Real topological Hochschild homology of schemes

We prove that real topological Hochschild homology THR for schemes with involution satisfies base change and descent for the Z/2-isovariant \'etale topology. As an application, we provide computations for the projective line (with and without involution) and the higher dimensional projective spaces.


Introduction
Hochschild and cyclic homology and their refinements THH and TC have been extensively studied for many decades, both for their own sake and for their deep connections with algebraic K-theory via traces, see e.g. [10]. Standard textbook references include [35] and [17]. More recently, Bhatt-Morrow-Scholze [6] have introduced a filtration on THH and TC that is strongly related to integral p-adic Hodge theory.
Both topological and algebraic K-theory have real or hermitian refinements. This is also true for THH and TC, but the serious study of their real variants THR and TCR has just started, see e.g. [15], [14], and [43]. These theories apply to very different branches in algebra and geometry: commutative rings, group rings, schemes, ring spectra, ... all with or without a non-trivial involution. Concerning the Z/2-equivariant cyclotomic trace, we understand that there is work in progress by Harpaz, Nikolaus, and Shah studying this map in the very general setting of stable Poincaré ∞-categories. (When studying traces, beware of the difference between rings, which is what most algebraic K-theorists and algebraic geometries look at, and ring spectra, which are the input of THH and TC.) The current article contributes to a better understanding of THR in algebraic geometry, although some of our results also apply to other settings. Our first main result is the base change result in Theorem 3.2.3 for the isovariantétale topology. This then is one of the main ingredients in the proof of the following isovariant etale descent theorem for THR. satisfies isovariantétale descent, where Aff Z/2 denotes the category of affine schemes with involutions, and Sp Z/2 denotes the ∞-category of Z/2-spectra.
In the non-equivariant case, similar results have been established for THH and TC by Weibel-Geller [54] and Geisser-Hesselholt [19].
This descent theorem implies in particular that THR satisfies the equivariant Zariski-Mayer-Vietoris property for affine schemes with involutions. This can be used to extend THR to non-affine schemes with involutions, see Definition 3.4.6, in a way that is compatible with existing definitions of THH, see Proposition 3.4.7.
Using this Zariski-Mayer-Vietoris theorem and various explicit computations of THR of products of monoid rings and maps between them, we are able to compute THR(X) for X = P 1 , P σ and more generally P n with trivial involution, see Theorems 5.1.2, 5.1.4 and 5.2.6. Here P σ denotes the projective line with the involution switching the homogeneous coordinates. We recall that Blumberg and Mandell [9] compute THH(P n ). For P n with the trivial involution, we obtain the following: Theorem 5.2.6. For any separated scheme with involution X and integer n ≥ 0, there is an equivalence of Z/2-spectra if n is even, THR(X) ⊕ ⌊n/2⌋ j=1 i * THH(X) ⊕ Σ n(σ−1) THR(X) if n is odd. For a Z/2-spectra E, i * E := E ⊕ E with the obvious involution, see (A.15). We refer to Remarks 2.0.1, 5.1.1, and 5.2.7 for a comparison with hermitian and real algebraic K-theory.
Recall that unlike algebraic and hermitian K-theory, THH and THR are not A 1invariant even on regular schemes. On the other hand, THH and TC do extend to log schemes, and using descent, trivial P 1 -bundle formula, and computations on the log schemes (P n , P n−1 ) can be shown to be representable in the log-variant of the P 1stable motivic Morel-Voevodsky homotopy category. This is done in the very recent joint work [7] of the second author with Federico Binda and Paul Arne Østvaer. We refer to [8] for an overview of their work. The results of this article are expected to provide most of the necessary ingredients for showing stable representability of THR or at least its fixed points THO in the corresponding equivariant log-homotopy categories. Furthermore, the work of Quigley-Shah [43] allows us to extend many results of this article including Theorems 3.4.3 and 5.2.6 to TCR. The second author hopes to carry out motivic representability of THR and TCR in forthcoming work.
We conclude with a short overview of the different sections. Section 2 discusses some generalities about commutative ring spectra with involutions and their THR. This includes the study of several functors between Z/2-equivariant and nonequivariant categories, including the norm and equivariant notion of flatness. The necessary background on G-stable equivariant homotopy theory for finite groups G, both ∞and model categorical, is provided in Appendix A. Section 3 reviews and extends several definitions for Grothendieck topologies on schemes with involutions, notably Thomason's isovariantétale topology. Base change and descent for THR are established in subsections 3.2 and 3.4. Section 4 recollects and extends material on real and dihedral nerves, which is crucial when computing THR of spherical monoid rings and of monoid rings over Eilenberg MacLane spectra, and where the monoids may have involutions. These monoid ring computations together with isovariant Zariski descent are then used in the computations for projective spaces in section 5.
Acknowledgements. This research was conducted in the framework of the DFGfunded research training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology.

Real topological Hochschild homology of rings with involutions
Remark 2.0.1. THR is to real algebraic K-theory KR what THH is to algebraic K-theory. Hence we recall some recent results on real algebraic K-theory KR. This is a Z/2-equivariant motivic spectrum constructed in [29] and [11]. We recover hermitian K-theory when restricting KR to schemes with trivial Z/2-action. (This is one incarnation of the philosophy "the fixed points of KR are hermitian K-theory" in the world of presheaves on Z/2-schemes). Forgetting the action, we recover Voevodsky's algebraic K-theory spectrum KGL. As observed in [55, section 7.3], see also [11], Schlichting's techniques generalize to show that KR is representable in SH Z/2 (k). In particular, KR satisfies equivariant Nisnevich descent. Here and below, following [11] and [29], we consider KR as a motivic spectrum with respect to the circle T ρ ≃ P 1 ∧ P σ , where ρ is the regular representation of Z/2, and P σ denotes P 1 with involution switching the homogeneous coordinates. It might be more consistent with the notations for THH to denote the (motivic) hermitian K-theory spectrum on schemes with involution by KO, and to reserve the notation KR for a (motivic) spectrum with involution whose fixed points are KO. For further results on KR and projective spaces we refer to Remarks 5.1.1 and 5.2.7 below.
Throughout this section, we fix morphisms of finite groupoids where BG denotes the finite groupoid consisting of a single set * with Hom BG ( * , * ) := G. According to section A.2, we often use the alternative notation N Z/2 = i ⊗ , Φ Z/2 = p ⊗ , ι = p * , and (−) G = p * instead of the notation in section A.1. In particular, all functors in the sequel are ∞-functors, which admit lifts to Quillen functors between model categories. We then have adjoint pairs (2.2) i * : Sp Z/2 ⇄ Sp : i * and ι : Sp ⇄ Sp Z/2 : (−) G and functors N Z/2 : Sp → Sp Z/2 and Φ Z/2 : Sp Z/2 → Sp.
