RO(S^1)-graded TR-groups of F_p, Z and ell

RO ( S 1 )-graded TR–groups of F p , Z and ℓ Vigleik Angeltv eit ∗ Dep artment of Mathematics, University of Chic ago, 5734 S University Ave, Chic ago IL 60637 T eena Gerhardt A-218 Wel ls Hal l, Michigan State University, East L ansing, MI 48824 Abstract W e giv e an algorithm for calculating the RO ( S 1 )-graded TR–groups of F p , completing the calculation started b y the second author. W e also calculate the RO ( S 1 )-graded TR–groups of Z with mo d p co efficien ts and of the Adams summand ℓ of connectiv e complex K -theory with V (1)-co efficien ts. Some of these calculations are used elsewhere to compute the algebraic K -theory o f certain Z -alg ebras. 1. In tro duction Higher algebraic K -theory asso ciates to a ring or ring s p ectrum A a sp ec- trum K ( A ) and a sequenc e o f ab elian gro ups K i ( A ) whic h are the homotopy groups of this sp ectrum. Although higher algebraic K -theory w as defined more than 30 years ago, computational progress has b een slo w. While the definition of a lg ebraic K -theory is not inheren tly equiv arian t, the to ols of equiv arian t stable homotopy theory hav e prov en useful for K -theory compu- tations via trace metho ds [4]. The equiv a r ian t stable homotopy computa- tions in this pap er serv e as input for these metho ds. In particular they hav e b een used in the computatio ns of the relativ e algebraic K -theory gro ups K ∗ ( Z [ x ] / ( x m ) , ( x )) and K ∗ ( Z [ x, y ] / ( xy ) , ( x, y )) up to extensions (see [2] and [1] resp ectiv ely). ∗ Partially supp orted b y the NSF Email addr esses: vigl eik@mat h.uchicago.edu (Vigleik Angeltv eit), teena@ math.m su.edu (T eena Gerhardt) Pr eprint submitte d to Elsevier Novemb er 11, 2018 The idea b ehind the t race metho ds is to approximate algebraic K -theory b y inv ar ia n ts of ring sp ectra whic h are more computable. The first ap- pro ximation is top o lo gical Ho c hsc hild homology [6], T ( A ). This is signifi- can tly easier t o compute than algebraic K -theory a nd t here is a t r ace map K ( A ) → T ( A ) called the top ological D ennis trace. A refinemen t of to p olog- ical Ho chsc hild homology called top ological cyclic ho mology , TC( A ), serv es as a n ev en b etter approxim ation o f algebraic K -theory . Indeed, there is a map tr c : K ( A ) → TC( A ) called the cyclotomic tr a ce [4] whic h is o ften close to an equiv alence [12, 15, 7]. So in g o o d cases trace metho ds reduce the computation o f algebraic K -theory , K q ( A ), to that of top o lo gical cyclic homology , TC q ( A ). T op ological cyclic homology is defined as a homotopy limit o f certain fixed p o in ts of to p ological Ho c hsc hild homology . Let p b e a prime. The circle S 1 acts on T ( A ) and w e define TR n ( A ; p ) = T ( A ) C p n − 1 to b e t he fixed p oin t sp ectrum under the action of the cyclic group o f order p n − 1 considered as a subgroup o f S 1 . It is imp ort a n t that T ( A ) is a genuine S 1 -equiv arian t sp ectrum, i.e., the spaces of T ( A ) are indexed on a complete univ erse of S 1 - represen tatio ns. F or a gen uine G -sp ectrum E , the H -fixed p oint sp ectrum E H for H ⊂ G has n ’th space E ( R n ) H . These sp ectra are connected by maps R , F , V and d [14], a nd a homo- top y limit ov er R a nd F giv es us the top ological cyclic homology sp ectrum TC( A ; p ). Therefore to compute to p ological cyclic homology , and hence alge- braic K -theory in go o d cases, it is sufficien t to understand TR n ( A ; p ) together with R , F : TR n +1 ∗ ( A ; p ) → TR n ∗ ( A ; p ) fo r eac h p and n . The homotopy groups o f these sp ectra a re denoted TR n q ( A ; p ) = [ S q ∧ S 1 /C p n − 1 + , T ( A )] S 1 . Throughout this pap er the prime p will b e implicit. Hence w e will write TR n q ( A ) f or TR n q ( A ; p ) and TC( A ) fo r TC( A ; p ). One t yp e of singular ring for which the algebraic K -theory is particularly approac hable is a p oin ted monoid a lgebra, A (Π). This a pproac h w as fir st used b y Hesselholt and Madsen [11] to compute the a lgebraic K -theory of F p [ x ] /x m . T o compute the K - theory of A (Π) using the approac h outlined ab o ve one first needs to understand the top o lo gical Ho c hsc hild homology T ( A (Π)). Hesselholt and Madsen [12] prov ed that there is a n equiv alence o f S 1 -sp ectra T ( A (Π)) ≃ T ( A ) ∧ B cy (Π) , (1.1) 2 where B cy (Π) denotes the cyclic bar construction on the p ointed monoid Π. As ab ov e, tr a ce metho ds essen tially reduce the computation of K q ( A (Π)) to that of TR n q ( A (Π)) = π q ( T ( A (Π)) C p n − 1 ) = [ S q ∧ S 1 /C p n − 1 + , T ( A (Π))] S 1 . Using Equation 1.1 w e can r ewrite this as TR n q ( A (Π)) = [ S q ∧ S 1 /C p n − 1 + , T ( A ) ∧ B cy (Π)] S 1 . If one can understand how B cy (Π) is built out of S 1 -represen ta tion spheres this gives a for mula for these TR- g roups in t erms of groups of the form TR n q − λ ( A ) = [ S q ∧ S 1 /C p n − 1 + , T ( A ) ∧ S λ ] S 1 . Here λ is a finite-dimensional S 1 -represen ta tion and S λ denotes the one-p oint compactification of this represen tation. These g roups are R O ( S 1 )-graded equiv arian t homoto p y groups of the S 1 -sp ectrum T ( A ). Recall that RO ( S 1 ) is the ring o f virtual real represen tations of S 1 , meaning that an elemen t α ∈ R O ( S 1 ) can b e written as α = [ β ] − [ γ ] where β and γ are finite- dimensional real S 1 -represen ta tions. F or α = [ β ] − [ γ ] in R O ( S 1 ) t he TR-gro up TR n α ( A ) is defined b y TR n α ( A ) = π α T ( A ) C p n − 1 = [ S β ∧ S 1 /C p n − 1 + , S γ ∧ T ( A )] S 1 , generalizing the integer-graded TR-gr o ups. As described ab ov e, these R O ( S 1 )- graded TR-groups a rise naturally in the computation of the algebraic K - theory of some singular rings. Indeed, in some cases the computation of the algebraic K -theory groups K q ( A (Π)) can b e reduced to the computation of the R O ( S 1 )-graded TR-g roups TR n q − λ ( A ). Ho w ev er, few computations of these RO ( S 1 )-graded TR-groups hav e b een done. The gr o ups TR n α ( A ) are only know n in general when A = F p and the dimension of α is eve n [8]. The curren t pap er broadly extends what is kno wn ab out RO ( S 1 )-graded TR-groups, making computatio ns for A = F p , Z , and ℓ . W e use the results of this pap er in [2], whic h is joint work with Lars Hesselholt, to compute the relativ e K -groups K ∗ ( Z [ x ] / ( x m ) , ( x )) up to ex- tensions, and in [1] to compute the relativ e K -groups K ∗ ( Z [ x, y ] / ( xy ) , ( x, y )) 3 up to extensions. Theorem 1.4 b elo w is t he necessary input to the trace metho d appro a c h described ab ov e, allowing us to mak e suc h computations. F or example, we compute the relative TC-g roups TC ∗ ( Z [ x ] / ( x m ) , ( x ); Z /p ) . Com bined with a rational computation this tells us the rank and the num b er of torsion summands in eac h degree and in particular that TC 2 i +1 ( Z [ x ] / ( x m ) , ( x )) ∼ = Z m − 1 is torsion free. An Euler c haracteristic argument then giv es the order o f the torsion groups. The computations in this pap er are also motiv a ted by our in terest in understanding the algebraic structure satisfied by the RO ( S 1 )-graded TR- groups. The a lgebraic structure satisfied by the ordinary ( Z -gra ded) TR- groups is v ery rigid and this has prov en quite useful [12, 13], for example by considering the univ ersal example. A b etter understanding of t he a lgebraic structure of the RO ( S 1 )-graded TR- groups should b e similarly