On the algebraic K-theory of Witt vectors of finite length

ON THE ALGEBRAIC K -THEOR Y OF WITT VECTOR S OF FINITE LENGTH VIGLEIK ANGEL TVEIT Abstract. Let k be a perfect ﬁe ld of characteristic p and let W n ( k ) denote the p -typical Witt v ectors of length n . F or example, W n ( F p ) = Z /p n . W e study the algebraic K -theory of W n ( k ), and prov e that K ( W n ( k )) s atisﬁes “Galois descen t”. W e a lso compute the K -groups throug h a r ange of degrees, and sho w that the ﬁrst p -torsion elemen t i n the stable homotop y groups of spheres is detected i n K 2 p − 3 ( W n ( k )) f or all n ≥ 2. 1. I ntroduction Let k b e a p erfect ﬁeld of characteristic p . Then the algebra ic K -theo ry of k is well understo o d, at least after p -completion. Indeed, the p -completed K -theor y of k is co ncentrated in degr ee 0. The situation is far mo r e complicated, but still well understo o d, if we lift to characteristic 0 using the Witt vector c o nstruction. B¨ okstedt and Madsen computed the p -completed algebraic K -theory of the p - adic integers Z p = W ( F p ) for p o dd in [9], and of W ( F p s ) (still for p o dd) in [1 0]. Rognes [35, 36, 34] computed the K - theory of the 2 -adic int egers, and Mitchell [33] ca lc ulated the K -theor y o f W ( F 2 s ). Later Hesse lho lt and Ma dsen [23] computed the K -theor y of complete discrete v a luation ﬁelds with residue ﬁeld k , at least in o dd characteris tic. It fo llows from the work o f Hesselholt and Madsen ( lo c. cit. ) that K ( W ( k )) satisﬁes Galois descent. B y this we mean that if k → k ′ is a G -Galois extension of per fect ﬁelds of characteris tic p with G ﬁnite then the ca nonical map K ( W ( k )) → K ( W ( k ′ )) hG to the homotopy ﬁxed p o int s of K ( W ( k ′ )) is an equiv alence on con- nective c overs after p -completion. (This is one version of the Lich tenbaum-Quillen conjecture.) But for W n ( k ) for n < ∞ the usual to ols from alg ebraic geometry do not work, b ecause W n ( k ) is not a regular ring. Despite considera ble eﬀo r t, v ery little is known ab out K ( W n ( k )). Our ﬁrst main theorem establishes that K ( W n ( k )) satisﬁes Galois descent. Theorem A. Su pp ose k → k ′ is a G -Galois extension of p erfe ct ﬁelds of char ac- teristic p for a ﬁnite gr oup G . Then the c anonic al map K ( W n ( k )) → K ( W n ( k ′ )) hG induc es an e quivalenc e on c onne ctive c overs after p -c ompletion for any n . While the ta sk o f understa nding K ( W n ( k )) completely a ppea rs insurmo un table with current technology , w e do hav e some par tia l results. The ﬁrst of thos e is the following. 1 Theorem B. Supp ose k = F q is a ﬁ nite ﬁeld of char acteristic p . Then K i ( W n ( k ) , ( p )) is a ﬁnite p - gro u p for al l i ≥ 0 and | K 2 i − 1 ( W n ( k ) , ( p )) | | K 2 i − 2 ( W n ( k ) , ( p )) | = q ( n − 1) i for al l i ≥ 1 . Combining this with Quillen’s calculation of K ( F q ) we get a simila r res ult for non-relative K - theo ry , s ee Co rollary 6 .7. W e can b e more explic it in low deg rees, explicitly determining the groups up to degree 2 p − 2. Theorem C. Supp ose k is a p erfe ct ﬁeld of char acteristic p . Then K i ( W n ( k ) , ( p )) for i ≤ 2 p − 2 c an b e describ e d as fol lows. In o dd de gr e e we have K 2 i − 1 ( W n ( k ) , ( p )) ∼ = W ( n − 1) i ( k ) for 2 i − 1 ≤ 2 p − 5 while K 2 p − 3 ( W n ( k ) , ( p )) is the dir e ct s u m of Z /p and a maximal ly nontrivial ext en sion of coker( Z /p → k ) by W ( n − 1)( p − 1) − 1 ( k ) . In even de gr e e we have K 2 i ( W n ( k ) , ( p )) ∼ = ( 0 for 0 ≤ 2 i ≤ 2 p − 4 cok er ( φ − 1) for 2 i = 2 p − 2 Mor e over, the u nit map fr om the spher e sp e ctrum sends the ﬁrs t p -t orsion ele- ment α 1 ∈ π 2 p − 3 ( S ) to a gener ator of Z /p ⊂ K 2 p − 3 ( W n ( k ) , ( p )) . Remark 1 .1. The ab ove isomorphisms ar e isomorph isms of ab elian gr oups only. In fact, t hr ough the ab ove r ange of de gr e es the non-unital gr ade d ring K ∗ ( W n ( k ) , ( p )) has trivial multiplic ation for de gr e e r e asons. 