Fixed-Point Spectra

Table of Contents

• 1. Three Flavors at a Glance
• 2. Families of Subgroups and Classifying Spaces
  • 2.1 Families of Subgroups
  • 2.2 Classifying Spaces and the Cofiber Sequence
  • 2.3 The Universal Space for a Family
• 3. Categorical Fixed Points
  • 3.1 Definition and Levelwise Construction
  • 3.2 Adjunction Properties
  • 3.3 Failure of Homotopy-Invariance
  • 3.4 Exercises
• 4. Homotopy Fixed Points
  • 4.1 Definition
  • 4.2 Homotopy Invariance
  • 4.3 The Homotopy Fixed Point Spectral Sequence
  • 4.4 Exercises
• 5. Geometric Fixed Points
  • 5.1 Definition via the Cofiber Sequence
  • 5.2 Monoidality and the Sphere Spectrum
  • 5.3 Change of Group for Geometric Fixed Points
  • 5.4 Exercises
• 6. The Tate Construction
  • 6.1 Historical Origins: Tate Cohomology
  • 6.2 The Norm Map: ∞-Categorical Construction
  • 6.3 Definition of the Tate Spectrum
  • 6.4 Tate Acyclicity and the Verdier Quotient Perspective
  • 6.5 Canonical Lax Symmetric Monoidal Structure
  • 6.6 The Tate Orbit and Fixpoint Lemmas
  • 6.7 The Tate Diagonal
  • 6.8 The Tate Diagram
  • 6.9 The Segal Conjecture
  • 6.10 Exercises
• 7. The Adams Isomorphism
  • 7.1 Statement
  • 7.2 Proof Sketch
  • 7.3 Consequences and the tom Dieck Splitting
  • 7.4 Exercises
• 8. Examples: C2-Spectra
  • 8.1 Fixed Points of HZ
  • 8.2 Geometric Fixed Points of MU_R
  • 8.3 The C2-Tate Sequence for HZ
• References

1. Three Flavors at a Glance 🔑

Given a finite group \(G\), a subgroup \(H \leq G\), and a genuine \(G\)-spectrum \(E\), there are three natural ways to extract an “H-fixed part” of \(E\), each landing in \(\mathrm{Sp}^{W_G H}\) where \(W_G H = N_G H / H\) is the Weyl group:

Here \(EH\) is a contractible free \(H\)-space and \(\widetilde{E\mathcal{P}(H)}\) is the cofiber of \(E\mathcal{P}(H)_+ \to S^0\) for the family of proper subgroups of \(H\). Each functor sees different aspects of the \(H\)-equivariant structure of \(E\).

The three functors are related by the Tate diagram, a homotopy-pushout square

where \(E^{tH}\) is the Tate spectrum, the map \(\rho\) is the natural restriction from categorical to homotopy fixed points, and \(\pi\) is the comparison from categorical to geometric. The rows and columns of this square are cofiber sequences.

2. Families of Subgroups and Classifying Spaces 📐

2.1 Families of Subgroups

Definition (Family of Subgroups). A family \(\mathcal{F}\) of subgroups of \(G\) is a collection of subgroups that is: 1. closed under conjugation: if \(H \in \mathcal{F}\) and \(g \in G\), then \({}^g H = g H g^{-1} \in \mathcal{F}\); 2. closed under passage to subgroups: if \(H \in \mathcal{F}\) and \(K \leq H\), then \(K \in \mathcal{F}\).

The most important families for fixed-point theory are:

2.2 Classifying Spaces and the Cofiber Sequence

Definition (\(E\mathcal{F}\) and \(\widetilde{E\mathcal{F}}\)). For a family \(\mathcal{F}\), the classifying space \(E\mathcal{F}\) is a \(G\)-CW complex (unique up to \(G\)-homotopy equivalence) satisfying: \[ (E\mathcal{F})^H \simeq \begin{cases} * & \text{if } H \in \mathcal{F} \\ \emptyset & \text{if } H \notin \mathcal{F}. \end{cases} \]

The reduced classifying space \(\widetilde{E\mathcal{F}}\) is defined by the cofiber sequence of based \(G\)-spaces: \[E\mathcal{F}_+ \longrightarrow S^0 \longrightarrow \widetilde{E\mathcal{F}},\] where \(E\mathcal{F}_+\) denotes \(E\mathcal{F}\) with a disjoint basepoint, and the map \(E\mathcal{F}_+ \to S^0\) collapses \(E\mathcal{F}\) to the non-basepoint.

Lemma. The fixed-point spaces of \(\widetilde{E\mathcal{F}}\) satisfy: \[ (\widetilde{E\mathcal{F}})^H \simeq \begin{cases} S^0 & \text{if } H \notin \mathcal{F} \\ * & \text{if } H \in \mathcal{F}. \end{cases} \]

Proof sketch. From the cofiber sequence \((E\mathcal{F}_+)^H \to (S^0)^H \to (\widetilde{E\mathcal{F}})^H\). When \(H \in \mathcal{F}\), \((E\mathcal{F})^H \simeq *\), so \((E\mathcal{F}_+)^H \simeq S^0\), and the map \(S^0 \to S^0\) is the identity, making the cofiber \(*\). When \(H \notin \mathcal{F}\), \((E\mathcal{F})^H = \emptyset\), so \((E\mathcal{F}_+)^H \simeq S^0_+\mid_{H\text{-fixed}} = \{*\}_+ = S^0\)… more precisely \((E\mathcal{F}_+)^H = (E\mathcal{F})^H_+ = \emptyset_+ = \{*\} = S^{-1}_+\)… let me be careful: \((E\mathcal{F}_+)^H = (E\mathcal{F})^H \sqcup \{*_{\text{base}}\}\). When \((E\mathcal{F})^H = \emptyset\), we get \((E\mathcal{F}_+)^H = \{*\}\), so the map \(\{*\} \to S^0\) includes the basepoint, and the cofiber is \(S^0/\{*\} = S^0\). \(\square\)

2.3 The Universal Space for a Family

For the family \(\mathcal{F}_0 = \{e\}\) (just the trivial subgroup), \(E\mathcal{F}_0 = EG\) is the universal free \(G\)-space — the total space of the universal principal \(G\)-bundle \(EG \to BG\). The reduced version \(\widetilde{EG}\) satisfies:

\[(\widetilde{EG})^H \simeq \begin{cases} S^0 & \text{if } H \neq e \\ * & \text{if } H = e. \end{cases}\]

For the family \(\mathcal{P}(H) = \{K \leq H : K \neq H\}\) of proper \(H\)-subgroups, \(E\mathcal{P}(H)\) is contractible as an \(H\)-space with the property that its \(K\)-fixed points are contractible for all \(K \subsetneq H\) and empty for \(K = H\). This is sometimes denoted \(\widetilde{EG}\) when \(H = G\) and \(G\) is understood.