We refer sections A.1 and A.2 for further properties of all these functors. We now define real topological Hochschild homology for commutative ring spectra with involution.
Since N Z/2 is left adjoint to i * , we have the counit map A ∧Z/2 → A. We use this map to define THR(A) : which is called the real topological Hochschild homology of A. The first ∧ in the formulation of THR is the pushout in NAlg Z/2 . Under a certain flatness condition, this is equivalent to the Bökstedt model of the real topological Hochschild homology, see [14,Theorem,p. 65]. The map to the second smash factor in A ∧ A ∧Z/2 A gives a canonical map In analogy with hermitian K-theory KO and real algebraic K-theory KR, we define An object X of Fun(B(Z/2), C) is called an object of C with involution, where B(Z/2) is the finite groupoid associated with Z/2. Explicitly, X is an object of C equipped with an automorphism w : X → X such that w • w = id.
In particular, we have the notions of commutative rings with involutions, commutative monoids with involutions, etc.
We do not discuss definitions of HR(A) refining HH(A) and comparison results between THR(A) and HR(A), similarly to e.g. [41,Proposition IV.4.2], but see Remark 4.2.3 below. The adjoint functors π 0 and H, which are studied in the appendix, map some of the adjunctions before the Proposition to the adjunctions, see e.g. Definition 2.3.2.
where C(G/e) is an abelian group with an involution w, C(G/G) is an abelian group with the trivial involution, res and tran are homomorphisms of abelian groups with involutions, and the equality res • tran = id + w is satisfied (i.e. the double coset formula holds). A morphism of Mackey functors C → D is a diagram of abelian groups with involutions where tran maps x ∈ M to x + w(x), and res is the inclusion. In this way, we obtain a fully faithful functor from the category of abelian groups with involutions to the category of Mackey functors Mack Z/2 . We often regard an abelian group with involution as a Mackey functor if no confusion seems likely to arise. the pair (f, g) commutes with res. We have Since res for L is injective, we deduce that the pair (f, g) commutes with tran. This constructs an inverse of (2.6).
2.3. Equivariant Eilenberg-MacLane spectra. In this subsection, we explain basic properties of Equivariant Eilenberg-MacLane spectra. We also explain how to define THR of commutative rings.
be the functors induced by (2.1). Let (−) Z/2 denote the right adjoint of ι if exists. Let us give some examples. For a commutative ring A, ιA is the commutative ring A with the trivial involution. For an A-module M , ιM is the ιA-module M with the trivial involution. By abuse of notation we sometimes denote the constant Mackey functors by ιA and ιM as well.
For a commutative ring B with involution, i * B is the commutative ring obtained by forgetting the involution, and B Z/2 is the Z/2-fixed point ring. For an B-module L, i * L is the i * B-module obtained by forgetting the involution. For a commutative ring A with involution, we can regard HA as an object of NAlg Z/2 as explained in [48,Example 11.12].
Note that for a given commutative ring B the commutative ring spectra HιB and ιHB are quite different. E.g., applying π 0 to the first one yields the constant Mackey functor associated with B whereas for the second one a tensor product with the Burnside ring Mackey functor of Z/2 appears.
is an isomorphism for X = Z/2, e and integer n ∈ Z. If n = 0, then both sides are vanishing. Assume n = 0. More concretely, it remains to show that the composite of the induced maps is an isomorphism.
If X = Z/2, then (2.10) can be written as the homomorphisms given by (x, y) → (x, 0, y, 0) and (x, y, z, w) → (x + y, z + w). The composite is an isomorphism. If X = e, then (2.10) can be written as the homomorphisms given by x → (x, 0) and (x, y) → x + y. The composite is also an isomorphism. If M is a commutative ring, then (2.9) is an equivalence in NAlg Z/2 . Hence we obtain (2.8) Observe that there is an isomorphism N Z/2 A A ∼ = ιA of commutative rings with involutions. We prefer to use the notation ιA instead of N  29), we obtain a map in CAlg G N Z/2 HA → Hπ 0 (N Z/2 HA) and a map of N Z/2 HA-modules N Z/2 HM → Hπ 0 (N Z/2 HM ).
Hence to construct (2.13), it suffices to construct a map of π 0 (N Z/2 HA)-modules By Lemma 2.2.3 and Definition 2.4.1, this is equivalent to constructing a morphism of π 0 (i * N Z/2 HA)-modules by Proposition A.2.7 (6). The canonical assignment x ⊗ y → x ⊗ y finishes the construction.
The class of M -modules such that (2.12) is an equivalence is closed under filtered colimits. By Lazard's theorem, every flat A-module is a filtered colimit of finitely generated free A-modules. Hence to show that (2.12) is an equivalence if M is flat, we may assume M = A n . In this case, there is an equivalence where V n is the set [n] × [n] with the Z/2-action given by (a, b) → (b, a). Hence we obtain equivalences Combining this with (2.12) we obtain a map (2.15) f : To show that f is an equivalence, it suffices to show that π 0 (f ) is an equivalence. Since V n × ιA and N Z/2 A A n are rings with involutions, by Lemma 2.2.3 it suffices to show that π 0 (f ) is an equivalence of modules over rings with involution. Hence to show that (2.12) is an equivalence, it suffices to show that it is an equivalence after applying π 0 , i.e., the induced map is an isomorphism. From the description of (2.14), we see that g is given by g((x ⊗ y) ⊗ a) = ax ⊗ y for x, y ∈ M and a ∈ A. One can readily check that this g is an isomorphism.
3. Descent properties of THR 3.1. Some equivariant topologies. We refer to [22] for the definition of the stable Z/2-equivariant motivic homotopy category SH Z/2 (k), which is compatible with the later work of Hoyois [28], but differs from Hermann's. See his Corollary 2.13 and Example 3.1, as well as [22, Example 2.16 and section 6.1], for a comparison. The following definitions are taken from [22]. Throughout this section, we assume that G is an abstract finite group, which we will identify with its associated finite group scheme over a fixed base scheme. We are mostly interested in the case G = Z/2 = C 2 .
Definition 3.1.1. Let x be a point of a G-scheme X. The set-theoretic stabilizer of X at x is defined to be The scheme-theoretic stabilizer of X at x is defined to be G x := ker(S x → Aut(k(x))).
Let f : Y → X be an equivariant morphism of G-schemes. We say that (i) f is (equivariant)étale if its underlying morphism of schemes isétale, (ii) f is an equivariantétale cover if f isétale and surjective, is an isomorphism, (iv) f is an isovariantétale cover if f is isovariant and an equivariantétale cover, (v) f is a fixed pointétale cover if it is anétale cover and for every point Nisnevich cover if f is anétale cover and for every point x ∈ X, there exists a point y ∈ f −1 (x) such that k(x) ≃ k(y) and S x ≃ S y . These covers define equivariantétale, isovariantétale, fixed pointétale, and equivariant Nisnevich topologies on the category of G-schemes.