1.1. Main pro of ide as. This pap er uses sp ectral s equences extens ively , b oth for abstract ar guments and fo r concrete calcula tions. W e stro ngly encourage the reader to do a few sample ca lculations on her own to g et a feel for how these s pectr al sequences b ehave. W e use the cyclotomic tra c e map [8] trc : K ( A ) → TC( A ) , which for A = W n ( k ) is a n equiv alence on connective cov er s after p -completion [22]. The s tarting p oint o f our calculation is the top ologic a l Ho ch schild ho mology of k , whic h lo ok s like T H H ( F p ) “tenso r ed up” to k . W e can b o otstra p fro m that to T H H ∗ ( W n ( k )) by ﬁltering W n ( k ) by p owers o f p . This induces a ﬁltration of T H H ( W n ( k )), and w e g et a sp ectra l seq uence E ∗ , ∗ 1 = T H H ∗ ( k [ x ] / ( x n )) = ⇒ T H H ∗ ( W n ( k )) . The E 1 term is known by work of Hesselholt a nd Mads en [2 1]. F rom this we r ecov er Brun’s calculation o f T H H ∗ ( Z /p n ) [13]. This ﬁltration o f T H H ( W n ( k )) is S 1 -equiv ariant, and as a result w e get a co r- resp onding spectral sequence con verging to TF ∗ ( W n ( k )). I f we use the relativ e version (1.2) E ∗ , ∗ 1 = TF ∗ ( k [ x ] / ( x n ) , ( x )) = ⇒ TF ∗ ( W n ( k ) , ( p )) , the E 1 term is conce n tr ated in o dd total degre e s o the sp ectral se quence collapses (Corollar y 4 .7). 2 The restriction map R do es not resp ect the ﬁltration, s o we canno t hop e to g et such a sp ectral seque nc e conv er ging to top olog ical cyc lic homology . Instead, the restriction map divides the ﬁltra tio n by p , meaning that we hav e a map R : F s TF( W n ( k )) → F ⌈ s/p ⌉ TF( W n ( k )) , where ⌈ s/p ⌉ deno tes s/p rounded up to the neares t int eger. Expanding on ideas of Brun [14] we co nstruct a sp ectral seq ue nce conv erging to TC ∗ ( W n ( k )) built from t wo copies of TF ∗ ( W n ( k )). The pro o f of Theor em A go es as follows. W e ﬁrst show tha t TF ( W n ( k ) , ( p )) satisﬁes Galois descen t. This follows b ecause TF( k [ x ] / ( x n ) , ( p )) satisﬁes Galois descent, plus the colla psing sp ectral seq ue nce in E quation 1.2. Then, bec ause homotopy ﬁxed p oints commute with homotopy equalizer s , the sa me is true for TC, a nd the statement for K -theory follows by taking connective cov er s. The pro of of Theor e m B uses the sp ectra l se q uence for TC discussed ab ov e. T he necessary input is Hesselho lt and Ma dsen’s co mputation o f TC ∗ ( k [ x ] / ( x n ) , ( x )) [21], which implies that for k = F q we hav e | TC 2 i − 1 ( k [ x ] / ( x n ) , ( x )) | = q ( n − 1) i and | TC 2 i − 2 ( k [ x ] / ( x n ) , ( x )) | = 0. F or Theorem C we use another idea o f Brun [14] of compa r ing with the homotopy orbit spectrum of T H H ( W n ( k )), which is more readily under sto o d. Even though we do not hav e a ma p, it is p ossible to co mpare TC( A ) to Σ T H H ( A ) hS 1 bec ause b oth are the homotopy ﬁb er of a map TF( A ) → TF( A ). In fact, if A is a complete ﬁltered ring with I = F 1 A w e hav e a sp ectral sequence converging to π ∗ Σ T H H ( A, I ) hS 1 with the same E 1 term a s the a b ove sp ectral sequence co nv erging to TC ∗ ( A, I ). F or ( A, I ) = ( W n ( k ) , ( p )) we can compare diﬀerentials and extensio ns thro ugh a range of degr ees. 1.2. Re l ations to previous w ork. The ca lculation of K 1 and K 2 is classical, and the observ ation that K 1 ( W n ( k )) ∼ = W n ( k ) × behaves diﬀerently in c har acteristic 2 is o f cour se even more classica l (compare ( Z / 2 n ) × to ( Z /p n ) × ). Theor e m C c a n b e thought o f as an ex tension o f that phenomenon to o dd degr ees. In characteristic 3 this was also obser ved by Geisser [1 8], w ho computed K 3 ( W 2 ( F 3 s )) when (3 , s ) = 1 and found the extr a Z / 3 summand coming from the 3-torsio n in π 3 ( S ). Evens and F riedlander [17] computed K 3 and K 4 of Z /p 2 for p ≥ 5, but the most general calculation to date, and the only o ne we know of that go es b eyond degree 4, is due to Brun [14] who computed K i ( Z /p n ) for i ≤ p − 3. 1.3. Conv en tio ns. Throughout the pap er k will b e a p erfect ﬁeld of c ha racteristic p for a ﬁxed prime p > 0. W e will us e the notation x . = y to mea n x = λy for a unit λ . W e will us e P ( x ), E ( x ) and Γ( x ) for a p olyno mia l algebr a, exterior algebra , and divided p ow e rs algebra, resp ectively . In Γ( x ) we will sometimes write x for γ 1 ( x ). 1.4. Ac kno wle dgements. This pap er would never ha ve been started without Mike Hill, with who m I had extensive discussio ns a b out the top ologica l Ho chsc hild homology spe ctral sequence c oming fr om a ﬁltr ation o f a r ing. At the time we did not k now that Morten Brun had alr e ady constr ucted such a sp ectra l seq uence, a nd we r eprov ed some of his results and did several sample computations together. In addition I w ould like to thank Tyler L awson, T eena Ger hardt, and Lars Hes- selholt for helpful conv er sations. I would also like to thank a n a nonymous r e feree for constructive criticis m and for s p otting a serio us mista ke in an ear lier version of this pap er . 3 Because of its long gestation p erio d this work has b een supported by several grants: An NSF All-Ins titutes Postdocto ral F ellowship administered by the Mathe- matical Sciences Research Institute through its core gr ant DMS-0441 170, NSF grant DMS-08059 17, and Australian Research Council Disco very Grant No. DP120101 399. 