3. Categorical Fixed Points 📐

3.1 Definition and Levelwise Construction

Definition (Categorical Fixed Points). For a genuine \(G\)-spectrum \(E \in \mathrm{Sp}^G_O\) and a subgroup \(H \leq G\), the categorical fixed-point spectrum \(E^H\) is the \(W_G H\)-spectrum defined levelwise:

\[(E^H)(V^{W_G H}) = E(V)^H,\]

where \(V^{W_G H}\) denotes the \(W_G H\)-representation obtained from a \(W_G H\)-representation \(V\) (viewed as a \(G\)-representation via \(N_G H \twoheadrightarrow W_G H\)), and \((-)^H\) denotes the \(H\)-fixed-point subspace.

More concretely, for an orthogonal \(G\)-spectrum \(E\): \[E^H = (E)^H \in \mathrm{Sp}^{W_G H}_O, \quad (E^H)(V) = E(V)^H,\] with structure maps induced by the \(H\)-fixed parts of the structure maps of \(E\).

3.2 Adjunction Properties

The categorical fixed-point functor \((-)^H: \mathrm{Sp}^G \to \mathrm{Sp}^{W_G H}\) is right adjoint to the induction functor \(G_+ \wedge_{N_G H} -: \mathrm{Sp}^{N_G H} \to \mathrm{Sp}^G\) (after restriction from \(G\) to \(N_G H\) and then to \(H\)). More precisely:

Proposition (Fixed Points as Right Adjoint). For \(E \in \mathrm{Sp}^G\) and \(F \in \mathrm{Sp}^{W_G H}\), there is a natural isomorphism

\[[G/H_+ \wedge F, E]_G \cong [F, E^H]_{W_G H},\]

where \(F\) is viewed as a \(G\)-spectrum via the quotient \(G \to W_G H\) on the left, and \(E^H\) has its \(W_G H\)-action on the right.

This adjunction is the categorical (non-derived) adjunction; it holds at the point-set level.

3.3 Failure of Homotopy-Invariance

The categorical fixed-point functor is not homotopy-invariant: a \(G\)-weak equivalence \(f: E \to E'\) does not in general induce a weak equivalence \(f^H: E^H \to (E')^H\) unless \(E\) and \(E'\) are fibrant.

Proposition. The categorical fixed-point functor \((-)^H\) is homotopy-invariant on the full subcategory of \(H\)-fibrant \(G\)-spectra — that is, \(G\)-spectra \(E\) for which \(E^K\) is an \(\Omega\)-spectrum for all \(K \leq H\).

In practice, to compute \(\pi_*^H(E)\) one computes \(\pi_*(R_f(E)^H)\) where \(R_f E\) is a fibrant replacement of \(E\) in the genuine model structure.

3.4 Exercises

4. Homotopy Fixed Points 💡

4.1 Definition

Definition (Homotopy Fixed Points). For \(E \in \mathrm{Sp}^G\) and \(H \leq G\), the homotopy fixed-point spectrum is

\[E^{hH} = F(EH_+, E)^H,\]

the categorical \(H\)-fixed points of the function spectrum from \(EH_+\) to \(E\). Here \(EH = E\mathcal{F}_0 H\) is any contractible free \(H\)-CW complex (the choice does not matter up to homotopy), and \(F(-, -)\) denotes the internal hom in \(\mathrm{Sp}^G_O\).

Equivalently, viewing \(E\) as an \(H\)-spectrum via restriction: \[E^{hH} = \mathrm{Map}_H(EH_+, E) = \mathrm{holim}_{H} E,\] the homotopy limit of the constant \(H\)-diagram with value \(E\).

4.2 Homotopy Invariance

Proposition. The homotopy fixed-point functor \((-)^{hH}: \mathrm{Sp}^G \to \mathrm{Sp}^{W_G H}\) is homotopy-invariant: a \(G\)-weak equivalence \(E \xrightarrow{\sim} E'\) induces a weak equivalence \(E^{hH} \xrightarrow{\sim} (E')^{hH}\).

Proof sketch. Since \(EH\) is a free \(H\)-CW complex, \(EH_+\) is cofibrant as an \(H\)-space. The functor \(F(EH_+, -)\) preserves weak equivalences between fibrant objects (it is right Quillen), and the categorical fixed-point functor \((-)^H\) preserves equivalences between fibrant \(H\)-spectra. Combining: \((-)^{hH} = F(EH_+, -)^H\) is homotopy-invariant on all objects (since \(EH_+\) is cofibrant and the functor is derived). \(\square\)

The contrast with categorical fixed points is sharp: the whole point of \(EH_+\) is to provide a “fattening” of the point that makes the fixed-point construction homotopy-invariant.

4.3 The Homotopy Fixed Point Spectral Sequence

The homotopy fixed point spectral sequence (HFPSS) is the main computational tool for \(E^{hH}\).

Theorem (HFPSS). For \(E \in \mathrm{Sp}^H\) and a finite group \(H\), there is a strongly convergent spectral sequence

\[E_2^{p,q} = H^p(H;\, \pi_q(E)) \;\Longrightarrow\; \pi_{q-p}(E^{hH}),\]

where \(H^p(H; \pi_q(E))\) is group cohomology with coefficients in the \(H\)-module \(\pi_q(E)\) (with \(H\)-action induced by the \(H\)-action on \(E\)).

