Homology, Homotopy and Applications, vol.3(2), 2001, pp.417–451

BASE CHANGE FUNCTORS IN THE 𝔸^{1}-STABLE HOMOTOPY CATEGORY

PO HU

(communicated by Gunnar Carlsson)

Abstract We relate base change functors of sheaves in 𝔸^{1}-homotopy theory to group change functors from equivariant homotopy theory, and use these functors to construct elements of the Picard group of the 𝔸^{1}-stable homotopy category. We also prove an analogue of the Wirthmüller isomorphism from equivariant homotopy theory in the 𝔸^{1}-context.

Introduction

In this note, we will discuss certain functors between the 𝔸^{1}-stable homotopy categories over fields. These are in a sense analogous to the change of groups functors in equivariant stable homotopy theory. Section 1 contains some preliminaries in doing 𝔸^{1}-stable homotopy theory. In Section 2, we define the change of base field functors between the 𝔸^{1}-homotopy categories over fields L and k, where L is a finite extension of k. In Section 3, We give a application of these functors in constructing objects of the 𝔸^{1}-stable homotopy category over k that are invertible under the smash product, by constructing a functor from the category of equivariant spaces with respect to the Galois group of k to the category of algebraic spaces over k. In Section 4, we prove an analogue of the Wirthmüller isomorphism, a classical result from algebraic topology. We will also show that in the present setting, this result is in fact not only an analogue, but a consequence of the equivariant result. This gives a very explicit formulation of the Wirthmüller isomorphism map. It is also analogous to classical results from the theory of sheaves. Finally, in Section 5, we give some technical details of the definitions of join and smash powers of a k-space, which are needed in showing that the examples constructed in Section 3 are indeed invertible.

1. Preliminaries

We recall some fundamental constructions needed to do stable homotopy theory in the algebraic geometrical setting, due to Morel and Voevodsky [11]. Let S be a

The author is supported by a NSF Postdoctoral Fellowship and the Institute for Advanced Study Received April 18, 2001, revised August 19, 2001; published on September 19, 2001. 2000 Mathematics Subject Classification: Primary 14F42, 14F05; Secondary 55P91. Key words and phrases: 𝔸^{1}-homotopy theory, Verdier duality, Wirthmüller isomorphism, Picard group. \begin{matrix}c \\ ◯ \end{matrix} 2001, Po Hu. Permission to copy for private use granted.

Noetherian scheme of finite dimension, and Sm/S the category of smooth schemes of finite type over S. The Nisnevich topology on Sm/S is defined to be the subtopology of the étale topology generated by diagrams of the following form.

\xymatrix{p^{−1}(U) \ar[r] \ar[d] & Y \ar[d]^{p} \\ U \ar[r]_{i} & X.}

Here, we require i to be an open embedding, p to be an étale map, such that p|_{p^{−1}(X\{0})} is an isomorphism.

The Nisnevich topology makes Sm/S into a small Grothendieck site [12]. The category of S-spaces

Spc(S) = Δ^{op}Sh(Sm/S)_{Nis}

is the category of simplicial sheaves of sets over Sm/S with respect to the Nisnevich topology. In particular, a smooth scheme X over S is the sheaf represented by X, concentrated in simplicial degree 0. The category Spc(S) has all small colimits and limits, and is the analogue of topological spaces. Also, it is generated by Sm/S in the sense that every S-space is a colimit of smooth schemes over S in Spc(S).

To do homotopy theory in Spc(S), one defines a closed model structure on the category Spc(S) in two steps as follows. First, one says that a map f is a simplicial cofibration if it is a monomorphism, a simplicial weak equivalence if it is a weak equivalence of simplicial sets at every point of the site Sm/S, and a simplicial fibration if it has the right lifting property with respect to all acyclic simplicial cofibrations. This defines the simplicial model structure on Spc(S). Let H_{s}(S) be the homotopy category on Spc(S) associated to this model structure. The simplicial model structure has too few weak equivalences for our purposes. In particular, we would like the affine line 𝔸_{S}^{1} over S to be weakly equivalent to the S-space S, which is the analogue in the category Spc(S) of a single point. To obtain this, one localizes in the sense of Bousfield [2] at all projection maps 𝔸^{1} × X → X. Specifically, an S-space Y is said to be 𝔸^{1}-local, if for all X, the map of morphism sets in H_{s}

H_{s}(X, Y) → H_{s}(X × 𝔸^{1}, Y)

is a bijection. A map f : X → Y is an 𝔸^{1}-weak equivalence if for all 𝔸^{1}-local objects Z, the map of morphism sets

H_{s}(Y, Z) → H_{s}(X, Z)

is a bijection. A map f is an 𝔸^{1}-cofibration if it is a simplicial cofibration, and it is an 𝔸^{1}-fibration if it satisfies the right lifting property with respect to all acyclic 𝔸^{1}- weak equivalences. These three classes of maps define the 𝔸^{1}-local model structure on Spc(S), the model structure with which one works in homotopy theory.

One can also consider the category Spc(S)_{•} of based S-spaces, an object of which is a S-space X together with a given map from S to X, called the basepoint of X. The 𝔸^{1}-local model structure on Spc(S)_{•} is defined in the unbased category Spc(S) [11, 6]. We also have analogues of certain basic constructions from topology.

In particular, we have the disjoint basepoint functor (−)_{+} : Spc(S) → Spc(S)_{•}, which takes an unbased S-space X to the based S-space X ⨿S, where the basepoint is the disjoint copy of S. Also, for X, Y ∈ Spc(S)_{•}, the smash product of X and Y over S is given by

X ∧_{S} Y = (X ×_{S} Y)/_{S}(X ∨_{S} Y)

where X ∨_{S} Y is the pushout in Spc(S) of the basepoint maps S → X and S → Y, and /_{S} is the quotient space functor in Spc(S), obtained by collapsing X ∧_{S} Y to S.

To do stable homotopy theory, one stabilizes with respect to the one-dimensional projective space ℙ_{S}^{1} over S. The category of S-spectra is defined in a similar manner as in topology. Namely, an S-prespectrum D is a sequence of based S-spaces {D_{n}}, along with a give structure map

Σ^{ℙ^{1}}D_{n} → D_{n+1}

for each n. Here, Σ^{ℙ^{1}} denotes the suspension functor ℙ_{S}^{1}∧−. Equivalently, a structure map is a map

D_{n} → Ω^{ℙ^{1}}D_{n+1}  (1.1)

where Ω^{ℙ^{1}} = \underline{Hom}_{•}(ℙ_{S}^{1}, −) is the internal Hom object from 𝔸_{S}^{1}, i. e. the right adjoint to the functor Σ^{ℙ^{1}} in the category of based S-spaces. An S-prespectrum is an S-spectrum if its structure maps (1.1) are isomorphisms in Spc(S)_{•}. There is a spectrification functor L from the category of S-prespectra to the category of Sspectra, which is analogous to the spectrification functor from inclusion prespectra to spectra in topology. Namely, given an S-prespectrum D = {D_{n}},

(LD)_{n} = colim_{k}Ω^{(ℙ^{1})^{∧k}}D_{n+k}.

We denote the category of S-spectra by Spectra(S). Stabilizing the 𝔸^{1}-local model structure on Spc(S)_{•} in the manner of Bousfield and Friedlander [3] gives the stable 𝔸^{1}-local model structure on the category of S-spectra (see [6, 14]). In particular, for any S-prespecturm D, the unit map D → LD is always an 𝔸^{1}-weak equivalence [6]. We will call the homotopy category associated with this model structure the stable homotopy category over S, denoted by SH(S). It is the algebraic analogue of the stable homotopy category in topology. As in topology, we can think of S-spectra as indexed on an universe U ≅ 𝔸_{S}^{∞} over S, then for two S-spectra E and E′, the internal smash product E ∧ E′ is given by first taking the external smash product E\overline{∧}E′, which is an S-spectrum indexed on U^{⊕2}, then taking the change of universe functor back to spectra indexed on U via an linear injection U^{⊕2} → U [6]. The operad of such linear injections is 𝔸_{S}^{1}-contractible. Hence, the smash product of spectra are well-defined in SH(S). In this note, we will work only with the case where S = Spec(k) for an arbitrary field k.

2. Change of Bases Functors

Let k be an arbitrary field, and let L be a finite separable extension of k. We have a canonical map

i : Spec(L) → Spec(k)  (2.1)

from the inclusion of k in L.

Definition 2.2. Define the functor

i^{∗} : Spc(k) → Spc(L)

by

i^{∗}(X) = X ×_{Spec(k)} Spec(L).

The structure map i^{∗}(X) → Spec(L) is the pullback of the structure map X → Spec(k) along i.

On the level of affine schemes, i^{∗} corresponds to the extension of scalars functor

i^{∗} = L ⊗_{k} − : Algebras(k) → Algebras(L).

Considering k-spaces as simplicial Nisnevich sheaves over Sm/k, Morel and Voevodsky [11] defined the inverse image functor with respect to a map of base schemes. If f : S_{1} → S_{2} is a morphism of schemes, it gives a continuous map of the Nisnevich sites (Sm/S_{1})_{Nis} → (Sm/S_{2})_{Nis}. Thus, there is an inverse image of sheaves functor f^{∗} : Spc(S_{2}) → Spc(S_{1}). In our case, f = i : Spec(L) → Spec(k) is a smooth map. So for a smooth scheme X ∈ Sm/k, the inverse image of the sheaf represented by X is the sheaf on Sm/L represeted by X×_{Spec(k)}Spec(L). Also, recall that every k-space is a colimit of sheaves represented by smooth schemes (see [6], Appendix). Thus, our functor i^{∗} is the same as the inverse image functor for all objects of Sm/k.

The functor i^{∗} is analogous to a change of groups functor from equivariant homotopy theory in the following sense. Recall from Lewis, May and Steinberger [9] that if we have a compact Lie group G, and a (closed) subgroup H ⊆ G, then there is a forgetful functor from the category of G-equivariant topological spaces to the category of H-equivariant topological spaces. On the other hand, we can also consider the category of G-equivariant spaces parametrized over a given G-equivariant based space X, i. e. the comma category of G-equivariant spaces Z, together with a given continuous G-map to X. In particular, there is a equivalence of categories between the category of G-equivariant spaces parametrized over the homogenous G-space G/H, and the category of H-equivariant spaces. Namely, given an H-space T, we have a G-space

G ×_{H} T = {(g, t) | g ∈ G, t ∈ T }/(gh, x) ∼ (g, hx)  (2.3)

where the action of G is induced by the multiplication of G on itself from the left. This has a natural G-map to G/H, induced from collapsing T to a single fixed point. Conversely, for a G-space Z with a G-map Z → G/H, the fiber Z_{eH} of Z over the coset eH of G/H is an H-equivariant space. It is straightforward to check that these two functors are inverse equivalences. Also, let f : G/H → ∗ be the collapse map. Then similarly as in the algebraic case, we also have the functor f^{∗} from the category of G-spaces to the category of G-spaces over G/H, give by f^{∗}(T) = G/H × T, with the diagonal G-action, and mapping to G/H via the first projection. Also, f^{∗} corresponds to the forgetful functor from G-spaces to Hspaces, with respect to the equivalence of categories between G-spaces over G/H and H-spaces. Now let G = Gal(E/k) for some finite Galois extension E of L, and let H = Gal(E/L). Then the E-points of a k-space has a natural G-action, and the E-points of an L-space has a natural H-action. The counterpart of the homogenous space G/H is Spec(L), whereas L-spaces, i. e. k-spaces over Spec(L), corresponding to G-spaces over G/H, and i : Spec(L) → Spec(k) corresponds to the G-map G/H → ∗. In this sense, i^{∗} is analogous to the forgetful functor with respect to the inclusion map H → G. For instance, i^{∗}Spec(k) = Spec(L), which is analogous to the fact that the G-space consisting of a single fixed point forgets to the H-space consisting of a single fixed point.

In the equivariant topological context, the forgetful functor has both a left adjoint and a right adjoint. The left adjoint is G ×_{H} − from H-equivariant spaces to Gequivariant spaces, as given in (2.3). The right adjoint is Maps_{H}(G, −), the space of H-equivariant maps from G, with the G-action induced by the action of G on itself from the right.

In the algebraic context, we also have both a left and a right adjoint to i^{∗}. We denote the left adjoint to i^{∗} by

i_{♯} : Spc(L) → Spc(k).

An L-space X has a structure map X → Spec(L), which is also a map over Spec(k). Composition with i gives a structure map X → Spec(k), which completely determines a k-space structure on X. This gives i_{♯}X. If we have a map f : i_{♯}X → Y over Spec(k) for an L-space X and a k-space Y, then taking the pullback of f along the i : Spec(L) → Spec(k) gives a map over Spec(L) from X to i^{∗}(Y) = Spec(L) ×_{Spec(k)} Y. Conversely, given a map g : X → Spec(L) ×_{Spec(k)} Y, composition with the map Spec(L) ×_{Spec(k)} Y gives a map from i_{♯}X to Y over Spec(k). It is clear that these corrspondances are inverse to each other, so i_{♯} is the left adjoint to i^{∗}. In particular, on the level of affine schemes, i_{♯} corresponds to a functor

Algebras(L) → Algebras(k).