The discussion in [22, p. 1223] shows that the equivariant Nisnevich topology is coarser than the fixed pointétale topology. As observed in the proof of [22,Corollary 6.6]. the fixed pointétale topology is equivalent to the isovariantétale topology. Hence we have the following inclusions of topologies: For a G-scheme X, let X/G denote the geometric quotient, which is an algebraic space. For the existence, see e.g. [46,Corollary 5.4]. If S is a locally noetherian scheme with the trivial G-action and X → S is a quasi-projective G-equivariant morphism, then X/G is representable by an S-scheme according to [ (Thomason). Let X be a G-scheme. Then there exists an equivalence of sites Proof. We refer to [22,Proposition 6.11].
Definition 3.1.5. Let X be a separated G-scheme. The presheaf X G on the category of separated schemes Sch is defined to be for Y ∈ Sch. By [13, Proposition XII.9.2], X G is representable by a closed subscheme of X. Furthermore, the points of the topological space underlying the scheme X G are in canonical bijection with the points of the topological space of fixed points (recall G is finite).
Hence ιB is a filtered colimit of free N We will show that this map is an equivalence as claimed by applying Proposition A.5.6. By Proposition A.3.7, N Z/2 HA and N Z/2 HB are (−1)-connected. Hence we may use Proposition A.4.4, and are reduced to showing that the induced morphism π 0 (THR(ιA)) ιA ιB → π 0 (THR(ιB)) is an isomorphism. This follows easily from applying Proposition 2.3.5 to A and B.
We now establish isovariantétale base change for commutative rings with possibly non-trivial involution. Proof. We have isomorphisms Spec(A Z/2 ) ≃ Spec(A)/(Z/2) and Spec(B Z/2 ) ≃ Spec(B)/(Z/2). Hence by Proposition 3.1.2, the induced homomorphism A Z/2 → B Z/2 isétale, and there is an isomorphism of commutative rings with involution Now Proposition 3.2.2 implies the left equivalence of the Proposition after canceling out one smash factor THR(ι(A Z/2 )). Applying the isovariance condition once more gives the right hand side equivalence.

Presheaves of equivariant spectra.
Zariski and other sheaves and completely determined by their behaviors on affine schemes. This is known to be true in some homotopical settings as well. The purpose of this section is to establish a rather general result, namely Proposition 3.3.12, which applies to our setting, that is the isovariantétale site and Z/2-equivariant spectra. For this, a result of [2] will be very useful. Definition 3.3.1. Let C be a category with a Grothendieck topology t, and let V be a presentable ∞-category. Let denote the ∞-category of presheaves on C with values in V. We say that a presheaf F ∈ Psh(C, V) satisfies t-descent if the induced map is an equivalence for every t-hypercover X → X. Let Shv t (C, V) denote the full subcategory of Psh(C, V) consisting of presheaves satisfying t-descent. We often omit t in the above notation if it is clear from the context. The above condition is sometimes called t-hyperdescent, in order to distinguish it from the weaker descent condition only for covering sieves rather than all hypercovers. We refer to [36, section 6.5] and [16] for a careful comparison.
If V → V ′ is a functor of ∞-categories, then this induces a functor Psh(C, V) → Psh(C, V ′ ). In this way, we obtain functors i * , (−) Z/2 , etc for the presheaf categories.
Proof. Let X → X be a hypercover. Since i * and (−) Z/2 preserve limits, (3.5) is an equivalence if and only if for all hypercover X → X and integer n. In particular, there is an adjoint pair LG is an equivalence. Proof. We need to show that the maps are local equivalences for all hypercover X → X and integer n, which follow from Proposition A.2.7(2),(7). Proposition 3.3.5. Let C be a site. Then the functor preserves local equivalences.
Proof. We need to show that the maps is an equivalence. The formulation (A.4) means that ι commutes with Σ ∞ , and there is an equivalence (ιX) Z/2 ≃ X. Together with the tom Dieck splitting [20, Theorem 3.10], we have an equivalence We have a similar equivalence for X too. Since Proof. By considering the fiber of f , we reduce to the case when G = 0. This means that i * F and F Z/2 are local equivalent to 0. The adjoint pair (2.4) gives a local equivalence F → F ′ with F ′ ∈ Shv(C, Sp Z/2 ). By Propositions 3.3.4 and 3.3.5, i * F ′ and F ′Z/2 are local equivalent to 0. Since i * F ′ , F ′Z/2 ∈ Shv(C, Sp) by Proposition 3.3.2, it follows that i * F ′ and F ′Z/2 are equivalent to 0. Hence F ′ is equivalent to 0, i.e., F is local equivalent to 0.
is a weak equivalence (resp. fibration) for all X ∈ C. One can form a projective model structure based on these, see [2,Definition 4.4.18].
If V is the underlying ∞-category of M, then the underlying ∞-category of Psh(C, M) with respect to the projective model structure is equivalent to Psh(C, V) by [ becomes an isomorphism after sheafification for all subgroup H of G and integer n. There is a local projective model structure, see [2,Definition 4.4.34]. This is a Bousfield localization of the projective model structure with respect to local weak equivalences.
Proof. Let f : F → G be a morphism of fibrant objects in Psh(C, Sp O G ) with respect to the projective model structure. We need to show that f is a local weak equivalence if and only if the corresponding map g in Psh(C, Sp G ) is a local equivalence.
By adjunction, f is a local weak equivalence if and only if the induced morphism of presheaves becomes an isomorphism after sheafification for H = e, Z/2 and integer n. By [16,Theorem 1.3], this is equivalent to saying that i * g and g Z/2 are local equivalences. Proposition 3.3.6 finishes the proof. Proposition 3.3.12. Let C and C ′ be sites. If there is an equivalence of topoi Shv(C) ≃ Shv(C ′ ), then there is an equivalence of ∞-categories Proof. Apply [2,Proposition 4.4.56] to the canonical (compose Yoneda and sheafification) functors C → Shv(C) and C ′ → Shv(C ′ ), where the right hand side categories are equipped with the topology described in [2, after Théorème 4.4.51]. Hence we obtain left Quillen equivalences We obtain the desired equivalence of ∞-categories thanks to the equivalence of topoi (and hence sites) Shv(C) ≃ Shv(C ′ ).  Proof. By Proposition A.3.6, H preserves filtered colimits. The two functors ι also preserve filtered colimits since they are left adjoints. Every flat A-module is a filtered colimit of finitely generated free A-modules by Lazard's theorem, so we reduce to the case when M is a finitely generated free A-module. In this case, the claim is clear.
