Var S/G ) for every X ∈ (Sch S,G ). If the G-action on S is tame, it factors through a group homomorphism Proof. For all X ∈ (Sch S,G ), the quotient X/G exists by [Gro67, Exposé V, Corollaire 1.4], because the G-action on X is assumed to be good. As the structure map X → S is G-equivariant, X/G is an S/G-scheme. As S/G is locally Notherian and S → S/G is finite and hence of finite type, by [Gro67, Corollaire 1.5], X/G is of finite type over S/G. Moreover, the fact that X is separated over S/G implies the same for X/G. Hence we get a well defined map from (Sch S,G ) to K mod 0 (Var S/G ) and K 0 (Var S/G ), respectively. So in order to show the proposition, we need to show that this map factors through K G 0 (Var S ). Hence we need to show that Relation (1) and Relation (2) from Definition 4.1 still hold after taking the quotient.
Take any X ∈ (Sch S,G ) and a closed subscheme Y of X, closed with respect to the action of G. Set U := X \ Y . Denote by π : X → X/G the quotient map. As If the action of G on S, and hence also the action of G on X is tame, then by Lemma 5.6 we get Equation (9) also in K 0 (Var S/G ). It remains to show that Relation (2) still holds after taking the quotient. Take any B ∈ (Sch S,G ), and let V → B and W → B be G-equivariant affine bundle of rank d with affine G-action. By Lemma 7.1 we have that and if the action of G on S is tame, Equation (10) holds in K 0 (Var S ). Altogether, taking the quotients gives us well defined maps as in the claim. (Var k ). Note that V → Spec(K) is an affine bundle of rank d with affine G-action in the category (Sch S,G ), and the same holds for A d K → Spec(K) with a G-action induced by the Galois action on K.
, so there cannot be a well defined map from K G 0 (Var k ) to K mod 0 (Var k ) as in Theorem 8.1. Hence Remark 6.4 and Remark 6.5 show that Theorem 8.1 in general does not hold without the assumption on the residue field F s for all s ∈ S/G or for a non-abelian group G. S/G for all X ∈ (Sch S,G ). Note that the fiber product of X and A 1 S is taken in the category (Sch S,G ). Note that A 1 S × S X = A 1 S/G × S/G X, and the action on A 1 S/G is trivial. As A 1 S/G is flat over S/G, [Gro67, Proposition 1.9] implies that (A 1 S/G × S/G X)/G = A 1 S/G × S/G X/G. Hence Equation (11) holds.
Consider now a profinite groupĜ Let S be a separated scheme with aĜ-action factorizing through a good action of a finite quotient G j ofĜ. Assume that the quotient S/Ĝ = S/G j is a separated, locally Notherian scheme, and that the quotient map π S : S → S/Ĝ is finite. For all s ∈ S and i ≥ j, set q si := |G si |, with G si ⊂ G i the stabilizer of a point s ′ ∈ π −1 s (s) ⊂ S under the action of the G i . Throughout this section, if not mentioned otherwise, let k be a field of characteristic zero containing all roots of unity, let X be an irreducible algebraic variety over k, and let f : X → A 1 k be a non-constant morphism. We denote by X 0 ⊂ X the zero locus of f in X, and assume that X η : We are now going to use the results from the previous section to show that the quotient of the motivic nearby fiber is a well defined invariant with values in M X0 . The motivic nearby fiber can be attached to a map f : X → A 1 k as above. It was constructed in [DL01] as a limit of the motivic Zeta function, and was investigated in more details in [Bit05]. Moreover we show that modulo L, this quotient is equal to motivic reduction R(f ) in the image of K 0 (Var X0 ) in M k , see Proposition 9.5. Here R(f ) is the class of the special fiber of a smooth modification of f in K 0 (Var X0 )/L. A smooth modification of f : X → A 1 k is a smooth irreducible algebraic variety Y over k, together with a proper morphism h : Y → X such that the restriction h : Y \ h −1 (X 0 ) → X \ X 0 is an isomorphism. Due to weak factorization, the definition of the motivic reduction does not depend on the choice of such a modification, see Lemma 9.3. This result implies the following: if f : X → A 1 k is proper and X is smooth, and the generic fiber X η of f is equal to 1 modulo L in K 0 (Var A 1 k \{0} ), then the same holds for the special fiber of f in the image of K 0 (Var k ) in M k , see Corollary 9.6. This can be seen as a motivic analog of the main theorem in [Esn06, Theorem 1.1], which says that if V is an absolutely irreducible smooth projective variety over a local field K with finite residue field F such that the m-thétale cohomology of V × KK ,K the algebraic closure of K, has coniveau 1 for all m ≥ 1, then the number of rational points of the special fiber of any regular projective model of V is congruent 1 modulo |F |.