2. A topological Hochschild homol ogy spectral sequence In this section we study a sp ectral sequence E s,t 1 = π s + t T H H ( Gr A ; s ) = ⇒ π s + t T H H ( A ) asso ciated to a ﬁltra tion of a ring A . W e will call this the “ﬁltered ring s pectr al sequence” for T H H . The existence of this sp ectr al sequence w as ﬁrst noted b y Brun [13], thoug h he only used it in an indir ect way in his co mputation of T H H ∗ ( Z /p n ). W e will demonstrate that this spec tr al s equence is an excellent to ol for computations by s implifying a nd extending k nown c a lculations o f T H H . F or conv e n tio ns and s tandard results ab out sp ectra l seq uences, see [7 ]. Most of the spectra l s equences in this paper will b e conditionally conv erg ent . If the sp ectral sequence satisﬁes some Mittag-Leﬄer condition it conv erges s trongly . This is typically ea sy to verify , in most of our exa mples it follows b ecause the E 1 -term is ﬁnite (or has ﬁnite length) ov er k in each bideg ree. Because o f the la rge num b er of sp ectra l seq uences a ppea ring we will not discuss conv ergence in each case . 2.1. A Ho c hsc hil d homolo gy sp ectral sequence. W e start with Hochschild homology , which is easier, in order to intro duce so me key ideas. Recall that for a ring A , the Ho chsc hild homolog y H H ∗ ( A ) is the homology of a c hain complex C ∗ ( A ) with A ⊗ q +1 in degree q a nd d ( a 0 ⊗ . . . ⊗ a q ) = X 0 ≤ i ≤ q − 1 ( − 1) i a 0 ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a q + ( − 1) q a q a 0 ⊗ a 1 ⊗ . . . ⊗ a q − 1 . It can a lso b e describ ed as the homolog y o f the cyc lic bar cons tr uction B cy ⊗ ( A ). If A is a DGA rather than a ring the ab ov e deﬁnition yields a bicomplex, and the Ho chsc hild homolog y o f A is the homolo gy of the asso ciated total complex. As alwa ys we fo llow the usual s ign rule, multiplying by ( − 1) whenever we mov e tw o things (elements, or op erators like d ) o f o dd degre e past each other. If the ring A is not ﬂat as a Z -mo dule, the ab ov e deﬁnition of Ho chsc hild homol- ogy will give the “wro ng ” result b ecause we were using underived tensor pro ducts while we sho uld hav e b een using derived tensor pro ducts. Shukla homology is the appropria te version of Ho chsc hild homolo gy with derived tensor pr o ducts every- where. Alternatively , we can repla ce A by a DGA e A which is degree-wise ﬂat a nd satisﬁes H ∗ ( e A ) ∼ = A . O ne wa y to do this is to put a mo del categ ory structure o n the category of DGAs [26] and let e A be a coﬁbra nt r e placement of A . Then e A is degree-wise pro jectiv e, hence ﬂat. It is standard that H H ∗ ( e A ) is independent of the replacement e A and agrees with the Shukla ho mology o f A . An alternative description of H H ∗ ( A ), at le a st whe n A is ﬂat, is as the homolo g y of the derived tensor pr o duct A ⊗ L A ⊗ A op A , or a s T or A ⊗ A op ∗ ( A, A ). The equiv alence betw een these last tw o deﬁnitions follows by replacing one of the A ’s by the 2- sided bar co nstruction B ( A, A, A ), which is a coﬁbrant replace men t of A a s an A -bimo dule. 4 Example 2.1 . L et A = Z / p . Be c ause Z /p is not pr oje ctive, we r eplac e it by the DGA g Z /p deﬁne d as fol lo ws. As a chain c omplex it is given by Z { a } d → Z { 1 } . Her e t he gener ator a is in de gr e e 1 , the gener ator 1 is in de gr e e 0 , and the b oundary map is given by d ( a ) = p · 1 . Ther e is only one way to deﬁne the mult iplic ation: we must have a 2 = 0 . We c an then c ompute the Ho chschild homolo gy of g Z /p . We ﬁnd t hat g Z /p ⊗ g Z /p ∼ = g Z /p ⊗ E Z ( b ) , wher e E Z ( b ) denotes an exterior algebr a (over Z ) on a gener ator b in de gr e e 1 . We c an take b = 1 ⊗ a − a ⊗ 1 . It fol low s that H ∗ ( g Z /p ⊗ g Z /p ) ∼ = E Z /p ( b ) , an exterior algebr a over Z /p on one gener ator b in de gr e e 1 . I t then fol low s fr om standar d homolo gic al algebr a that H H ∗ ( g Z /p ) ∼ = T or E Z /p ( b ) ∗ ( Z /p, Z /p ) ∼ = Γ Z /p ( µ 0 ) is a divide d p owers algebr a over Z /p on a class µ 0 in de gr e e 2 . T he class µ 0 is r epr esente d by 1 ⊗ a − a ⊗ 1 in the Ho chschild c omplex; it has de gr e e 2 b e c ause it has internal de gr e e 1 and Ho chschild de gr e e 1 . Suppo se A = L i ∈ Z A i is a graded ring. In the examples this grading will usu- ally b e indep endent of the homological g r ading. Then we get a splitting of the Ho ch schild ho mology o f A . Lemma 2.2 . Supp ose A is a gr ade d ring. Then the Ho chschild homolo gy H H ∗ ( A ) of A splits as a dir e ct sum H H ∗ ( A ) ∼ = M s H H ∗ ( A ; s ) , wher e H H ∗ ( A ; s ) is t he homolo gy of the sub c omplex of C ∗ ( A ) of internal de gr e e s . Her e we giv e a 0 ⊗ . . . ⊗ a q in C q ( A ) , with e ach a i homo gene ous, internal de gr e e | a 0 | + . . . + | a q | . Pr o of. This is clear, bec a use the Ho chsc hild diﬀerent ial preserves the internal de- gree.  