For a scheme X, letÉtAff /X denote the category of affine schemesétale over X. We start by considering the affine case X = Spec(A).
for all integers q. In particular, π q F is a quasi-coherent sheaf onÉtAff / Spec(A). According to the paragraph preceding [19, Proposition 3.1.2], there is a conditionally convergent spectral sequence is a quasi-coherent sheaf for all integer t, the cohomology H s (X, aé t π −t F ) vanishes for all integer s = 0. It follows that π −t F → π −t Lé t F is an isomorphism for all integer t, i.e., F satisfiesétale descent.
Suppose A is a commutative ring with involution. Let isoÉtAff / Spec(A) denote the category of affine schemes with involutions isovariantétale over Spec(A).
satisfies isovariantétale descent, where Aff Z/2 denotes the category of affine schemes with involutions.
Proof. We only need to show that the restriction of THR to isoÉtAff / Spec(A) satisfies isovariantétale descent. The functor sending Spec(B) to Spec(A⊗ ιA Z/2 ιB) is an equivalence of sites by Proposition 3.
Hence it remains to check that F satisfiesétale descent. There is an equivalence By Lemmas A.2.9 and 3.4.1, there are equivalences Lemma 3.4.2 implies that i * F and F Z/2 satisfyétale descent, which means that F satisfiesétale descent.
Proposition 3.4.4. For every separated scheme X with involution, there exists an isovariantétale covering {U i → X} i∈I such that each U i is an affine scheme with involution.
Proof. Let X Z/2 denote the closed subscheme of X obtained by Definition 3.1.5. We set Y : is again an affine open neighborhood of x since X is separated, and w can be restricted to V x . It follows that Choose a Zariski covering {W j → Y } j∈J after forgetting involution such that each W j is an affine scheme, and then is an isovariantétale covering.
Let Sch Z/2 denote the category of separated schemes with involutions.
There is an equivalence of topoi Hence there is an equivalence of ∞-categories Proof This definition immediately implies that THR(X) that satisfies isovariantétale descent for all X ∈ Sch Z/2 .
Below, we carry out computations of THR(X) for projective spaces X = P n , even with involution for n = 1. Let us first check that extending Proposition 2.1.3 to schemes yields a definition of THH for schemes equivalent to the one of [19] as follows. See e.g. [9, p. 1055 and chapter 3] for a discussion of other equivalent definitions of THH(X).
where the involution on Y ∐ Y switches the components.
Proof. The question is isovariantétale local on X andétale local on Y , so we reduce to the case when X = Spec(A) for some commutative ring A with involution and Y = Spec(B) for some commutative ring B. We obtain equivalences is cartesian.
For the computations about THR(P n ) we provide in the next sections, we may simply use appropriate equivariant Nisnevich covers by affine schemes and Corollary 3.4.9, which e.g. exhibits THR(P 1 ) as part of a homotopy cocartesian square in which all other entries are of the form THR(Y ) for Y = Spec(A[M ]) corresponding to some commutative monoid rings for M = N or M = Z over the commutative base ring A. We could even go one step further and define THR(P n S , σ) of THR for projective spaces, possibly with non-trivial involution σ, over the sphere spectrum S rather than over A or HA, using the homotopy pushout of the appropriate diagram of THR of the corresponding spherical monoid rings S[M ], compare the proofs in section 5.1.

The dihedral replete bar construction
4.1. Crossed simplicial groups. Hochschild and cyclic homology and their real refinements are closely related to cyclic, real and dihedral nerves. We now present a uniform treatment of these constructions. The original references are [18] and [35], parts of this are also explained e.g. in [15] and [14]. ) and g ∈ G op m . Observe that every crossed simplicial group is a (or rather has an underlying) simplicial set, see [18,Lemma 1.3]. A G-set is a functor ∆G op → Set, which by restriction has an underlying simplicial set.
Construction 4.1.2. Let X be a simplicial set. For a crossed simplicial group G, the G-set F G (X) is defined in [18,Definition 4.3]. In simplicial degree q, we have The G q action on F G (X) q is the left multiplication on G q . The faces and degeneracy maps are given by According to [18,Proposition 5.1], we have the projection p 1 : F G (X) → G given by p 1 (g, x) := g. We also have the projection p 2 : |F G (X)| → |X| given by for u ∈ ∆ q top . Furthermore, the two projections define a homeomorphism |F G (X)| ∼ = |G| × |X|.
If X is a G-set, then we have the evaluation map ev : F G (X) → X given by ev(g, x) := gx.
According to [18,Theorem 5.3], the composite map defines a |G|-action on |X|. The G-geometric realization of X is |X| with this |G|-action. (1) [18, Example 2, section 1.5] introduces a crossed simplicial group G with G n := Z/2. For this G, a G-set is called a real simplicial set. The G-geometric realization is called the real geometric realization. Explicitly, a real simplicial set X is a simplicial set equipped with isomorphisms w n : X n → X n with w 2 n = id for all integer n ≥ 1 satisfying the relations d i w n = w n−1 d n−i and s i w n = w n+1 s n−i for 0 ≤ i ≤ n.
(2) [18, Example 4, section 1.5] introduces a crossed simplicial group G with G n := C n+1 . For this G, a G-set is called a cyclic set, which was defined by Connes. The G-geometric realization is called the cyclic geometric realization, and comes with an action of S 1 = SO(2) (see also [35,Theorem 7.1.4]). Explicitly, a cyclic set X is a simplicial set equipped with isomorphisms t n : X n → X n with (t n ) n+1 = id for all integers n ≥ 1 satisfying the relations (3) [18, Example 5, section 1.5] introduces a crossed simplicial group G with G n := D n+1 , where D n denotes the dihedral group of order 2n. For this G, a G-set is called a dihedral set. The G-geometric realization is called the dihedral geometric realization.
Explicitly, a dihedral set X is a simplicial set equipped with t n and w n for all integer n ≥ 1 satisfying w n t n = t −1 n w n and all the above conditions for t n and w n . There are obvious forgetful functors from dihedral to real and to cyclic sets.   (1). For a simplicial set X, the bijection G × X q = X q ∐ X q induces a canonical isomorphism of G-sets where w n : X n ∐ X op n → X n ∐ X op n is the switching map. If X is a real simplicial set, then the evaluation map ev : F G (X) → X sends x ∈ X op n to w n (x). It follows that the Z/2-action on |X| is given by the composite map where the first map is the canonical identity. This is Connes' cyclic n-complex. We warn the reader that F G (X) for a simplicial set is different from the cyclic bar construction.

4.2.
Real and dihedral nerves. We now recall how commutative monoids with involution give rise to real and dihedral simplicial sets, refining nerves and cyclic nerves for commutative monoids without involutions.
