The Frobenius and the Tate diagonal eCHT Reading Seminar on THH - PowerPoint PPT Presentation

The Frobenius and the Tate diagonal eCHT Reading Seminar on THH Yuqing Shi University of Utrecht 10.03.2020

Outline 1 The Tate diagonal 2 The Frobenius 3 The Tate construction and the p -completion

Motivation from Algebra Let A be an abelian group. The diagonal map A ⊗ p � C p , a �→ a ⊗ a ⊗ · · · ⊗ a � ∆ C p : A → is not a group homomorphism. One way to fix this: Definition Let M be an abelian group with finite group G -action. The norm map is � Nm G ( M ): M G → M G , m �→ g . m g ∈ G

Motivation from Algebra Proposition The composition ∆ C p A ⊗ p � C p ։ A ⊗ p � C p � A ⊗ p � � � � ∆ p : A − − − → Nm C p is a homomorphism of abelian groups. Moreover, the target of this map is p -torsion, and the induced map A ⊗ p � C p � A ⊗ p � � � A / p → Nm C p is an isomorphism.

Tate diagonal for spectra Recall that for a spectrum Y ∈ Sp G with G a finite group. We have a cofiber sequence Nm G ( Y ) → Y h G → Y t G . Y h G − − − − − The analogues statements in Sp : Theorem ([NS18, Theorem III.1.7, Proposition III.3.1]) For every X ∈ Sp , there is a unique map X ⊗ S p � tC p ∈ Sp � ∆ p ( X ): X → such that it is natural in X and is symmetric monoidal. If X is bounded below, then ( X ⊗ p ) tC p is weak equivalent to the p -completion of X .

Tate diagonal for spectra Definition For every X ∈ Sp , we call the map ∆ p ( X ) the Tate diagonal . Remark Let R be an E n -ring spectra, with 0 ≤ n ≤ ∞ . The map ∆ p ( R ) is a map of E n -ring spectra, because ∆ p ( − ) is symmetric monoidal. Example For the sphere spectrum S , the statement that the map ∆ p ( S ): S → S t C p is a p -completion is equivalent to the Segal conjecture.

The Frobenius Use the Tate diagonals to construct extra structures on the spectra THH( R ) , for R an E n -ring spectrum with 1 ≤ n ≤ ∞ . For simplicity, we do this in detail for E ∞ -ring spectra and sketch the case for E 1 -ring spectra.

A universal property of THH( R ) for R an E ∞ -ring spectra Recall fron last talk that there is a natural T -action on THH( R ) , which is compatible with the lax symmetric monoidal structure on the functor THH( − ) . Proposition (McClure–Schwänzl–Vogt) ι Let R be an E ∞ -ring spectrum. The canonical map R − → THH( R ) induces an equivalence E ∞ (THH( R ) , Z ) ι ∗ Map T − → Map E ∞ ( R , Z ) for every Z ∈ Alg T E ∞ ( Sp ) .

Sketch of the proof of MSV It suffices to see a natural isomorphism THH( R ) ≃ R ⊗ T ∈ Sp E ∞ . Indeed, the adjunction −⊗ T Alg T Fr : Alg E ∞ ( Sp ) E ∞ ( Sp ) : U ⇄ forget implies that the canonical map R → R ⊗ T is initial among all the E ∞ -ring maps from R to Z ∈ Alg T E ∞ ( Sp ) .

Sketch the equivalence THH( R ) ≃ R ⊗ S 1 R ⊗ T := colim T R , where the colimit is taken in the ∞ -category Alg E ∞ ( Sp ) . The standard simplicial model (∆ 1 � ∂ ∆ 1 ) • for the circle has n + 1 simplices in dimension n . R ⊗ T ≃ | R ⊗ (∆ 1 � ∂ ∆ 1 ) • | ≃ THH( R ) . The first weak equivalence is [MSV97, Proposition 4.3]. For the second equivalence, we use the fact that the coproduct in Alg E ∞ ( Sp ) is the tensor product.

The Frobenius for E ∞ -ring spectra Analogously, the canonical map R → THH( R ) induces an E ∞ -ring map R ⊗ C p → THH( R ) which is C p -equivariant.Note that R ⊗ C p ≃ R ⊗ S p . Applying the Tate construction and precompose with the Tate diagonal map, we obtain a morphism ∆ p → ( R ⊗ C p ) tC p → THH( R ) tC p − − R of E ∞ -ring spectra. Note that THH( R ) tC p has a residue T / C p ∼ = T action.

The Frobenius for E ∞ -ring spectra Definition Let R be an E ∞ -ring spectrum. The Frobenius of THH( R ) is the unique T -equivariant, E ∞ -ring spectra morphism ϕ p , such that the diagram ι R THH( R ) ∆ p ϕ p ( R ⊗ C p ) tC p THH( R ) tC p

Sketch: The Frobenius for E 1 -ring spectra Let R be a E 1 -ring spectrum. R ⊗ S 3 R ⊗ S 2 · · · R ∆ p ∆ p ∆ p ( R ) R ⊗ S 3 p � tC p R ⊗ S 2 p � tC p ( R ⊗ S p ) tC p � � · · · The colimit of the upper simplicial ring spectrum is THH( R ) . The colimit of the lower one is THH( R ) tC p . We can extend the Tate diagonal ∆ p ( R ) for R to a map from the upper to the lower simplicial ring spectra. See [NS18, Section III.2] for details.