Now s uppos e A is a complete ﬁltered r ing. By this we mea n that A comes with a dec reasing ﬁltration . . . ⊂ F s +1 A ⊂ F s A ⊂ . . . ⊂ F 0 A = A. W e as sume the ﬁltration is compatible with the multiplicativ e structure, meaning that the multiplication o n A induces maps F i A ⊗ F j A → F i + j A . Complete means that the cano nical map A → lim s A/F s A is an isomor phism. The canonical example comes from an ideal I ⊂ A . If A is I -co mplete then F s A = I s A deﬁnes a complete ﬁltration on A . Let Gr i A = F i A/F i +1 A and let GrA = L i Gr i A . Then Gr A is a graded r ing, a nd we can co mpute H H ∗ ( GrA ) as a bove. T o get homotopically mea ning ful results we do the following. First, we replace A by a pro jective DGA e A and deﬁne F s e A by basechange along F s A → A . Then each F s e A is degr eewise pro jective, hence ﬂat, and the multiplication map on e A induces maps F i e A ⊗ F j e A → F i + j e A . ( F i e A ⊗ F j e A maps to b oth e A a nd to F i + j A , hence maps to the pullback F i + j e A .) Second, we deﬁne e F s e A = hoco lim t ≥ s F t e A, 5 which is chain homotopic to F s e A . The multiplication ma p F i e A ⊗ F j e A → F i + j e A then induces a ma p e F i e A ⊗ e F j e A → e F i +1 e A , and w e deﬁne f Gr s e A = e F s e A/ e F s +1 e A. Then f Gr s e A is chain homotopic to F s e A/F s +1 e A , but has the adv antage that each f Gr s e A is ﬂat. The direct sum f Gr e A = M s ≥ 0 f Gr s e A is a gr aded r ing, s o we get a splitting of H H ∗ ( f Gr e A ) a s in Lemma 2.2. Next we deﬁne a corr esp onding ﬁltra tion o f C ∗ ( A ), or ra ther of C ∗ ( e A ). T o g e t a homotopically meaningful result w e do this by deﬁning e F s C q ( e A ) = ho colim s 0 + ... + s q ≥ s F s 0 e A ⊗ . . . ⊗ F s q e A. Then e F 0 C q ( e A ) is c ha in homoto p y equiv alent to C q ( e A ), s o we can think o f . . . ⊂ e F s +1 C q ( e A ) ⊂ e F s C q ( e A ) ⊂ . . . ⊂ e F 0 C q ( e A ) as a ﬁltra tion o f C q ( e A ). Then we deﬁne f Gr s C q ( e A ) = e F s C q ( e A ) / e F s +1 C q ( e A ) and f Gr C q ( e A ) = M s ≥ 0 f Gr s C q ( e A ) . The Ho chsch ild diﬀerential maps the diagram deﬁning e F s C q ( e A ) to the diagr am deﬁning e F s C q − 1 ( e A ), so it induces a map e F s C q ( e A ) → e F s C q − 1 ( e A ). Hence we have a ﬁltration of e F 0 C ∗ ( e A ), and this ﬁltra tion gives us a sp ectral sequence. W e can ident ify the E 1 term o f this s pectr al s equence a s follows: Theorem 2.3. Supp ose A is a c omplete ﬁlter e d ring with asso ciate d gr ade d GrA . Then ther e is a we akly c onver gent sp e ctr al se quenc e E s,t 1 = H H s + t ( g GrA ; s ) = ⇒ H H s + t ( e A ) . The diﬀer ential d r has bide gr e e ( r, − r − 1) . If A is c ommutative this is an algebr a sp e ctr al se qu enc e. W e only g et weak convergence in gener al b ecaus e we hav e no guarantee that e F ∞ C q ( e A ) is con tractible even if F ∞ A = 0. Pr o of. It follows that with the ab ov e ﬁltration of e F 0 C q ( e A ), f Gr s C q ( e A ) is chain homotopy eq uiv a lent to M s 0 + ... + s q = s f Gr s 0 e A ⊗ . . . ⊗ f Gr s q ( e A ) . This co mputes the E 1 term o f the s p ectr a l seq uence. If A is commutativ e then we can choo se e A to b e commutativ e as well, and each f Gr C q ( e A ) b ecomes a commut ative DGA. Then the shuﬄe pro duct induces a m ultiplication on the s pectr al sequence which makes it a n a lgebra sp ectral sequence in the standar d way .  6 Note that b eing a n alg ebra sp ectra l sequence means that the diﬀerentials are graded der iv ations: d r ( xy ) = d r ( x ) y + ( − 1) | x | xd r ( y ) for all r ≥ 1 a nd x, y ∈ E ∗ , ∗ r . Next we lo ok at some examples to show tha t this sp ectral seque nc e can b e used quite eﬀectiv ely . W e ﬁx a per fect ﬁeld k of c har acteristic p . Then b y a gener alization of Exa mple 2.1 (use [22, Lemma 5.5 ]) the Ho chsc hild homology o f e k is a divided powers algebra ov er k on one generator µ 0 in degree 2 . Example 2. 