useful, a nd this is an area for further study . The computations in this pap er provide imp ortan t examples that we hop e will b e helpful in t his regard. Note that in cases where computing TR n ∗ ( A ) with in tegra l co efficien ts pro v es to b e to o difficult one can instead consider the groups TR n ∗ ( A ; V ) = π ∗ ( T ( A ) C p n − 1 ∧ V ) for a suitable finite complex V . F or instance, smashing with the mo d p Mo ore sp ectrum V (0) = S/p w as used in [5 ] to compute t he mo d p groups TR n ∗ ( Z ; V (0)) = TR n ∗ ( Z ; Z /p ) for p ≥ 3. Similarly , smashing with the Smith- T o da complex V (1) = S/ ( p, v 1 ) was used in [3] to compute TR n ∗ ( ℓ ; V (1)) f or p ≥ 5. Here ℓ is the Adams summand of connectiv e comple x K -theory lo calized at p . In b oth o f t hese cases, the ∗ refers to an integer grading. W e will use this tec hnique of smashing with a finite complex in our computations, whic h are R O ( S 1 )-graded. In this pap er w e calculate TR n α ( F p ), the RO ( S 1 )-graded TR- g roups of F p , TR n α ( Z ; V (0)), the RO ( S 1 )-graded TR-groups o f Z with mo d p co efficien ts, and TR n α ( ℓ ; V (1)) , the RO ( S 1 )-graded TR-groups of ℓ with V (1) co efficien ts. F or the last case w e assume p ≥ 5, as V (1 ) do es not exist at p = 2 and is not a ring sp ectrum at p = 3 . If V is a ring sp ectrum, TR n α + ∗ ( A ; V ) for fixed α will b e a mo dule ov er t he integer-graded TR n ∗ ( A ; V ). While V (0) is not a ring spectrum a t p = 2, our computation of TR n α ( Z ; V (0)) is still v alid additiv ely . This dep ends on a clev er extension of the integer-graded computation of TR n ∗ ( Z ; V (0)) to the case p = 2 tha t w as carr ied o ut b y Rognes in [16], using that V (0) is a mo dule ov er the mo d 4 Mo ore sp ectrum S/ 4. 4 The calculations in these three cases are essen tially iden tical. T o treat all three cases sim ult a neously , w e intro duce an intege r c ≥ 0, the c hromatic lev el. If c = 0 we let A = F p and use integral co efficien ts. If c = 1 we let A = Z and use mo d p co efficien ts. If c = 2 w e let A = ℓ and use V (1)- co efficien ts. Giv en a prime p suc h that the sp ectrum B P h c i with ho mo t o p y groups Z ( p ) [ v 1 , . . . , v c ] (or its p - completion) is E ∞ and the Smith-T o da com- plex V ( c ) exists and is a ring sp ectrum, the ob vious generalization o f the calculations in the pap er applies. In ligh t of the problems with V (1) men tioned ab ov e at p = 2 and p = 3, the following restriction on p will b e in force throughout the pap er: Assumption 1.2. I f c = 0 or c = 1 , p c an b e any p rime. If c = 2 , we assume p ≥ 5 . The case c = 1 , p = 2 is sp ecial, and in tho se argumen ts where w e would normally use a ring structure (e.g. the pro of of Theorem 6.1) we hav e to instead use a mo dule structure o v er the corresp onding ob ject with mo d 4 co efficien ts. T o state some of t hese results, w e must first intro duce some no t ation. Giv en a virtual real represen tation α ∈ RO ( S 1 ), w e define a prime op eration b y α ′ = ρ ∗ p α C p where ρ p : S 1 → S 1 /C p is the isomorphism g iv en b y the p ’th ro ot [8]. W e let α ( n ) denote the n -fo ld it erat ed prime op eration a pplied to α . A real S 1 -represen ta tion can b e decomp osed as a direct sum of copies of the trivial represen tation R and the 2-dimensional represen tations C ( n ) with action given b y λ · z = λ n z for n ≥ 1 . The prime op eration acts o n these summands as follows : C ( n ) ′ = ( C ( n p ) if p | n , 0 otherwise. and R ′ = R . Giv en a virtual real r epresen tation µ , w e often write µ = α + q as a sum of a complex represen ta t io n α ∈ R ( S 1 ) a nd a trivial represen t a tion q ∈ Z . Let d i ( α ) = dim C ( α ( i ) ). The RO ( S 1 )-graded TR- g roups considered in this pap er all hav e the prop erty that TR n α + ∗ for ∗ ∈ Z is determined by the sequence of in tegers d 0 ( α ) , . . . , d n − 1 ( α ) . Giv en any sequen ce of in tegers d 0 , . . . , d n − 1 it is p ossible to find a virtual represen tatio n α with d i = d i ( α ) for eac h i . Conv ersely , the p - homotop y 5 t yp e of S α as a C p n − 1 -equiv arian t sp ectrum is determined b y the in tegers d 0 ( α ) , . . . , d n − 1 ( α ), so the fact that TR n α + ∗ is determined b y these in tegers is p erhaps not surprising. If α = λ or α = − λ for an actual represen tation λ , t his sequence of in tegers is non- increasing o r non-decreasing, resp ective ly , and the TR-calculations simplify . Fix an in teger c ∈ { 0 , 1 , 2 } , and define δ n c ( α ) = − d 0 ( α ) + X 1 ≤ k ≤ n − 1  d k − 1 ( α ) − d k ( α )  p ck . (1.3) If c = 0, let A = F p and V = S 0 . If c = 1, let A = Z and V = V (0). If c = 2, let A = ℓ and V = V (1) . W e pro v e in Theorem 4.2 b elow that in the stable range, i.e. for q sufficien tly lar g e w ith resp ect to the in tegers − d i ( λ ), w e ha v e TR n α + q ( A ; V ) ∼ = TR n q − 2 δ n c ( α ) ( A ; V ) . A similar result w as obtained b y Tsalidis [17] in the case c = 1 f o r α = − λ where λ is a n actual S 1 -represen ta tion. W e highligh t the follow ing result, whic h is ess en tial to the K -theory com- putations in [2] and [1]: Theorem 1.4. L et λ b e a fi nite c omplex S 1 -r epr es e ntation. Th e n fo r any prime p the finite Z ( p ) -mo dules TR n q − λ ( Z ; Z /p ) have the f o l l o wing structur e: 1. F or q ≥ 2 d 0 ( λ ) , TR n q − λ ( Z ; Z /p ) has length n , if q i s c ongruent to 2 δ n 1 ( λ ) or 2 δ n 1 ( λ ) − 1 mo dulo 2 p n , an d n − 1 otherwise . 2. F or 2 d s ( λ ) ≤ q < 2 d s − 1 ( λ ) with 1 ≤ s < n, TR n q − λ ( Z ; Z /p ) has length n − s if q is c ongruent to 2 δ n − s 1 ( λ ( s ) ) or 2 δ n − s 1 ( λ ( s ) ) − 1 mo dulo 2 p n − s and n − s − 1 , otherwise. 3. F or q < 2 d n − 1 ( λ ) , TR n q − λ ( Z ; Z /p ) is zer o . A t an o dd prime p , TR n α ( Z ; Z /p ) is a utomatically a Z /p -v ector space. It follo ws a p osteriori tha t when p = 2, TR n q − λ ( Z ; Z / 2 ) is a Z / 2-v ector space; see [2, Corollary 2.7]. 1.1. Or ganization W e b egin in § 2 by recalling the fundamen tal diagram of TR-theory , whic h will b e essen tial to t he computations throughout the pap er. In § 3 we set up a sp ectral sequenc e f r o m the homotop y gr o ups o f a homotop y orbit sp ectrum to the TR-gr o ups w e a re aiming to compute. In § 4 w e study the T ate sp ectral 6 sequence in the R O ( S 1 )-graded setting, whic h is essen tial to understanding the homotop y orbit sp ectrum which serv es as input for our computatio ns. W e handle the cases of F p , Z , and ℓ sim ultaneously . W e find in Theorem 4.2 b elo w that in eac h case the T at e sp ectral sequence is a shifted vers ion of the corresp onding Z -graded sp ectral sequence. In § 5 w e study the effect of truncating the T ate sp ectral sequence to obtain sp ectral sequences con v erging to the homotopy orbits and the homotopy fixed p oin ts. This pro vides the induction step neede d to prov e Theorem 4.2 fr om the previous section. In § 6 we describ e the homotop y o rbit to TR sp ectral sequence fro m § 3 in our examples for a general virtual represen tation α . In § 7 w e consider the case A = F p and use the homo t op y orbit to TR sp ectral sequenc e with Z /p l co efficien ts for all l ≥ 1 t o giv e an algorithm f o r computing TR n +1 α + ∗ ( F p ) for an y virtual represen tat io n α . In § 8 w e sp ecialize to represen ta t io ns of the form − λ , where λ is an a ctual S 1 -represen ta tion. W e sho w that in this case the homotop y orbit to TR sp ectral sequence simplifies, and prov e Theorem 1.4. 