This is the right adjoint to the extension of scalars functor, i. e. the forgetful functor. When there is no possibility of confusion, we will omit i_{♯} from the notation. Since i is a smooth morphism, then by Proposition 3.1.23 of [11], when we think of i^{∗} as the inverse image of sheaves functor, we can also define the left adjoint i_{♯} of i^{∗} to be an “extension by zero” functor of sheaves. By the uniqueness of adjoints, this definition of i_{♯} coincide with our definitions.

There is also a right adjoint

i_{∗} : Spc(L) → Spc(k)

to i^{∗}. To write down i_{∗}, recall that the category of k-spaces is closed, i. e. for any Y, the functor

Y ×_{Spec(k)} − : Spc(k) → Spc(k)

has a right adjoint

\underline{Hom}_{Spec(k)}(Y, −) : Spc(k) → Spc(k).

Let X be an L-space, we think of Spec(L) and X as k-spaces via the forgetful functor i_{♯}. We define i_{∗}X to be Maps_{L}(Spec(L), X), the k-space of maps Spec(L) → X over L. More precisely, the structure map

p_{X} : X → Spec(L)

of X gives a map of k-spaces

(p_{X})_{∗} : \underline{Hom}_{Spec(k)}(Spec(L), X) → \underline{Hom}_{Spec(k)}(Spec(L), Spec(L)).

Also, the identity map Id : Spec(k) ×_{Spec(k)} Spec(L) → Spec(L) over Spec(k) is adjoint to

ι : Spec(k) → \underline{Hom}_{Spec(k)}(Spec(L), Spec(L)).

We define i_{∗}X = Maps_{L}(Spec(L), X) by the pullback diagram over Spec(k)

\xymatrix{Maps_{L}(Spec(L), X) \ar[r] \ar[d] & \underline{Hom}_{Spec(k)}(Spec(L), X) \ar[d]^{(p_{X})_{∗}} \\ Spec(k) \ar[r]_{ι} & \underline{Hom}_{Spec(k)}(Spec(L), Spec(L)).}  (2.4)

Lemma 2.5. The functor i_{∗} = Maps_{L}(Spec(L), −) is the right adjoint to i^{∗}.

Proof. For a k-space Y and an L-space X, a map over Spec(k)

f : Y → Maps_{L}(Spec(L), X)

is equivalent to a commutative diagram over Spec(k)

\xymatrix{Y \ar[r]^{f} \ar[d] & \underline{Hom}_{Spec(k)}(Spec(L), X) \ar[d]^{(p_{X})_{∗}} \\ Spec(k) \ar[r]_{ι} & \underline{Hom}_{Spec(k)}(Spec(L), Spec(L)).}

By naturality of the adjunction (Spec(L) ×_{Spec(k)} −, \underline{Hom}_{Spec(k)}(Spec(L), −)) in Spc(k), this is equivalent to a commutative diagram

\xymatrix{Y ×_{Spec(k)} Spec(L) \ar[r] \ar[d] & X \ar[d]^{p_{X}} \\ Spec(k) ×_{Spec(k)} Spec(L) \ar[r]_{Id} & Spec(L)}

i. e. a map Y ×_{Spec(k)} Spec(L) → X over Spec(L).

Again, by the uniqueness of adjoints, i_{∗} is the same as the direct image functor of sheaves [11], if we think of spaces over k and over L as simplicial Nisnevich sheaves. Remark: For any L-spaces X and Y, there is another mapping space from X to Y over Spec(L), namely

\underline{Hom}_{Spec(L)}(X, Y)

which is the right adjoint to the functor

X ×_{Spec(L)} − : Spc(L) → Spc(L).

In particular, for X = Spec(L), Spec(L) ×_{Spec(L)} − is the identity functor, thus, so is \underline{Hom}_{Spec(L)}(Spec(L), −). For a general L-space Y, the k-space

i_{∗}Y = Maps_{L}(Spec(L), Y)

is not the same as i_{♯}(\underline{Hom}_{Spec(L)}(Spec(L), Y)) = i_{♯}Y.

On the level of affine schemes,

i_{∗} : Algebras(L) → Algebras(k)

is the left adjoint to the extension of scalars functor. If [L : k] = n, and we choose a basis α_{1}, …, α_{n} of L as a vector space over k, then for a finitely generated L-algebra

R = L[y_{1}, …, y_{r}]/I

we can write down the k-algebra i_{∗}R in terms of generators and relations to be

i_{∗}R = k[x_{1,1}, …, x_{1,r}, x_{2,1}, …, x_{n,r}]/J

where J is the following ideal of k[x_{1,1}, …, x_{1,r}, x_{2,1}, …, x_{n,r}]. For f a polynomial in L[y_{1}, …, y_{r}], we have unique g_{1}(f), …, g_{n}(f) in k[x_{1,1}, …, x_{1,r}, x_{2,1}, …, x_{n,r}] such that

f (∑_{j=1}^{n} x_{j,1}α_{j}, …, ∑_{j=1}^{n} x_{j,r}α_{j}) = ∑_{j=1}^{n} g_{j}(f)(x_{1,1}, …, x_{n,r})α_{j}.

Then set J = {g_{j}(f) | f ∈ I}. It is routine to check that this is independent of the choice of basis, and is indeed the left adjoint to the functor L ⊗_{k} −.

Examples:

1. Consider 𝔸_{L}^{1} = Spec(L[x]) over Spec(L). Then on the level of algebras,

i_{∗}(L[x]) = k[x_{1}, …, x_{[L:k]}]

so i_{∗}(𝔸_{L}^{1}) = 𝔸_{k}^{[L:k]}.

2. Suppose L = k[\sqrt{a}] is an extension of degree 2 over k, for some a ∈ k^{×}, a ∉ (k^{×})^{2}, and choose the basis \{1, \sqrt{a}\} of L as a vector space over k. Consider the affine variety

(𝔾_{m})_{L} = Spec(L[x, y]/xy = 1).

Then on the level of algebras

i_{∗}(L[x, y]/xy = 1) = k[x_{1}, x_{2}, y_{1}, y_{2}]/ ∼

If we think of x = x_{1} + x_{2}\sqrt{a}, y = y_{1} + y_{2}\sqrt{a}, then the relation ∼ is given by

xy = (x_{1} + x_{2}\sqrt{a})(y_{1} + y_{2}\sqrt{a}) = 1,

so x_{1}y_{1} + ax_{2}y_{2} = 1, and x_{1}y_{2} + x_{2}y_{1} = 0. Hence, the k-space i_{∗}(𝔾_{m})_{L} is

Spec(k[x_{1}, x_{2}, y_{1}, y_{2}]/(x_{1}y_{1} + ax_{2}y_{2} − 1, x_{1}y_{2} + x_{2}y_{1}).

This gives the adjoint functors i_{♯}, i^{∗} and i_{∗} between the categories of unbased spaces Spc(k) and Spc(L). We would also like to have these functors on the categories of based k-spaces and L-spaces. In the equivariant context, the forgetful functor from based G-spaces to based H-spaces is given by applying the unbased forgetful functor to the diagram of G-spaces ∗ → X, where ∗ denotes a single fixed point. This suggests that

i^{∗} : Spc(k)_{•} → Spc(L)_{•}

should be defined similarly. Thus, for X ∈ Spc(k)_{•}, we define i^{∗}(X) by applying the unbased i^{∗} to the diagram Spec(k) → X. We have i^{∗}(Spec(k)) = Spec(L), i^{∗}X = Spec(L)×_{Spec(k)}X, and the induced basepoint Spec(L) → Spec(L)×_{Spec(k)}X is the pullback along i of the basepoint map Spec(k) → X.

Given a based L-space X, we can define the based k-space i_{♯}X by the following pushout diagram

\xymatrix{Spec(L) \ar[r] \ar[d]^{i} & X \ar[d] \\ Spec(k) \ar[r] & i_{♯}X}

where the top horizontal map is the basepoint of X. This is a diagram over k, so strictly speaking, the top right corner of the square is the unbased version of i_{♯}X. It is routine to check that the functor i_{♯} : Spc(L)_{•} → Spc(k)_{•} is the left adjoint to i^{∗} : Spc(k)_{•} → Spc(L)_{•}.

The based right adjoint i_{∗} : Spc(L)_{•} → Spc(k)_{•} is also defined, same as in the unbased case, to be Maps_{L}(Spec(L), −). For a based L-space Y, the basepoint map Spec(k) → Maps_{L}(Spec(L), Y) is the adjoint to the basepoint map Spec(L) → Y of Y. The functor i_{∗} : Spc(L)_{•} → Spc(k)_{•} is the right adjoint to i^{∗} since it is the right adjoint to i^{∗} in the categories of unbased spaces.

This gives the functors i_{♯}, i^{∗} and i_{∗} for based spaces. We would also like the spectra versions of these functors. For this, we need the following results. First, note that i^{∗}(ℙ_{k}^{1}) = ℙ_{L}^{1}.

Lemma 2.6. If X is a based space over Spec(k), then

i^{∗}(Σ^{ℙ_{k}^{1}}X) ≅ Σ^{ℙ_{L}^{1}}i^{∗}X

naturally.

Proof. Let i_{X} : Spec(k) → X be the basepoint of X, and i_{ℙ_{k}^{1}} : Spec(k) → ℙ_{k}^{1} be the basepoint of ℙ_{k}^{1}. We have that

Σ^{ℙ_{k}^{1}}X = \frac{ℙ_{k}^{1} ×_{Spec(k)} X}{ℙ_{k}^{1} ∨_{Spec(k)} X}

where the quotient is in the category of k-spaces, and the map of ℙ_{k}^{1} ∨_{Spec(k)} X into ℙ_{k}^{1} ×_{Spec(k)} X is induced by i_{X} and i_{ℙ_{k}^{1}}. Since i^{∗} is a left adjoint, it preserves pushouts, so

i^{∗}Σ^{ℙ_{k}^{1}}X ≅ \frac{i^{∗}ℙ_{k}^{1} ×_{Spec(k)} X}{i^{∗}ℙ_{k}^{1} ∨_{Spec(k)} X}  (2.7)

naturally, where the quotient takes place in Spc(L). We have that i^{∗}(Spec(k)) = Spec(L), and i^{∗}(ℙ_{k}^{1}) = ℙ_{L}^{1}, so

i^{∗}ℙ_{k}^{1} ∨_{Spec(k)} X ≅ ℙ_{L}^{1} ∨_{Spec(L)} i^{∗}X

naturally. Also, since i^{∗} is a right adjoint, it preserves pullbacks, so

i^{∗}ℙ_{k}^{1} ×_{Spec(k)} X ≅ ℙ_{L}^{1} ×_{Spec(L)} i^{∗}X

naturally. Finally, as L-spaces, the basepoints of ℙ_{L}^{1} and i^{∗}X are i^{∗} applied to i_{ℙ_{k}^{1}} and i_{X} respectively. Hence, (2.7) is naturally isomorphic to

Σ^{ℙ_{L}^{1}}i^{∗}X = \frac{ℙ_{L}^{1} ×_{spec(L)} i^{∗}X}{ℙ_{L}^{1} ∨_{Spec(L)} i^{∗}X} .

Proposition 2.8. Let X be a based k-space, and Y a based L-space. Then there are natural isomorphisms of based k-spaces

ζ : i_{♯}(Y ∧_{Spec(L)} i^{∗}X) ≅ (i_{♯}Y) ∧_{Spec(k)} X
φ : i_{∗}(\underline{Hom}_{Spec(L)}(i^{∗}X, Y)) ≅ \underline{Hom}_{Spec(k)}(X, i_{∗}Y).

Proof. The first statement follows by explicitly checking the definitions of the smash products and of the based i_{♯}. For the second statement, note that the functor i_{∗}(\underline{Hom}_{Spec(L)}(i^{∗}X, −)) from Spc(L)_{•} to Spc(k)_{•} is right adjoint to the functor i^{∗}(−) ∧_{Spec(L)} i^{∗}X, and the functor Hom_{Spec(k)}(X, i_{∗}(−)) is right adjoint to the functor i^{∗}(−∧_{Spec(k)}X). By explicitly checking the definitions of the smash product, we have that for any based k-space Z,

i^{∗}Z ∧_{Spec(L)} i^{∗}X ≅ i^{∗}(Z ∧_{Spec(k)} X).

So the two left adjoints coincide, and the statement follows by the uniqueness of adjoints.

Taking the adjoint of Lemma 2.6, we get that for a based L-space Y,

Ω^{ℙ_{k}^{1}}i_{∗}Y ≅ i_{∗}(Ω^{ℙ_{L}^{1}}Y)  (2.9)

naturally. Also, taking X = ℙ_{k}^{1} in the first statement of Proposition 2.8 gives that for a based L-space Y,

i_{♯}(Σ^{ℙ_{L}^{1}}Y) ≅ Σ^{ℙ_{k}^{1}}i_{♯}Y  (2.10)

naturally. Taking the adjoint of this gives that for a based k-space X,

Ω^{ℙ_{L}^{1}}i^{∗}X ≅ i^{∗}(Ω^{ℙ_{k}^{1}}X)  (2.11)

naturally. Hence, we can define i^{∗} and i_{∗} on the level of spectra, and use the spectrification to define i_{♯} on spectra. More precisely, we have the following definition.