4 . First we c onsider W ( k ) ﬁ lter e d by p owers of p . Th en t he asso ciate d gr ade d is GrW ( k ) ∼ = k [ x ] , and we have a sp e ctr al se quenc e E s,t 1 = H H s + t ( e k [ x ]; s ) = ⇒ H H ∗ ( ^ W ( k )) . A priori this sp e ctr al se quenc e only c onver ges we akly b e c ause e F ∞ C q ( e A ) is a lar ge r ational ve ctor sp ac e. But the p -c ompletion of e F ∞ C q ( e A ) is trivial, so the sp e ctr al se quenc e c onver ges str ongly to the p -c ompletion of H H ∗ ( ^ W ( k )) . We ﬁ nd that E ∗ , ∗ 1 = H H ∗ ( e k [ x ]) ∼ = P ( x ) ⊗ E ( σ x ) ⊗ Γ( µ 0 ) , wher e µ 0 c omes fr om H H ∗ ( e k ) and σ x is r epr esente d by 1 ⊗ x − x ⊗ 1 . Her e the tensor pr o duct is over k . This is bigr ade d, with | µ 0 | = (0 , 2) , | x | = (1 , − 1) and | σ x | = (1 , 0 ) . We have an imme diate diﬀer ential d 1 ( γ j ( µ 0 )) = γ j − 1 ( µ 0 ) σ x for e ach j ≥ 1 , le aving E ∗ , ∗ 2 = E ∗ , ∗ ∞ = P ( x ) c onc entr ate d in homolo gic al de gr e e 0 . If in addition we use that t her e is a c omultiplic ation on E ∗ , ∗ 1 with ψ ( γ j ( µ 0 )) = P a + b = j γ a ( µ 0 ) ⊗ γ b ( µ 0 ) as in [5] we ﬁnd that the d 1 -diﬀer ential is gener ate d by the single diﬀer ential d 1 ( µ 0 ) = σ x . Sinc e x r epr esents multiplic at ion by p , this r e c overs (at le ast up t o p - c ompletion) the classic al r esult that H H 0 ( ^ W ( k )) = W ( k ) and H H i ( ^ W ( k )) = 0 for i > 0 . Example 2. 5. Next we c onsider W n ( k ) ﬁlter e d by p owers of p . Then the asso ciate d gr ade d is GrW n ( k ) = k [ x ] / ( x n ) , and in this c ase ther e ar e no c onver genc e pr oblems. L et E ∗ , ∗ 0 = P n ( x ) ⊗ E ( σx ) ⊗ Γ( x n ) ⊗ Γ( µ 0 ) , wher e the new gener ator x n has bide gr e e | x n | = ( n, 2 − n ) . Now deﬁne a diﬀer ential d 0 on E ∗ , ∗ 0 , gener ate d m u ltiplic atively by d 0 ( γ j ( x n )) = nx n − 1 γ j − 1 ( x n ) σ x for j ≥ 1 . Then E ∗ , ∗ 1 = H H ∗ ( e k [ x ] / ( x n )) ∼ = H ∗ ( E ∗ , ∗ 0 , d 0 ) . If p divides n then d 0 = 0 , and E ∗ , ∗ 1 = E ∗ , ∗ 0 with a d 1 -diﬀer ential gener ate d multi- plic atively by d 1 ( γ j ( µ 0 )) = γ j − 1 ( µ 0 ) σ x for j ≥ 1 , le aving E ∗ , ∗ 2 = E ∗ , ∗ ∞ = P n ( x ) ⊗ Γ( x n ) . This is the asso ciate d gra de d of H H ∗ ( ^ W n ( k )) ∼ = W n ( k ) ⊗ Γ( x n ) . As ab ove, if we use that ther e is a c omultiplic ation on E ∗ , ∗ 1 with ψ ( γ j ( µ 0 )) = P a + b = j γ a ( µ 0 ) ⊗ γ b ( µ 0 ) we c an say that the d 1 -diﬀer ential is gener ate d by the single diﬀer ential d 1 ( µ 0 ) = σ x . 7 If p do es not divide n then the E 1 -term is somewhat smal ler. We stil l have a d 1 -diﬀer ential gener ate d by d 1 ( µ 0 ) = σ x , bu t now the E 2 -term is somewhat lar ger. In this c ase we also have d 2 -diﬀer entials d 2 ( γ i ( x n ) x n − 1 γ j ( µ 0 )) . = γ i +1 ( x n ) γ j − 2 ( µ 0 ) σ x for j ≥ 2 . We il lustr ate this with the zig-zag γ i ( x n ) x n − 1 γ j ( µ 0 ) ✙ , , ❨ ❨ ❨ ❨ ❨ ❨ ❨ γ i ( x n ) x n − 1 γ j − 1 ( µ 0 ) σ x γ i +1 ( x n ) γ j − 1 ( µ 0 ) ✪ 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ✙ , , ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ γ i +1 ( x n ) γ j − 2 ( µ 0 ) σ x This le aves E ∗ , ∗ 3 = E ∗ , ∗ ∞ = P n ( x ) { 1 } ⊕ M j ≥ 1  k { x n − 1 µ 0 γ j − 1 ( x n ) } ⊕ P n − 1 ( x ) { xγ j ( x n ) }  . Ther e is a hidden multiplic ation by p extension, so again we r e c over that H H ∗ ( ^ W n ( k )) ∼ = W n ( k ) ⊗ Γ( x ′ n ) . Now γ j ( x ′ n ) is r epr esente d by x n − 1 µ 0 γ j − 1 ( x n ) , while pγ j ( x ′ n ) is r epr esente d by xγ j ( x n ) . Remark 2.6. Note that in the ab ove example the c ase p ∤ n is mor e c omplic ate d. It is p ossible to ﬁlter away this adde d c omplexity, as fol lows. In the Ho chschild chain c omplex C ∗ ( ^ W n ( k )) , intr o duc e a thir d gr ading by giving the class r epr esenting γ j ( x n ) de gr e e − j with asso ciate d gr ade d c Gr C ∗ ( ^ W n ( k )) . Then we get a sp e ctr al se quenc e ( c Gr E ∗ , ∗ r , b d r ) c onver ging to c Gr H H ∗ ( \ W n ( k )) . The asso ciate d gr ade d c Gr E ∗ , ∗ 0 is the ring E ∗ , ∗ 0 ab ove, now trigr ade d. Then we get the same d 1 -diﬀer ential as in the c ase p | n , at which p oint t he sp e ctr al se quenc e onc e again c ol lapses. We now have another sp e ctr al se quenc e E ∗ , ∗ 1 = W n ( k ) ⊗ Γ( x n ) = ⇒ H H ∗ ( ^ W n ( k )) , which c ol la pses at the E 1 -term, giving us t he desir e d r esult without having to c om- pute higher diﬀer ent ials. 2.2. T op ologi cal Ho c h s c hild homolo gy. F or a naive deﬁnition of T H H we hav e a wide c hoice of frameworks with which to work. F or exa mple, we could deﬁne T H H ( A ) as the geometric realiza tion of a simplicial spectr um with q 7→ A ∧ q +1 , the ( q + 1)-fold s mash pro duct o f A with itself. But to build T H H ( A ) as a cyclotomic sp ectrum (see Section 3.1 below for the deﬁnition of a cyclotomic spectr um) we need a more sophisticated deﬁnition. A v a r iant o f this deﬁnition go e s back to B¨ okstedt [11], see a lso [22]. Since this technology is well established, w e will b e brief. Our deﬁnitio n will tak e as input a symmetric ring s pec tr um and g ive us ba ck an o rthogona l S 1 -sp ectrum; our deﬁnition should be compar ed to that given by Hesselho lt and Madsen in [22]. See [1] fo r a mo dern deﬁnition of T H H which ta kes place entirely in or thogonal sp ectra. 