How does this analogy work? In both cases one deduces a property of the special fiber from a property of the generic fiber of some proper and smooth scheme. We are now going to outline the connection between the properties on the generic fibers and the properties on the special fibers, respectively.
As remarked in [Esn06], if the characteristic of K is equal to zero, then the fact that theétale cohomology of V has coniveau 1 implies that the Hodge type of the de Rham cohomology is ≥ 1, or equivalently that H q (V, O V ) = 0 for all q ≥ 1. Now consider the Hodge-Deligne polynomial HD : K 0 (Var K ) → Z[u, v], see [NS11, Example 4.1.6]. This is a ring morphism sending the class of a projective and smooth K-variety X to p,q (−1) p+q dim K (H q (X, Ω p X ))u p v q . We have that Hence, if the class of V is equal to 1 modulo L in K 0 (Var K ), HD(V ) = 1 modulo uv, hence in particular H q (V, O V ) = 0 for q ≥ 1, so the de Rham cohomology of V has Hodge type ≥ 1. Note furthermore that if we have a finite field F , and the class of a variety V F over F in K 0 (Var F ) is equal to 1 modulo L, then #(V F (F )) = 1 modulo |F |.
The Motivic nearby fiber. Recall the following notation from [DL01]. Let h : Y → X be an embedded resolution of f , i.e. h is a proper morphism inducing an isomorphism Y \ h −1 (X 0 ) → X \ X 0 , Y is smooth, and h −1 (X 0 ) is a simple normal crossing divisor. Such an embedded resolution always exists due to resolution of singularities in characteristic zero. Denote by E i , i ∈ J, the irreducible components of h −1 (X 0 ). For each i ∈ J, denote by N i the multiplicity of E i in the divisor f • h on Y . For I ⊂ J, we consider nonsingular varieties Remark 9.2. Let X be a smooth and proper variety over K = k((t)). Let X be proper sncd-model of X, i.e. X is a proper and flat k[[t]]-scheme with generic fiber isomorphic to X such that its special fiber X 0 is a simple normal crossing divisor. By [NS07,Definition 8.3] the motivic volume S(X,K) with values in M k , which we can associate to the t-adic completionX of X , does only depend on the generic fiber ofX, and thus only on X and not on the model X . By [NS07, Proposition 8.2] we have that where theẼ o I are constructed from X 0 as done above. If X is actually coming from some f : X → A 1 k as studied above, by [NS07,Theorem 9.13], we have that If we could show that actually S(X,K) was well defined in Mμ k (this is work in progress), where we consider theẼ o I withμ-action as done above, then also would be well defined by Corollary 8.5, and we could proof results analogous to Proposition 9.5 and Corollary 9.6 also in this context.
Connection with the motivic reduction. To be able to define the motivic reduction of f , we first need to proof the following lemma. Proof. Let h 1 : Y 1 → X, h 2 : Y 2 → X be two smooth modifications of f . We have shown that (14) [h −1 1 (X 0 )] = [h −1 2 (X 0 )] ∈ K 0 (Var X0 )/L. Consider the fiber product Y 12 := Y 1 × X Y 2 . Note that the projection maps p i : Y 12 → Y i are proper, and induce isomorphisms between Y 12 \ p −1 i (h −1 i (X 0 )) and Y i \ h −1 i (X 0 ). As the h i are smooth modifications of f , the Y i \ h −1 i (X 0 ) are isomorphic to X η = X \ X 0 , which is smooth by assumption.
Let b :Ỹ → Y 12 be an embedded resolution of in K 0 (Var k )/L. By the symmetry of the construction it suffices to show it for i = 1.
Note that g 1 :Ỹ → Y 1 is a proper birational morphism over X between two smooth varieties. By [Bit04, Remark 2.3], the Weak Factorization Theorem, see [AKMW02, Theorem 0.1.1], holds for g 1 , i.e. there exist a sequence of birational maps as follows and for all i, either φ i : V i−1 V i or φ −1 i : V i V i−1 is a morphism obtained by blowing up a smooth center. By [Bit04, Remark 2.4], the factorization is a factorization over X, i.e. there are structure maps ϕ i : V i → X, with ϕ 0 = h 1 • g 1 and ϕ l = h 1 , and the φ i are maps over X.
Take any i ∈ {1, . . . , l}. If φ i : V i−1 → V i is a blowup in the smooth center C i ⊂ V i , then we get using the scissors relation that ] ∈ K 0 (Var X0 )/(L). Analogously, we get the same statement if φ −1 i is a blowup. Hence it follows that the classes of g −1 1 (h −1 1 (X 0 )) and h −1 1 (X 0 ) coincide in K 0 (Var k )/(L), and hence the claim follows as observed above.
Definition 9.4. The motivic reduction R(f ) of f : X → A 1 k is defined as the class of h −1 (X 0 ) in K 0 (Var X0 )/L of any smooth modification h : Y → X of f .