2. The fundamen tal diagram The TR-groups a re connected by sev era l op erators: R , F , V and d . In the ordinary (in teger-gra ded) case, there are maps as follo ws ( see [14] fo r more details). Inclusion o f fixed p oin ts induces a map F : TR n +1 q ( A ) → TR n q ( A ) called the F rob enius. This map has an asso ciated tra nsfer, V : TR n q ( A ) → TR n +1 q ( A ) , the V ersc hiebung. The differen tial d : TR n q ( A ) → TR n q +1 ( A ) is give n b y m ultiplying with the fundamen tal class of S 1 /C p n − 1 using the circle action. T op olo gical Ho c hsc hild homology is a cyclotomic sp ectrum [12], whic h gives a map R : TR n +1 q ( A ) → TR n q ( A ) called the restriction. The iden tification of the target of the restriction map with TR n ( A ) uses this cyclotomic structure of T ( A ), whic h identifie s the 7 geometric fixed p oints T ( A ) g C p with T ( A ). T o mak e this iden tification w e need t o c hange unive rses, b ecause the S 1 acting on T ( A ) is no t the same a s the S 1 acting on T ( A ) g C p . As a sp ecial case, consider T ( G ) fo r G a top o- logical group. Then T ( G ) ≃ Σ ∞ M ap ( S 1 , B G ) + is the suspension sp ectrum of the free lo op space on B G . The geometric fixed p oints ar e then giv en by T ( G ) g C p ≃ Σ ∞ M ap ( S 1 /C p , B G ) + , the free lo op space o n loops parametrized b y S 1 /C p . The primary approach used to compute TR-gro ups is t o compare the fixed p oint spectra to the homot o p y fixed p oint spectra. Let E denote a free con tractible S 1 -CW complex. Recall that the homotopy fixed p oint spectrum is defined by T ( A ) hC p n := F ( E + , T ( A )) C p n , and the TR-sp ectrum is defined b y TR n +1 ( A ) := T ( A ) C p n . The map E + → S 0 giv en b y pro jection on to the non-basep oin t induces a ma p Γ n : TR n +1 ( A ) → T ( A ) hC p n . The g eneral strategy for computing the homotop y gro ups TR n +1 q ( A ) is to compute π q ( T ( A ) hC p n ) and the map Γ n . This is facilitated thro ug h the use of a fundamental diagram of horizontal cofiber sequences , see [5, § 1-2] or [1 2, Equation 25 ]: T ( A ) hC p n N / / =   TR n +1 ( A ) R / / Γ n   TR n ( A ) ∂ / / ˆ Γ n   Σ T ( A ) hC p n =   T ( A ) hC p n N h / / T ( A ) hC p n R h / / T ( A ) tC p n ∂ / / Σ T ( A ) hC p n (2.1) Let ˜ E denote the cofib er of E + → S 0 . Then T ( A ) hC p n := ( E + ∧ T ( A )) C p n is the homotopy orbit sp ectrum and T ( A ) tC p n := ( ˜ E ∧ F ( E + , T ( A )) C p n is the T ate sp ectrum, see [9]. A theorem of Tsalidis [18, Theorem 2.4] c haracterizes situations when this map Γ n is an isomorphism. The computation of R O ( S 1 )-graded TR-gro ups can b e a pproac hed sim- ilarly . As befo re, w e ha v e the F rob enius F : TR n +1 α ( A ) → TR n α ( A ), the V ersc hiebung V : TR n α ( A ) → TR n +1 α ( A ), the differen tia l d : TR n α ( A ) → TR n α +1 ( A ), and t he restriction R : TR n +1 α ( A ) → TR n α ′ ( A ) . Note tha t the tar- get of R is the group in dimension α ′ , not α (see [12] for a detailed explanation of the restriction in this context). The fundamen tal diag ram a lso extends to this RO ( S 1 )-graded con text. Let T denote T ( A ) and let T [ − α ] = T ( A ) ∧ S − α denote the desusp ension 8 of T by α . Then w e ha v e the f o llo wing fundamen tal diagram o f horizontal cofib er sequences, see [12, Equation 49 ]: T [ − α ] hC p n N / / =   TR n +1 ( A )[ − α ] R / / Γ n   TR n ( A )[ − α ′ ] ∂ / / ˆ Γ n   Σ T [ − α ] hC p n =   T [ − α ] hC p n N h / / T [ − α ] hC p n R h / / T [ − α ] tC p n ∂ / / Σ T [ − α ] hC p n (2.2) Notice that TR n ( A )[ − α ′ ] app ears, rather than TR n ( A )[ − α ]. W e can tak e homotop y groups of the top row and get a long exact sequence . . . → π q T [ − α ] hC p n → TR n +1 α + q ( A ) → TR n α ′ + q ( A ) → π q − 1 T [ − α ] hC p n → . . . (2.3) This is t he fundamen ta l long exact sequence of RO ( S 1 )-graded TR-theory . The strat egy for computing TR n α + ∗ ( A ) is to use Diagr a m 2.2 and induction. One can attempt to understand the b ottom row via sp ectral sequences, see [9] and [12, Equation 26]. In this case the sp ectral sequences lo ok as follows : ˆ E 2 s,t ( α ) = ˆ H − s ( C p n , V t ( T [ − α ])) ⇒ V s + t ( T [ − α ] tC p n ) E 2 s,t ( α ) = H − s ( C p n , V t ( T [ − α ])) ⇒ V s + t ( T [ − α ] hC p n ) E 2 s,t ( α ) = H s ( C p n , V t ( T [ − α ])) ⇒ V s + t ( T [ − α ] hC p n ) Note that in general w e hav e ˆ H k ( C p n , M ) ∼ = H k ( C p n , M ) for k > 0 and ˆ H k ( C p n , M ) ∼ = H − ( k + 1) ( C p n , M ) for k < − 1, and that when M = Z /p w e ha v e ˆ H 0 ( C p n , M ) ∼ = H 0 ( C p n , M ) and ˆ H − 1 ( C p n , M ) ∼ = H 0 ( C p n , M ). This means that the res triction of t he T ate sp ectral sequence to the first quadra nt, meaning filtra t ion ≥ 1, give s the ho mo t op y or bit sp ectral sequence with the filtration shifted b y 1. T his corresp o nds to the connecting ho mo mor phism T [ − α ] tC p n → Σ T [ − α ] hC p n in D iagram 2.2 ab ov e. Similarly , the restriction of the T ate sp ectral sequence to the second quadran t, meaning filtratio n ≤ 0, giv es the homot o p y fixed p oint sp ectral sequence. W e use these sp ectral sequences to make computations of the homotop y groups on the b o ttom of the diagram. Understanding the maps Γ n and ˆ Γ n is also k ey to our arguments. Theorem 5.1 b elo w, whic h is due to Tsalidis [18] in the non-equiv ariant case, sa ys that if ˆ Γ 1 is an isomorphism in sufficien tly high degrees then so a re Γ n and ˆ Γ n for all n . If w e know TR n α ′ + ∗ w e can 9 then use ˆ Γ n to understand the T ate sp ectrum T [ − α ] tC p n and the rest of the b ottom row. This give s TR n +1 α + ∗ in sufficien tly high degrees. W e are then left to compu te TR n +1 α + ∗ in the unstable range. In the following section w e dev elop a sp ectral sequence that allo ws us to do the computat ions in the unstable range. This sp ectral sequence starts with the homotop y groups of v arious homotop y orbit sp ectra and conv erges to the TR-gro ups w e w ould lik e to compute. The sp ectral sequence a llo ws us to tr eat the cases of F p , Z , and ℓ sim ultaneously . How ev er, in the case of F p there are additional extension issues whic h need to b e resolve d. In the Z -graded case, it is useful to first compute TR n ∗ ( F p ; Z /p ). This sho ws that TR n 2 q ( F p ) is cyclic and TR n 2 q +1 ( F p ) = 0, and fr o m this w e conclude that the relev an t extensions are maximally non trivial. In the R O ( S 1 )-graded case, TR n α + q ( F p ) could ha v e sev eral summands, and indeed, fo r many α it do es. It is p ossible to compute the order of TR n α + q ( F p ) inductive ly using Diagram 2.2, and computations with Z /p -co efficien ts determine the n um b er of summands, but this informatio n is not enough to determine the group. W e solv e this problem by using Z /p l co efficien ts for all l ≥ 1, calculating the asso ciated g raded of TR n α + q ( F p ; Z /p l ), and this is enough to solv e the extension problem. No suc h extension problems arise in our computat ions of TR n α + ∗ ( Z ; V (0)) and TR n α + ∗ ( ℓ ; V (1)) as graded ab elian groups. Ho w ev er, it is con v enien t to consider these not only as gra ded ab elian groups but as mo dules ov er F p [ v 1 ] using the map v 1 : Σ 2 p − 2 V (0) → V (0) in the first case and o v er F p [ v 2 ] using the map v 2 : Σ 2 p 2 − 2 V (1) → V (1) in the second case. T his simplifies the b o okk eeping, and by writing Z /p n as F p [ v 0 ] /v n 0 w e can treat all three cases sim ult a neously . In the stable range the mo dule structure ov er F p [ v c ] is clear, but there could b e hidden v c -m ultiplications in low degree. One could then consider using S ( p, v l 1 ) or S/ ( p, v 1 , v l 2 ) as co efficien ts, and although w e believ e this would give a similar algo r it hm for resolving the extensions as the one w e find f o r TR n α + ∗ ( F p ) w e will not pursue that a v en ue here. W e will express TR n α + ∗ ( Z ; V (0)) as an F p [ v 1 ]-mo dule and TR n α + ∗ ( ℓ ; V (1)) as an F p [ v 2 ]-mo dule, with the ca v eat t hat there mig ht b e additional hidden extensions. A t p = 2 there is no map v 1 : Σ 2 V (0) → V (0), so it do es not make sense to express TR n α + ∗ ( Z , V (0)) as a F 2 [ v 1 ]-mo dule. So when we write down TR n α + ∗ ( Z ; V (0)) the result should b e in terpreted additively , or as a mo dule o v er F 2 [ v 4 1 ] using the map v 4 1 : Σ 8 V (0) → V (0), when p = 2 . 