8 Let A be a symmetric ring sp ectrum in the sense o f [25], but with to p olo gical spaces instead of simplicial sets. F or con venince w e will assume that all our symmet- ric ring sp ectra are s trictly c o nnective and conv ergent, i.e., A n is ( n − 1)-connected and there exists a sequence of nondecrea sing integers α n tending to inﬁnity such that Σ A n → A n +1 is ( n + α n )-connected. If A is a ring, w e can rega rd A as a symmetric ring sp ectrum by setting A i = K ( A, i ) for a particular choice of K ( A, i ), see [2 5, Example 1 .2.5]. F or each simpli- cial degr ee q and ﬁnite-dimensional real inner pro duct space V we ca n consider the space T H H ( A ) q ( V ) = ho co lim I q +1 Ω i 0 + ... + i q ( A i 0 ∧ . . . ∧ A i q ∧ S V ) . Here I is the ca tegory whose o b jects are n = { 1 , . . . , n } for n ≥ 0 and whose mo r- phisms a re all injective ma ps. By v arying n we ge t an orthogo nal sp ectrum (in the sense of [29]) T H H ( A ) q for each q , and by v ar ying q we get a simplicial or thogonal sp ectrum. W e deﬁne the orthog onal sp ectrum T H H ( A ) as the g eometric r ealiza- tion o f this simplicial orthog o nal sp ectrum. Beca use it is the geometric realization of a cyclic ob ject we ge t a n S 1 -action. It is p ossible to deﬁne an orthogonal S 1 -sp ectrum as a n orthogona l sp ectr um with an S 1 -action. T o distinguish betw een “naive” and “genuine” equiv ariant sp ectra one can v ar y the mo del structure. If we wish to ev alua te T H H ( A ) on an S 1 - representation V , we ca n deﬁne T H H ( A )( V ) as ab ov e with S 1 acting diagonally . W e will discuss the mo del structure on o rthogonal S 1 -sp ectra in Section 3 b elow. In unpublished work [12], B ¨ okstedt computed T H H ( F p ) and T H H ( Z ), a nd in [22] Hesselho lt and Madsen extended the ﬁrst of these ca lc ulations to T H H ( k ) fo r any p erfect ﬁeld k of ch aracter istic p . The y found that π ∗ T H H ( k ) ∼ = P ( µ 0 ) , a p olynomial algebra over k on one v a riable µ 0 in degr ee 2. Here µ 0 is r epresented by the image o f ¯ τ 0 ∈ π 1 ( H F p ∧ H F p ) in π 1 ( H k ∧ H k ), where τ 0 is the mo d p Bo ckstein and ¯ τ 0 = − τ 0 is its conjugate. If we put a CW structure on H F p with one 0-c ell denoted 1 and one 1- cell denoted a then µ 0 is r epresented by 1 ∧ a − a ∧ 1 in the cellula r c ha in co mplex o f H F p ∧ H F p . It follows that the class µ 0 maps to the class with the same name in H H 2 ( e k ). Let ν p ( i ) deno te the p -a dic v aluation of i . W e will s ee in Ex a mple 2.11 below that π j T H H ( W ( k )) ∼ =      W ( k ) if j = 0 W ν p ( i ) ( k ) if j = 2 i − 1 is o dd 0 if j 6 = 0 is ev e n These sp ectra ar e s o metimes ea sier to understa nd if we use mo d p co e ﬃcie nts. Let V (0) denote the mo d p Mo ore sp ectrum. Then V (0) ∗ T H H ( W ( k )) ∼ = E ( λ 1 ) ⊗ P ( µ 1 ) , where the g round ring is k and | λ 1 | = 2 p − 1, | µ 1 | = 2 p . W e can then recov er T H H ∗ ( W ( k )) by r unning the Bo ckstein s pectr al s equence E ∗ , ∗ 1 = V (0) ∗ T H H ( W ( k ))[ v 0 ] = ⇒ T H H ∗ ( W ( k )) . This sp ectr al se quence is g enerated multiplicativ ely by the diﬀere ntials d j +1 ( µ p j 1 ) = v j +1 0 µ p j − 1 1 λ 1 9 for j ≥ 0. If in addition we use the “ Leibniz r ule” d j +1 ( y p ) = v 0 y p − 1 d j ( y ) then the Bo ckstein sp ectra l seq ue nce is gener ated by the single diﬀerential d 1 ( µ 1 ) = v 0 λ 1 . Remark 2.7. The “L eibniz rule” in the Bo ckstein sp e ctr al se qu enc e go ing fr om mo d p homolo gy to inte gr al homolo gy is discusse d in [30, Pro po sition 6 .8] ; at p = 2 ther e is a c orr e ct ion term for d 2 but otherwise it holds. While we have mo d p and inte gr al homotopy inste ad of homolo gy, a similar r esu lt holds. The c orr e ction term for d 2 at p = 2 is Q 4 ( λ 1 ), and an ex plicit c omputation shows that this is inde e d 0 . Returning to the general theory , supp o se A is a g raded ring. Then we get a splitting of T H H ( A ) int o homogeneous pieces in the same wa y as for Ho chschild homology . Lemma 2.8. Supp ose A is a gr ade d ring or symmetric ring sp e ct r u m. Then T H H ( A ) ∼ = _ s T H H ( A ; s ) , wher e T H H ( A ; s ) is the ge ometric r e alization of the su b c omplex T H H ( A ; s ) • of internal de gr e e s . Pr o of. W rite Gr s A for the s ’th gr aded piec e of A , and deﬁne T H H ( A ; s ) q ( V ) = _ s 0 + ... + s q = s ho colim I q +1 Ω i 0 + ... + i q ( Gr s 0 A i 0 ∧ . . . ∧ Gr s q A i q ∧ S V ) . The face and dege ne r acy maps resp ect this splitting, hence we get a co rresp onding splitting after geometric r ealization.  