The degeneracy maps are The rotation maps are The involution is We have the map of dihedral sets sending (x 0 , . . . , x q ) to x 0 +· · ·+x q in simplicial degree q, where we regard M as the constant dihedral set whose rotation maps are the identities and whose involutions are given by σ. For every σ-orbit I in M , we set Proof. The two projections M × L ⇒ M, L induce (4.5). To show that it is an isomorphism, observe that it is given by the shuffle homomorphism in simplicial degree q. Then there is an isomorphism of dihedral sets for all x ∈ M and y ∈ L.
Proof. Follows from the commutativity of the square where the upper horizontal homomorphism is the shuffle homomorphism, and the vertical homomorphisms are the summation homomorphisms.  If j = w(j), then there is an isomorphism of real simplicial sets As a consequence, there is an isomorphism of real simplicial sets Proof. If j = w(j), the assignment in simplicial degree q constructs the isomorphism (4.6). If j = w(j), the assignment in simplicial degree q constructs the isomorphism (4.7), where a := 0 (resp. a := 1) if x 0 + · · · + x q = j (resp. x 0 + · · · + x q = −j).
Definition 4.2.7. Let S σ be the Z/2-space whose underlying space is S 1 and whose Z/2-action is given by the reflection (x, y) ∈ S 1 ⊂ R 2 → (x, −y). This is the real geometric realization of the real simplicial set whose underlying simplicial set is the simplicial circle ∆ 1 /∂∆ 1 and whose involution on the non-degenerate simplicies is the identity. One easily checks that the relations in Example 4.1.3 uniquely determines the higher w n on ∆ 1 /∂∆ 1 icies is the identity. To understand the involution on the real realization, the reader is advised to look at the explanations in Example 4. Proof. Consider the element 1 ∈ (N σ Z) 1 ≃ Z in simplicial degree 1. This is fixed by the involution on N σ Z, so the real geometric realization of the real simplicial subset of N σ Z whose only non-degenerate simplicies are 1 and the base point is S σ . Hence we obtain a map of Z/2-spaces S σ → B σ Z. This is the usual homotopy equivalence after forgetting the involutions. It remains to check that the induced map is a homotopy equivalence as well. By [15, Proposition 2.1.6], (B σ Z) Z/2 is the classifying space of the category Sym Z, whose objects are integers, and whose morphisms are given by The classifying space of Sym Z is obviously homotopy equivalent to S 0 , and hence we deduce a homotopy equivalence This implies that π n ((B σ Z) Z/2 ) is trivial for all integer n > 0 and π 0 ((B σ Z) Z/2 ) consists of two points.
Together with (4.6), we obtain an equivalence For every integer j, let S σ (j) denote the space S 1 with the O(2)-action, whose SO(2)-action is given by (t, x) → t j x for t ∈ SO(2) and x ∈ S 1 , and whose Z/2action is given by the complex conjugate x → x. Here, we regard S 1 as the unit circle in C. For j ≥ 0, let ∆ j σ be the Z/2-space whose involution is the reflection mapping the vertex i to j − i for all 0 ≤ i ≤ j.
In this formulation, the values are calculated modulo j. Let Λ j−1 σ be the dihedral set whose underlying cyclic set is Λ j−1 and whose involution is given by Then (4.16) becomes a morphism of dihedral sets The cyclic geometric realization of (4.16) is S 1 -homeomorphic to the quotient map σ . Combine these two facts to deduce that the dihedral geometric realization of (4.17) is O(2)-homeomorphic to the quotient map σ )/C j In particular, we obtain (4.14). Since ∆ j−1 σ is D j -contractible to its barycenter, we obtain (4.15), Let us review what the proof of [44,Proposition 3.21] contains. For every integer r ≥ 1, let sd r denote the r-fold edgewise subdivision functor in [10]. For every integer j, the r-fold power map N di (Z; j) → sd r N di (Z; j) given by If r ∤ j, then we have Suppose j ≥ 0. We similarly have a homeomorphism If r ∤ j, then we have we have If H is a closed subgroup of SO (2), then the induced map is a homotopy equivalence.

5.
Properties of real topological Hochschild homology 5.1. THR of the projective line. We now establish the first computations of THR for non-affine schemes, namely P 1 and P σ .  [29] for different proofs. In particular, this leads to an equivariant motivic spectrum KR which is P 1 ∧P σ -periodic. For further periodicities of KR see [29, Theorem 10]. In their notation, we have P 1 ≃ S 1 ∧S α and P σ ≃ S γ ∧S γα ≃ P 1 − . For the definition of S γ = S σ in the motivic setting and a proof of the last equivalence, we refer to [11, section 2.5]. Although THR is not A 1 -invariant, it seems reasonable to expect that the formulas for THR(P n ) are similar to those for KR. In the cases considered below this is indeed the case: the following Proposition implies that Ω 1+α THR ≃ Σ γ−1 THR. Smashing with S 1 , we obtain the same periodicity as (35) in [29]. Similarly, the next proposition corresponds to (36) of loc. cit.
The following computations rely on Proposition 4.2.13 and the computations for dihedral nerves in the previous section.
Theorem 5.1.2. For any X ∈ Sch Z/2 , there is an equivalence of Z/2-spectra Proof. For notational convenience, we will write the proof as if everything takes place over S rather than X. Using the description of THR for spherical groups rings from Proposition 4.2.13 and its extension to log schemes with involution from Proposition 4.2.15, we are reduced to consider the following homotopy cocartesian square, corresponding to the standard Zariski cover of P 1 by two copies of A 1 and using Corollary 3.4.9 and the description of THR(S[M ]) from Proposition 4.2.13: Here the notation N and −N indicates the two different embeddings of A 1 in P 1 . It is crucial to notice that even as G m has trivial involution, the involution given by the dihedral nerve (see Definition 4.2.2 above) yields nontrivial involutions. Using the decomposition of (4.4) and Propositions 4.2.6, 4.2.8, and 4.2.11, we obtain the following Z/2-equivariant (homotopy) cocartesian square: An obvious cancellation, using Proposition 4.2.12 on the relevant maps, yields the following Z/2-equivariant (homotopy) cocartesian square and the result follows.
Lemma 5.1.3. There is a natural equivalence of functors Proof. Consider the cocartesian square of Z/2-spaces Together with the explicit descriptions of i ♯ and i * in Construction A.2.4, we obtain a natural equivalence By adjunction, we obtain the desired natural equivalence.