Notation. Let R ∈ K 0 (Var X0 ) any element in the inverse image of R(f ) under the quotient map. We denote with R(f ) also the class of the image of R in M X0 modulo L.
Using the forgetful map K 0 (Var X0 ) → K 0 (Var k ) induced by the structure morphism X 0 → Spec(k), we can view R(f ) also as an element in K 0 (Var k )/L. We denote with R(f ) also its image in K 0 (Var k )/L. Now we can combine the definition of the motivic reduction with the motivic nearby fiber, and get the following proposition.
Proposition 9.5. The class of R(f ) and S f /μ in the image of K 0 (Var X0 ) in M X0 modulo L coincide.
Proof. Let h : Y → X be a embedded resolution of f , and let E o I be constructed from h : Y → X as done above. Then by Equation (12) we have One observes that S f /μ lies in the image of K 0 (Var X0 ). Hence S f /μ is equal to [h −1 (X 0 )] modulo L in the image of K 0 (Var X0 ) in M X0 . This is equal to R(f ), because h : Y → X is an embedded resolution and hence a smooth modification of f : X → A 1 k . Corollary 9.6. Let X be a smooth variety over k, and let f : X → A 1 k be a proper morphism. If the class of X η is equal to 1 modulo L in K 0 (Var A 1 k \{0} ), then the class of f −1 (X 0 ) is equal to 1 modulo L in the image of K 0 (Var k ) in M k .
Proof. If [X η ] = 1 mod L ∈ K 0 (Var A 1 k \{0} ), we can write [X η ] = 1 + L[V ] with [V ] ∈ K 0 (Var A 1 k \{0} ), also for the class of X η in M A 1 k \{0} . Consider the map ψ : M A 1 k \{0} → M k , see Equation (13). On the one hand side,ψ([X η ]) = 1 + Lψ(V ) ∈ M k , becauseψ is M k -linear and maps 1 to 1. On the other hand,ψ([X η ]) = S f /μ, and we have already seen that this is equal to R(f ) in the image of K 0 (Var k ) in M k . Here S f /μ and R(f ) are elements in K 0 (Var k ) via the map K 0 (Var X0 ) → K 0 (Var k ) induced by the structure map X 0 → Spec(k). Moreover R(f ) = [f −1 (X 0 )], because X is smooth, and hence id is a smooth modification of f .
All together [f −1 (X 0 )] is equal to 1 modulo L in the image of K 0 (Var k ) in M k .
Remark 9.7. It should be possible to proof Corollary 9.6 without using S f , by showing that there exists a well defined map K 0 (Var A 1 k ) → K 0 (Var k )/L, which is K 0 (Var k )-linear, and sends [X] to R(f ) if the structure map f : X → A 1 k is proper and non-constant, and to 0 if f is proper and constant. To do this, one would need to consider the blowup relations in the Grothendieck ring of varieties, see [Bit04,Theorem 5.1].
Remark 9.8. As we do not know whether K 0 (Var X0 ) is a subgroup of M X0 , see Remark 4.2, we cannot show Proposition 9.5 in K 0 (Var k )/L. If we could show that the motivic nearby fiber was well defined in Kμ 0 (Var X0 )/L, it would follow from the fact that taking the quotient also gives a well defined map from Kμ 0 (Var X0 )/(L) to K 0 (Var X0 )/(L), that S f /μ would be well defined in K 0 (Var X0 )/(L). Like this we could also show that R(f ) is well define without using the Weak Factorization Theorem.
In order to avoid the problem that K 0 (Var X0 ) is maybe not a subgroup of M X0 , we can work in the Grothendieck ring of effective motives. To make things easier we work over k right away.
Let Mot eff k be the additive category of effective motives with rational coefficients, let K 0 (Mot eff k ) its Grothendieck ring, and let L mot be the class of the Lefschetz motive in this ring. Let Mot k be the category of motives with rational coefficients, let K 0 (Mot k ) its Grothendieck ring, and let L mot be the image of the Lefschetz motive. For the precise definitions of these objects we refer to [NS11, Example 2.1.3]. By [NS11, Theorem 4.1.11] we get commuting maps as follows: Here χ eff mot maps the class of a projective k-variety X to the class of the effective motive (X, id). L is mapped to L mot . By [Nic11, Proposition 2.7] ρ is injective if one assumes the following standard conjecture: Assume now that Conjecture 9.9 is true. Hence ρ is injective, and as S f /μ lies in the image of K 0 (Var k ) in M k , the inverse image of χ mot (S f /μ) under ρ has precisely one element, which we denote by S f /μ mot . Set R(f ) mot := χ eff mot (R(f )). Proposition 9.5 then implies the following: Corollary 9.10. S f /μ mot and R(f ) mot coincide in K 0 (Mot eff k ).