2.3. A top ol ogical Ho chsc hi ld homo logy sp ectral sequence. Now supp ose A is a complete ﬁltered ring or s y mmetric ring sp ectrum. Reca ll that we ar e assuming that A is strictly connective and conv ergent. W e will assume that each F s A is strictly connective and co n vergent as well. W e can then deﬁne a corresp onding ﬁltration on T H H ( A ), by setting F s T H H ( A ) q ( V ) = hoco lim i 0 + ... + i q ≥ s ho colim I q +1 Ω i 0 + ... + i q  F s 0 A i 0 ∧ . . . ∧ F s q A i q ∧ S V  . W e note that ther e is an induced ma p F s T H H ( A ) q ( V ) ∧ S W → F s T H H ( A ) q ( V ⊕ W ), so each F s T H H ( A ) q is indeed a n or thogonal s pectr um. Moreover, this ﬁltra- tion is compatible with the face and degenera cy maps, so we can deﬁne F s T H H ( A ) as the geometric realization of the simplicial orthogona l sp ectrum q 7→ F s T H H ( A ) q . W e can now pr ove a result due to Brun, stated in mor e mo der n lang uage. Theorem 2.9 (Br un [13]) . Su pp ose A is a c omplete ﬁlter e d ring or symmet ric ring sp e ctrum with asso ciate d gr ade d Gr A . Then ther e is a we akly c onver gent sp e ctr al se quenc e E s,t 1 = T H H s + t ( GrA ; s ) = ⇒ T H H s + t ( A ) . If A is c ommu tative this is an algebr a sp e ctr al s e quenc e. Pr o of. (Compare [1 3, P rop osition 5.2.2].) W e hav e F s T H H ( A ) q ( V ) = ho colim s 0 + ... + s q ≥ s ho colim I q +1 Ω i 0 + ... + i q  F s 0 A i 0 ∧ . . . ∧ F s q A i q ∧ S V  . W e can exchange the tw o homotopy colimits and consider ho colim s 0 + ... + s q ≥ s Ω i 0 + ... + i q  F s 0 A i 0 ∧ . . . ∧ F s q A i q ∧ S V  . 10 Because A is co nnective, it follows that the map ho colim s 0 + ... + s q ≥ s Ω i 0 + ... + i q  F s 0 A i 0 ∧ . . . ∧ F s q A i q ∧ S V  → Ω i 0 + ... + i q  ho colim s 0 + ... + s q ≥ s F s 0 A i 0 ∧ . . . ∧ F s q A i q ∧ S V  bec omes inc r easingly connective as the dimens ion o f V go es to inﬁnity . It follows that the ﬁltration q uotient F s T H H ( A ) q ( V ) / F s +1 T H H ( A ) q ( V ) is equiv alent to Ω i 0 + ... + i q  _ s 0 + ... + s q = s Gr s 0 A i 0 ∧ . . . ∧ Gr s q A i q ∧ S V  through a r ange that go e s to inﬁnity as the dimension of V go es to inﬁnity . If A is comm utative the sh uﬄe pro duct induces an algebra structure on the sp ectral sequence in the usua l way .  Remark 2.10. T o get a mu ltiplic ation on the sp e ctr al se qu en c e it suﬃc es to assume that A is an E 2 ring sp e ct r u m. This is r elate d to how T H H ( A ) is an S -algebr a as long as A is an E 2 ring sp e ctrum, se e [15] . We omit the details, as we wil l not ne e d them. 2.4. Sample com putations. In this section we us e Theor em 2.9 to compute T H H ( A ) in some ex amples. Example 2.11. We start by c omputing T H H ∗ ( W ( k )) fr om T H H ∗ ( k [ x ]) . We ﬁnd that E ∗ , ∗ 1 = T H H ∗ ( k [ x ]) ∼ = P ( x ) ⊗ E ( σ x ) ⊗ P ( µ 0 ) , wher e µ 0 c omes fr om T H H ∗ ( k ) and the tensor pr o duct is over k . The only diﬀer- enc e fr om t he Ho chschild homolo gy c alculation in Example 2.4 is that her e µ 0 is a p olynomial gener ator r ather than a divide d p owers gener ator. We have an imme diate diﬀer ential d 1 ( µ 0 ) = σ x, b e c ause µ 0 is r epr esente d by 1 ⊗ ¯ τ 0 wher e τ 0 is the mo d p Bo ckstein and σ x is r epr esente d by 1 ⊗ x . Henc e E ∗ , ∗ 2 = P ( x ) ⊗ E ( λ 1 ) ⊗ P ( µ 1 ) , wher e µ 1 = µ p 0 and λ 1 = µ p − 1 0 σ x . Next we use the L eibniz ru le to get a diﬀer ential d 2 ( µ 1 ) = xλ 1 , so E ∗ , ∗ 3 = P ( x ) ⊗ E ( λ 2 ) ⊗ P ( µ 2 ) ⊕ { x -tors ion } . In gener al we ﬁnd that E ∗ , ∗ r +1 = P ( x ) ⊗ E ( λ r ) ⊗ P ( µ r ) ⊕ { x -tors ion } , wher e µ r = µ p r − 1 and λ r = µ p − 1 r − 1 λ r − 1 , and we r e c over T H H ∗ ( W ( k )) . Note t hat the E 2 -term of this sp e ct r al se qu enc e is isomorphic to the E 1 -term of the Bo ck- stein sp e ctr al se quenc e which c omputes T H H ∗ ( W ( k )) fr om V (0 ) ∗ T H H ( W ( k ))[ v 0 ] , discusse d imme diately b efor e R emark 2.7. 