4.4 Exercises

5. Geometric Fixed Points 🔑

5.1 Definition via the Cofiber Sequence

Definition (Geometric Fixed Points). For \(E \in \mathrm{Sp}^G\) and \(H \leq G\), the geometric fixed-point spectrum is

\[\Phi^H E = \left(\widetilde{E\mathcal{P}(H)} \wedge E\right)^H,\]

where \(\mathcal{P}(H) = \{K \leq H : K \subsetneq H\}\) is the family of proper subgroups of \(H\), and \((-)^H\) denotes categorical fixed points (applied after fibrant replacement).

From the cofiber sequence \(E\mathcal{P}(H)_+ \to S^0 \to \widetilde{E\mathcal{P}(H)}\), smashing with \(E\) and taking \(H\)-fixed points gives a cofiber sequence:

\[\left(E\mathcal{P}(H)_+ \wedge E\right)^H \longrightarrow E^H \longrightarrow \Phi^H E.\]

This displays \(\Phi^H E\) as the cofiber of the natural map from \(E\mathcal{P}(H)\)-homotopy-orbits to the categorical fixed points.

Equivalently, one characterizes \(\widetilde{E\mathcal{P}(H)}\) by its fixed points: \[(\widetilde{E\mathcal{P}(H)})^K \simeq \begin{cases} S^0 & \text{if } K = H \\ * & \text{if } K \subsetneq H. \end{cases}\]

So \(\widetilde{E\mathcal{P}(H)} \wedge E\) “kills” the \(K\)-equivariant information for all \(K \subsetneq H\), and the \(H\)-fixed points see only the “purely \(H\)-equivariant” part.

5.2 Monoidality and the Sphere Spectrum

Theorem (Monoidality of Geometric Fixed Points). The functor \(\Phi^H: \mathrm{Sp}^G \to \mathrm{Sp}^{W_G H}\) is strong symmetric monoidal:

\[\Phi^H(E \wedge F) \simeq \Phi^H E \wedge \Phi^H F, \quad \Phi^H \mathbb{S} \simeq \mathbb{S}.\]

Proof sketch. By the formula \(\Phi^H E = (\widetilde{E\mathcal{P}(H)} \wedge E)^H\) and the fact that \(\widetilde{E\mathcal{P}(H)}\) is a commutative \(H\)-ring spectrum (as the cofiber of a unit map for a Thom isomorphism), the smash product distributes: \[\Phi^H(E \wedge F) = (\widetilde{E\mathcal{P}(H)} \wedge E \wedge F)^H \simeq (\widetilde{E\mathcal{P}(H)} \wedge E)^H \wedge (\widetilde{E\mathcal{P}(H)} \wedge F)^H,\] where the last step uses the diagonal \(\widetilde{E\mathcal{P}(H)} \to \widetilde{E\mathcal{P}(H)} \wedge \widetilde{E\mathcal{P}(H)}\) (since \(\widetilde{E\mathcal{P}(H)}\) is idempotent for the smash product: \(\widetilde{E\mathcal{P}(H)} \wedge \widetilde{E\mathcal{P}(H)} \simeq \widetilde{E\mathcal{P}(H)}\)). For the sphere spectrum: \(\Phi^H \mathbb{S} = (\widetilde{E\mathcal{P}(H)})^H = S^0\) (from the fixed-point formula), and \((\Phi^H \mathbb{S})_V = (S^V)^H = S^{V^H}\), which is exactly the level spaces of \(\mathbb{S}\) as a \(W_G H\)-spectrum. \(\square\)

Corollary. \(\Phi^H\) sends commutative \(G\)-ring spectra to commutative \(W_G H\)-ring spectra.

Key computations of geometric fixed points:

\[\Phi^H(\Sigma^\infty_+ G/K) \simeq \begin{cases} \Sigma^\infty_+ W_G H & \text{if } K \sim_G H \text{ (conjugate to } H) \\ * & \text{if } K \not\sim_G H. \end{cases}\]

Proof: \(G/K\) is a transitive \(G\)-set. Its \(H\)-fixed points \((G/K)^H = \{gK : g^{-1}Hg \leq K\}\); this is non-empty iff \(H\) is conjugate (in \(G\)) into \(K\). The geometric fixed points are controlled by \(\Phi^H(\Sigma^\infty_+ G/K) = \Sigma^\infty_+(G/K)^H_{\text{geometric}} = \Sigma^\infty_+ (G/K)^H\) when \(K \sim_G H\) and \(*\) otherwise (since all proper-\(H\)-subgroup fixed points are killed). \(\square\)

5.3 Change of Group for Geometric Fixed Points

For a subgroup \(K \leq H \leq G\), there is a transitivity formula:

\[\Phi^H \simeq \Phi^{H/K} \circ \Phi^K\]

when \(K \trianglelefteq H\) is normal in \(H\) (so \(H/K\) is a group acting on \(\Phi^K E\)). More generally, geometric fixed points satisfy:

Proposition (Wirthmuller compatibility). For \(H \leq G\) and \(E \in \mathrm{Sp}^H\), there is a natural equivalence

\[\Phi^H(G_+ \wedge_H E) \simeq W_G H_+ \wedge_{\{e\}} \Phi^H E,\]

where \(W_G H\) acts on \(\Phi^H E\) via the residual Weyl group action. In particular, when \(H = G\): \[\Phi^G(G_+ \wedge_G E) = \Phi^G(E).\]

5.4 Exercises

6. The Tate Construction 📐

6.1 Historical Origins: Tate Cohomology

The name “Tate construction” traces directly to Tate cohomology, a \(\mathbb{Z}\)-graded cohomology theory for finite groups introduced by J. Tate in the early 1950s in the context of class field theory. Tate’s insight was to “complete” ordinary group cohomology into a theory that works in all integer degrees — including negative ones — and that is naturally periodic for cyclic groups.