10 3. The homotopy orbit to TR sp ectral sequence It is p o ssible to glue t o gether the long exact sequences in Equation 2.3 to obtain a sp ectral sequence conv erging to TR n +1 α + ∗ ( A ; V ) with co efficien ts in V . F or this section A can b e an y connectiv e S - algebra and V can b e an y sp ectrum. Let T = T ( A ). The E 1 term is giv en by E 1 s,t ( α ) = ( V t T [ − α ( n − s ) ] hC p s for 0 ≤ s ≤ n , 0 otherwise. This sp ectral sequence conv erges to TR n +1 α + t ( A ; V ). Note that w e use a slightly non-standard gra ding con v en tion here; w e find it more con v enien t not to deal with V t − s T [ − α ( n − s ) ] hC p s . The reason this spectral sequence ha s not b een introduced b efore is t ha t in previously compute d examples, one can unde rstand TR n +1 ∗ ( A ; V ) completely b y comparing with V ∗ T hC p n . In the RO ( S 1 )-graded case, there is a range of degrees where this comparison is less useful. The d r differen tial has bidegree ( r, − 1), d r : E r s,t ( α ) → E r s + r,t − 1 ( α ) , and can be defined as follows: F or x ∈ V t T [ − α ( n − s ) ] hC p s , d r ( x ) is giv en by lifting N ( x ) up to TR s + r α ( n − s − r + 1) + t ( A ; V ) and then a pplying ∂ : TR s + r α ( n − s − r + 1) + t ( A ; V ) ∂ / / R   V t − 1 T [ − α ( n − s − r ) ] hC p s + r . . . R   TR s +2 α ( n − s − 1) + t ( A ; V ) R   ∂ / / V t − 1 T [ − α ( n − s − 2) ] hC p s +2 V t T [ − α ( n − s ) ] hC p s N / / TR s +1 α ( n − s ) + t ( A ; V ) ∂ / / V t − 1 T [ − α ( n − s − 1) ] hC p s +1 Observ at ion 3.1. We note that if A and V ar e ( − 1) -c onne cte d the filtr ation s pie c e V ∗ T [ − α ( n − s ) ] hC p s is zer o in de g r e e ∗ < − 2 d n − s ( α ) . 11 Definition 3.2. Consider the short exact se quenc e 0 → cok er ( R h )[ − 1] → V ∗ T [ − α ] hC p n → ker ( R h ) → 0 obtaine d by taking V ∗ ( − ) of the b ottom r o w of Diagr am 2.2 . We c al l the image of cok er ( R h )[ − 1] i n V ∗ T [ − α ] hC p n the T ate pie c e and denote it by V t ∗ T [ − α ] hC p n . If the se quenc e is s p lit we cho ose a splitting an d c al l the im- age of k er ( R h ) und e r the splitting the homotopy fixe d p oint pie c e , denote d V h ∗ T [ − α ] hC p n . Henc e if the ab ove short exact se quenc e splits we have a de c omp osition V ∗ T [ − α ] hC p n ∼ = V t ∗ T [ − α ] hC p n ⊕ V h ∗ T [ − α ] hC p n . The purp ose of the ab o v e definition is to get a b etter handle on the differen tials in the homoto p y orbit to TR sp ectral sequence: Lemma 3.3. In the homotopy orbit to TR sp e ctr a l se quen c e, eve ry class in the T ate pie c e V t ∗ T [ − α ( n − s ) ] hC p s is a p ermanent cycle, and the image of any differ e n tial is c ontaine d in the T ate pie c e . I f the short ex a ct se quenc e in Definition 3.2 splits then al l differ entials g o fr om a sub gr oup of the homotopy fixe d p oint pie c e to a quotient of the T ate pie c e. Pr o of. This is a straightforw ard diag ram chase , using the construction of the sp ectral sequence and Diagram 2.2. W e will denote classes in V t ∗ T [ − α ] hC p n b y their name in V ∗ T [ − α ] tC p n and classes in V h ∗ T [ − α ] hC p n b y their name in V ∗ T [ − α ] hC p n . 4. The T ate sp ectral sequence In order to use the sp ectral sequence from the previous section, we mus t first understand the homoto p y orbit sp ectrum. The homo t op y orbit sp ectral sequence computing V ∗ T [ − α ] hC p n is the restriction of t he cor r esp onding T ate sp ectral sequence to p ositiv e filtration, so w e fir st need to study the T ate sp ectral sequence con v erging to V ∗ T [ − α ] tC p n . F or c ∈ { 0 , 1 , 2 } , let V a nd A b e as in the introduction, and let T = T ( A ). Recall [12, 5, 3] that the homot o p y groups o f top ological Ho c hsc hild ho- mology with these co efficien ts are giv en as follows: 12 π ∗ T ( F p ) = P ( µ 0 ) , V (0) ∗ T ( Z ) = E ( λ 1 ) ⊗ P ( µ 1 ) , V (1) ∗ T ( ℓ ) = E ( λ 1 , λ 2 ) ⊗ P ( µ 2 ) . Here P ( − ) denotes a p o lynomial algebra and E ( − ) denotes an exterior al- gebra, b oth o v er F p . The degrees are giv en by | λ i | = 2 p i − 1 and | µ c | = 2 p c , with λ i represen ted b y σ ¯ ξ i and µ c represen ted b y σ ¯ τ c in the B¨ okstedt sp ectral sequence . At p = 2, λ i is represen ted by σ ¯ ξ 2 i and µ c is represen ted by σ ¯ ξ c +1 . The ab ov e formula for V (0) ∗ T ( Z ) can b e interpre ted multiplicativ ely ev en though V (0) is not a ring sp ectrum a t p = 2, by using that V (0) ∧ T ( Z ) ≃ T ( Z ; Z / 2), top olo g ical Ho c hsc hild homology of Z with co efficien ts in t he bi- mo dule Z / 2. (A similar tric k gives an in terpretation of V (1) ∗ T ( ℓ ) at p = 2 and p = 3, but w e will not need this.) But note that there is no S 1 -action on top ological Ho c hsc hild homology with co efficien ts in a bimo dule, so there is no corresponding ring structure on the TR-groups if the coefficien t sp ectrum is not a ring sp ectrum. Rognes [1 6] has sho wn that a t p = 2 ev erything still w orks, by sho wing the T ate sp ectral sequence conv erging to V (0) ∗ T ( Z ) tC p n has a form al algebra structure, so w e can pro ceed a s if V (0) ∧ T ( Z ) tC p n w as a ring sp ectrum. W e hav e V ∗ T [ − α ] ∼ = V 2 d 0 ( α )+ ∗ T , and we know from [12, Lemma 9.1] that the T a t e sp ectrum T [ − α ] tC p n only dep ends on α ′ . With the usual grading con v en tions the T ate sp ectral se- quence will depend on α , and not just on α ′ . In fact, b y considering the T ate sp ectral sequence for some β with α ′ = β ′ the pattern of differen tials will c hange in the following w a y . If we hav e a differential d 2 r ( t k x ) = t k + r y in the sp ectral sequence conv erging to V ∗ T [ − α ] tC p n w e get a differen tial d r ( t k − d 0 ( β )+ d 0 ( α ) x ) = t k − d 0 ( β )+ d 0 ( α )+ r y in the sp ectral sequence conv erging to V ∗ T [ − β ] tC p n . T o get a T ate sp ectral sequence that only dep ends on α ′ , w e do the follo wing. W r it e V ∗ T [ − α ] = t d 0 ( α ) V ∗ T (4.1) 13 where | t | = − 2. Then the T ate sp ectral sequence con v erging t o V ∗ T [ − α ] tC p n has E 2 term giv en by ˆ E 2 ( α ) = ˆ H ∗ ( C p n ; V ∗ T [ − α ]) ∼ = V ∗ T ( A ) ⊗ P ( t, t − 1 ) ⊗ E ( u n )[ − α ] , a free mo dule ov er the corresp onding non-equiv aria nt sp ectral sequence on a generator [ − α ]. Here | u n | = − 1 and | t | = − 2 ar e in negativ e filtration degree ( s ) and zero fib er degree ( t ), while V ∗ T ( A ) is concen trated in filtrat io n degree 0. With a factor of t d 0 ( α ) coming from V ∗ T [ − α ], the T ate sp ectral sequence no w only dep ends on α ′ and the E 2 term