Theorem 5.1.4. For any X ∈ Sch Z/2 , there is an equivalence of Z/2-spectra Proof. As before, we will write the proof as if everything takes place over S. Consider the cartesian equivariant Nisnevich square as in [11,Lemma 2.23], where the Z/2-action on the upper right corner is induced by the action on P σ S . Propositions 2.1.3 and 3.4.7 give equivalences (5.2), identifications similar to those in the proof of Theorem 5.1.2 -but using (4.13) rather than Proposition 4.2.8 -yield the following homotopy cocartesian square: , where the right vertical map (resp. lower horizontal map) is obtained by applying i * i * to the left (resp. right) of the unit map id → i * i * . Since Φ Z/2 i * ≃ 0 by Proposition A.2.7(3),(5), Φ Z/2 Q is cartesian. The objects in the square i * Q are direct sums of j>0 Σ ∞ + S 1 , and the right vertical and lower horizontal maps are the matrix multiplications given by  for certain choices of bases. From this, one can check that i * Q is cartesian too. It follows that Q is cartesian.
Hence the other direct summand of the square (5.3) is also cartesian. It follows that THR(P σ S ) is equivalent to the direct sum Together with Lemma 5.1.3, we obtain the desired equivalence.

THR of projective spaces.
Definition 5.2.1. As usual, for any integer n ≥ 1, we consider the n-cube (∆ 1 ) n as a partially ordered set, and use the same symbol for the associated category. For an ∞-category C, an n-cube in C is a functor The following ∞-categorical result can be shown by dualizing the arguments from [7, Appendix A].
Proposition 5.2.2. Let Q be an n-cube in an ∞-category C with small limits, where n is a nonnegative integer. Then for every integer 1 ≤ i ≤ n, there exists a fiber sequence Proposition 5.2.3. Let C be a symmetric monoidal stable ∞-category with small limits such that the tensor product operation on C preserves fiber sequences in each variable. If Q is an n-cube and f : X 0 → X 1 is a map in C, then there is a canonical equivalence where Q ⊗ f is the associated (n + 1)-cube sending (a 1 , . . . , a n+1 ) ∈ (∆ 1 ) n+1 to Q(a 1 , . . . , a n ) ⊗ X an+1 .
Proof. This is again obtained by dualizing the arguments for the corresponding Proposition in [7]. An intermediate step is to show tfib(Q) ⊗ X ≃ tfib(Q ⊗ X) for any X ∈ C, where Q ⊗ X is the associated n-cube sending (a 1 , . . . , a n ) ∈ (∆ 1 ) n to Q(a 1 , . . . , a n ) ⊗ X. Then one can use Proposition 5.2.2.
Proposition 5.2.4. Let C be a symmetric monoidal stable ∞-category with small limits such that the tensor product operation on C preserves fiber sequences in each variable. If i 1 : X 0,1 → X 1,1 , . . ., i n : X 0,n → X 1,n are maps in C, then there is a canonical equivalence where i 1 ⊗ · · · ⊗ i n is the associated n-cube sending (a 1 , . . . , a n ) ∈ (∆ 1 ) n to X a1,1 ⊗ · · · ⊗ X an,n and arrows given by tensor products of i j s and identities.
Together with Proposition 5.2.2, we reduce to showing that the induced square is cartesian. This follows from Corollary 3.4.9.
Theorem 5.2.6. For any X ∈ Sch Z/2 and integer n ≥ 0, there is an equivalence of Z/2-spectra Proof. Proposition A.2.7(3) allows us to replace i * by i ♯ in the claim. We set for j = 1, . . . , n and Observe that there is an isomorphism of commutative monoids M j ∼ = Z n−1 ⊕ N for all j = 1, . . . , n + 1. We set M I := M i1 ∩ · · · ∩ M iq for all nonempty subset is an equivalence for all integer 0 ≤ s ≤ n − 1 if the first coordinate of v is greater than 0. By a change of coordinates in the target, we see that the induced map where O denotes the origin in Z n . Now consider the standard cover of P n S by (n + 1) copies of A n S , and recall that A n S is the spherical monoid ring of N n . (We continue switching between spherical monoid rings and honest schemes over rings as before. Also, note that by Proposition 4.2.15 products of affine schemes correspond to products of monoids when computing THR.) Choosing suitable coordinates, the intersections of s elements of the cover with 0 < s ≤ n + 1 are given by (the spherical monoid rings of) M I above such that |I| = n + 1 − s. By Proposition 5.2.5, we have an equivalence ..,n} = 0, we also have an equivalence Combine these two to obtain an equivalence . Together with (5.5), we have an equivalence Consider the canonical decomposition of (n + 1)-cubes Together with Proposition 4.2.11 and (4.12), we obtain equivalences where (1, 0) is the base point of S σ . Proposition 5.2.2 gives a fiber sequence If d is odd, then we obtain a fiber sequence Since π 0 (Σ −1 ) = 0, the map Σ −1 → i * Σ dσ−d ≃ Σ 0 obtained by adjunction is equivalent to 0. It follows that f is equivalent to 0, i.e., (5.7) splits. This completes the induction argument for odd d.
If d is even, we obtain a fiber sequence As above, the induced map ⌊d/2⌋−1 j=1 i ♯ Σ −1 → Σ dσ−d is equivalent to 0. It follows that we have an equivalence We now analyze the non-trivial map g : On the level of commutative monoids, this corresponds to the inclusion where u i := e 1 − e d+1 for i ∈ {1, . . . , n} − {d + 1} and u d+1 := e d+1 . As the 0-entry for B di N is just a point, we only need to study the homomorphism Z d−1 → Z d given by (a 1 , . . . , a d−1 ) → (a 1 , . . . , a n−1 , −a 1 − · · · − a d−1 ).
In the degree (0, . . . , 0), this is easily seen to induce via Proposition 4.2.8 the map (S σ ) ×d−1 → (S σ ) ×d given by . After a further cancellation of base points, we are left with studying the map h : (S σ ) ∧d−1 → (S σ ) ∧d . This is an equivariant cofibration, as it is the (S σ ) ∧d−1suspension of the push-out of the cofibration G × S 0 → G × I along the projection G × S 0 → (G/G) × S 0 . Hence unstably the equivariant (homotopy) cofiber of f is given by (S σ ) ∧d /h((S σ ) ∧d−1 ) ≃ S d ∨ S d where G = Z/2 acts on the latter by switching the spheres. Thus the stable equivariant (homotopy) fiber fib(g) is given by 15). This completes the induction argument for even d.
Together with (5.6), we obtain an equivalence Use Propositions 4.2.15 and 5.2.5 for the standard cover of P n to obtain an equivalence THR(X × P n ) ≃ THR(X) ∧ THR(P n S ) whenever X is an affine scheme with involution. Use Proposition 5.2.5 again to generalize this equivalence to the case when X is a separated scheme with involution. Hence to obtain the desired equivalence, it suffices to obtain an equivalence This follows from Propositions A.1.10 and 2.1.3.