Definition (Tate cohomology). Let \(G\) be a finite group, \(M\) a \(\mathbb{Z}[G]\)-module. The norm element is \(N = \sum_{g \in G} g \in \mathbb{Z}[G]\), giving an additive map \(N: M \to M\) with \(\mathrm{Im}(N) \subset M^G\). Tate cohomology is the \(\mathbb{Z}\)-graded abelian group defined by:

\[\hat{H}^n(G; M) = \begin{cases} H^n(G; M) & n \geq 1 \\ M^G / N(M) & n = 0 \\ \ker(N: M \to M) / I_G \cdot M & n = -1 \\ H_{-(n+1)}(G; M) & n \leq -2, \end{cases}\]

where \(I_G = \ker(\mathbb{Z}[G] \to \mathbb{Z})\) is the augmentation ideal and \(H_k(G; M)\) is ordinary group homology.

The definition splices together cohomology and homology via the norm map, which in the derived sense is the map

\[N: M_{hG}[-1] \longrightarrow M^G,\]

or more precisely the natural map between coinvariants and invariants induced by \(\sum_{g \in G} g\). The Tate cohomology groups are precisely the homotopy groups of the cofiber of this norm map:

\[\hat{H}^n(G; M) = \pi_{-n}\left(\mathrm{cofib}\!\left(HM_{hG} \xrightarrow{N} HM^{hG}\right)\right).\]

Key algebraic properties of Tate cohomology that the spectral construction lifts: 1. Periodicity for cyclic groups: \(\hat{H}^n(C_m; M) \cong \hat{H}^{n+2}(C_m; M)\) for all \(n\), with \(\hat{H}^0(C_m; \mathbb{Z}) = \mathbb{Z}/m\). 2. Tate acyclicity: \(\hat{H}^n(G; \mathbb{Z}[G]) = 0\) for all \(n\) — Tate cohomology vanishes on induced (free) modules. 3. Long exact sequences: a short exact sequence \(0 \to M' \to M \to M'' \to 0\) gives a long exact sequence in \(\hat{H}^*\). 4. Künneth formula: \(\hat{H}^*(G; M \otimes N) \cong \hat{H}^*(G; M) \otimes_{H^*(G;\mathbb{Z})} H^*(G; N)\) in favorable cases.

The spectral Tate construction \(E^{tG}\) is the direct categorification of all four of these properties to the \(\infty\)-category of spectra.

6.2 The Norm Map: ∞-Categorical Construction

The modern treatment by Nikolaus–Scholze (2018) rebuilds the Tate construction from scratch in the \(\infty\)-categorical setting, giving a clean universal characterization. Their key move is to construct the norm map as a natural transformation in any preadditive \(\infty\)-category.

Setup. Let \(\mathcal{C}\) be a stable \(\infty\)-category (additive, all finite limits and colimits exist, \(\Omega: \mathcal{C} \to \mathcal{C}\) is an equivalence). For a finite group \(G\), a \(G\)-equivariant object in \(\mathcal{C}\) is a functor \(BG \to \mathcal{C}\), where \(BG\) is the one-object \(\infty\)-groupoid with \(\pi_1 = G\). The \(\infty\)-category of \(G\)-equivariant objects is \(\mathcal{C}^{BG} = \mathrm{Fun}(BG, \mathcal{C})\).

The homotopy orbit and homotopy fixed point functors are the colimit and limit over \(BG\): \[X_{hG} = \mathrm{colim}_{BG}\, X, \qquad X^{hG} = \lim_{BG}\, X.\]

Definition (Norm Map, NS Definition I.1.10). For a finite group \(G\) and \(X \in \mathcal{C}^{BG}\), the norm map is the natural transformation

\[\mathrm{Nm}_G: X_{hG} \longrightarrow X^{hG}\]

constructed as follows. Since \(BG\) is a finite \(\infty\)-groupoid, the diagonal \(\delta: BG \to BG \times BG\) is a relatively finite map. The norm is the composite

\[\mathrm{colim}_{BG}\, X \xrightarrow{\delta_!(\mathrm{id})} \mathrm{colim}_{BG \times BG}(p_1^* X) \xrightarrow{\sim} \lim_{BG}\, \mathrm{colim}_{BG}\, p_1^* X \xrightarrow{\varepsilon} \lim_{BG}\, X,\]

where the middle step uses the Beck–Chevalley isomorphism (base change along \(BG \to *\)), and the last step uses the counit of the \((\mathrm{colim}, \mathrm{res})\) adjunction along the fibers.

BG × BG ─→ BG
   │           │
   ↓           ↓
  BG  ──→  *

Proposition (NS Lemma I.1.8–I.1.9). In any stable \(\infty\)-category \(\mathcal{C}\): 1. The norm map \(\mathrm{Nm}_G: X_{hG} \to X^{hG}\) exists naturally in \(X \in \mathcal{C}^{BG}\). 2. When \(\mathcal{C} = \mathrm{Ab}\) (discrete stable category) and \(X = M\) is a \(G\)-module, \(\mathrm{Nm}_G\) recovers the classical norm map \(M_G \xrightarrow{\bar{N}} M^G\), \([m] \mapsto \sum_{g \in G} g \cdot m\). 3. \(\mathrm{Nm}_G\) is natural in \(\mathcal{C}\): any exact functor \(F: \mathcal{C} \to \mathcal{D}\) carries \(\mathrm{Nm}_G^{\mathcal{C}}\) to \(\mathrm{Nm}_G^{\mathcal{D}}\).

Comparison with the smash-product definition. For \(\mathcal{C} = \mathrm{Sp}\) (spectra), the norm map \(\mathrm{Nm}_G: X_{hG} \to X^{hG}\) coincides with the classical map induced by the \(G\)-equivariant map \(EG_+ \to S^0\) (collapse of the free space):

\[X_{hG} = (EG_+ \wedge X)^G \xrightarrow{(f \wedge \mathrm{id})^G} (S^0 \wedge X)^G = X^G \xrightarrow{\rho} X^{hG} = F(EG_+, X)^G,\]

where \(f: EG_+ \to S^0\) and \(\rho: X^G \to X^{hG}\) is the categorical-to-homotopy comparison.