is isomorphic as a bigraded ab elian group to t he corresp onding non-equiv ariant E 2 term. The price w e pay is that w e ha v e to redefine what we mean b y the first and second quadrant of this sp ectral sequence. Now first quadran t means filtration ≥ − 2 d 0 ( α ) + 1 and second quadran t means filtration ≤ − 2 d 0 ( α ). The class v c ∈ π 2 p c − 2 V (recall tha t v 0 = p ) maps to a class in V ∗ T hS 1 represen ted b y tµ c in the E 2 term of the homot o p y fixed p oint sp ectral se- quence (see e.g. [3, Prop o sition 4.8]) , so by abuse of notation w e will denote the class tµ c in the C p n T ate sp ectral sequence b y v c . Recall [12, 5, 3] that V ∗ T tC p n is 2 p cn -p erio dic and the definition of δ n c ( α ) in Equation 1.3 in t he introduction. Theorem 4.2. The R O ( S 1 ) -gr ade d TR gr oups of A satisfy TR n α + ∗ ( A ; V ) ∼ = TR n ∗− 2 δ n c ( α ) ( A ; V ) for ∗ sufficiently la r ge, and the V -homo topy gr oups of T [ − α ] tC p n satisfy V ∗ T [ − α ] tC p n ∼ = V ∗− 2 δ n c ( α ′ ) T tC p n for al l ∗ . W e prov e t his theorem in the next section, aft er ana lyzing the restriction of the T ate sp ectral sequence to the first and second quadra nt. The pro of go es b y induction, using a vers ion o f Tsalidis’ theorem (Theorem 5.1). The p oin t is that knowing TR n α ′ + ∗ ( A ; V ) in t he stable range tells us ab out the b eha vior of the T at e sp ectral sequence con v erging to V α + ∗ T ( A ) tC p n , which b y restriction to the second quadrant tell us ab out V α + ∗ T ( A ) hC p n and hence ab out TR n +1 α ( A ; V ). 14 W e sp ell out the b eha vior of the T ate sp ectral sequence in eac h case. The pro of of Theorem 4.2, a s w ell as the following form ula s, are prov ed after Theorem 5 .1 in the next section. D efine r ( n ) by r ( n ) = X 1 ≤ k ≤ n p ck . (4.3) As in the non-equiv arian t case the classes λ i and v c are p ermanen t cycles, and the T ate sp ectral sequence is determined by the follow ing (compare [12, 5, 3]): In each case we hav e a fa mily of differen t ials giv en b y d 2 r ( n )+1 ( t − k u n [ − α ]) = v r ( n − 1)+1 c t p cn − k [ − α ] if ν p ( k − δ n c ( α ′ )) ≥ cn . If c = 0 this condition is empt y , and this is the only family of differen tials. F or c ≥ 1 w e hav e, for eac h 1 ≤ j ≤ n , a differen tial d 2 r ( j ) ( t − k [ − α ]) = v r ( j − 1) c t p cj − k λ c [ − α ] if ν p ( k − δ n c ( α ′ )) = cj − 1. Finally , if c = 2 w e hav e, for each 1 ≤ j ≤ n , a differen tial d 2 r ( j ) /p ( t − k [ − α ]) = v r ( j − 1) /p 2 t p 2 j − 1 − k λ 1 [ − α ] if ν p ( k − δ n 2 ( α ′ )) = 2 j − 2. 5. The homotopy orbit and homotopy fixed p oin t sp ectra T o find V ∗ T [ − α ] hC p n and V ∗ T [ − α ] hC p n w e restrict the T ate sp ectral se- quence from the prev ious section t o the first or sec ond quadran t. Recall that b ecause o f our g rading con v en tions, in pa r ticular Equation 4.1 ab ov e, the first quadrant means filt r a tion greater tha n − 2 d 0 ( α ). Hence the homotopy orbit sp ectral sequence has E 2 -term V ∗ T ⊗ E ( u n ) { t k [ − α ] : k < d 0 ( α ) } [ − 1] and the homotop y fixed p oin t sp ectral sequence has E 2 -term V ∗ T ⊗ E ( u n ) { t k [ − α ] : k ≥ d 0 ( α ) } . 15 Analyzing these sp ectral sequenc es is straightforw ard, but requires some amoun t of b o okk eeping. W e will write do wn V ∗ T [ − α ] hC p n completely b e- cause it is the input to the homo t o p y orbit t o TR sp ectral sequence. W e will partia lly describe V ∗ T [ − α ] hC p n b y explaining how some v c -tow ers in the homotop y fixed p oint piece of V ∗ T [ − α ] hC p n b ecome divisible by some p o we r of v c in V ∗ T [ − α ] hC p n . The rest of V ∗ T [ − α ] hC p n consists of those v c -tow ers that are concen trated in negativ e to t a l degree, and these are isomorphic to the corresp onding v c -tow ers in V ∗ T [ − α ] tC p n . W e separate V ∗ T [ − α ] hC p n in to the T a te piece and the homotopy fixed p oin t piece as in Definition 3.2, and each piece comes in c + 1 families, eac h of whic h can b e split in to a stable part and an unstable part. In sufficien tly high degrees the ma p R h in Diag r am 2.2 is zero, so N h is an isomorphism b et w een the homotop y fixed p o int piece of V ∗ T [ − α ] hC p n and V ∗ T [ − α ] hC p n in the stable ra nge. This isomorphism can b e describ ed in terms of those dif- feren tials in the T ate sp ectral sequence whic h go from the first to the second quadran t. Suc h a differen tial lea v es one class in V ∗ T [ − α ] hC p n and one class in V ∗ T [ − α ] hC p n , neither of whic h has a corresp onding class in V ∗ T [ − α ] tC p n . T o describ e the first f a mily , whic h is the one “created” b y the longest differen tial d 2 r ( n )+1 in the T a te sp ectral sequence, let E = F p for c = 0, E ( λ 1 ) for c = 1 and E ( λ 1 , λ 2 ) for c = 2. Then the T ate piece o f the first family splits as the following direct sum: M k ≥ r ( n − 1)+1 − d 0 ( α ) ν p ( k − δ n c ( α ′ )) ≥ cn E ⊗ P r ( n − 1)+1 ( v c ) { t − k [ − α ] } [ − 1] M 1 ≤ k + d 0 ( α ) ≤ r ( n − 1) ν p ( k − δ n c ( α ′ )) ≥ cn E ⊗ P k + d 0 ( α ) ( v c ) { t − k [ − α ] } [ − 1] In par ticular, in the stable r a nge w e ha v e v c -tow ers of heigh t r ( n − 1) + 1 starting in degree 2 δ n c ( α ′ ) + mp cn . Similarly , the homotopy fixed p o in t piece splits as a direct sum as follo ws: M k ≥ r ( n − 1)+1 ν p ( k − d 0 ( α ) − δ n c ( α ′ )) ≥ cn E ⊗ P r ( n )+1 ( v c ) { t d 0 ( α ) µ k c [ − α ] } M 1 ≤ k ≤ r ( n ) ν p ( k − d 0 ( α ) − δ n c ( α ′ )) ≥ cn E ⊗ P k ( v c ) { v r ( n )+1 − k c t d 0 ( α ) µ k − p cn c [ − α ] } 16 In part icular, in the stable range w e ha ve v c -tow ers of heigh t r ( n ) + 1 starting in degree − 2 d 0 ( α ) + 2 p c ( d 0 ( α ) + δ n c ( α ′ ) + mp cn ) = 2 δ n +1 c ( α ) + mp c ( n +1) . Next w e compare this to V ∗ T [ − α ] hC p n . F or the v c -tow ers of maximal heigh t, the map N h in Equation 2 .2 is an isomorphism. No w consider a generator x of P k ( v c ) { v r ( n )+1 − k c t d 0 ( α ) µ k − p cn c [ − α ] } = P k ( v c ) { t r ( n )+1 − k + d 0 ( α ) µ r ( n − 1)+1 c [ − α ] } and its image N h ( x ) in V ∗ T [ − α ] hC p n . W e ha v e t w o cases, with the first case only applicable if c ≥ 1. First, if k < p cn then N h ( x ) is divisible by v r ( n − 1)+1 c and w e get a v c -tow er E ⊗ P r ( n − 1)+1+ k ( v c ) { t p cn − k + d 0 ( α ) [ − α ] } . If k ≥ p cn then N h ( x ) is divisible b y v r ( n )+1 − k c and we get a v c -tow er E ⊗ P r ( n )+1 ( v c ) { t d 0 ( α ) µ k − p cn c [ − α ] } . If c ≥ 1 the second f a mily is “created” b y the differentials d 2 r ( j ) for 1 ≤ j ≤ n . Let E ′ n = E ( u n ) if c = 1 and E ( λ 1 , u n ) if c = 2. Then the T ate piece of the second f a mily splits a s the follow ing direct sum: M 2 ≤ j ≤ n M k ≥ r ( j − 1) − d 0 ( α ) ν p ( k − δ n c ( α ′ ))= cj − 1 E ′ n ⊗ P r ( j − 1) ( v c ) { t − k λ c [ − α ] } [ − 1] M 2 ≤ j ≤ n M 1 ≤ k + d 0 ( α ) ≤ r ( j − 1) − 1 ν p ( k − δ n c ( α ′ ))= cj − 1 E ′ n ⊗ P k + d 0 ( α ) ( v c ) { t − k λ c [ − α ] } [ − 1] Similarly , t he homotopy fixed p oin t piece splits a s a direct sum as fo llo ws: M 1 ≤ j ≤ n M k ≥ r ( j − 1) ν p ( k − d 0 ( α ) − δ n c ( α ′ ))= cj − 1 E ′ n ⊗ P r ( j ) ( v c ) { t d 0 ( α ) µ k c λ c [ − α ] } M 1 ≤ j ≤ n M 1 ≤ k ≤ r ( j ) − 1 ν p ( k − d 0 ( α ) − δ n c ( α ′ ))= cj − 1 E ′ n ⊗ P k ( v c ) { v r ( j ) − k c t d 0 ( α ) µ k − p cj c λ c [ − α ] } 17 Consider a generator x o f P k ( v c ) { v r ( j ) − k c t d 0 ( α ) µ k − p cj c λ c [ − α ] } and its image N h ( x ) in