Note that this result is compatible with the projective bundle theorem for the oriented theory THH, see [9], after applying i * and using Propositions 3.4.7 and A.2.7.
For all X ∈ Sch Z/2 and integer m, we set Recall that (−) Z/2 commutes with Σ 1 , but unlike Φ G not with Σ σ .
Looking at fixed points, the result becomes (5.9) Remark 5.2.7. As for n = 1 in the previous subsection, the formula for THO(X × P n ) corresponds to the one for KO(X × P n ). Indeed, for higher dimensional projective spaces P n with trivial involution, KO = KR has been recently computed by [45] and [31]. Analyzing the arguments of [45], one sees that for even n the results for KR(P n ) with and without involution are the same.

Appendix A. Equivariant homotopy theory
The purpose of this appendix is to review equivariant homotopy theory. Throughout this section, G is a finite group.
A.1. ∞-categories of equivariant spectra. In this subsection we review the ∞categorical formulation of equivariant homotopy theory following Bachmann and Hoyois [3]. We restrict to finite groupoids, although Bachmann and Hoyois deal more generally with profinite groupoids . This approach will be compared to more classical references as [24] in section A.2 below.
Definition A.1.1. Let FinGpd denote the 2-category of finite groupoids, that is those with only finitely many objects and morphisms. A morphism in FinGpd is called a finite covering if its fibers (which by our assumptions have only finitely many objects) automatically are sets, i.e., do not have non-trivial automorphisms. Recall that the fiber of a 1-morphism f : Y → X over a point * in X is Y × X * . For X ∈ FinGpd, let Fin X denote the category of finite coverings of X. There is an equivalence between Fin BG and the category of finite G-sets by [3,Lemma 9.3].
For a morphism f : Y → X in FinGpd, there is an adjunction where f * sends V ∈ Fin X to V × X Y . If f is a finite covering, then there is an adjunction where f ♯ sends V ∈ Fin Y to V , compare the paragraph preceding [3, section 9.2].
Example A.1.2. If i is the obvious morphism of groupoids pt → BG, then i ♯ E = G E and (i ♯ , i * ) is the usual free-forgetful adjunction between sets and G-sets. On the other hand, for any finite set E the G-set i * E is isomorphic to G E with G acting by permuting the indices.
For p : BG → pt we have p * E = E G for every finite G-set E. Note that p ♯ E is not defined as p is not a finite covering, although for finite G-sets E the left adjoint to p * exists, and is given by the orbit set E/G. Definition A.1.3. For a category C with pull-backs, let Span(C) denote the category of spans, whose objects are the same as C, whose morphisms are given by the diagrams (X f ← − Y p − → Z) in C, and whose compositions of morphisms are given by is called a forward morphism (resp. backward morphism) if f = id (resp. p = id). The notion of spans can be generalized to the case when C is an ∞-category, see [4, section 5] for the details.
together with natural transformations Let us explain parts of their construction. For an ∞-category C with finite coproducts, let P Σ (C) denote the ∞-category of presheaves of spaces that transforms finite coproducts into finite products. For X ∈ FinGpd, we set H(X) := P Σ (Fin X ). Then we set H • (X) := H(X) * , which is the ∞-category of pointed objects in H(X). As claimed in [3, p. 81], for X = BG these yield the usual ∞-categories of G-spaces and pointed G-spaces.
For a morphism f : Y → X, the functor f * for H and H • is induced by (A.1), and f * admits a right adjoint f * . For H • , f ⊗ is a symmetric monoidal functor preserving sifted colimits such that f ⊗ (V + ) ≃ f * (V ) + for V ∈ Fin X . If f is a finite covering, f * for H and H • admits a left adjoint f ♯ .
We obtain SH(X) from H • (X) by ⊗-inverting p ⊗ (S 1 ) for all finite covering p : Y → X. The functor f * for SH is induced by that for H • . This admits a right adjoint f * , and this admits a left adjoint f ♯ if f is a finite covering. The functor f ⊗ for SH is a unique symmetric monoidal functor preserving sifted colimits such that the square commutes. Furthermore, if f has connected fibers, then f ⊗ preserves colimits.
Proposition A.1.6. Suppose X ∈ FinGpd. Then the family compactly generates SH(X). In other words, the functor Map(Σ n Σ ∞ V + , −) preserves filtered colimits for all V ∈ Fin X and n ∈ Z, and the family of functors Let Fold X denote the full subcategory of Fin X consisting of the finite fold maps (id, . . . , id) : X ∐ · · · ∐ X → X, Proposition A.1.7. Let f : X ∐n → X be the n-fold map, where X ∈ FinGpd and n ≥ 1 is an integer. Then the composite is the n-fold smash product. A normed Xspectrum is a section of SH ⊗ over Span(Fin X ) that is cocartesian over the backward morphisms. Let NAlg(SH(X)) denote the ∞-category of normed X-spectra.
Mapping X to pt yields an equivalence between Fold X and Fold pt , and forgetting the map to pt Fold pt is obviously equivalent to the category of finite sets. By [3,Corollary C.2], CAlg(SH(X)) is equivalent to the ∞-category of sections of SH ⊗ over Span(Fold X ) that is cocartesian over the backward morphisms. Hence there is a forgetful functor NAlg(SH(X)) → CAlg(SH(X)), which is conservative and preserves colimit and limits as in [ . Note that the symmetric monoidal structure on CAlg(−) is given by the coproduct. The monoidal product in SH(X) (resp. Mod R ) is denoted by ∧ (resp. ∧ R ). Then we have the induced monoidal products on CAlg(SH(X)) and CAlg(Mod R ).
There is an equivalence between CAlg(Mod R ) and the ∞-category of R-algebras CAlg(SH(X)) /R by [37, Corollary 3.4. 1.7]. This means that we have a canonical equivalence where ∧ in the right hand side means the coproduct in CAlg(SH(X)). The notation ∧ R on Mod R is compatible with the notation ∧ in CAlg(SH(X)) in this sense.
be a cartesian square in FinGpd such that f is a finite covering. For SH, the natural transformation f ′ ♯ g ′ * → g * f ♯ given by the composite is an equivalence.