Definition 2.12. The functors

i_{♯} :Spectra(L) → Spectra(k)
i^{∗} :Spectra(k) → Spectra(L)
i_{∗} :Spectra(L) → Spectra(k)

are given as follows. Suppose D is a k-spectrum, and E is an L-spectrum. For i_{♯}E, let the k-prespectrum i_{♯}^{pre}E be given by

i_{♯}^{pre}(E)_{n} = i_{♯}(E_{n})

with structure maps

Σ^{ℙ_{k}^{1}}i_{♯}(E_{n}) ≅ i_{♯}(Σ^{ℙ_{L}^{1}}E_{n}) →^{i_{♯}\overline{ρ}_{n}} i_{♯}E_{n+1}

where \overline{ρ}_{n} : Σ^{ℙ_{L}^{1}}E_{n} → E_{n+1} is the adjoint structure map of E. Define i_{♯}E to be the spectrification of i_{♯}^{pre}(E).

We define the L-spectrum i^{∗}D by

(i^{∗}D)_{n} = i^{∗}(D_{n})

with structure maps

i^{∗}(D_{n}) →^{i^{∗}r_{n}} i^{∗}(Ω^{ℙ_{k}^{1}}D_{n+1}) ≅ Ω^{ℙ_{L}^{1}}i^{∗}D_{n+1}

where r_{n} : D_{n} → Ω^{ℙ_{k}^{1}}D_{n+1} is the structure map of D. Similarly, define the kspectrum i_{∗}E by

(i_{∗}E)_{n} = i_{∗}(E_{n}).

The structure maps are

i_{∗}(E_{n}) →^{i_{∗}ρ_{n}} i_{∗}(Ω^{ℙ_{L}^{1}}E_{n+1}) ≅ Ω^{ℙ_{k}^{1}}i_{∗}E_{n+1}

where ρ : E_{n} → Ω^{ℙ_{L}^{1}}E_{n+1} is the structure map of E.

The following proposition follows from the adjunction relations between the functors i_{♯}, i^{∗} and i_{∗} on based spaces.

Proposition 2.13. The functor i_{♯} : Spectra(L) → Spectra(k) is the left adjoint to i^{∗} : Spectra(k) → Spectra(L), and the functor i_{∗} : Spectra(L) → Spectra(k) is the right adjoint to i^{∗}.

Remark: In the equivariant category, the analogues of Proposition 2.8 are the homeomorphisms of G-spaces

G_{+} ∧_{H} (Y ∧ X) ≅ (G_{+} ∧_{H} Y) ∧ X
F_{H}(G_{+}, F (X, Y)) ≅ F (X, F_{H}(G_{+}, Y))

for X a based G-space and Y a based H-space (see [9]). As we will see in Section 5, the analogues of these statements in the theory of derived categories of sheaves into an abelian category can also be stated, as a version of Verdier duality [1, 8].

By the composition of adjoints, (i_{♯}i^{∗}, i_{∗}i^{∗}) form a adjoint pair of functors from Spectra(k) to Spectra(k). For X ∈ Spc(k)_{•}, let

F (X, −) : Spectra(k) → Spectra(k)

be the function spectrum functor, i. e. the right adjoint of the functor X ∧ − : Spectra(k) → Spectra(k). The following corollary is analogous to the fact that in equivariant topology, for a G-spectrum E, we have G_{+} ∧_{H} E ≅ (G/H)_{+} ∧ E, and F_{H}(G_{+}, E) ≅ F ((G/H)_{+}, E).

Corollary 2.14. For a k-spectrum E, we have isomorphisms of k-spectra

i_{♯}i^{∗}E ≅ Spec(L)_{+} ∧ E
i_{∗}i^{∗}E ≅ F (Spec(L)_{+}, E).

Proof. Since the k-spectrum Spec(L)_{+} ∧ E is defined spacewise, it suffices to show that i_{♯}i^{∗}X ≅ Spec(L)_{+}∧X for a based k-space X. We have that for S_{L}^{0} = Spec(L)⨿ Spec(L) in Spc(L)_{•}, i_{♯}S_{L}^{0} = Spec(L)_{+} = Spec(L) ⨿ Spec(k) in Spc(K)_{•}. Thus, applying Proposition 2.8 to S_{L}^{0} gives the first statement. The second statement follows from the uniqueness of adjoints, and the fact that F (Spec(L)_{+}, −) is the right adjoint to Spec(L)_{+} ∧ −.

Remarks:

1. In general, for any smooth Noetherian scheme S of finite dimension over k, consider the category of Spc(S) of S-spaces. For any smooth map of schemes over k

f : S′ → S

the functors f_{♯}, f^{∗} and f_{∗} between the categories Spc(S) and Spc(S′) can be defined as for an extension of fields. The based and stable versions of these functors are also defined accordingly. If f is smooth and finite, then f^{∗} is the same as the inverse image functor. Its left adjoint f_{♯} is the “extension by zero” functor constructed in Proposition 3.1.23 of [11], and its right adjoint f_{∗} is the direct image functor.

2. If f is smooth and finite, such as f : Spec(L) → Spec(k), then by Corollary 3.1.24 and Proposition 3.2.9 of [11], f^{∗} preserves simplicial and 𝔸^{1}-weak equivalences. Also, By Propositions 3.2.9 and 3.2.12 of [11], the left derived functor of f_{♯} and the right derived functor of f_{∗} in the simplicial homotopy categories preserve 𝔸^{1}-weak equivalences. Thus, f_{♯} preserves 𝔸^{1}-weak equivalences between simplicially cofibrant objects, and f_{∗} preserves 𝔸^{1}-weak equivalences between simplicially fibrant objects.

3. An application to P ic(SH(k))

We give an application of the base change functors towards constructing elements of the Picard group P ic(SH(k)) of the stable homotopy category over k. This is the group of objects that are invertible with respect to the smash product. For instance, there are two versions of the circle in Spc(k)_{•}

S_{s}^{1} = 𝔸^{1}/{0, 1}
𝔾_{m} = 𝔸^{1} \ {0}.

We have that in Spc(k)_{•},

S_{s}^{1} ∧ 𝔾_{m} ≃ ℙ^{1}.

Hence, in SH(k), S_{s}^{1} and 𝔾_{m} are both in P ic(SH(k)). In this sense, ℙ^{1} is a “mixed 2-sphere”.

There are several known classes of elements in P ic(SH(k)) not generated by S_{s}^{1} and 𝔾_{m} [5]. For example, let a ∈ k^{×}, a ∉ (k^{×})^{2}. The affine variety S^{a} given by the equation x^{2} − ay^{2} = 1 has the property that

S^{a} ∧ { \atop Spec(k[\sqrt{a}])} ≅ ℙ^{1}

where denotes the unreduced suspension. Thus, S^{a} and { \atop Spec(k[\sqrt{a}])} are in P ic(SH(k), and motivic cohomology calculations show that they are not in the subgroup generated by S_{s}^{1} and S_{t}^{1}.

We would like to find other ways of constructing these objects. One such way is via the category of equivariant topological spaces. For this section, let L be a finite Galois extension of k, and G = Gal(L/k). For the category Sh(Sm/k)_{Nis} of sheaves on the site Sm/k with the Nisnevich topology, we define a functor

F_{L/k} : Finite G−sets → Sh(Sm/k)_{Nis}

which takes a homogenous G-set G/H to Spec(L^{H}), and passes to disjoint unions. To give F_{L/k} on morphisms of finite G-sets, it suffices to give F_{L/k}(α) for any Gequivariant map G/H → G/K, where H and K are subgroups of G. By adjunction, we have that the nonequivariant space of G-equivariant maps from G/H to G/K is naturally isomorphic to the nonequivariant space of H-equivariant maps ∗ → G/K, where the single point ∗ is thought of as a fixed H-space. In turn, this is the same as the space of nonequivariant maps ∗ → (G/K)^{H}, i. e. the nonequivariant space (G/K)^{H} of the H-fixed points of the homogenous G-set G/K. We have that

(G/K)^{H} = {gK |g^{−1}Hg ⊆ K}.

In particular, it is empty if H is not subconjugate to K. But G = Gal(L/k), so an element g ∈ G gives g : L → L. If g^{−1}Hg ⊆ K, then for any x ∈ L^{K} and h ∈ H, we have that g^{−1}hg(x) = x, so hg(x) = g(x). Thus, such a g takes L^{K} to L^{H}. Also, for every k ∈ K, we have that gk|_{L^{K}} = g|_{L^{K}}. So for any gK ∈ (G/K)^{H}, we get a well-defined map L^{K} → L^{H}, which gives

F_{L/k}(gK) : F_{L/k}(G/H) = Spec(L^{H}) → Spec(L^{K}) = F_{L/k}(G/K).

Recall that the category of k-spaces is just the category of simplicial sheaves on the site Sm/k with the Nisnevich topology. So taking the simplicial categories of both the source and the target, F_{L/k} extends to a functor

F_{L/k} : Finite G−simplicial sets → Spc(k).  (3.1)

Since F_{L/k} takes a single fixed point to Spec(k), it also passes to a functor from the category of based G-simplicial sets to Spc(k)_{•}. Namely, a based G-simplicial set is a G-simplicial set X, together with a G-map i_{X} : ∗ → X, where ∗ = G/G is a single fixed point. We have that F_{L/k}(∗) = Spc(L^{G}) = Spec(k), since the extension L/k is Galois. Hence, applying F_{L/k} to i_{X} gives a map Spec(k) → F_{L/k}(X), which makes F_{L/k}(X) a based space over k. Also, if X is a G-space with a triangulation, then applying F_{L/k} to the simplicial model of X gives F_{L/k}(X) as a k-space. Two different triangulations of X are simplicially equivalent, so the k-space F_{L/k}(X) is well-defined up to simplicial weak equivalences. When there is no possibility of confusion, we write just F for F_{L/k}.

Suppose L is a finite Galois extension of k, H is a subgroup of G = Gal(L/k), and E = L^{H}. Then the functors F_{L/E} from H-equivariant spaces to Spc(E) and F_{L/k} from G-equivariant spaces to Spc(k) are related in the following manner. Let i : Spec(E) → Spec(k) be the map corresponding to the inclusion k ⊆ E, and let U denote the forgetful functor from G-equivariant topological spaces to H-equivariant topological spaces. In equivariant topology, recall that there is a natural equivalence of categories between the category of G-equivariant spaces over the homogenous Gspace G/H and the category of H-equivariant spaces. Let f : G/H → ∗ be the map collapsing G/H to a single fixed point. Then we have a functorf^{∗} from G-spaces to G-spaces over G/H, given by f^{∗}(T) = G/H × T. Then the diagram of functors

\xymatrix{ & G−spaces \ar[dl]_{U} \ar[dr]^{f^{∗}} \\ H−spaces \ar[rr]^{≃}_{G×_{H}−} &  & G−spaces over G/H}  (3.2)

commutes. An analogous equivalence of categories hold between the categories of G-simplicial sets over G/H and H-simplicial sets.

Now for G = Gal(L/k), F_{L/k}(G/H) = Spec(L^{H}) = Spec(E). So by passing to comma categories, F_{L/k} induces a functor from G-spaces over G/H to k-spaces with a structure map to Spec(E), i. e. Spc(E). We will denote this functor on comma categories also by F_{L/k}. If N is a subgroup of H, then

F_{L/E}(H/N) = Spec(L^{N}) = F_{L/k}(G/N).

But the homogenous G-space G/N has a natrural map to G/H, and it is G×_{H}(H/N) as a G-space over G/H. Hence, F_{L/E} is the same as the composition

H−spaces ⟶^{G×_{H}−} G−spaces over G/H ⟶^{F_{L/k}} Spc(E).  (3.3)

Lemma 3.4. In the above situation, the diagram of functors

\xymatrix{G−simplicial sets \ar[d]_{U} \ar[r]^{F_{L/k}} & Spc(k) \ar[d]^{i^{∗}} \\ H−simplicial sets \ar[r]_{F_{L/E}} & Spc(E)}

commutes up to natural equivalence.

Proof. By diagram (3.2) and (3.3), it suffices to show that the diagram of functors

\xymatrix{G−simplicial sets \ar[d]_{f^{∗}} \ar[r]^{F_{L/k}} & Spc(k) \ar[d]^{i^{∗}} \\ G−simplicial sets over G/H \ar[r]_{F_{L/k}} & Spc(E)}

commutes up to natural equivalence. Since f^{∗} and i^{∗} commute with simplicial structures, it suffices to show this for a homogenous G-set G/K, where K is a subgroup of G. We have that F_{L/k}(G/K) = Spec(L^{K}), so

i^{∗}F_{L/k}(G/K) = Spec(E) ×_{Spec(k)} Spec(L^{K})

as an E-space. On the other hand, f^{∗}(G/K) = G/H × G/K, which maps to G/H by collapsing G/K. For any (g_{1}H, g_{2}K) ∈ G/H × G/K, an element g ∈ G fixes (g_{1}H, g_{2}K) if and only if g ∈ H ∩ K. Thus, the isotropy subgroup of every point of G/H × G/K is H ∩ K, i. e. as a G-set,

G/H × G/K ≅ ∐ G/(H ∩ K)

is the disjoint union of n copies of G/(H ∩ K), where n = ([G : H][G : K])/[G : H ∩ K]. Thus,

F_{L/k}f^{∗}(G/K) = ∐ Spec(L^{H∩K}) = ∐ Spec(EL^{K}).

This is naturally isomorphic to Spec(E) ×_{Spec(k)} Spec(L^{K}) since the extension L over k is Galois.