V ∗ T [ − α ] hC p n . Again we ha v e tw o cases. If k < p cj then N h ( x ) is divisible by v r ( j − 1) c and w e get a v c -tow er E ′ n ⊗ P r ( j − 1)+ k ( v c ) { t p cj − k + d 0 ( α ) λ c [ − α ] } . If k ≥ p cj then N h ( x )is divisible by v r ( j ) − k c and w e get a v c -tow er E ′ n ⊗ P r ( j ) ( v c ) { t d 0 ( α ) µ k − p cj c λ c [ − α ] } . Finally , if c = 2 t he thir d family is “created” b y the differen tials d 2 r ( j ) /p for 1 ≤ j ≤ n . Let E ′′ n = E ( λ 2 , u n ). Then the T at e piece of the third family splits a s the following direct sum: M 2 ≤ j ≤ n M k ≥ r ( j − 1) /p − d 0 ( α ) ν p ( k − δ n 2 ( α ′ ))=2 j − 2 E ′′ n ⊗ P r ( j − 1) /p ( v 2 ) { t − k λ 1 [ − α ] } [ − 1] M 2 ≤ j ≤ n M 1 ≤ k + d 0 ( α ) ≤ r ( j − 1) /p − 1 ν p ( k − δ n 2 ( α ′ ))=2 j − 2 E ′′ n ⊗ P k + d 0 ( α ) ( v 2 ) { t − k λ 1 [ − α ] } [ − 1] Similarly , t he homotopy fixed p oin t piece splits a s a direct sum as fo llo ws: M 1 ≤ j ≤ n M k ≥ r ( j − 1) /p ν p ( k − d 0 ( α ) − δ n 2 ( α ′ ))=2 j − 2 E ′′ n ⊗ P r ( j ) /p ( v 2 ) { t d 0 ( α ) µ k 2 λ 1 [ − α ] } M 1 ≤ j ≤ n M 1 ≤ k ≤ r ( j ) /p − 1 ν p ( k − d 0 ( α ) − δ n 2 ( α ′ ))=2 j − 2 E ′′ n ⊗ P k ( v 2 ) { v r ( j ) /p − k 2 t d 0 ( α ) µ k − p 2 j − 1 2 λ 1 [ − α ] } Once ag a in, consider the image N h ( x ) o f a generator x of the v 2 -tow er P k ( v 2 ) { v r ( j ) /p − k 2 t d 0 ( α ) µ k − p 2 j − 1 2 λ 1 [ − α ] } in V ∗ T [ − α ] hC p n . If k < p 2 j − 1 then N h ( x ) is divisible b y v r ( j − 1) /p 2 and we get a v 2 -tow er E ′′ n ⊗ P r ( j − 1) /p + k ( v 2 ) { t p 2 j − 1 − k + d 0 ( α ) λ 1 [ − α ] } . If k ≥ p 2 j − 1 then N h ( x ) is divisible by v r ( j ) /p − k 2 and we g et a v 2 -tow er P r ( j ) /p ( v 2 ) { t d 0 ( α ) µ k − p 2 j − 1 2 λ 1 [ − α ] } . W e will use the followin g theorem, whic h with integral co efficien ts is due to Tsalidis [18, Theorem 2.4 ] in the Z -gr a ded case and Hess elholt-Madsen [12, Addendum 9.1] in a sp ecial case of the RO ( S 1 )-graded case: 18 Theorem 5.1. L et A b e a c onne ctive rin g sp e ctrum of fi nite typ e. Sup- p ose the map ˆ Γ 1 : T ( A ) → T ( A ) tC p induc es an isom o rphism π q ( T ( A ); V ) → π q ( T ( A ) tC p ; V ) for q ≥ i . Then, for any n ≥ 1 , ˆ Γ n induc es an isomorphism TR n α ′ + q ( A ; V ) → V q T [ − α ] tC p n for q ≥ 2 max( − d 1 ( α ) , . . . , − d n ( α )) + i. Equivalently, Γ n induc es an isomo rp hism TR n +1 α + q ( A ; V ) → π q ( T [ − α ] hC p n ; V ) in the same r ange. Pr o of. The pr o of in [12, Addendum 9.1] go es through v erbatim with T ( F p ) replaced by V ( c ) ∧ T ( A ). Pr o of of T he or em 4.2. In eac h case Theorem 5.1 applies, see e.g. [12, Prop o- sition 5.3] for F p , [5, Lemma 6.5] for Z , and [3, Theorem 5.5] for ℓ . F o r c = 0 w e hav e i = 0, for c = 1 w e ha ve i = 0 and for c = 2 w e hav e i = 2 p − 1 (the class t p 2 λ 1 λ 2 in V (1) ∗ T ( ℓ ) tC p is in degree 2 p − 2). Supp ose by induction that the statemen t of the Theorem holds for TR n α ′ + ∗ ( A ; V ). Then the ma p ˆ Γ n : TR n α ′ + ∗ ( A ; V ) → V ∗ T [ − α ] tC p n is co connectiv e, so V ∗ T [ − α ] tC p n is shifted b y 2 δ n c ( α ′ ) degrees in the stable rang e. Using that V ∗ T [ − α ] tC p n is a mo dule o v er V ∗ T tC p n and that V ∗ T tC p n is 2 p n -p erio dic the statemen t f or V ∗ T [ − α ] tC p n follo ws. The pat t ern of differen tials in the T ate sp ectral sequence describ ed after the statemen t of Theorem 4 .2 also follow s from this. Restricting the T a t e sp ectral sequence t o t he second quadran t give s a sp ectral sequence computing V ∗ T [ − α ] hC p n , a nd eac h differen tial on a class t − k in the T ate sp ectral sequence gives a class t d 0 ( α ) µ k + d 0 ( α ) c . The differen tials on t − k for v arious k are shifted by 2 δ n c ( α ′ ) degrees, whic h means that the classes in the homotop y fixed p oin t sp ectrum are shifted by − 2 d 0 ( α ) + 2 p c ( d 0 ( α ) + δ n c ( α ′ )) = 2 δ n +1 c ( α ) degrees. Using that Γ n : TR n +1 α + ∗ ( A ; V ) → V ∗ T [ − α ] hC p n is co connectiv e the statemen t then holds for TR n +1 α + ∗ ( A ; V ). 19 6. A splitt ing of the homotop y orbit to TR sp ectral sequence In this section w e describ e the homotopy orbit t o TR sp ectral sequence in the three cases o f in terest. W e sho w that the sp ectral sequence splits as the direct sum o f “small” sp ectral seque nces, with no differen tials b et w een differen t summands. W e first describ e the small sp ectral sequences. Conside r the fo llowing diagram: P r (0)+1 ( v c ) { t d n ( α ) µ k c } s s v v } } P r (0)+1 ( v c ) { t − p c k − δ 1 c ( α ( n ) ) } [ − 1] P r (1)+1 ( v c ) { t d n − 1 ( α ) µ p c k + d n − 1 ( α )+ δ 1 c ( α ( n ) ) c } s s z z P r (1)+1 ( v c ) { t − p 2 c k − δ 2 c ( α ( n − 1) ) } [ − 1] P r (2)+1 ( v c ) { t d n − 2 ( α ) µ p 2 c k + d n − 2 ( α )+ δ 2 c ( α ( n − 1) ) c } v v . . . . . . P r ( n − 1)+1 ( v c ) { t − p cn k − δ n c ( α ′ ) } [ − 1] P r ( n )+1 ( v c ) { t d 0 ( α ) µ p cn k + d 0 ( α )+ δ n c ( α ′ ) c } F or eac h k , there is a summand of the E 1 term of the homotop y or bit to TR sp ectral sequen ce whic h lo oks like t he ab o v e diagram tensored with E (recall that E = F p , E ( λ 1 ) or E ( λ 1 , λ 2 )), with submo dules of the modules in t he righ t hand column and quotien t mo dules of the mo dules in the left hand column (the summands are allow ed to b e 0 ). If c = 0, this describ es the whole E 1 term. If c = 1 there is one more family of diagrams to consider and if c = 2 there are tw o mor e families of diagr a ms to consider. F or c = 1 or 2 the second family of small sp ectral sequences lo oks as follo ws. Recall that E ′ j = E ( u j ) if c = 1 and E ( u j , λ 1 ) if c = 2. F o r eac h 0 ≤ j ≤ n − 1 and each k with ν p ( k − d j ( α ) + d j +1 ( α )) = c − 1 we ha v e a corresp onding diag ram, where the righ t hand side consists of submo dules of E ′ n − j + m ⊗ P r ( m +1) ( v c ) { t d j − m ( α ) µ p cm k + d j − m ( α )+ δ m c ( α ( j − m +1) ) c λ c } 20 for 0 ≤ m ≤ j and t he left hand side consists of quotient mo dules of E ′ n − j + m ⊗ P r ( m ) ( v c ) { t − p cm k − δ m c ( α ( j − m +1) ) λ c } [ − 1] for 1 ≤ m ≤ j . Finally , if c = 2 the third family of small sp ectral sequences lo oks as follo ws. Recall that E ′′ j = E ( u j , λ 2 ). F or eac h 1 ≤ j ≤ n and each k with ν p ( k − d j ( α ) + d j +1 ( α )) = 0 w e hav e a corresp onding diagram, where the righ t hand side consists of submo dules o f E ′′ n − j + m ⊗ P r ( m +1) /p ( v 2 ) { t d j − m ( α ) µ p 2 m k + d j − m ( α )+ δ m 2 ( α ( j − m +1) ) 2 λ 1 } for 0 ≤ m ≤ j and t he left hand side consists of quotient mo dules of E ′′ n − j + m ⊗ P r ( m ) /p ( v 2 ) { t − p 2 m k − δ m 2 ( α ( j − m +1) ) λ 1 } [ − 1] for 1 ≤ m ≤ j . The following theorem gives an algorithm for computing the homotop y orbit to TR sp ectral sequence. The expres sion for d ρ ( x ) lo oks unpleasant, but for c = 1 or 2 the form ula, in the case when d ρ ( x ) is nontrivial, can b e obtained simply from degree considerations. Theorem 6.1. The hom otopy orbit to TR sp e c tr al s e quenc e E 1 s,t ( α ) = V ∗ T [ − α ( n − s ) ] hC p s = ⇒ TR n +1 α + ∗ ( A ; V ) splits as a dir e ct sum of the ab ove sp e ctr al se quenc es, wi th no diff er entials b etwe en summands. The diffe r entials a r e determin e d by the fol lowing data. L et e j = u ǫ 0 j λ ǫ 1 1 λ ǫ 2 2 and supp ose x = v i c t d j ( α ) e n − j µ k c [ − α ( j ) ] is a nontrivial cla ss in the ho m otopy fixe d p oint pie c e of E 1 n − j, ∗ . L et y h = v i ( h ) c t − p hc k − δ h c ( α ( j − h +1) ) e n − j + h [ − α ( j − h ) ] , wher e i ( h ) = i − r ( h − 1) k − X 0 ≤ k ≤ h − 2  d j − h +1 ( α ) − d j ( α )  p ck . If x survive s to E ρ n − j, ∗ and the classes y h ∈ V ∗ T [ − α ( j − h ) ] tC p n − j + h ar e nonzer o for 1 ≤ h ≤ ρ then d ρ ( x ) = ∂ h ( y ρ ) c onsider e d as a class in E ρ n − j + ρ, ∗ . If at le ast one of the clas ses y h for 1 ≤ h ≤ ρ is zer o then d ρ ( x ) = 0 . 