6.3 Definition of the Tate Spectrum

Definition (Tate Spectrum, NS Definition I.1.13). For \(\mathcal{C}\) a stable \(\infty\)-category, \(G\) a finite group, and \(X \in \mathcal{C}^{BG}\), the Tate construction is the cofiber of the norm map:

\[X^{tG} := \mathrm{cofib}\!\left(\mathrm{Nm}_G : X_{hG} \longrightarrow X^{hG}\right).\]

In the case \(\mathcal{C} = \mathrm{Sp}\), this gives a cofiber sequence of spectra:

\[X_{hG} \xrightarrow{\mathrm{Nm}_G} X^{hG} \longrightarrow X^{tG}\]

and there is a natural equivalence

\[X^{tG} \simeq X^{hG} \wedge \widetilde{EG} = F(EG_+, X)^G \wedge \widetilde{EG},\]

where \(\widetilde{EG}\) is the cofiber of \(EG_+ \to S^0\). These two descriptions agree because smashing \(X^{hG}\) with the cofiber sequence \(EG_+ \to S^0 \to \widetilde{EG}\) gives a cofiber sequence whose first map \(X^{hG} \wedge EG_+ \to X^{hG}\) is null in the homotopy category (since \(EG\) is contractible, so \(EG_+\) is \(S^0\)-equivalent after inverting the free action), with cofiber \(X^{hG} \wedge \widetilde{EG} = X^{tG}\).

Homotopy groups. For \(E = HM\) (an Eilenberg–Mac Lane spectrum for a \(\mathbb{Z}[G]\)-module \(M\)), the long exact sequence of the cofiber sequence \(HM_{hG} \xrightarrow{N} HM^{hG} \to HM^{tG}\) gives:

\[\cdots \to H_n(G; M) \xrightarrow{N_*} H^n(G; M) \to \pi_{-n}(HM^{tG}) \to H_{n-1}(G; M) \to \cdots\]

where \(N_*\) is the classical norm map. Since \(\pi_{-n}(HM^{tG}) = \hat{H}^n(G; M)\) (Tate cohomology), this long exact sequence is exactly the defining long exact sequence of Tate cohomology. The spectral construction thus lifts Tate’s 1952 definition verbatim.

6.4 Tate Acyclicity and the Verdier Quotient Perspective

The most conceptually clarifying result of Nikolaus–Scholze §I.3 is the identification of the Tate construction as the Verdier quotient that kills induced spectra.

Definition (Induced Spectra). Let \(\mathrm{Sp}^{BG}_{\mathrm{ind}} \subset \mathrm{Sp}^{BG}\) be the full stable subcategory generated by induced spectra — \(G\)-spectra of the form \(\bigoplus_{g \in G} X\) (the direct sum of \(|G|\) copies of a non-equivariant spectrum \(X\), with \(G\) acting by permuting summands). In terms of functors \(BG \to \mathrm{Sp}\), these are the objects in the image of the left Kan extension \(\mathrm{Sp} \xrightarrow{\mathrm{ind}} \mathrm{Sp}^{BG}\).

Lemma (Tate Acyclicity, NS Lemma I.3.8). For any induced spectrum \(X \in \mathrm{Sp}^{BG}_{\mathrm{ind}}\), the Tate spectrum vanishes: \[X^{tG} \simeq 0.\]

Proof. An induced spectrum is of the form \(\mathrm{ind}(Y) = G_+ \wedge Y\) (with permutation \(G\)-action). By direct computation, \((\mathrm{ind}(Y))_{hG} = Y\) and \((\mathrm{ind}(Y))^{hG} = Y\) (both collapse to \(Y\)), and the norm map \(\mathrm{Nm}_G\) is the identity \(Y \xrightarrow{\mathrm{id}} Y\), so \(\mathrm{cofib}(\mathrm{id}) = 0\). \(\square\)

Proposition (NS Lemma I.3.8). \(\mathrm{Sp}^{BG}_{\mathrm{ind}}\) is a tensor ideal in \(\mathrm{Sp}^{BG}\): for any \(X \in \mathrm{Sp}^{BG}\) and \(Y \in \mathrm{Sp}^{BG}_{\mathrm{ind}}\), the smash product \(X \wedge Y \in \mathrm{Sp}^{BG}_{\mathrm{ind}}\).

Proof. \(X \wedge (G_+ \wedge Z) = G_+ \wedge (X \wedge Z)\) (distributing the \(G\)-action), which is induced. \(\square\)

Definition (Verdier Quotient). The Verdier quotient of \(\mathrm{Sp}^{BG}\) by \(\mathrm{Sp}^{BG}_{\mathrm{ind}}\) is the stable \(\infty\)-category

\[\mathrm{Sp}^{BG} \big/ \mathrm{Sp}^{BG}_{\mathrm{ind}},\]

defined as the localization at the class of morphisms \(f: X \to Y\) whose cofiber lies in \(\mathrm{Sp}^{BG}_{\mathrm{ind}}\). By the tensor ideal property, this quotient inherits a symmetric monoidal structure (see §6.5).

Theorem (NS, implicit in §I.3). The Tate construction \(X \mapsto X^{tG}\) factors through the Verdier quotient:

\[\mathrm{Sp}^{BG} \twoheadrightarrow \mathrm{Sp}^{BG}\big/\mathrm{Sp}^{BG}_{\mathrm{ind}} \xrightarrow{\;\sim\;} \mathrm{Sp}^{BG},\]

and the functor \((-)^{tG}: \mathrm{Sp}^{BG} \to \mathrm{Sp}\) is the unique exact functor that: 1. Vanishes on all induced spectra. 2. Agrees with the homotopy fixed-point functor on non-equivariant spectra (i.e., \(X^{tG}|_{X = \text{const}} \simeq 0\)… actually more precisely: the composite \(\mathrm{Sp} \xrightarrow{\mathrm{triv}} \mathrm{Sp}^{BG} \xrightarrow{(-)^{tG}} \mathrm{Sp}\) factors through the Tate construction on the trivially-acted spectrum).