Consider a finite-demensional real representation V of the group G. We will denote the unit sphere of V by S(V), and the one-point compactification of V by S^{V}. The following theorem give a class of invertible objects in SH(k).

Theorem 3.5. For V a finite-dimensional real representation of G = Gal(L/k), F_{L/k}(S^{V}) is invertible in SH(k).

To prove the theorem, we introduce the notion of join powers and smash powers of a k-space X to the power of T, where T is an étale scheme over Spec(k). Recall that for topological spaces X and Y, the join X ∗ Y is the homotopy pushout of the diagram

\xymatrix{X × Y \ar[r] \ar[d] & X \ar[d] \\ Y \ar[r] & X ∗ Y}

where X ×Y maps to X and Y by the projections. For k-spaces X and Y, their join X ∗ Y is defined in the same way. In particular, a model for X ∗ Y is the quotient space of X × Y × 𝔸^{1}, obtained by collapsing X × Y × {0} to X and X × Y × {1} to Y. Also, recall that the unreduced suspension X̃ of a k-space X is defined by the cofiber sequence

X_{+} → S^{0} → X̃.

The join product has the property that

{ \atop X ∗ Y} ≃ X̃ ∧ Ỹ.

Now as a simplicial set, the 1-simplex in the topological category realizes to the unit interval I, whereas the 1-simplex in the category Spc(k) realizes to the affine line 𝔸^{1}. Thus, for a G-space X with a triangulation, F_{L/k}(X × I) = F_{L/k}(X) × 𝔸^{1}. Also, as shown in the proof of Lemma 3.4, for subgroups H and K of G,

F_{L/k}(G/H × G/K) ≅ F_{L/k}(G/H) × F_{L/k}(G/K)

naturally. Passing to the simplicial categories, we get that for G-equivariant spaces X and Y with triangulations,

F_{L/k}(X × Y) ≃ F_{L/k}(X) × F_{L/k}(Y)

naturally. Using these facts, and the fact that F_{L/k} preserves pushouts, it is easy to check that F commutes with the join product and the unreduced suspension. Also, the based version of F_{L/k} commutes with the smash product. This is because for based G-spaces X and Y, X ∧ Y = (X × Y)/(X ∨ Y), and similarly for based k-spaces. Since X ∨ Y is a pushout of the basepoint maps ∗ → X and ∗ → Y, and F_{L/k}(∗) = Spec(k), we get that F_{L/k}(X ∨ Y) = F_{L/k}(X) ∨_{Spec(k)} F_{L/k}(Y). But F_{L/k} also preserves products and quotient by a subspace, so it preserves the smash product.

For the rest of this section, we will abbreviate F_{L/k} to just F. For X ∈ Spc(k), and T → Spec(k) étale, X^{∗T} is an analogue of the join power X^{∗n}, which takes into account the “Galois action” on the parametrizing k-space T. Likewise, if X ∈ Spc(k)_{•}, we have the smash power X^{∧T}. We will defer the exact definitions of the join and smash powers to T to Secton 5. The properties of the usual join and smash powers apply to (−)^{∗T} and (−)^{∧T}. For instance, similarly as in the case of X^{∗n} and X^{∧n}, the join and smash powers to T has the property that for X ∈ Spc(k),

X̃^{∗T} ≃ (X̃)^{∧T} .

Likewise, for a G-space X and a G-set T_{G}, we can define X^{∗T_{G}}. From the definitions of (−)^{∗T_{G}} and (−)^{∗T}, we will see in Section 5 that if G = Gal(L/k), and X is a triangulated G-space, then there is a weak equivalence of k-spaces

F (X^{∗T_{G}}) ≃ (F (X))^{∗F_{L/k}(T_{G})}.

Let ℝ[G] be the regular real representation of G. We have the following lemma.

Lemma 3.6. For the triagulated G-space S(ℝ[G]), we have a natural isomorphism

F (S(ℝ[G])) ≅ (S^{0})^{∗Spec(L)}.

Proof. It suffices to show that S(ℝ[G]) is naturally isomorphic to (S^{0})^{∗G}, where S^{0} is ∗ ⨿ ∗ in the G-equivariant category. In particular, S^{0} = S(ℝ) is the unit sphere of the one-dimensional trivial representation ℝ of G. For the nonequivariant join product, we have that for two representations V and W of G, there is a natural G-equivariant homeomorphism S(V)∗S(W) ≅ S(V ⊕W). Similarly, for (−)^{∗G}, one has that

S(ℝ)^{∗G} ≅ S(ℝ^{G})

naturally as G-equivariant spaces. But ℝ^{G} is just ℝ[G] as a G-representation.

We also have the following lemma, whose proof we defer to Section 5.

Lemma 3.7. For an étale scheme T over k, the functor (−)^{∧T} has the property that for X, Y ∈ Spc(k)_{•},

X^{∧T} ∧ Y^{∧T} ≃ (X ∧ Y)^{∧T} .

Proof of Theorem 3.5. Since F (S(ℝ[G])) ≃ (S^{0})^{∗Spec(L)}, and F preserves unreduced suspensions, by taking the unreduced suspension of both sides, we get that

F (S^{ℝ[G]}) ≃ { \atop (S^{0})^{∗Spec(L)}} ≃ (S_{s}^{1})^{∧Spec(L)}

since S_{s}^{1} = S̃^{0} by definition. Recall also from [11] that there is a 𝔸^{1}-weak equivalence of k-spaces

S_{s}^{1} ∧ 𝔾_{m} ≃ ℙ^{1}.

So by Lemma 3.7,

(S_{s}^{1})^{∧Spec(L)} ∧ (𝔾_{m})^{∧Spec(L)} ≃ (ℙ^{1})^{∧Spec(L)}.

Also, note that

(𝔸_{k}^{1})^{Spec(L)} = i_{∗}i^{∗}𝔸_{k}^{1} = i_{∗}𝔸_{L}^{1} = 𝔸_{k}^{n}.

But we also have an 𝔸^{1}-homotopy equivalence ℙ^{1} ≃ 𝔸^{1}/𝔸^{1} \ {0} ([11]). Thus,

(ℙ^{1})^{∧Spec(L)} ≃ (𝔸^{1}/𝔸^{1} \ {0})^{∧Spec(L)}
≅ (𝔸^{1})^{Spec(L)}/(𝔸^{1})^{Spec(L)} \ {0}
≅ (𝔸^{1})^{n}/(𝔸^{1})^{n} \ {0}
≃ (ℙ^{1})^{∧n}.

This is invertible in SH(k), so F (S^{ℝ[G]}) = (S_{s}^{1})^{∧Spec(L)} is invertible in SH(k). But for any irreducible representation V, V is a direct summand of ℝ[G], so F (S^{V}) is a smash summand of F (S^{ℝ[G]}), and is therefore also invertible in SH(k). Finally, for any finite-dimensional representation V of G, we have V = ⊕_{i=1}^{n}V_{i} as a finite direct sum of irreducible representations V_{i} of G, so S^{V} is a finite smash product of S^{V_{i}}’s. Therefore, F (S^{V}) is in P ic(SH(k)) for any finite-dimensional representation V of G.

Example: Let L be a cyclic extension of degree p over k, and let γ be a generator of Gal(L/k) = ℤ/p. Define O_{+} to be the cofiber in the stable homotopy category over k of the map

Spec(L)_{+} ⟶^{Id−γ} Spec(L)_{+}.

It is the suspension spectrum of the homotopy coequalizer O of the maps

Id, γ : Spec(L) → Spec(L)

together with a disjoint basepoint. Consider the 2-dimensional real representation V of ℤ/2 given by multiplication by e^{2πi/p} in ℝ^{2}. The simpicial decompositions of S(V) is that it has one 0-cell ℤ/p × ∗, and one 1-cell ℤ/p × I. By the definition of the homotopy coequalizer, we see that the simplicial definition of O is the same, with ℤ/p replaced by Spec(L), and I replaced by 𝔸_{k}^{1}. Since F (ℤ/p) = Spec(L), we get

F (S(V)) = O.

Hence, O = F (S^{V}) is invertible in SH(k). (More on O will be in [7].)

4. The Wirthmüller Isomorphism

The main result of this section is the following theorem, which is an analogue of the Wirthmüller isomorphism for the 𝔸^{1}-setting ([15], see also [9, 4]).

Theorem 4.1. For L a finite separable extension of k, and E an L-spectrum, we have a natural 𝔸^{1}-weak equivalence of k-spectra

i_{♯}E ≃ i_{∗}E.

We will give an explicit construction of the equivalence, in terms of the smashinvertible objects considered in the last section.

We begin by definition a natural map of k-spectra

ψ : i_{♯}E → i_{∗}E.  (4.2)

By adjunction, this is equivalent to a map of L-spectra

\overline{ψ} : i^{∗}i_{♯}E → E.  (4.3)

Let X be a based L-space. Let

F_{Spectra(L)}(X, −) : Spectra(L) → Spectra(L)

denote the right adjoint to the functor X ∧ − : Spectra(L) → Spectra(L).

To construct \overline{ψ}, we will first consider i^{∗}i_{♯} on the level of based spaces. Let X be a based space over Spec(L), with basepoint i_{X} : Spec(L) → X and structure map p_{X} : X → Spec(L). We have that i_{♯}(X) = X/_{Spec(k)}Spec(L), where /_{Spec(k)} denotes taking quotient of i_{X} in the category of spaces over Spec(k). So

\begin{aligned}i^{∗}i_{♯}(X) & = Spec(L) ×_{Spec(k)} (X/_{Spec(k)}Spec(L)) \\ & ≅ (Spec(L) ×_{Spec(k)} X)/_{Spec(L)}(Spec(L) ×_{Spec(k)} Spec(L)) \end{aligned}

since i^{∗} commutes with pushouts as a left adjoint. Note that on the right hand side, both Spec(L) ×_{Spec(k)} X and Spec(L) ×_{Spec(k)} Spec(L) are spaces over Spec(L) via the first projection map.

We have the following property of Spec(L) ×_{Spec(k)} X.

Lemma 4.4. If X is a space over Spec(L), then for Spec(L) ×_{Spec(k)} X as a space over Spec(L) via the first projection, consider the embedding over Spec(L)

Δ_{X} = p_{X} ×_{Spec(k)} Id : X → Spec(L) ×_{Spec(k)} X

which is the structure map of X on the first coordinate, and the identity on X on the second. Then we have

Spec(L) ×_{Spec(k)} X = Δ_{X} ⨿ ((Spec(L) ×_{Spec(k)} X) \ Δ_{X})

as spaces over Spec(L), where (Spec(L) ×_{Spec(k)} X) \ Δ_{X} is a space over Spec(L) by the first projection.

Proof. For the case when X is an affine scheme of finite type over Spec(L), say X = Spec(R) for a finitely generated L-algebra R, we have that Spec(L) ×_{Spec(k)} X = Spec(L ⊗_{k} R), and the map Δ_{X} corresponds to the map of L-algebras

L ⊗_{k} R → R

which is the multiplication. Hence, passing to the level of L-algebras, we get that the lemma holds in the case when X is an affine scheme of finite type over Spec(L).

For general X, recall that every space X over Spec(L) is a colimit of affine schemes of finite type, say X = colim_{i}X_{i} over Spec(L), where each X_{i} = Spec(R_{i}) for some finitely generated L-algebra R_{i}. Then

Spec(L) ×_{Spec(k)} X ≅ colim_{i} Spec(L) ×_{Spec(k)} X_{i}

naturally, since Spec(L) ×_{Spec(k)} − = i^{∗} commutes with colimits. Suppose f : X_{i} → X_{j} is a map of the colimit. Then the diagram

\xymatrix{X_{i} \ar[d]_{f} \ar[r]^{Δ_{X_{i}}} & Spec(L) ×_{Spec(k)} X_{i} \ar[d]^{Spec(L)×_{Spec(k)}f} \\ X_{j} \ar[r]_{Δ_{X_{j}}} & Spec(L) ×_{Spec(k)} X_{j}}  (4.5)

Homology, Homotopy and Applications, vol. 3(2), 2001 435

commutes, since f is a map over Spec(L). Hence, Δ_{X} is

colim_{i}Δ_{i} : X = colim_{i}X_{i} → colim_{i}(Spec(L) ×_{Spec(k)} X_{i}) = Spec(L) ×_{Spec(k)} X. Also, by passing to L-algebras, it is straightforward to check that diagram (4.5) is in fact a pullback square. So the map

Spec(L) ×_{Spec(k)} f : Spec(L) ×_{Spec(k)} X_{i} → Spec(L) ×_{Spec(k)} X_{j}

restricts to Δ_{X_{i}}(X_{i}) → Δ_{X_{j}}(X_{j}) and

(Spec(L) ×_{Spec(k)} X_{i}) \ Δ_{X_{i}}(X_{i}) → (Spec(L) ×_{Spec(k)} X_{j}) \ Δ_{X_{j}}(X_{j}) and the diagram

\xymatrix{Spec(L) ×_{Spec(k)} X_{i} \ar[d]^{Spec(L)×_{Spec(k)}f} \ar[r]^{≅} & Δ_{X_{i}}(X_{i}) ⨿ ((Spec(L) ×_{Spec(k)} X_{i}) \ Δ_{X_{i}}(X_{i})) \ar[d] \\ Spec(L) ×_{Spec(k)} X_{j} \ar[r]_{≅} & Δ_{X_{j}}(X_{j}) ⨿ ((Spec(L) ×_{Spec(k)} X_{j}) \ Δ_{X_{j}}(X_{j}))}

commutes. So passing to colimits, we get that Spec(L) ×_{Spec(k)} X ≅ (colim_{i}Δ_{X_{i}}(X_{i})) ⨿ (colim_{i}(Spec(L) ×_{Spec(k)} X_{i}) \ Δ_{X_{i}}(X_{i})) naturally. We have colim_{i}Δ_{X_{i}}(X_{i}) ≅ Δ_{X}(X), so colim_{i}(Spec(L) ×_{Spec(k)} X_{i}) \ Δ_{X_{i}}(X_{i}) is (Spec(L) ×_{Spec(k)} X) \ Δ_{X}(X).