21 Pr o of. If w e are not in the case c = 1, p = 2 , then TR k β + ∗ ( A ; V ) is a mo dule o v er TR k ∗ ( A ; V ), whic h con tains an elemen t µ N c for N a m ultiple of p c ( k − 1) . In the case c = 1, p = 2 , TR k β + ∗ ( Z , V (0)) is a mo dule ov er TR k ∗ ( Z , S/ 4), whic h con tains an elemen t µ N 1 for N a multiple of 2 k . This follo ws by induction, using the results in [16] and Tsalidis’ theorem. In all cases we hav e a w a y of comparing with the stable ra nge b y m ultiplying by µ N c for an appropriate N . The class x in V ∗ T [ − α ( j ) ] hC p n − j maps to a class with the same name in TR n − j +1 α ( j ) + ∗ ( A ; V ). By comparing with the stable range w e find tha t ˆ Γ n − j +1 ( x ) = y 1 . By construction of the sp ectral sequen ce this imp lies that d 1 ( x ) = ∂ h ( y 1 ). If d 1 ( x ) = 0, then x lifts to a class x 1 in TR n − j +2 α ( j − 1) + ∗ ( A ; V ). Let z 1 = Γ n − j +1 ( x 1 ) in V ∗ T [ − α ( j − 1) ] hC p n − j +1 . While x 1 , and hence z 1 , ma y not b e unique, we ha v e a cano nical choice for a represen tative for z 1 in the homo t o p y fixed p oint sp ectral sequence giv en by taking a represen ta tiv e for ˆ Γ n − j +1 ( x ) in the T ate sp ectral sequence and restricting to the second quadrant. W e then hav e tw o cases. Case 1: The class z 1 m ultiplies nontrivially by µ N c to the stable range. Be- cause V ∗ T [ − α ( j − 1) ] hC p n − j +1 is isomorphic to V ∗ T [ − α ( j − 2) ] tC p n − j +2 in the stable range, this happ ens exactly when y 2 = ˆ Γ n − j +2 ( x 1 ) 6 = 0. Ag ain it fo llo ws b y construction of the sp ectral sequence that d 2 ( x ) = ∂ h ( y 2 ). The formula for d ρ ( x ) assuming y 1 , . . . , y ρ are a ll nonzero follo ws b y induction. Case 2: The class z 1 m ultiplies trivially b y µ N c to the stable range. In this case w e find that ˆ Γ n − j +2 ( x 2 ) = 0, so d 2 ( x ) = 0. By induction, x lifts t o a class x h in TR n − j + h +1 α ( j − h ) + ∗ ( A ; V ) which m ultiplies trivially to the stable range for all h . Hence d ρ ( x ) = 0 for all ρ . The same argumen t applies as so on as some y h is zero. 7. The R O ( S 1 )-graded TR–groups of F p While Theorem 6.1 ab ov e t ells us all t he differentials in the sp ectral se- quence con v erging to TR n +1 α + ∗ ( F p ), w e need some additional information to resolv e the extension problems. As show n in [8], the extension problem is in fact quite delicate. W e observ e that if w e know the order of TR n +1 α + ∗ ( F p ; Z /p l ) f or eac h l ≥ 1 , w e can reconstruct TR n +1 α + ∗ ( F p ). Let T = T ( F p ). W e find that π ∗ ( T [ − α ]; Z /p l ) ∼ = t d 0 ( α ) E ( β l ) ⊗ P ( µ 0 ) , 22 where β l is in degree 1, a nd the T ate sp ectral sequence b ehav es as follows : Lemma 7.1. Consider the sp e c tr al se quenc e c onv er ging to π ∗ ( T [ − α ] tC p n ; Z /p l ) . If n < l ther e is a differ ential d 2 n +1 ( u n ) = tv n 0 and if n ≥ l ther e is a differ- ential d 2 l ( β l ) = v l 0 . Pr o of. This is clear b ecause the mo d p l Bo c kstein β l will alwa ys kill the represen tative f or p l if p ossible. W e can then record π ∗ ( T [ − α ] hC p n ; Z /p l ). As b efore, w e split it into the T ate piece and t he homotop y fixed p oin t piece. If n < l we find that the T ate piece is M k ≥ n − d 0 ( α ) E ( β l ) ⊗ P n ( v 0 ) { t − k [ − α ] } [ − 1] M 1 ≤ k + d 0 ( α ) ≤ n − 1 E ( β l ) ⊗ P k + d 0 ( α ) ( v 0 ) { t − k [ − α ] } [ − 1] . Similarly , t he homotopy fixed p oin t piece is as follows : M k ≥ n E ( β l ) ⊗ P n +1 ( v 0 ) { t d 0 ( α ) µ k 0 [ − α ] } M 1 ≤ k ≤ n E ( β l ) ⊗ P k ( v 0 ) { v n +1 − k 0 t d 0 ( α ) µ k − 1 0 [ − α ] } If n ≥ l we find that the T ate piece is M k ≥ l − d 0 ( α ) E ( u n ) ⊗ P l ( v 0 ) { t − k [ − α ] } [ − 1] M 1 ≤ k + d 0 ( α ) ≤ l − 1 E ( u n ) ⊗ P k + d 0 ( α ) ( v 0 ) { t − k [ − α ] } [ − 1] . Similarly , t he homotopy fixed p oin t piece is as follows : M k ≥ l E ( u n ) ⊗ P l ( v 0 ) { t d 0 ( α ) µ k 0 [ − α ] } M 1 ≤ k ≤ l − 1 E ( u n ) ⊗ P k ( v 0 ) { v l − k 0 t d 0 ( α ) µ k 0 [ − α ] } 23 Theorem 7.2. Consider the sp e ctr al se quenc e E 1 ( α ) = M 0 ≤ s ≤ n π ∗ ( T [ − α ( n − s ) ] hC p s ; Z /p l ) = ⇒ TR n +1 α + ∗ ( F p ; Z /p l ) . The differ entials ar e determine d by the fo l l o wing data. S upp ose x = p i t d j ( α ) u ǫ n − j µ k 0 [ − α ( j ) ] is a nontrivia l class in the hom o topy fixe d p oin t p i e c e of E 1 n − j, ∗ . Th en d ρ ( x ) is given as in The or em 6.1 . Now supp ose ¯ x = p i t d j ( α ) β l µ k 0 [ − α ( j ) ] is a nontrivial cla ss in the ho m otopy fixe d p oint pie c e of E 1 n − j, ∗ , an d ¯ y h = ( p i ( h ) t − k − δ h 0 ( α ( j − h +1) ) β l [ − α ( j − h ) ] if n − j + h − l < 0 , p i ( h ) − ( n − j + h − l ) t − k − 1 − δ h 0 ( α ( j − k +1) ) u n − j + h [ − α ( j − h ) ] if n − j + h − l ≥ 0 . If ¯ x survives to E ρ n − j, ∗ and the c l a sses ¯ y h ∈ π ∗ ( T [ − α ( j − h ) ] tC p n − j + h ; Z /p l ) ar e nonzer o for 1 ≤ h ≤ ρ then d ρ ( ¯ x ) = ∂ h ( ¯ y ρ ) c onsider e d as a class in E ρ n − j + ρ, ∗ . If at le ast one of the classes ¯ y h is zer o then d ρ ( ¯ x ) = 0 . Pr o of. The pro of is similar to the pro of o f Theorem 6.1. The extra factor of p − ( n − j + h − l ) comes from ha ving n − j + h − l homoto p y o rbit sp ectral sequences with a differen tial on β l rather than a differen tial on some u j + h . F or eac h one, the p ossible differen tial, a nd p ossible success ive lift of ¯ x , b eha v es as if w e had star t ed with a m ultiple of u n − j µ k +1 0 [ − α ( j ) ] ra ther than a multiple o f β l µ k 0 [ − α ( j ) ]. 8. The T R groups in degree q − λ It is the TR–groups indexed b y represen tations of the form α = q − λ that are most applicable to computatio ns of algebraic K -theory . See, for example, Hesselholt and Madsen’s computation of K q ( F p [ x ] / ( x m ) , ( x )) in [10] and results of the a uthors and Hesselholt on K q ( Z [ x ] / ( x m ) , ( x )) in [2]. Prop osition 8.1. Consider the sp e ctr a l se quenc e E 1 s,t ( − λ ) = M 0 ≤ s ≤ n V ∗ T [ − λ ( n − s ) ] hC p s = ⇒ TR n +1 ∗− λ ( A ; V ) for an actual r epr esen tation λ . Then e very nonzer o c l a ss in the T ate pie c e is kil le d by a differ ential. 