Proof. As usual, Proposition A.1.6 allows us to reduce to showing that the induced map f ′ ♯ g ′ * Σ n Σ ∞ W + → g * f ♯ Σ n Σ ∞ W + is an equivalence for all W ∈ Fin Y and integer n. This follows from the fact that the composite of the induced morphisms Proposition A.1.10. Let f : Y → X be a finite covering. Then for F ∈ SH(Y ) and G ∈ SH(X), there exists a canonical equivalence Proof. As usual, Proposition A.1.6 allows us to reduce to the case when F = Σ m Σ ∞ V + and G = Σ n Σ ∞ W + for some V ∈ Fin Y , W ∈ Fin X , and m, n ∈ Z. In this case, the canonical isomorphism gives the desired equivalence. Furthermore, the two squares in (A.14) NAlg(SH(X)) NAlg(SH(Y )) Let SH ⊗ |Span(Fin X ) be the restriction of SH ⊗ to Span(Fin X ). For every V ∈ Span(Fin X ), let f V : V × X Y → V be the projection. The functor f * V : SH(V ) → SH(V × X Y ) admits the right adjoint f V * . Apply [3,Proposition 8.16] to the cocartesian fibration SH ⊗ |Span(Fin X ) to obtain the desired adjunction.
Let us review the descriptions of f * and f * for NAlg in this reference. The functor f * for NAlg used here is the same as the functor f * for NAlg in Proposition A.1.11. Suppose B ∈ NAlg(SH(Y )). For V ∈ Span(Fin X ), the section (f * B)(V ) is given by f V * (B(V × X Y )). In particular, the section (f * B)(X) is given by f * (B(X)). Hence the two squares in (A.14) commute.
This notation is further justified by Remark A.2.3.
A.2. Equivariant orthogonal spectra. The purpose of this section is to review equivariant homotopy theory using model categories. Our references for that are [20], [39], [24], and [48]. We will also review the comparison between ∞-categorical and model categorical constructions of equivariant spectra. Consequently, we may apply certain known constructions and results for equivariant orthogonal spectra to ∞-categories as discussed in the previous subsection.
Let BG denote the associated finite groupoid. In this subsection, we are interested in the obvious morphisms where H → G is an inclusion.
Definition A.2.1. Let Sp O denote the category of orthogonal spectra. For a finite group G, let Sp O G denote the category of orthogonal G-spectra. Recall that an orthogonal G-spectrum is an orthogonal spectrum with a G-action. A morphism of orthogonal G-spectra is a morphism of underlying orthogonal spectra that is compatible with the G-actions.
The definition of orthogonal G-spectra in [24] is different from the above one, but the two categories are equivalent. See [ [24,Proposition B.129], Comm G has a nice model structure. A morphism in Comm G is a weak equivalence (resp. fibration) precisely when its underlying morphism in Sp O G is a weak equivalence (resp. fibration). The (underived) coproduct in Comm G is denoted by ∧.
Remark A.2.3. As observed in the preceding paragraphs of [3, Lemma 9.6], Sp G is equivalent to the underlying ∞-category of the model category of symmetric G-spectra. This is equivalent to the underlying ∞-category of Sp O G by [38]. See also [3,Remark 9.12] for another ∞-description. Furthermore, as observed in [3, after Definition 9.14], NAlg G is equivalent to the underlying ∞-category of the model category of G-E ∞ -rings, which is equivalent to the underlying ∞-category of Comm G . We refer to [21] for a comparison of different models, rectification results and further references. In short, comparing the adjunctions between Comm G with those for the underlying spectra, N G H already exists for spectra, but only becomes a left adjoint to i * in Comm G , replacing i ♯ .
Compare the diagram in [24,Proposition A.56] with (A.12) and use the conservativity of the forgetful functor NAlg G → Sp G to show that N G H : Comm H → Comm G is a model for the functor i ⊗ : NAlg H → NAlg G . Then i * : Comm G → Comm H is a model for the functor i * : NAlg G → NAlg H by adjunction.
Proposition A.2.7. We have the following equivalences of functors between ∞categories Sp G for appropriate G: Proof. The first two follow from pi = id. The next three follow from [ Proof. We have maps where the second map is induced by the counit map p * p * L → L. By adjunction, we obtain M ∧ R p * L → p * (p * M ∧ p * R L). We only need to show that this is an equivalence in Sp after forgetting the module structures. Let F be the class of R-modules M such that this map is an equivalence. The functors p * , ∧ R p * L, and ∧ p * R L preserve colimits. As explained after [ For all X ∈ Sp G , M ∈ Fin BG , and integer n, we set (A. 17) π n (X)(M ) := Hom Ho( where the isomorphism comes from [24, Example 2.6]. The first (resp. second) formulation is contravariant (resp. covariant) in M , and these two can be combined to produce the equivariant homotopy group functor We refer to [24, section 3.1] for the details. For X ∈ Sp G and an integer n, we say that X is n-connected if π k (X) = 0 for all integer k ≤ n. Definition A.3.3. Suppose n is an integer. Let (Sp G ) ≥n (resp. (Sp G ) ≤n ) denote the full subcategory of Sp G spanned by X ∈ Sp G such that π k (X) = 0 for all integer k < n (resp. k > n). Observe that there are equivalences (A.21) (Sp G ) ≥n ≃ Σ n (Sp G ) ≥0 and (Sp G ) ≤n ≃ Σ n (Sp G ) ≤0 .
Definition A.4.1. For Mackey functors M and L, the box product of M and L is defined to be M L := π 0 (HM ∧ HL).
There is a purely algebraic definition of the box product of Mackey functors at least for G = Z/p and some prime p, see [32, p. 61]. This is expected to coincide with the above Definition, but we won't need this.
Definition A.4.2. A Green functor A is a commutative monoid in the category Mack G , i.e., A is equipped with morphisms A A → A and π 0 (S) → A satisfying the unital, associative, and commutative axioms, where S denotes equivariant sphere spectrum. Let Green G denote the category of Green functors.
An A-module M is an object of Mack G equipped with an action morphism A M → M satisfying the module axioms. Proof. We refer to [5, section 6]. With the stronger assumption A ∈ NAlg G , this result is due to Lewis and Mandell [33, Theorem 6.6].
Apply Proposition A.4.4 to the case when A is the equivariant sphere spectrum to obtain the symmetric monoidal structure on (Sp G ) ≥0 that is the restriction of the symmetric monoidal structure on Sp G . Furthermore, the functor Definition A.5.2. Recall from [33, section 2] that the Burnside category B G is defined to be the additive category whose objects are the finite G-sets and whose hom groups are given by Hom BG (X, Y ) := Hom Ho(Sp G ) (Σ ∞ X + , Σ ∞ Y + ) for all finite G-sets X and Y .
For a finite G-set X, let B X denote the Mackey functor Hom BG (−, X). As explained in [33, p. 519], there is an isomorphism for all finite G-sets X and Y .  Proof. Since ∧ commutes with colimits in each variable, we have a canonical equivalence colim i∈I (HM i ∧ HA HL) ≃ (colim i∈I HM i ) ∧ HA HL.
Proposition A.5.9. Let A be a Green functor, and let Together with the fact that cofiber sequences and exact sequences coincide in the heart of a t-structure, we deduce the claim.