6.5 Canonical Lax Symmetric Monoidal Structure

Theorem (NS Theorem I.3.1 — Uniqueness of Lax Monoidal Structure). The space of pairs: - a lax symmetric monoidal structure on \((-)^{tG}: \mathrm{Sp}^{BG} \to \mathrm{Sp}\), and - a lax symmetric monoidal refinement of the natural transformation \((-)^{hG} \to (-)^{tG}\),

is contractible. In other words, the Tate construction carries a unique lax symmetric monoidal structure compatible with homotopy fixed points.

Proof strategy. Apply the Multiplicative Verdier Localization theorem (NS Theorem I.3.6): if \(\mathcal{D} \subset \mathcal{C}\) is a tensor ideal in a symmetric monoidal stable \(\infty\)-category \(\mathcal{C}\), the Verdier quotient \(\mathcal{C}/\mathcal{D}\) inherits a unique symmetric monoidal structure making the projection \(\mathcal{C} \to \mathcal{C}/\mathcal{D}\) symmetric monoidal. Since \(\mathrm{Sp}^{BG}_{\mathrm{ind}} \subset \mathrm{Sp}^{BG}\) is a tensor ideal, the quotient \(\mathrm{Sp}^{BG}/\mathrm{Sp}^{BG}_{\mathrm{ind}}\) is uniquely symmetric monoidal. The Tate construction factors through this quotient, inheriting the lax monoidal structure for free. The contractibility of the space of structures follows from the universal property of the localization. \(\square\)

Consequence. For an \(\mathbb{E}_\infty\)-ring spectrum \(R\) with \(G\)-action, \(R^{tG}\) is canonically an \(\mathbb{E}_\infty\)-ring spectrum, and the map \(R^{hG} \to R^{tG}\) is a map of \(\mathbb{E}_\infty\)-rings.

6.6 The Tate Orbit and Fixpoint Lemmas

These two lemmas from NS §I.2 are the key technical inputs for comparing genuine and naive cyclotomic spectra. They are stated for \(C_{p^2}\)-spectra and control Tate spectra of fixed-point spectra.

Lemma (Tate Orbit Lemma, NS Lemma I.2.1). Let \(X\) be a spectrum with \(C_{p^2}\)-action that is bounded below (\(\pi_i(X) = 0\) for \(i \ll 0\)). Then:

\[(X_{hC_p})^{t(C_{p^2}/C_p)} \simeq 0.\]

Explanation. The homotopy orbits \(X_{hC_p}\) of \(X\) over the subgroup \(C_p \leq C_{p^2}\) inherit a residual \((C_{p^2}/C_p) \cong C_p\)-action. The Tate construction of this action on \(X_{hC_p}\) vanishes for bounded-below spectra.

Lemma (Tate Fixpoint Lemma, NS Lemma I.2.2). Let \(X\) be a spectrum with \(C_{p^2}\)-action that is bounded above (\(\pi_i(X) = 0\) for \(i \gg 0\)). Then:

\[(X^{hC_p})^{t(C_{p^2}/C_p)} \simeq 0.\]

Proof of Tate Orbit Lemma (sketch). The key is to reduce to Eilenberg–Mac Lane spectra \(HM\) for \(\mathbb{F}_p[C_{p^2}]\)-modules \(M\). In this case, \((HM_{hC_p})^{t(C_{p^2}/C_p)} = H\hat{H}^*(C_p; M)^{t(C_p)}\) (Tate cohomology of \(C_p\) acting on \(M\), then Tate again). One then computes:

• The group cohomology of \(C_p\) acting on \(\mathbb{F}_p\) is \(\mathbb{F}_p[u, u^{-1}]\) with \(|u| = 2\).
• The outer \(C_p\)-action on \(H^*(C_p; \mathbb{F}_p) = \mathbb{F}_p[u, u^{-1}]\) factors through a module over \(\mathbb{F}_p[C_p]\), and one checks that the \(C_p\)-Tate cohomology of \(\mathbb{F}_p[u, u^{-1}]\) (with appropriate \(C_p\)-action) vanishes.
• The general bounded-below case reduces to the \(\mathbb{F}_p\)-case by the Postnikov truncation and a five-lemma argument.

Corollary (for \(TC\)-theory). For \(X = \mathrm{THH}(R)\) (topological Hochschild homology of a connective ring spectrum \(R\)), both lemmas apply (since \(\mathrm{THH}(R)\) is bounded below). This is why NS can define \(\mathrm{TC}(R)\) purely in terms of naive cyclotomic structure (the Frobenius \(\varphi_p\)) without needing genuine \(C_{p^\infty}\)-equivariant structure.

6.7 The Tate Diagonal

The Tate diagonal is one of the most remarkable constructions in NS — a multiplicative map from a non-equivariant spectrum to its \(p\)-fold Tate construction, with no equivariant input required.

Definition (Tate Diagonal, NS Definition III.1.4). For a prime \(p\) and a spectrum \(X \in \mathrm{Sp}\), the Tate diagonal is the natural map

\[\Delta_p: X \longrightarrow (X^{\otimes p})^{tC_p},\]

where \(C_p\) acts on \(X^{\otimes p}\) by cyclic permutation of tensor factors.

The existence of \(\Delta_p\) is non-obvious: there is no diagonal map \(X \to X^{\otimes p}\) for a general spectrum (unlike for spaces, where the diagonal \(X \to X^p\) is available). The Tate construction “corrects” for the failure of commutativity: while \(X^{\otimes p}\) does not have a canonical \(X\)-algebra map from \(X\), after taking the \(C_p\)-Tate construction the correction terms (coming from the failure of \(X\) to be strictly commutative) are killed.

Construction. By NS Proposition III.1.2, exact functors \(\mathrm{Sp} \to \mathrm{Sp}\) are classified by their value on the sphere spectrum: \(\mathrm{Fun}^{\mathrm{ex}}(\mathrm{Sp}, \mathrm{Sp}) \simeq \mathrm{Sp}\), \(F \mapsto F(\mathbb{S})\). Applied to the functor \(T_p(X) = (X^{\otimes p})^{tC_p}\) (which is exact by NS Proposition III.1.1), the Tate diagonal is the map corresponding to

\[\mathbb{S} \longrightarrow T_p(\mathbb{S}) = (\mathbb{S}^{\otimes p})^{tC_p} = \mathbb{S}^{tC_p},\]

which is the composite \(\mathbb{S} \to \mathbb{S}_{hC_p} \xrightarrow{N} \mathbb{S}^{hC_p} \to \mathbb{S}^{tC_p}\) (the norm cofiber sequence applied to the sphere).