This allows us to define a map of unbased spaces over Spec(L)

\overline{ψ}_{u} : Spec(L) × X ≅ Δ_{X}(X) ⨿ ((Spec(L) ×_{Spec(k)} X) \ Δ_{X}(X)) → X. This is the identity on Δ_{X}(X) ≅ X, and on the other component, it is

(Spec(L) ×_{Spec(k)} (X) \ Δ_{X}(X) →^{π_{1}} Spec(L) →^{i_{X}} X.

For X ∈ Spc(L)_{•}, we define

\overline{ψ} : i^{∗}i_{♯}(X) ≅ (Spec(L) ×_{Spec(k)} X)/_{Spec(L)}(Spec(L) ×_{Spec(k)} Spec(L)) → X to be induced from \overline{ψ}_{u}. To check that this is a well-defined map in Spc(L)_{•}, we need that \overline{ψ}_{u} maps Spec(L) ×_{Spec(k)} Spec(L) into the basepoint of X, i. e. the diagram

\xymatrix{Spec(L) ×_{Spec(k)} Spec(L) \ar[r]^{Id×i_{X}} \ar[d]_{π_{1}} & Spec(L) ×_{Spec(k)} X \ar[d]^{_{\overline{ψ}_{u}}} \\ Spec(L) \ar[r]_{i_{X}} & X}  (4.6)

commutes. This follows because we have

\begin{matrix}Spec(L) ×_{Spec(k)} Spec(L) \\ = Δ_{Spec(L)}(Spec(L)) ⨿ ((Spec(L) ×_{Spec(k)} Spec(L)) \ Δ_{Spec(L)}(Spec(L))) \end{matrix}

and

Spec(L) ×_{Spec(k)} X = Δ_{X}(X) ⨿ ((Spec(L) ×_{Spec(k)} X) \ Δ_{X}(X)).

It is straightforward to check that diagram (4.6) commutes on Δ_{Spec(L)}(Spec(L)) and on (Spec(L) ×_{Spec(k)} Spec(L)) \ Δ_{Spec(L)}(Spec(L)).

Passing to the stable categories, we can now define \overline{ψ} for a spectrum E over Spec(L)

i^{∗}i_{♯}(E) → E

to be given by first applying spacewise \overline{ψ} for based spaces, then taking the spectrification functor. To check that applying \overline{ψ} spacewise gives a map of L-prespectra, we need that the diagram

\xymatrix{Σ^{ℙ_{L}^{1}}(i^{∗}i_{♯}(E_{n})) \ar[r]^{Σ^{ℙ^{1}_{L}} \overline{ψ}} \ar[d]_{≅} & Σ^{ℙ_{L}^{1}}E_{n} \ar[d]^{=} \\ i^{∗}i_{♯}Σ^{ℙ_{L}^{1}}E_{n} \ar[d]_{i^{∗}i_{♯}ρ_{n}} & Σ^{ℙ_{L}^{1}}E_{n} \ar[d]^{ρ_{n}} \\ i^{∗}i_{♯}E_{n+1} \ar[r]_{\overline{ψ}} & E_{n+1}}

commutes, where ρ : Σ^{ℙ_{L}^{1}}E_{n} → E_{n+1} is the structure map of E. This follows since ρ is a map over Spec(L). This gives the map ψ of (4.2).

To define the inverse to ψ, we use the constructions of [9]. Let E be the normal closure of L, and let G = Gal(E/k), and H = Gal(E/L). Consider the functor F given by (3.1) from the category of finite G-simplicial sets to the category of k-spaces. In particular, F ((G/H)_{+}) = Spec(E^{H})_{+} = Spec(L)_{+}. Let V be a finitedimensional representation of G. such that G/H embeds in V. This embedding extends to an open tubular neighborhood Uof G/H in V. Now the quotient space S^{V} /(S^{V} \ U) is the Thom space of the normal bundle of the embedding of G/H_{+} in S^{V}, so it is G-equivariantly homotopy equivalent to (G/H)_{+} ∧ S^{V}. Thus, we have a Pontrjagin-Thom map

t : S^{V} → S^{V} /(S^{V} \ U) ≃ (G/H)_{+} ∧ S^{V}

in the category of G-equivariant spaces. Taking simplicial approximation and applying the functor F and then the suspension spectrum functor to t gives a map of k-spectra

F (t) : F (S^{V}) → Spec(L)_{+} ∧ F (S^{V}).

Now let D be any k-spectrum, we define a pretransfer map

\overline{t} : D → Spec(L)_{+} ∧ D

as follows. By Theorem 3.5, F (S^{V}) is invertible in the stable homotopy category SH(k) over k. By formal arguments, in SH(k), F (S^{V})^{−1} must be DF (S^{V}) = F (S^{0}, S^{V}), the Spanier-Whitehead dual of F (S^{V}). In fact, bythe proof of Theorem 3.5, we have a rigid model of F (S^{V})^{−1} in the category Spectra(k), which underlies SH(k). Namely, let G = Gal(E/k), and let ℝ[G] − V be the complement of V in the regular representation ℝ[G] of V, and let 𝔾_{m} be the multiplicative group over k. Then we have a natural 𝔸^{1}-weak equivalence

F (S^{V}) ∧ F_{E/k}(S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(E)} ≅ (ℙ_{k}^{1})^{n}

where n = [E : k]. Thus, we can define the model of F (S^{V})^{−1} to be

Σ^{−n}Σ^{∞}(F (S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(E)}).  (4.7)

Here, Σ^{−n}Σ^{∞} denotes the n-th shift desuspension of the suspension spectrum of the space over k. For the smash product of spectra F (S^{V}) ∧ F (S^{V})^{−1}, we choose any linear injection α : U^{⊕2} → U, where U ≅ 𝔸_{k}^{∞} is the universe over k [6]. Then using the rigid model (4.7) of F (S^{V})^{−1}, and α for the smash product of spectra, similarly as in topology, we get a natural homotopy equivalence

F (S^{V}) ∧ F (S^{V})^{−1} ≃ Σ^{−n}Σ^{∞}(F (S^{V}) ∧ F (S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(L)}) ≅ S_{k}^{0}. (4.8)

In particular, in SH(k), we have the coevaluation map

c : S_{k}^{0} → F (S^{V})^{−1} ∧ F (S^{V})

and the evaluation map

e : F (S^{V})^{−1} ∧ F (S^{V}) → S_{k}^{0}.

Then e and c are inverse isomorphisms in SH(k). Again, we define rigid models in Spectra(k) for c and e, using (4.7) for F (S^{V})^{−1}. Namely, define the rigid coevaluation map to be

\begin{aligned}c : S_{k}^{0} & ≅ Σ^{−n}Σ^{∞}(ℙ^{1})^{∧n} \\ & →^{≃} Σ^{−n}Σ^{∞}(F (S^{V}) ∧ (F (S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(E)})) \\ & →^{≃} F (S^{V}) ∧ Σ^{−n}Σ^{∞}(F (S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(E)}) \\ & = F (S^{V}) ∧ F (S^{V})^{−1}. \end{aligned}  (4.9)

Similarly, the rigid model of the evaluation map is

\begin{aligned}e : F (S^{V}) ∧ F (S^{V})^{−1} & = F (S^{V}) ∧ Σ^{−n}Σ^{∞}(F (S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(E)}) \\ & →^{≃} Σ^{−n}Σ^{∞}(F (S^{V}) ∧ (F (S^{ℝ[G]−V}) ∧ (𝔾_{m})^{∧Spec(E)})) \\ & →^{≃} Σ^{−n}Σ^{∞}(ℙ^{1})^{∧n} \\ & ≅ S_{k}^{0}. \end{aligned} (4.10)

Then the rigid models of c and e pass to c and e in SH(k), so they are inverse 𝔸^{1}-weak equivalences in Spectra(k). We define

\overline{t} : D →^{c} F (S^{V})^{−1} ∧ D ∧ F (S^{V})

→^{F (t)} F (S^{V})^{−1} ∧ D ∧ (Spec(L)_{+} ∧ F (S^{V}))

→^{τ} Spec(L)_{+} ∧ (F (S^{V})^{−1} ∧ D ∧ F (S^{V}))

→^{e} Spec(L)_{+} ∧ D

≅ i_{♯}i^{∗}D.

The last (natural) isomorophism is by Corollary 2.14. Also, the transposition map τ is a natural homotopy equivalence, since it is naturally homotopic to taking Σ^{−n}Σ^{∞} of a transposition of spaces. By arguments similar to that of [9], when we pass to SH(k), \overline{t} is independent of the choice of V.

Let D = i_{∗}E for an L-spectrum E. We define a map ω of k-spectra by

ω : i_{∗}E →^{\overline{t}} i_{♯}i^{∗}(i_{∗}E) →^{i_{♯}ϵ} i_{♯}E

where the second map ϵ is the counit of the adjunction (i^{∗}, i_{∗}).

To prove Theorem 4.1, we first make the following reduction.

Lemma 4.11. If ψ : i_{♯}E → i_{∗}E is an 𝔸^{1}-weak equivalence for all L-spectra E that are shift desuspensions of suspension spectra, then it is an 𝔸^{1}-weak equivalence for all L-spectra.

Proof. For any L-spectrum E = {E_{n}}, we have a canonical 𝔸^{1}-weak equivalence

E ≃ colim_{n}Σ^{−n}Σ^{∞}E_{n}

(see [6]). Since i_{♯} : Spectra(L) → Spectra(k) is a left adjoint, it commutes with all small colimits. Also, recall that any smooth scheme Y over k is an small object in Spc(k), i. e.

Hom_{Spc(k)}(Y, colim_{j}X_{j}) ≅ colim_{j}Hom_{Spc(k)}(Y, X_{j})

for all directed systems {X_{j}} of k-spaces. Now let Y ∈ Sm/k, then Y ×_{Spec(k)} Spec(L) ∈ Sm/L, so it is also small in Spc(L). Hence, for any directed system {X_{j}} in Spc(L),

Hom_{Spc(k)}(Y, i_{∗}(colim_{j}X_{j})) ≅ Hom_{Spc(L)}(Y ×_{Spec(k)} Spec(L), colim_{j}X_{j})
≅ colim_{j}Hom_{Spc(L)}(Y ×_{Spec(k)} Spec(L), X_{j})
≅ colim_{j}Hom_{Spc(k)}(Y, i_{∗}X_{j})
≅ Hom_{Spc(k)}(Y, colim_{j}(i_{∗}X_{j})).

Also, recall that every object of Spc(k) is a colimit of smooth schemes (see Appendix of [6]). Thus, for any Y ∈ Spc(k), we have Y = colim_{r}Y_{r}, where each Y_{r} is a smooth scheme over k.

Hom_{Spc(k)}(Y, i_{∗}(colim_{j}X_{j})) ≅ lim_{r} Hom_{Spc(k)}(Y_{r}, i_{∗}(colim_{j}X_{j}))

≅ lim_{r} Hom_{Spc(k)}(Y_{r}, colim_{j}(i_{∗}X_{j}))

≅ Hom_{Spc(k)}(Y, colim_{j}(i_{∗}X_{j}))

So i_{∗} : Spc(L) → Spc(k) commutes with all small directed colimits. The same holds for the case of based spaces. Since colimits of spectra are formed spacewise, i_{∗} : Spectra(L) → Spectra(k) also commutes with all small directed colimits. Further, since i : Spec(L) → Spec(k) is a smooth finite morphism, by Propositions 3.2.9 and 3.2.12 of [11], i_{♯} and i_{∗} preserve 𝔸^{1}-weak equivalences. Thus, we have canonical 𝔸^{1}-weak equivalences of k-spectra

i_{♯}E ≃ colim_{n}i_{♯}(Σ^{−n}Σ^{∞}E_{n})
i_{∗}E ≃ colim_{n}i_{∗}(Σ^{−n}Σ^{∞}E_{n}).

By the naturality of ψ, the map ψ_{E} : i_{♯}E → i_{∗}E is the directed colimit of the maps

ψ_{Σ^{−n}Σ^{∞}E_{n}} : i_{♯}(Σ^{−n}Σ^{∞}E_{n}) → i_{∗}(Σ^{−n}Σ^{∞}E_{n}).

Also, recall that directed colimits of k-spectra preserve 𝔸^{1}-weak equivalences, since they coincide with homotopy colimits of k-spectra ([11, 6]). Thus, if each ψ_{Σ^{−n}Σ^{∞}E_{n}} is an 𝔸^{1}-weak equivalence, then so is ψ_{E}.

The heart of the proof of Theorem 4.1 is the following lemma. Let V be a representation of G = Gal(E/k) as above, and let H = Gal(E/L). We will write

U : G−spaces → H−spaces

for the forgetful functor.