The Tate construction and the p -completion In this section we want to prove the following Lemma ([NS18, Lemma II.4.2]) Let X be a bounded below spectrum with T -action. The canonical map X t T → X tC p � h ( T / C p ) � is a p -completion. Nm � � X h T X t T Σ X h T ≃ X hC p h T / C p Nm � h T / C p X hC p � h T / C p X tC p � h T / C p Nm � � � Σ X hC p

Direct consequences Let R ∈ Alg ≥ 0 E 1 ( Sp ) . Then THH( R ) satisfies the hypothesis of the above lemma. The maps THH( R ) h T ϕ h T THH( R ) t T � ∧ THH( R ) tC p � h T Lemma � p � − − → ≃ p , for all primes p , induce a map THH( R ) t T � ∧ THH( R ) t T � ∧ ϕ : THH( R ) h T → � � � := p , p where ( − ) ∧ denotes the profinite completion. Furthermore, the map ϕ fits into a fibre sequence TC( R ) → TC( R ) − ϕ → TP( R ) ∧ , − see [NS18, Remark II.4.3].

Proof of the Lemma Lemma Let X be a bounded below spectrum with T -action. The canonical map X t T → X tC p � h ( T / C p ) � is a p -completion. The proof has two main steps: 1 Reduce to the case where X = HM with trivial T -action, where M is a torsion free abelian group. HM hC p � h T / C p and HM tC p � h T / C p � � 2 Compare the HFPSS’s of

Reduction to HM with M torsion free abelian Reduce to the case of Eilenberg–MacLane spectra: Let Y ∈ Sp G . We have commutative diagram: Y h G Y t G Y h G ≃ ≃ ≃ − ( τ ≤ n Y ) h G − ( τ ≤ n Y ) t G lim − ( τ ≤ n Y ) h G lim lim ← ← ← Limits of p -complete objects are p -complete. Reduction to the case HM with M a torsion free abelian group X tC p � h T / C p is p-complete X tC p is p -torsion. In particular, � Use a 2-term resolution

Reformulate To prove: Lemma For a torsion free abelian group M and the spectrum HM with trivial T -action, the canonical map HM t T → HM tC p � h T / C p � is a p -completion.

Reduction again Convergence of the Tate spectral sequences of HM t T and HM tC p leads to: HM t T ≃ Σ 2 i HM Σ 2 i H M / p HM tC p ≃ � � and i ∈ Z i ∈ Z In other words, HM t T is a periodised HM h T . Thus, it suffices to prove Lemma The induced map on homotopy groups on each negative even HM hC p � h T → HM tC p � h T is degrees, by the natural map HM h T ≃ � � the p -completion M → M ∧ p .

HFPSS HM hC p � h T and � To prove the lemma, we compare the HFPSS’s of HM tC p � h T , respectively: � �� HM hC p � h T � E 2 p , q = H − p ( B T ; π q ( HM hC p )) ⇒ π p + q �� HM tC p � h T � E 2 p , q = H − p ( B T ; π q ( HM tC p )) ⇒ π p + q Remark Note that we obtain the HFPSS for X h G ≃ F( EG , X ) G by either filtering E G by a suitable filtration, or we consider the tower of fibration induced by the Postnikov tower of X .

HFPSS In the following picture, we set • = M and ◦ = M / p HM hC p � h T HM tC p � h T � � 2 2 0 0 -4 -2 -4 -2 -2 -2 -4 -4 Note that both spectral sequences collapse at E 2 -page.

HM hC p � h T � HFPSS of � ( HM ) h T � The spectral sequence computes π 2 k ≃ M �� HM hC p � h T � From the associated graded, we know π 2 k is endowed with the filtration p k M ⊆ p k − 1 M ⊆ · · · ⊆ pM ⊆ M Truncate the spectral sequence in x < − 2 n area, we obtain a spectral sequence computing � for − n ≤ i < 0 M , � S 2 n + 1 , HM hC p � T � ∼ � π 2 i F = M / p n + 1 for i < − n

HM tC p � h T � HFPSS of The spectral sequence is “periodic”, because HM tC p is. Truncate the spectral sequence in x < − 2 n area and compare HM hC p � h T , we have � it with the truncated HFPSS of � S 2 n + 1 , HM tC p � T � ≃ M / p n + 1 , for all i ∈ Z , � π 2 i F � � S 2 n + 1 , HM tC p � T �� � The filtration F n ≥ 0 of π 2 i �� HM tC p � h T � π i is complete (?). Thus �� HM tC p � h T � ≃ M ∧ π i p

End of proof HM hC p � h T → HM tC p � h T induces The natural map HM h T ≃ � � the p -completion map M → M ∧ p on each negative even degree homotopy groups. The canonical map HM t T → HM tC p � h T / C p � p , because HM t T is periodised is the p -completion M → M ∧ HM h T . Proof of [NS18, Lemma II.4.2] is complete by the previous reductions.

Summary For X ∈ Sp , we introduce the Tate diagonal X ⊗ S p � tC p . � ∆ p ( X ): X → We defined the Frobenius ϕ : THH( R ) → THH( R ) tC p , for R ∈ Alg E 1 ( Sp ) . We showed that for a bounded below spectrum X with T -action, the canonical map X t T → X tC p � h T / C p � is a p -completion.

References Achim Krause and Thomas Nikolaus. Lectures on topological hochschild homology and cyclotomic spectra. J. McClure, R. Schwänzl, and R. Vogt. = R ⊗ S 1 for E ∞ ring spectra. THH ( R ) ∼ J. Pure Appl. Algebra , 121(2):137–159, 1997. Thomas Nikolaus and Peter Scholze. On topological cyclic homology. Acta Math. , 221(2):203–409, 2018.