Theorem (NS Theorem III.1.7 — Tate Diagonal and \(p\)-Completion). For any spectrum \(X\) that is bounded below, the Tate diagonal

\[\Delta_p: X \longrightarrow (X^{\otimes p})^{tC_p}\]

exhibits \((X^{\otimes p})^{tC_p}\) as the \(p\)-completion of \(X\):

\[\Delta_p: X \xrightarrow{\;\sim\;} \left((X^{\otimes p})^{tC_p}\right)^\wedge_p.\]

Relevance to \(\mathbb{E}_\infty\)-rings. For an \(\mathbb{E}_\infty\)-ring spectrum \(R\), the multiplication gives a canonical Frobenius map:

\[\varphi_p: R \xrightarrow{\Delta_p} (R^{\otimes p})^{tC_p} \xrightarrow{\mu^{tC_p}} R^{tC_p},\]

where \(\mu: R^{\otimes p} \to R\) is the \(p\)-fold multiplication. This Frobenius \(\varphi_p: R \to R^{tC_p}\) is the structure map making \(R\) a cyclotomic spectrum in the naive sense.

flowchart TD
    A["R (E∞-ring)"] -->|"Δ_p (Tate diagonal)"| B["(R⊗p)^{tCp}"]
    B -->|"μ^{tCp} (mult. map)"| C["R^{tCp}"]
    A -->|"φ_p = Frobenius"| C
    C --> D["TC(R) = fib(φ_p - can: THH(R)^{hT} → THH(R)^{tT})"]

6.8 The Tate Diagram

The three fixed-point functors and the Tate spectrum are related by the Tate diagram, a commutative square that is simultaneously a homotopy pushout and (under favorable conditions) a homotopy pullback.

Theorem (Tate Diagram). For any \(G\)-spectrum \(E\) and \(H \leq G\), there is a commutative square

where: - \(\rho: E^H \to E^{hH}\) is the natural comparison from categorical to homotopy fixed points; - \(\pi: E^H \to \Phi^H E\) arises from \(S^0 \to \widetilde{E\mathcal{P}(H)}\); - the square is a homotopy pushout.

The two rows of the expanded diagram are cofiber sequences:

\[E_{hH} \longrightarrow E^H \xrightarrow{\pi} \Phi^H E \quad \text{(from } E\mathcal{P}(H)_+ \to S^0 \to \widetilde{E\mathcal{P}(H)}\text{)}\]

\[E_{hH} \xrightarrow{\mathrm{Nm}_H} E^{hH} \xrightarrow{} E^{tH} \quad \text{(norm cofiber sequence)}\]

These two cofiber sequences share the left term \(E_{hH}\) and assemble into the pushout square by the universal property. In the expanded \(3 \times 2\) diagram:

both rows are cofiber sequences, the left square commutes, and the right square is the Tate diagram (a homotopy pushout).

The Tate diagram as a pullback. The square is also a homotopy pullback when the Tate spectrum of the categorical-to-homotopy comparison is trivial. For \(p\)-complete spectra with \(p\)-group action:

6.9 The Segal Conjecture

The Segal conjecture (proven by Carlsson in 1984 for \(p\)-groups, with later generalizations) describes the Tate spectrum of the sphere spectrum and establishes that the Tate diagram is a homotopy pullback \(p\)-adically.

Theorem (Carlsson 1984). Let \(G\) be a finite \(p\)-group. The natural comparison map

\[\rho: \mathbb{S}^G \longrightarrow \mathbb{S}^{hG}\]

from categorical to homotopy \(G\)-fixed points of the sphere spectrum induces an isomorphism on \(p\)-completed homotopy groups. Equivalently:

1. The norm map \(\mathrm{Nm}_G: \mathbb{S}_{hG} \to \mathbb{S}^{hG}\) is a \(p\)-adic equivalence.
2. The Tate spectrum \(\mathbb{S}^{tG}\) is contractible after \(p\)-completion.

In the NS formalism, statement (2) says exactly that the Tate diagonal \(\Delta_p: \mathbb{S} \to \mathbb{S}^{tC_p}\) is a \(p\)-adic equivalence (when restricted to \(C_p\)-groups), which is the base case of their Theorem III.1.7.

Reformulation via the Tate diagonal (NS). For any bounded-below spectrum \(X\) (NS Theorem III.1.7 applied to \(X = \mathbb{S}\)):

\[\Delta_p: \mathbb{S} \xrightarrow{\;\sim_p\;} (\mathbb{S}^{\otimes p})^{tC_p} = \mathbb{S}^{tC_p}\]

is a \(p\)-adic equivalence. This is both a clean restatement of the Segal conjecture and a template for NS’s proof of the general statement: they reduce to \(X = H\mathbb{F}_p\) (where it’s explicit) and then bootstrap.

6.10 Exercises

7. The Adams Isomorphism 💡

7.1 Statement

The Adams isomorphism is a fundamental comparison between categorical fixed points and quotients for free spectra.

Theorem (Adams Isomorphism). Let \(H \leq G\) be a finite group and let \(X \in \mathrm{Sp}^H\) be an \(H\)-spectrum. If \(X\) is a free \(H\)-spectrum (i.e., \(X\) is built from cells of the form \(H_+ \wedge D^n\)), then there is a natural equivalence of non-equivariant spectra:

\[(G_+ \wedge_H X)^G \xrightarrow{\;\sim\;} X / H = X_{hH},\]

i.e., the \(G\)-fixed points of the induced \(G\)-spectrum agree with the \(H\)-homotopy-orbits of \(X\).