Lemma 4.12. The composition map in the category of based L-spaces

\begin{aligned}i^{∗}F_{E/k}(S^{V}) →^{i^{∗}(F_{E/k}(t))} i^{∗}(Spec(L)_{+} ∧ F_{E/k}(S^{V})) & ≅ i^{∗}(i_{♯}i^{∗}F_{E/k}(S^{V})) \\ & →^{\overline{ψ}} i^{∗}F_{G}(S^{V}) \end{aligned} (4.13)

is 𝔸^{1}-homotopic to the identity.

Proof. Since in the category of based G-spaces, G_{+} ∧_{H} S^{V} = G/H_{+} ∧ S^{V}, we have a map u : G/H_{+} ∧ S^{V} → S^{V} induced by the map G_{+} → H_{+}, which maps G \ H to the disjoint basepoint in H_{+}. By Lemma II.5.9 of [9], the following composition in the category of based G-spaces

S^{V} →^{t} G/H_{+} ∧ S^{V} →^{u} S^{V}  (4.14)

is H-homotopic to the identity. We will show that (4.13) is F_{E/k} ·f^{∗} of (4.14). Since F_{E/k}(G/H_{+}) = Spec(L)_{+}, and F_{E/k} preserves smash products, we get that

i^{∗}F_{E/k}(G/H_{+} ∧ S^{V}) = i^{∗}(Spec(L)_{+} ∧ F_{E/k}(S^{V})).

To see that the maps are correct, recall that there is a natural equivalence of categories between H-equivariant spaces and G-equivariant spaces over G/H. If we write f : G/H → ∗ for the collapse map, then the forgetful functor U corresponds to f^{∗} = G/H × −. By Lemma 3.4, We have that

F_{E/L} · U ≅ i^{∗} · F_{E/k} ≅ F_{E/k} · f^{∗}.

Here, F_{E/k} is thought of as a functor from the comma category of G-spaces over G/H to the comma category of k-spaces over F_{E/k}(G/H) = Spec(L), i. e. L-spaces. Hence, checking the definitions of \overline{ψ} and u, we get that (4.13) is indeed F_{E/k}f^{∗} applied to the sequence (4.14).

But F_{E/k} is a simplicial functor, so it takes a homotopy in based G-spaces to an 𝔸^{1}-homotopy in Spc(k)_{•}. Also, i^{∗}(X ×_{Spec(k)} 𝔸_{k}^{1}) = i^{∗}X ×_{Spec(L)} 𝔸_{L}^{1} for any k-space X, so it takes an 𝔸^{1}-homotopy of k-spaces to an 𝔸^{1}-homotopy of L-spaces.

Lemma 4.15. 1. For Y a based L-space, the diagram

\xymatrix{Σ^{∞}(i^{∗}i_{♯}Y) \ar[d]_{Σ^{∞}\overline{ψ}_{Y}} \ar[r]^{≅} & i_{∗}i_{♯}Σ^{∞}Y \ar[d]^{_{\overline{ψ}_{Σ^{∞}Y}}} \\ Σ^{∞}Y \ar[r]_{=} & Σ^{∞}Y}

commutes. Here, the left vertical map \overline{ψ}Y is thought of as in the category of based L-spaces, and the right vertical map \overline{ψ}_{Σ^{∞}Y} is in the category of L-spectra.

2. For D a k-spectrum and Y a based L-space, the diagram

\xymatrix{i^{∗}(i_{♯}(i^{∗}D ∧ Y)) \ar[d]_{_{\overline{ψ}_{i^{∗}D∧Y}}} \ar[r]_{≅}^{ζ} & i^{∗}(D ∧ (i_{♯}Y)) \ar[r]^{≅} & i^{∗}D ∧ (i^{∗}i_{♯}E) \ar[d]_{Id∧\overline{ψ}_{Y}} \\ i^{∗}D ∧ Y \ar[rr]_{=} &  & i^{∗}D ∧ Y}

commutes. Here, ζ is the isomorphism from Proposition 2.8.

Proof. The first part follows from the fact that the map \overline{ψ} on spectra is defined spacewise. The second part follows from the analogous statement for based spaces, and stabilizing with respect to D. Here, we use the fact that the functors i^{∗} and i_{♯} commute with the spectrification functor L, so for the L-spectrum i^{∗}D and the L-space Y,

i^{∗}i_{♯}(i^{∗D} ∧ E) ≅ L({i^{∗}i_{♯}(i^{∗}D_{n} ∧ Y)})

naturally, where {i^{∗}i_{♯}(i^{∗}D_{n} ∧ Y)} is the L-prespectrum obtained by applying i^{∗}i_{♯} to the L-prespectrum {D_{n} ∧ Y } spacewise.

Proof of Theorem 4.1. We will show that ω and ψ are inverse 𝔸^{1}-weak equivalences on all shift desuspensions of suspension spectra. For this, we use arguments similar to that of [9]. We first show that ψ · ω is homotopic to the identity on i_{∗}E, for any L-spectrum E. Let ϵ : i^{∗}i_{∗}E → E denote the counit of the adjunction (i^{∗}, i_{∗}), then by definition, ϵ · (i^{∗}ψ) = \overline{ψ}. Thus, it suffices to show that \overline{ψ} · i^{∗}ω is naturally homotopic to ϵ : i^{∗}i_{∗}E → E. Consider the following diagram of L-spectra.

\xymatrix{i^{∗}i_{∗}E \ar[r]^{i^{∗}\overline{t}} & i^{∗}i_{♯}i^{∗}i_{∗}E \ar[d]_{\overline{ψ}} \ar[r]^{i^{∗}i_{♯}ϵ} & i^{∗}i_{♯}E \ar[d]^{\overline{ψ}} \\  & i^{∗}i_{∗}E \ar[r]_{ϵ} & E.}

The top row of this diagram is just i^{∗}ω. By the naturality of ϵ and \overline{ψ}, the square commutes. Thus, it suffices to show that the composition

i^{∗}i_{∗}E ⟶^{i^{∗}\overline{t}} i^{∗}i_{♯}i^{∗}i_{∗}E ⟶^{\overline{ψ}} i^{∗}i_{∗}E

passes to the identity in the stable homotopy category SH(L).

We will show that the composition

i^{∗}D →^{i^{∗}\overline{t}} i^{∗}i_{♯}i^{∗}D →^{\overline{ψ}} i^{∗}D  (4.16)

is the identity in SH(L) for any k-spectrum D, and apply it to D = i_{∗}E. By part 1 of Lemma 4.15, Lemma 4.12 also holds for the suspension spectrum of F (S^{V}), by stabilizing the homotopy to Σ^{∞}F (S^{V}). We have the following diagram of L-spectra.

\xymatrix{i^{∗}D \ar[dd]_{i^{∗}\overline{t}} \ar[r]^{≃}_{c} & i^{∗}F (S^{V})^{−1} ∧ i^{∗}D ∧ i^{∗}F (S^{V}) \ar[d]_{i^{∗}F (t)} \ar[dr] \\  & \begin{matrix}i^{∗}F (S^{V})^{−1} ∧ i^{∗}D \\ ∧i^{∗}(Spec(L)_{+} ∧ F (S^{V})) \end{matrix} \ar[d]_{e}^{≃} \ar[r]_{Id∧\overline{ψ}} & \begin{matrix}i^{∗}F (S^{V})^{−1} ∧ i^{∗}D \\ ∧i^{∗}F (S^{V}) \end{matrix} \ar[d]^{e}_{≃} \\ i^{∗}i_{♯}i^{∗}D \ar[r]_{≅} & i^{∗}Spec(L)_{+} ∧ i^{∗}D \ar[r]_{\overline{ψ}} & i^{∗}D.}

The left square of this diagram commutes up to homotopy by the definition of \overline{t}, and keeping track of the transpositions. The lower right square of this diagram is

\xymatrix{i^{∗}F (S^{V})^{−1} ∧ i^{∗}D ∧ i^{∗}(i_{♯}i^{∗}F (S^{V})) \ar[d]_{≅} \ar[r]^{Id∧\overline{ψ}_{i^{∗}F (S^{V} )}} & i^{∗}F (S^{V})^{−1} ∧ i^{∗}D ∧ i^{∗}F (S^{V}) \ar[d]^{=} \\ \begin{matrix}i^{∗}i_{♯}(i^{∗}F (S^{V})^{−1} ∧ i^{∗}D \\ ∧i^{∗}F (S^{V})) \end{matrix} \ar[d]_{i^{∗}i_{♯}e} \ar[r]^{_{\overline{ψ}_{i^{∗}F (S^{V} )^{−1}∧i^{∗}D∧i^{∗}F (S^{V} )}}} & \begin{matrix}i^{∗}F (S^{V})^{−1} ∧ i^{∗}D \\ ∧i^{∗}F (S^{V}) \end{matrix} \ar[d]^{e} \\ i^{∗}i_{♯}(i^{∗}D) \ar[r]_{_{\overline{ψ}_{i^{∗}D}}} & i^{∗}D.}

The upper part of this commutes by part 2 of Lemma 4.15, and lower part commutes by the naturality of \overline{ψ}. Hence, the composition (4.16) is the composition

i^{∗}D →^{≃} i^{∗}F (S^{V})^{−1} ∧ i^{∗}D ∧ i^{∗}F (S^{V}) → i^{∗}F (S^{V})^{−1} ∧ i^{∗}D ∧ i^{∗}F (S^{V}) →^{≃} i^{∗}D.

The first and third maps are the inverse homotopy equivalences c and e, respectively. The middle map is homotopic to the identity by Lemma 4.12. Thus, (4.16) is homotopic to i^{∗}e · i^{∗}c, which passes to the identity in SH(L), since by the remark at the end of Section 2, i^{∗} preserves 𝔸^{1}-weak equivalences.

We still have to show that ω · ψ is the identity on i_{♯}E when we pass to the stable homotopy category over k, if E ∈ Spectra(L) is the shift desuspension of a suspension spectrum. Let η : E → i^{∗}i_{♯}E be the unit of the adjunction (i_{♯}, i^{∗}). Then it suffices to show that i^{∗}ω ·i^{∗}ψ ·η is homotopic to η over L. Recall the isomorphism ζ given in Proposition 2.8. For an L-spectrum E, we consider the following diagram of k-spectra.

\xymatrix{i_{♯}E \ar[r]^{ψ} \ar[d]_{\overline{t}} & i_{∗}E \ar[dr]_{ω} \ar[d]_{\overline{t}} \\ i_{♯}i^{∗}i_{♯}E \ar[r]_{i_{♯}i^{∗}ψ} & i_{♯}i^{∗}i_{∗}E \ar[r]_{i_{♯}ϵ} & i_{♯}E.}  (4.17)

This diagram commutes by the naturality of \overline{t}. By the fact that \overline{ψ} = ϵ · (i^{∗}ψ), we get that

ω · ψ = i_{♯}(\overline{ψ}) · \overline{t}.

We consider two more diagrams. First, let Y be a based L-space. We have the following diagram of based L-spaces.

\xymatrix{i^{∗}F (S^{V}) ∧ Y \ar[d]_{Id∧η} \ar[r]^{Id∧η} & i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}Y \ar[r]^{i^{∗}F (t)∧Id} \ar[d]_{=} & i^{∗}i_{♯}i^{∗}F (S^{V}) ∧ i^{∗}i^{♯}Y \ar[d]_{≅} \ar[dl]^{\overline{ψ}∧Id} \\ i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}Y \ar[r]^{=} & i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}Y \ar[d]_{≅} & i^{∗}(i_{♯}i^{∗}F (S^{V}) ∧ i_{♯}Y) \ar[d]^{i^{∗}ζ^{−1}}_{≅} \\ i^{∗}(F (S^{V}) ∧ i_{♯}Y) \ar[u]^{≅} & i^{∗}(F (S^{V}) ∧ i_{♯}Y) \ar[d]_{i^{∗}ζ^{−1}}^{≅} & i^{∗}i_{♯}(i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}Y) \ar[d]^{i^{∗}i_{♯}(Id∧\overline{ψ})} \ar[ul]_{\overline{ψ}} \\ i^{∗}i_{♯}(Y ∧ i^{∗}F (S^{V})) \ar[u]^{≅}_{i^{∗}ζ} & i^{∗}i_{♯}(i^{∗}F (S^{V}) ∧ Y) \ar[l]_{≅} & i^{∗}i_{♯}(i^{∗}F (S^{V}) ∧ Y). \ar[l]_{=}} (4.18)

The two squares on the left of the diagram commute. Also, the upper right triangle commutes up to homotopy by Lemma 4.12. The middle right triangle commutes by part 2 of Lemma 4.15. Also, the lower right square commutes when we restrict to the image of the composition around the top and right side of the diagram, starting with the upper left corner i^{∗}F (S^{V}) ∧ Y. Thus, the large square commutes up to homotopy in the category of based L-spaces.

Let E be an L-spectrum. We also consider the following diagram of k-spectra.