24 Pr o of. W e prov e this by induction, but with a slightly extended induction hy- p othesis. W e consider a represen ta t io n λ whic h is almost an actual represen- tation, by whic h we mean that d i ( λ ) ≥ d i +1 ( λ ) for i ≥ 1 and d 0 ( λ ) ≥ d 1 ( λ ) − 1. Consider the first f amily of sp ectral sequences describ ed in § 6. It is enough to show that z = t − p cn k + δ n c ( λ ′ ) [ − 1] in the T ate piece o f E 1 n, ∗ ( − λ ) is hit by a differential. F or z to b e nonzero w e m ust hav e p cn k − d 0 ( λ ) − δ n c ( λ ′ ) ≥ 0 . Consider x = t − d 1 ( λ ) µ p c ( n − 1) k − d 1 ( λ ) − δ n − 1 c ( λ ′′ ) c in the homotopy orbit piece of E 1 n − 1 , ∗ . If p c ( n − 1) k − d 1 ( λ ) − δ n − 1 c ( λ ′′ ) > r ( n − 2) then x is no nzero and d 1 ( x ) = z . No w supp o se p c ( n − 1) k − d 1 ( λ ) − δ n − 1 c ( λ ′′ ) ≤ r ( n − 2) . Consider the class y = v p c ( n − 1) k − d 1 ( λ ) − δ n − 1 c ( λ ′′ ) c t − p c ( n − 1) k + δ n − 1 c ( λ ′′ ) in the T ate sp ectral sequence con v erging to V ∗ T [ − λ ′ ] tC p n − 1 . Then y is in filtration 2 d 1 ( λ ), whic h means that y is not in the first quadran t of the sp ectral sequence and hence ∂ h ( y ) = 0. Note that 0 ≤ p c ( n − 1) k − d 1 ( λ ) − δ n − 1 c ( λ ′′ ) ≤ r ( n − 2) , so y is nonzero in V ∗ T [ − λ ′ ] tC p n − 1 . By assumption, d 1 ( λ ) ≥ d 2 ( λ ). Then w e can consider a r epresen tation µ with µ ′′ = λ ′′ and d 1 ( µ ) = d 1 ( λ ) − 1. Then ∂ h ( y ) 6 = 0 in E 1 n − 1 , ∗ ( − µ ). By induction ∂ h ( y ) = d ρ ( w ) f o r some w in E ρ ∗ , ∗ ( − µ ). But then Theorem 6.1 implies that d ρ +1 ( w ) = z in E ρ +1 ∗ , ∗ ( − λ ), prov ing t he result. The remaining t w o families of differentials can b e treated in a similar w a y . W e can now redo the calculation in [12]: 25 Corollary 8.2. I t fol low s that | TR n q − λ ( F p ) | =      p n for q = 2 m and d 0 ( λ ) ≤ m , p n − s for q = 2 m and d s ( λ ) ≤ m ≤ d s − 1 ( λ ) , 0 for q o dd. Pr o of. In the case of F p , if w e consider the sp ectral sequence E 1 s, ∗ ( − λ ) = π ∗ T [ − λ ( n − 1 − s ) ] hC p s = ⇒ TR n ∗− λ ( F p ) , the only elemen ts in o dd total degree are t ho se in the T ate piece. By Prop o- sition 8.1 , all those elemen ts are killed, hence | TR n q − λ ( F p ) | = 0 for q o dd. In ev en degrees, since the differen tials are surjectiv e | TR n 2 m − λ ( F p ) | = Q s | E 1 s, 2 m | Q s | E 1 s, 2 m − 1 | = ( p n for d 0 ( λ ) ≤ m, p n − s for d s ( λ ) ≤ m ≤ d s − 1 ( λ ) . F rom the sp ectral sequence for TR n q − λ ( F p ; Z /p ) in § 7 we conclude that TR n q − λ ( F p ) has just o ne summand. So w e get the following result: Theorem 8.3. L et λ b e a finite c omplex S 1 -r epr es e ntation. Then TR n q − λ ( F p ) ∼ =      Z /p n for q = em an d d 0 ( λ ) ≤ m , Z /p n − s for q = 2 m and d s ( λ ) ≤ m ≤ d s − 1 ( λ ) , 0 for q o dd. This agrees with the result of Hesselholt and Madsen [12]. In t he case of A = Z w e can then prov e Theorem 1.4: Pr o of of T he or em 1.4. As describ ed in § 6 t he E 1 -term o f the homotopy or- bit t o TR sp ectral sequence is comp osed o f tw o families of small sp ectral sequence s. In sufficien tly high degrees w e are left with the lo we r right-hand summands in the diagrams of § 6. W e first give the a r gumen t in high degrees and then describ e the mo difications needed in lo w degrees. In the E ∞ -term w e are left with E ( λ 1 ) ⊗ P r ( n − 1)+1 ( v 1 ) { t − d 0 ( λ ) µ p n − 1 k − d 0 ( λ ) − δ n − 1 1 ( λ ′ ) 1 } 26 from the first family of sp ectral sequences, and E ( u j ) ⊗ P r ( j +1) ( v 1 ) { t − d 0 ( λ ) µ p j k − d 0 ( λ ) − δ j 1 ( λ ′ ) 1 λ 1 } for 0 ≤ j ≤ n − 2 and k suc h that v p ( k + d j ( λ ) + δ n − 1 − j 1 ( λ ( j +1) )) = 0 from the second fa mily . Assum e q = 2 m is ev en. The length of TR n q − λ ( Z ; Z /p ) is the num b er of differen t w ays can 2 m b e written as 2 m = 2 d 0 ( λ ) + 2 p n k − 2 p ( d 0 ( λ ) + δ n − 1 1 ( λ ′ )) + a (2 p − 2) for 0 ≤ a ≤ r ( n − 1) or 2 m = 2 d 0 ( λ ) + 2 p j +1 k − 2 p ( d 0 ( λ ) + δ j 1 ( λ ′ )) + ( a + 1)(2 p − 2) for 0 ≤ j ≤ n − 2, 0 ≤ a < r ( j + 1), a nd v p ( k + d j ( λ ) + δ n − 1 − j 1 ( λ ( j +1) )) = 0. Noting that δ j 1 ( λ ′ ) = δ n − 1 1 ( λ ′ ) − p j d j ( λ ) − p j δ n − 1 − j 1 ( λ ( j +1) ) w e can rewrite these t w o equations as 2 m − 2 d 0 ( λ ) + 2 p ( d 0 ( λ ) + δ n − 1 1 ( λ ′ )) = 2 p n k + a (2 p − 2 ) or 2 m − 2 d 0 ( λ ) + 2 p ( d 0 ( λ ) + δ n − 1 1 ( λ ′ )) = 2 p j +1 ( k + d j ( λ ) + δ n − j − 1 1 ( λ ( j +1) )) + ( a + 1)(2 p − 2) with the same conditions on a, j , and k as ab ov e. It follows that the length of TR n 2 m − λ ( Z ; Z /p ) is the n umber of w a ys to write b = m − d 0 ( λ ) + p ( d 0 ( λ ) + δ n − 1 1 ( λ ′ )) as b = p n k + a ( p − 1) where 0 ≤ a ≤ r ( n − 1 ) or b = p j +1 k + a ( p − 1) where 0 ≤ j ≤ n − 2, 1 ≤ a ≤ r ( j + 1), and v p ( k ) = 0 . No w, if b = p n k + a ( p − 1) with 1 ≤ a ≤ r ( n − 1) w e can rewrite this as b = p n − 1 ( pk ) + a ( p − 1), and if b = p j +1 k + a ( p − 1 ) with 1 ≤ a ≤ r ( j ) w e can r ewrite it as b = p j ( pk )+ a ( p − 1). 27 Hence w e ha v e o ne class when c = 0 mo dulo p n and one class for eac h w ay to write b = p j +1 k + a ( p − 1) with 0 ≤ j ≤ n − 2 and r ( j ) < a ≤ r ( j + 1), with no condition on ν p ( k ). There is exactly one suc h pair ( k , a ) for each j , so w e get n − 1 classes, plus an additiona l class fr o m the first fa mily when m = δ n 1 ( λ ) mo dulo p n corresp onding to a = 0. The case q = 2 m + 1 o dd is similar. If q ≥ 2 d 0 ( λ ), but q is not sufficien tly high that the spectral seque nces degenerate with only the low er right hand summands in the E ∞ term, the result follows b y comparing with π ∗ ( T [ − µ ] tC p n ; Z /p ) for some µ with µ ′ = λ . Using tha t the mo d p homotopy gr oups of the T a te sp ectrum ar e 2 p n -p erio dic and Theorem 5.1, the result follows. P art 2 and 3 follo w b y using that if q < 2 d 0 ( λ ) w e hav e an isomorphism R : TR n q − λ ( Z ; Z /p ) ∼ = → T R n − 1 q − λ ′ ( Z ; Z /p ) . References [1] Vigleik Angeltve it and T eena G erhardt. On the algebraic K -theory of the co o r dinate axes ov er the integers . Pr eprin t, arXiv:0909.4 2 87 . [2] Vigleik Angeltv eit, T eena Gerhardt, and Lars Hesselholt. On the K- theory of truncated p olynomial algebras ov er the integers. J T op olo gy , 2(2):277–2 94, 2009. [3] Christian Ausoni and John Rognes. Algebraic K -theory of top o logical K -theory . A cta Math. , 188(1):1– 3 9, 2 0 02. [4] M. B¨ okstedt, W. C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K - t heory of spaces. Invent. Math. , 111 (3):465–539 , 1993. [5] M. B¨ o kstedt and I. Mads en. T op ological cyc lic homology of the inte gers. Ast ´ erisque , (226 ):7–8, 57–143 , 1994. K -theory (Strasb ourg, 1 9 92). [6] Marcel B¨ okstedt. T op o logical Ho c hsc hild homology . Unpublishe d . [7] Thomas Geisser and Lar s Hesselholt. Bi-r elative alg ebraic K -theory and top ological cyclic homolog y . Invent. Math. , 166 (2):359–395 , 2006. 28 [8] T eena G erhardt. The R S 1 -graded equiv ariant homo t o p y of T H H ( F p ). A lgebr aic and Ge ometric T op olo gy , 8(4 ):1961–1987 , 2008. [9] J. P . C. Gr eenlees and J. P . May . Generalized Tate coho molo gy . Mem. A mer. Math. So c. , 113(543) :viii+1 7 8, 1 995. [10] Lars Hesselholt. K -theory of truncated p olynomial alg ebras. In Hand- b o ok of K -the ory. V ol. 1, 2 , pages 71–110. Springer, Berlin, 200 5 . [11] Lars Hesselholt and Ib Madsen. Cyclic p olytop es and the K -theory o f truncated p olynomial algebras. Invent. Math. , 130( 1):73–97, 199 7 . [12] Lars Hesselholt and Ib Madsen. On the K -theory of finite alg ebras ov er Witt vec tors o f p erfect fields. T op olo g y , 36(1 ):29–101, 199 7 . [13] Lars Hesselholt and Ib Madsen. On the K -theory of lo cal fields. A nn. of Math. (2) , 158(1):1– 1 13, 2003. [14] Lars Hesselholt and Ib Madsen. On the De Rha m-Witt complex in mixed c haracteristic. Ann. Sci. ´ Ec ole Norm. Sup. (4) , 37 ( 1):1–43, 2004. [15] Randy McCarth y . R elativ e algebraic K -theory and t o p ological cyclic homology . A c ta Math. , 179(2):1 97–222, 1997. [16] John Rognes. T op olo g ical cyclic homology of the integers at tw o. J. Pur e Appl. Algebr a , 13 4 (3):219–286 , 1999. [17] Sta vros Tsalidis. On the algebraic K -theory o f truncated p olynomial algebras. Pr eprint . [18] Sta vros Tsalidis. T op ological Ho chs c hild homology and the homotop y descen t problem. T op olo gy , 37 (4):913–934 , 19 9 8. 29