More precisely: for \(G\)-free \(X\) (a \(G\)-spectrum with free underlying \(H\)-action):

\[X^G \simeq X_{hG}.\]

7.2 Proof Sketch

The key input is the equivariant Pontryagin–Thom construction. For a free \(H\)-spectrum \(X\), consider the diagram:

flowchart LR
    A["X (free H-spectrum)"] -->|"G_+ ∧_H -"| B["G_+ ∧_H X (G-spectrum)"]
    B -->|"(-)^G"| C["(G_+ ∧_H X)^G"]
    A -->|"(-)/H = (-)_{hH}"| D["X/H = X_{hH}"]
    C -->|"≃ Adams"| D

Proof: By the categorical fixed-point adjunction, \([Y, (G_+ \wedge_H X)^G]_{\mathrm{Sp}} \cong [G/G_+ \wedge Y, G_+ \wedge_H X]^G \cong [Y, X]^H\) for any non-equivariant spectrum \(Y\). On the other hand, \([Y, X_{hH}]_{\mathrm{Sp}} = [Y \wedge EH_+, X]^H\) (by the definition of homotopy orbits as the left adjoint to homotopy fixed points). Since \(X\) is \(H\)-free, there is a zig-zag of equivalences \([Y, X]^H \simeq [Y \wedge EH_+, X]^H\) (using the freeness to replace \(Y\) by \(Y \wedge EH_+\) without changing homotopy classes). This gives a natural equivalence of mapping spectra, hence of the spectra themselves. \(\square\)

7.3 Consequences and the tom Dieck Splitting

The Adams isomorphism, combined with the Wirthmuller isomorphism, gives the tom Dieck splitting — a structural theorem about the \(G\)-fixed points of the equivariant sphere spectrum.

Theorem (tom Dieck Splitting). For a finite group \(G\),

\[(\Sigma^\infty_+ X)^G \simeq \bigvee_{(H) \leq G} \Sigma^\infty_+(EW_GH \times_{W_GH} X^H),\]

where the wedge is over conjugacy classes of subgroups \(H \leq G\), \(W_G H = N_G H / H\) is the Weyl group, and \(X^H\) denotes the \(H\)-fixed points of the \(G\)-space \(X\).

Proof sketch. Filter \(X\) by skeleton. Each \(G\)-cell \(G/H_+ \wedge D^n\) contributes a piece. By the Adams isomorphism, the \(G\)-fixed points of \(G_+ \wedge_H F\) (for an \(H\)-free spectrum \(F\)) are \(F_{hH}\). Assembling these pieces using the cofiber sequences for each cell dimension gives the tom Dieck splitting.

7.4 Exercises

8. Examples: \(C_2\)-Spectra 🔑

8.1 Fixed Points of \(H\underline{\mathbb{Z}}\)

The Eilenberg–Mac Lane spectrum \(H\underline{\mathbb{Z}}\) for the constant Mackey functor \(\underline{\mathbb{Z}}\) is one of the most instructive examples. Here \(\underline{\mathbb{Z}}(C_2/C_2) = \mathbb{Z} = \underline{\mathbb{Z}}(C_2/e)\) with \(\mathrm{res} = \mathrm{id}\) and \(\mathrm{tr}(n) = 2n\).

The homotopy fixed point spectral sequence for \(H\underline{\mathbb{Z}}^{hC_2}\):

\[E_2^{p,q} = H^p(C_2; \pi_q(H\mathbb{Z})) = H^p(C_2; \mathbb{Z}) \cdot \delta_{q=0} \;\Longrightarrow\; \pi_{-p}(H\underline{\mathbb{Z}}^{hC_2}).\]

The \(E_2\)-page has \(\mathbb{Z}\) at \((0,0)\) and \(\mathbb{Z}/2\) at \((2k, 0)\) for \(k \geq 1\), with all other entries zero. There are no differentials, so:

\[\pi_n(H\underline{\mathbb{Z}}^{hC_2}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/2 & n = -2k, \; k \geq 1 \\ 0 & \text{otherwise.} \end{cases}\]

8.2 Geometric Fixed Points of \(MU_\mathbb{R}\)

The real bordism spectrum \(MU_\mathbb{R} \in \mathrm{Sp}^{C_2}\) is a genuine \(C_2\)-spectrum whose underlying spectrum is \(MU\) (complex cobordism), and the \(C_2\)-action is by complex conjugation. This spectrum plays a central role in the HHR theorem.

Theorem (Geometric fixed points of \(MU_\mathbb{R}\)). There is an equivalence of non-equivariant spectra:

\[\Phi^{C_2}(MU_\mathbb{R}) \simeq MO,\]

the real cobordism (unoriented bordism) spectrum.

Proof sketch. Geometrically, \(\Phi^{C_2}(MU_\mathbb{R})\) represents bordism classes of real manifolds (real loci of complex manifolds with \(C_2\)-action), which are unoriented manifolds. The monoidality \(\Phi^{C_2}(MU_\mathbb{R}) \simeq MO\) follows from the identification at the level of Thom spectra: the Thom space of the tautological real bundle over \(BO\) is exactly \(MO\).

8.3 The \(C_2\)-Tate Sequence for \(H\underline{\mathbb{Z}}\)

The Tate diagram for \(H\underline{\mathbb{Z}}\) with \(G = C_2\):

The bottom-left is \(\Phi^{C_2} H\underline{\mathbb{Z}} \simeq H\mathbb{Z}\), and \(H\mathbb{Z}^{tC_2}\) is the Tate spectrum with 2-periodic homotopy. The right vertical \(H\mathbb{Z}^{hC_2} \to H\mathbb{Z}^{tC_2}\) kills the \(\mathbb{Z}\) in degree 0 and keeps the \(\mathbb{Z}/2\)-periodic part.

The norm cofiber sequence reads:

\[H\mathbb{Z}_{hC_2} \xrightarrow{N} H\mathbb{Z}^{hC_2} \longrightarrow H\mathbb{Z}^{tC_2},\]

with \(\pi_*(H\mathbb{Z}_{hC_2}) = H_*(C_2; \mathbb{Z})\) (group homology) and \(\pi_*(H\mathbb{Z}^{hC_2}) = H^{-*}(C_2; \mathbb{Z})\) (group cohomology), so the long exact sequence of the cofiber sequence recovers Tate cohomology \(\hat{H}^*(C_2; \mathbb{Z})\).

References