\xymatrix{F (S^{V}) ∧ i_{♯}E \ar[r]^{Id∧\overline{t}} \ar[d]_{=} & F (S^{V}) ∧ i_{♯}i^{∗}(i_{♯}E) \ar[r]^{Id∧i_{♯}\overline{ψ}} & F (S^{V}) ∧ i_{♯}E \\ F (S^{V}) ∧ i_{♯}E \ar[d]_{Id∧F (t)} & i_{♯}(i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}E) \ar[u]^{ζ} \ar[r]_{i_{♯}(Id∧\overline{ψ})} & i_{♯}(i^{∗}F (S^{V}) ∧ E) \ar[u]^{ζ} \\ i_{♯}i^{∗}F (S^{V}) ∧ i_{♯}E \ar[r]_{ζ^{−1}} & i_{♯}(i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}E) \ar[u]^{=} & .}  (4.19)

The lower left corner of the diagram i_{♯}i^{∗}F (S^{V}) ∧ i_{♯}E is Spec(L)_{+} ∧ F (S^{V}) ∧ i_{♯}E, and the middle top term F (S^{V}) ∧ i_{♯}i^{∗}(i_{♯}E) is F (S^{V}) ∧ Spec(L)_{+} ∧ i_{♯}E. By writing out the definition of \overline{t} on i_{♯}E and keeping track of the transpositions, we get that the left square of the diagram commutes. The right upper square of the diagram commutes by the naturality of ζ and \overline{ψ}.

By part 1 of Lemma 4.15, if E = Σ^{−n}Σ^{∞}Y for a based L-space Y, then we can replace the composition around the right of (4.18) by i^{∗} applied to the composition around the bottom of (4.19), which is in turn the top row of (4.19). Hence, we have that

Id ∧ η : i^{∗}F (S^{V}) ∧ E → i^{∗}F (S^{V}) ∧ i^{∗}i_{♯}E

is 𝔸^{1}-homotopic to

Id ∧ (i^{∗}i_{♯}\overline{ψ} · i^{∗}\overline{t} · η)

in the category of L-spectra. Also, since i^{∗} preserves smash products, and i^{∗}S_{k}^{0} = S_{L}^{0}, we get that i^{∗}F (S^{V}) is invertible in the stable homotopy category over L, with inverse i^{∗}(F (S^{V})^{−1}). So smashing the homotopy with i^{∗}(F (S^{V})^{−1}) gives that η is the same as

i^{∗}(i_{♯}\overline{ψ}) · i^{∗}(\overline{t}) · η

in the stable homotopy category SH(L). But now by (4.17), we can replace i^{∗}(i_{♯}\overline{ψ})· i^{∗}(\overline{t}) by i^{∗}ω · i^{∗}(ψ). This shows that ω · ψ pass to the identity in SH(k), so ω and ψ are inverse 𝔸^{1}-weak equivalences of k-spectra.

5. The Join and Smash Powers to an Étale Scheme

In this section, we give the exact definition for X^{∗Spec(L)} for X ∈ Spc(k), and X^{∧Spec(L)} for X ∈ Spc(k)_{•}. In fact, for any scheme T over k, such that T → Spec(k) is étale, we will define X^{∗T} for unbased X and X^{∧T} for based X. For X^{∗T}, we begin by recalling the join power X^{∗n} for an integer n ⩾ 1. This can be described as follows. Given X^{n}, for each pair of subsets of \underline{n} = \{1, …, n\} of orders i and j, i > j, we take all possible projection maps X^{i} → X^{j}. We have a partially ordered set of such projections, and the homotopy pushout of this diagram is X^{∗n}. For motivation, recall that for a small category D and a functor F from D to the category of topological spaces, the homotopy colimit of F with respect to D can be described as the classifying space of the following topological category C(F). The objects space of C(F) is

∐_{D∈D} F (D)

and for x, y ∈ Obj(C(F)), x ∈ F (D) and y ∈ F (D′), the maps x → y in C(F) corresponds to maps f : D → D′ in D, such that F (f) : x ↦ y. This is topologized as

∐_{D′∈Obj(D)} ∐_{f:D→D′} F (D).

For a space X and n ⩾ 1, we can define the topological category C_{n}(X) as follows. Let A be the set of all nonempty subsets of \underline{n}, and for S ⊆ \underline{n}, denote by X^{S} the space of maps from S into X. Then

Obj(C_{n}(X)) = ∐_{S∈A} X^{S}  (5.1)

where the disjoint union runs over all pairs S, S′ in A with S′ ⊆ S. For a ∈ X^{S} and b ∈ X^{S′}, we have a map a → b in C_{n}(X) if and only if S′ ⊆ S, and a projects to b via the projection X^{S} → X^{S′}. Hence, the morphisms in C_{n}(X) are parametrized by pairs S′ ⊆ S, and points in X^{S}. So

Mor(C_{n}(X)) = ∐_{S′⊆S∈A} X^{S}.

Then X^{∗n} is the classifying space of C_{n}(X).

For an unbased k-space X and an étale scheme T over Spec(k), we define X^{∗Spec(L)} via the interpretation of the homotopy colimit as the classifying space of a category. For motivation, we first consider the case of equivariant topological spaces. Let G be a finite group, and let T be a G-set. Then for a G-equivariant topological space X, we can define X^{∗T} in the above manner. Define

A = {S ⊆ T | S nonempty}.

Then A has a natural G-action by the multiplication of G from the right. For each S ∈ A, let H_{S} be the stabilizer subgroup of S. Then S is an H_{S}-equivariant set. Define the H_{S}-equivariant space X^{S} = \underline{Hom}_{G}(S, X) to be the space of nonequivariant maps from S to X, with an H_{S}-action by conjugation. For each α ∈ G, and S ∈ A, we have H_{α(S)} = αH_{S}α^{−1}, so the stabilizer subgroups are isomorphic for elements in the same G-orbit of A. Also, note that if a ∈ X^{S}, then α · a · α^{−1} ∈ X^{α(S)}. We have a natural isomorphism of G-equivariant spaces

α_{∗} : G ×_{H_{S}} X^{S} →^{≅} G ×_{H_{α(S)}} X^{α(S)}  (5.2)

For g ∈ G and a ∈ X^{S}, α_{∗} takes (g, a) ∈ G ×_{H_{S}} (X^{S}) to (gα^{−1}, α · a · α^{−1}) in G ×_{H_{α(S)}} X^{α(S)}. It is straightforward to check that this is indeed a G-equivariant isomorphism. If α(S) = β(S), then α and β differ by h ∈ H_{S}. But h_{∗} = Id for h ∈ H_{S}. Hence, the map α_{∗} of (5.2) depends only on the sets S, α(S).

We define the following category C_{T} (X), which is enriched over the category of G-equivariant topological spaces. If {S_{1}, …, S_{n}} is a G-orbit in A, we will replace ∐_{i=1}^{n} X^{S_{i}} in (5.1) by G ×_{H(S_{i})} X^{S_{i}} for any choice of i between 1 and n, to get the G-action. So the object space of C_{T} (X) is

Obj(C_{T} (X)) = ∐_{S_{j}} G ×_{H_{S_{j}}} X^{S_{j}}

where the S_{j}’s range over a set of representatives of the orbits in A. This is a Gequivariant space. For g(S_{j}) in the orbit of S_{j} in A, and a ∈ X^{g(S_{j})}, a corresponds to (g^{−1}, g · a · g^{−1}) in G ×_{H_{S_{j}}} X^{S_{j}}. By (5.2), Obj(C_{T} (X)) is independent of the choices of orbit representatives.

For objects (g, a) ∈ G ×_{H_{S_{j}}} X^{S_{j}} and (g′, b) ∈ G ×_{H_{Sr}} X^{S_{r}}, which correspond to g^{−1} · a · g ∈ X^{g^{−1}(S_{j})} and (g′)^{−1} · b · g′ ∈ X^{(g′)^{−1}(S_{r})}, there is a morphism in C_{T} (X)

(g, a) → (g′, b)

if and only if (g′)^{−1}(S_{r}) ⊆ g^{−1}(S_{j}), and (g′)^{−1} · b · g′ ∈ X^{(g′)^{−1}(S_{r})} is the projection of g^{−1} · a · g ∈ X^{g^{−1}(S_{j})}. In other words, each morphism of C_{T} (X) corresponds uniquely to a pair of sets S′, S ∈ A, S′ ⊆ S, and a point in X^{S}. Let

A_{2} = {(S′, S) | S, S′ ∈ A, S′ ⊆ S}.

Then A_{2} has a G-action by multiplication from the right. For each (S′, S) ∈ A_{2}, let H_{(S′,S)} be the stabilizer subgroup of (S′, S). Let {(S′_{i}, S_{i})} be a set of representatives or the orbits in A_{2}. Then the G-space of morphisms in C_{G}(T) is

Mor(C_{T} (X)) = ∐_{(S′_{i},S_{i})} G ×_{H_{(S′_{i},S_{i})}} X^{S_{i}}.

For g ∈ G and a ∈ X^{S_{i}}, (g, a) ∈ G×_{H_{i}} X^{S} corresponds to the morphism that comes from the projection of g^{−1} · a · g ∈ X^{g^{−1}(S_{i})} to X^{g^{−1}(S′_{i})}. There is a natural G-space structure on Mor(C_{T} (X)). For S ∈ A, (S, S) ∈ A_{2}, H_{S} = H_{(S,S)}, so the

Identity : Obj(C_{T} (X)) → Mor(C_{T} (X))

is given by a disjoint union of identity maps

G ×_{H_{S}} X^{S} → G ×_{H_{(S,S)}} X^{S}

composed with isomorphisms of the form (5.2) to make (S, S) one of the chosen representatives of an orbit in A_{2}. For each pair (S′, S) ∈ A_{2}, H_{(S′,S)} ⊆ H_{S}, so define

Source : Mor(C_{T} (X)) → Obj(C_{T} (X))

to be a disjoint union of quotient maps

G ×_{H_{(S′,S)}} X^{S} → G ×_{H_{S}} X^{S}

composed with appropriate isomorphisms of the form (5.2). Also, the inclusion S′ → S induces an H_{(S′,S)}-equivariant map X^{S} → X^{S′}. so define

T arget : Mor(C_{T} (X)) → Obj(C_{T} (X))

to be a disjoint union of compositions

G ×_{H_{(S′,S)}} X^{S} → G ×_{H_{(S′,S)}} X^{S′} → G ×_{H_{S′}} X^{S′}

where the first map is induced by X^{S} → X^{S′}, and the second map is the quotient map. Finally, let A_{3} = {S′′ ⊆ S′ ⊆ S} with a G-action via multiplication from the right, and let H_{(S′′,S′,S)} be the stabilizer subgroup of (S′′, S′, S) in A_{3}. Then Mor(C_{T} (X)) ×_{Obj(C_{T} (X))} Mor(C_{T} (X)) is a disjoint union of G-spaces of the form

(G ×_{H_{(S′,S)}} X^{S}) ×_{G×_{H_{S′}} X^{S′}} (G ×_{H_{(S′′,S^{′)}}} X^{S′}) ≅ G ×_{H_{(S′′,S′,S)}} X^{S}

where (S′′, S′, S) ranges over a set of representatives of orbits in A_{3}. So

Composition : Mor(C_{T} (X)) ×_{Obj(C_{T} (X))} Mor(C_{T} (X)) → Mor(C_{T} (X))

is a disjoint union of quotient maps of the form

G ×_{H_{(S′′,S′,S)}} X^{S} → G ×_{H_{(S′,S)}} X^{S}.

By definition, the identity, source, target, and composition of morphisms are Gequivariant. Thus, C_{T} (X) is a category enriched over G-topological spaces. The classifying space of C_{T} (X) is X^{∗T}, which has a natural structure as a G-space.

The construction in the algebraic case is similar. Let X be a k-space, and T is a scheme over k, with étale map T → Spec(k). Then

T = ∐_{r} Spec(F_{r})

is a disjoint union of the spectra of finite extension fields F_{r} over k. Let L be a Galois extension that contains F_{r} for every r, such as the the algebraic closure of k, and let G = Gal(L/k). In particular, G may be a profinite group. For each F_{r}, let H(F_{r}) = Gal(L/F_{r}), so F_{r} = L^{H(F_{r})}. Thus, let the G-set T_{G} be given by T_{G} = ∐_{r} G/H(F_{r}), then T = F_{L/k}(T_{G}). Again, let A be the collection of nonempty subsets of T_{G}, with a G-action by the multiplication of G from the right. For S ∈ A, let H_{S} be the stabilizer subgroup of S, and let E_{S} = L^{H_{S}}, so Spec(E_{S}) = F_{L/k}(G/H_{S}). Also, let f_{S} : Spec(E_{S}) → Spec(k) be the map corresponding to the inclusion of fields. Then F_{L/E_{S}}(S) ∈ Spc(E_{S}), and the analogue of the H_{S}-equivariant topological space X^{S} is the E_{S}-space

X^{F_{L/E_{S}} (S)} = \underline{Hom}_{Spec(E_{S})}(F_{L/E_{S}}(S), (f_{S})^{∗}(X)).  (5.3)

In particular, if S = T_{G}, then E_{S} = k and F_{L/E_{S}}(S) = F_{L/k}(T_{G}) = T as a k-space, so we have

X^{T_{G}} = \underline{Hom}_{Spec(k)}(T, X)

by Lemma 2.14. For α ∈ G and S ∈ A, H_{α(S)} = αH_{S}α^{−1}, so E_{α(S)} = α(E_{S}). So similarly as for (5.2), we have an isomorphism of k-spaces

α_{∗} : (f_{S})_{♯}X^{F_{L/E_{S}} (S)} →^{≅} (f_{α(S)})_{♯}X^{F_{L/E_{α(S)}} (α(S))}.  (5.4)

For α ∈ H_{S}, the map (5.4) is the identity map, since f_{S} = f_{S} · α : Spec(E_{S}) → Spec(k). Thus, in general the map (5.4) depends only on the sets S and α(S). We define the category C_{T/k}(X), enriched over Spc(k), as follows. Define the k-space of objects to be

Obj(C_{T/k}(X)) = ∐_{S_{j}}(f_{S_{j}})_{♯}X^{S_{j}}

where S_{j} ranges over the representatives of the orbits in A. For (S′, S) ∈ A_{2}, set E_{(S′,S)} = L^{H_{(S′,S)}}, so k ⊆ E_{S} ⊆ E_{(S,S′)}. Let f_{(S′,S)} : Spec(E_{(S′,S)}) → Spec(k), and a_{(S′,S)} : Spec(E_{(S′,S)}) → Spec(E_{S}) be the maps corresponding to the inclusions of fields. Then the analogue of G ×_{H_{(S′,S)}} X^{S} is (f_{(S′,S)})_{♯}a_{(S′,S)}^{∗}X^{F_{L/E_{S}} (S)}. Thus, define the k-space of morphisms in C_{T/k}(X) to be

Mor(C_{T/k}(X)) = ∐_{(S′_{i},S_{i})}(f_{(S′_{i},S_{i})})_{♯}(a_{(S′_{i},S_{i})})^{∗}X^{F_{L/E_{S}} (S)}

where (S_{i}, S′_{i}) ranges over a set of representatives or the orbits in A_{2}.

The identity, source, target, and composition maps of C_{T/k}(X) are defined similarly as in the equivariant case. For S in A, E_{(S,S)} = E_{S}, a : Spec(E_{S}) → Spec(E_{(S,S)}) is the identity, and f_{S} = f_{(S,S)} : Spec(E_{S}) → Spec(k). So

Identity : Obj(C_{T/k}(X)) → Mor(C)_{T/k}(X)

is a disjoint union of identity maps on (f_{S})_{♯}X^{F_{L/E_{S}} (S)}, composed with isomorphisms of the form (5.4) to make (S, S) one of the chosen representatives of an orbit in A_{2}. For the source map, note that for each (S′, S) in A_{2}, f_{(S′,S)} = f_{S} · a_{(S′,S)}. Let c be the counit of the adjunction pair ((a_{(S′,S)})_{♯}, (a_{(S′,S)})^{∗}). So define

Source : Mor(C_{T/k}(X)) → Obj(C_{T/k}(X))

to be a disjoint union of maps

\begin{aligned}(f_{(S′,S)})_{♯}(a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)} & = (f_{S})_{♯}(a_{(S′,S)})_{♯}(a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)} \\ & →^{c} (f_{S})_{♯}X^{F_{L/E_{S}} (S)} \end{aligned}

composed with isomorphisms of the form (5.4) to make (S′, S) an orbit representative in A_{2}. To define the target map, consider the natural map b_{(S′,S)} : Spec(E_{(S′,S)}) → Spec(E_{S′}). So we have a commuatative diagram

\xymatrix{Spec(E_{(S′,S)}) \ar[d]_{a_{(S′,S)}} \ar[r]^{b_{(S,S′)}} \ar[dr]_{f_{(S′,S)}} & E_{S′} \ar[d]_{f_{S′}} \\ Spec(E_{S}) \ar[r]_{f_{S}} & Spec(k).}

Then by Lemma 3.4, the inclusion S′ → S induces a map of E_{(S′,S)}-spaces

(b_{(S′,S)})^{∗}F_{L/E_{S′}} (S′) = F_{L/E_{(S′,S)}}(S′) → F_{L/E_{(S′,S)}}(S) = (a_{(S′,S)})^{∗}F_{L/E_{S}}(S).

This in turn induces a map of E_{(S′,S)}-spaces

(a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)} → (b_{(S′,S)})^{∗}X^{F_{L/E_{S′}} (S′)}.

This is because by taking the adjoint of Proposition 2.8, (b_{(S′,S)})^{∗} and (a_{(S′,S)})^{∗} commute with the internal Hom functor. This gives that

X^{(a_{(S′,S)})^{∗}F_{L/E_{S}} (S)} = \underline{Hom}_{E_{(S′,S)}}((a_{(S′,S)})^{∗}F_{L/E_{S}}(S), (f_{(S′,S)})^{∗}X)
≅ (a_{(S′,S)})^{∗}\underline{Hom}_{E_{S}}(F_{L/E_{S}}(S), f_{S}^{∗}X)
= (a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)}

and similarly for X^{(b_{(S′,S)})^{∗}F_{L/E_{S′}} (S′)}. So

T arget : Mor(C_{T/k}(X)) → Obj(C_{T/k}(X))

is a disjoint union of compositions

(f_{(S′,S)})_{♯}(a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)} → (f_{(S′,S)})_{♯}(b_{(S′,S)})^{∗}X^{F_{L/E_{S′}} (S′)} → (f_{S′})^{∗}X^{F_{L/E_{S′}} (S′)} where the second map is the counit of the adjunction pair ((b_{(S′,S)})_{♯}, (b_{(S′,S)})^{∗}). Finally, for the composition of morphisms in C_{T/k}(X), consider (S′′, S′, S) ∈ A_{3}. Let E_{(S′′,S′,S)} = L^{H_{(S′′,S′,S)}}. We have natural maps

f_{(S′′,S′,S)} : Spec(E_{(S′′,S′,S)}) → Spec(k)

and

c_{(S′′,S′,S)} : Spec(E_{(S′′,S′,S)}) → Spec(E_{(S′,S)}).

Then Mor(C_{T/k}(X)) ×_{Obj(C_{T /k}(X))} Mor(C_{T/k}(X)) is a disjoint union of k-spaces of the form

(f_{(S′,S)})_{♯}(a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)} × (f_{(S′′,S′)})_{♯}(a_{(S′′,S′)})^{∗}X^{L/E_{S′} (S′)}

where the product is over (f_{S′})_{♯}X^{L/E_{S′} (S′)}, to which the two factors map by the target and source maps, respectively. This is isomorphic as a k-space to

(f_{(S′′,S′,S)})_{♯}(c_{(S′′,S′,S)})^{∗}(a_{(S′,S)})^{∗}X^{F_{L/E_{S}} (S)}.

Hence,

Composition : Mor(C_{T/k}(X)) ×_{Obj(C_{T /k}(X))} Mor(C_{T/K}(X)) → Mor(C_{T/k}(X))

is a disjoint union of counit maps

(f_{(S′′,S′,S)})_{♯}c_{(S′′,S′,S)}^{∗}a_{(S′,S)}^{∗}X^{F_{L/E_{S}} (S)} → (f_{(S′,S)})_{♯}a_{(S′,S)}^{∗}X^{F_{L/E_{S}} (S)}

for the adjunction pairs ((c_{(S′′,S′,S)})_{♯}, (c_{(S′′,S′,S)})^{∗}), where (S′′, S′, S) ranges over a set of orbit representatives in A_{3}. Then C_{T/k}(X) is a category enriched over Spc(k). The join power X^{∗T} is defined to be the classifying space of C_{T/k}(X), which has a natural structure as a k-space.

To show that X^{∗T} is independent of the choice of L, it suffices to consider the case where T = Spec(F), for some separable finite extension F of k. Suppose L and L^{prime} are two Galois extensions of k containing F. We can assume without loss of generality that L′ contains L. Let J = Gal(L′/L), G′ = Gal(L′/k) and G = Gal(L/k). So we have a short exact sequence of groups

1 → J → G′ →^{p} G → 1.

Let H = Gal(L/F), and H′ = Gal(L′/F). Then T = Spec(F) = F_{L/k}(G/H) = F_{L′/k}(G′/H′). But H′ = p^{−1}(H), so there is a canonical isomorphism of G′-sets

G′/H′ ≅ G/H

where G/H is thought of as a G′-set fixed by J. So let A_{G} be the collection of nonempty subsets of G/H, and A_{G′} be the collection of nonempty subsets of G′/H′. There is a canonical G′-equivariant bijection between A_{G} and A_{G′}. In particular, for any S ∈ A_{G′}, let H′_{S} ⊆ G′ be the isotropy subgroup of S in A_{G′}, and let H_{S} ⊆ G be the isotropy subgroup of S in A_{G}. Then H′_{S} = p^{−1}(H_{S}). Then

L^{H_{S}} = (L′)^{H′_{S}}

so the definition of E_{S} = L^{H_{S}} is independent of the choice of L. For a k-space X, the objects and morphisms of the category C_{T/k}(X) are build up out of k-spaces of the form X^{F_{L/E_{S}} (S)} for all S ∈ A_{G}. By arguments similar as above,

F_{L/E_{S}}(S) = F_{L′/E_{S}}(S)

as k-spaces, where S is thought of as a subset of G/H on the left hand side, and as a subset of G′/H′ on the right hand side. Hence, by the definition of X^{F_{L/E_{S}} (S)} in 5.3, we see that X^{F_{L/E_{S}} (S)} is independent of the choice of L.

If X is a triangulated G-equivariant space, T_{G} is a G-set, and T = F_{L/k}(T_{G}) is étale over Spec(k). Then from the completely analogous definitions of the categories C_{G}(X) and C_{T/k}(F_{T/k}(X)), we get a weak equivalence of k-spaces

F_{L/k}(X^{∗T_{G}}) = F_{L/k}(X)^{∗T} .

Likewise, for X ∈ Spc(k)_{•} and T → Spec(k) étale, we can also define X^{∧T}, the smash powerof X to T. As before, we have an extension field L of k, with G = Gal(L/k), such that T = F_{L/k}(T_{G}) for some G-set T_{G}. For any S ∈ A, we have f_{S} : Spec(E_{S}) → Spec(k). Then there is a map of unbased E_{S}-spaces

X^{F_{L/E_{S}} (S)} → (f_{S})^{∗}X^{T}

by inclusions of basepoints, induced by the inclusion map S → G, which is H_{S}- equivariant. This corresponds to a map of unbased k-spaces

(f_{S})_{♯}X^{F_{L/E_{S}} (S)} → X^{Spec(L)}.

In particular, for any α ∈ G, then E_{α(S)} = α(E_{S}), and the diagram

\xymatrix{(f_{α(S)})_{♯}X^{F_{L/E_{α(S)}} (α(S))} \ar[rr]^{α_{∗}}_{≅} \ar[dr] &  & (f_{S})_{♯}X^{F_{L/E_{S}} (α(S))} \ar[dl] \\  & X^{T}}

commutes, where α_{∗} is the map of (5.4). Let S_{j} range over a set of orbit representatives of A \ {G}. Then define the X^{∧T} ∈ Spc(k)_{•} to be

X^{∧T} = X^{T} /_{Spec(k)} ⋃_{S_{j}}(f_{S_{j}})_{♯}X^{F_{L/E_{S_{j}}} (S_{j})}.

Here, ⋃ denotes the union inside X^{T}.

It remains to prove Lemma 3.7.

Proof of Lemma 3.7. For any S ⊂ T_{G}, the functor

(−)^{F_{L/E_{S}} (S)} = \underline{Hom}_{Spec(E_{S})}(F_{L/E_{S}}(S), (f_{S})^{∗}−) : Spc(k) → Spc(E_{S})

commutes with fibered products of unbased k-spaces. For X, Y ∈ Spc(k)_{•},

(X ∧ Y)^{∧T}

= (X ∧ Y)^{T} / ⋃_{S_{j}}(f_{S_{j}})_{♯}(X ∧ Y)^{F_{E/S_{j}} (S_{j})}

= (X × Y/X ∪ Y)^{T} / ⋃_{S_{j}}(f_{S_{j}})_{♯}(X ∧ Y)^{F_{E/S_{j}} (S_{j})}.

On the other hand,

X^{∧T} ∧ Y^{∧T}

= (X^{T} / ⋃_{S_{j}}(f_{S_{j}})_{♯}X^{F_{E/S_{j}} (S_{j})}) ∧ (Y^{T} / ⋃_{S_{j}}(f_{S_{j}})_{♯}Y^{F_{E/S_{j}} (S_{j})})

≅ \frac{X^{T} × Y^{T}}{(⋃_{S_{j}}(f_{S_{j}})_{♯}X^{F_{E/S_{j}} (S_{j})}Y^{T}) ∪ (X^{T} × ⋃_{S_{j}}(f_{S_{j}})_{♯}Y^{F_{E/S_{j}} (S_{j})})}.

for all ordered pairs of subsets (S, S′) in T_{G}, (S, S′) ≠ (T_{G}, T_{G}), we have a map

((f_{S})_{♯}X^{F_{L/E_{S}} (S)}) × ((f_{S′})_{♯}Y^{F_{L/E_{S′} (S′)}}) → X^{T} × Y^{T} ≅ (X × Y)^{T} .

Both (X ∧ Y)^{∧T} and X^{∧T} ∧ Y^{∧T} are isomorphic to the quotient

(X × Y)^{T} / ⋃_{S_{i},S′_{j}}((f_{S})_{♯}X^{F_{L/E_{S}} (S)}) × ((f_{S′})_{♯}Y^{F_{L/E_{S′}} (S′)}).

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

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Po Hu pohu@math.uchicago.edu

Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637