11 Example 2.12. Next we c ompute T H H ∗ ( W n ( k )) , starting fr om T H H ∗ ( k [ x ] / ( x n )) . As for Ho chschild homolo gy, the c alculation is e asier if p | n . L et E ∗ , ∗ 0 = P n ( x ) ⊗ E ( σx ) ⊗ Γ( x n ) ⊗ P ( µ 0 ) and deﬁne a diﬀer ential d 0 on E 0 by d 0 ( x n ) = nx n − 1 σ x . Then E ∗ , ∗ 1 = T H H ∗ ( k [ x ] / ( x n )) ∼ = H ∗ ( E ∗ , ∗ 0 , d 0 ) . First supp ose p | n . Then d 0 = 0 , so E ∗ , ∗ 1 = E ∗ , ∗ 0 and we get t he same diﬀer en- tials d j +1 ( µ p j 0 ) = x j µ p j − 1 0 σ x as for W ( k ) , for 0 ≤ j ≤ n − 1 . Next supp ose p ∤ n . Then, just as in the c omputation of H H ∗ ( W n ( k )) , t his moves the diﬀer entials ar ound. The end r esult is that the E ∞ term is isomorphi c to what we get in the c ase p | n , exc ept that we have a class γ m − 1 ( x n ) x n − 1 µ 0 inste ad of γ m ( x n ) for e ach m ≥ 0 . Alternativel y, we c an fol low the appr o ach in Re mark 2.6. As for Ho chschild homolo gy, we intr o duc e another ﬁltr ation on T H H ( W n ( k )) so that t he asso ciate d gr ade d is the ring E ∗ , ∗ 0 ab ove, n ow trigr ade d. This r e duc es the c ase p ∤ n to the c ase p | n . Thi s pr oves the fol lowing: Theorem 2.13. The ring T H H ∗ ( W n ( k )) is the homolo gy of the DGA A = Γ W n ( k ) ( x ′ n ) ⊗ E W n ( k ) ( λ 0 ) ⊗ P W n ( k ) ( µ 0 ) wher e λ 0 is r epr esente d by σ x , x ′ n is r epr esente d by x n − np n − 1 µ 0 , and the t ensor pr o duct is over W n ( k ) . The diﬀer ential is multiplic atively gener ate d by d ( µ 0 ) = λ 0 . The homolo gy is given by T H H 2 i ( W n ( k )) ∼ = M 0 ≤ j ≤ i W max( ν p ( j ) ,n ) ( k ) fo r i ≥ 0 T H H 2 i − 1 ( W n ( k )) ∼ = M 1 ≤ j ≤ i W max( ν p ( j ) ,n ) ( k ) fo r i ≥ 1 This r e c overs Brun ’s c alculation of T H H ∗ ( Z /p n ) fr om [13] . We note t hat t he ﬁrst nonzer o o dd gr oup is T H H 2 p − 1 ( W n ( k )) ∼ = k , and t hat t he c anonic al map T H H ( W ( k )) → T H H ( W n ( k )) maps T H H 2 p − 1 ( W ( k )) ∼ = k isomorphic al ly ont o this k . W e include o ne mor e ex ample. This next e xample will not b e used in the res t of the pape r. Example 2.14 . Consider the Ad ams sum mand ℓ of c onne ct ive p -lo c al c omplex K -the ory k u ( p ) . We ﬁlter this by p owers of v 1 : . . . → Σ ( n +1)(2 p − 2) ℓ → Σ n (2 p − 2) ℓ → . . . → ℓ. This ﬁ lt r ation is multiplic ative, and the asso ciate d gr ade d is Grℓ ∼ = H Z ( p ) [ v 1 ] , wher e | v 1 | = 2 p − 2 . Now, c onsider the r esulting sp e ctr al se quenc e with mo d p c o eﬃcients. We ﬁnd that E ∗ , ∗ 1 = V (0) ∗ T H H ( Z ( p ) [ v 1 ]) ∼ = E ( λ 1 ) ⊗ P ( µ 1 ) ⊗ P ( v 1 ) ⊗ E ( σ v 1 ) , 12 and ther e is an imme diate diﬀer ent ial d 1 ( µ 1 ) = σ v 1 , le aving us with E ∗ , ∗ 2 = P ( µ 2 ) ⊗ E ( λ 1 , λ 2 ) ⊗ P ( v 1 ) . Her e µ 2 = µ p 1 and λ 2 = µ p − 1 1 σ v 1 . This c oincides with the E 1 -term of t he v 1 - Bo ckst ein sp e ctra l se quenc e c onsider e d in [32] . This sp e ctr al se qu en c e is also inter esting with int e gr al c o eﬃcients. R e c al l fr om [3] that in T H H ∗ ( ℓ ) t her e is an inﬁ nite v 1 -tower on λ 1 which b e c omes incr e asingly p - divisible . In T H H 2 p − 1 ( Z ( p ) [ v 1 ]) ther e is a Z /p gener ate d by λ 1 and a Z ( p ) gener ate d by σ v 1 , and t her e is a nont rivial extension p · λ 1 = σ v 1 in T H H ∗ ( ℓ ) . Henc e the class λ 1 is 1 /p times a natu r al ly deﬁne d class. We have not attempte d to understand t he gener al b ehavior of the sp e ctr al se qu en c e T H H ∗ ( Z ( p ) [ v 1 ]) = ⇒ T H H ∗ ( ℓ ) , t hough it is inter est ing that with the two sp e ctr al se quenc es in [3] we now have thr e e sp e ctr al se quenc es al l c onver ging to T H H ∗ ( ℓ ) . 2.5. Re l ativ e top ological Ho chsc hi ld homolog y . Given an ideal I ⊂ A , we deﬁne T H H ( A, I ) to be the homotop y ﬁb er of the canonica l map T H H ( A ) → T H H ( A/I ). If A is a ﬁltered ring a s ab ov e , I = F 1 A b ecomes an idea l and T H H ( A, I ) is ho motopy equiv alent to F 1 T H H ( A ). Hence we get a sp ectral se- quence c o nv erging to T H H ∗ ( A, I ) with E 1 term E s,t 1 = ( T H H s + t ( GrA ; s ) s ≥ 1 0 s = 0 W e analyze the eﬀect of removing ﬁltratio n 0 in some examples. Example 2.15. Consider T H H ( W ( k ) , ( p )) with W ( k ) ﬁlter e d by p owers of p . Then we have a sp e ctr al se quenc e E ∗ , ∗ 1 = k er  P ( x ) ⊗ E ( σ x ) ⊗ P ( µ 0 ) → P ( µ 0 )  = ⇒ T H H ∗ ( W ( k ) , ( p )) . We have essential ly the s ame diﬀer entials as b efor e, now with d j +1 ( xµ p j 0 ) = x j +2 µ p j − 1 0 σ x, and this tel ls us the fol lowing. Theorem 2.16. We have T H H q ( W ( k ) , ( p )) ∼ =      pW ( k ) if q = 0 W ν p ( i )+1 ( k ) if q = 2 i − 1 is o dd 0 if q ≥ 2 is even In p articular the long exact se qu enc e c oming fr om the ﬁb er se quenc e deﬁning T H H ( W ( k ) , ( p )) de gener ates into short exact se quenc es 0 → T H H 2 i ( k ) ∼ = k → T H H 2 i − 1 ( W ( k ) , ( p )) ∼ = W ν p ( i )+1 ( k ) → T H H 2 i − 1 ( W ( k )) ∼ = W ν p ( i ) ( k ) → 0 . Example 2.17 . Next we c onsider T H H ( W n ( k ) , ( p )) . L et E ∗ , ∗ 0 = k er  P n ( x ) ⊗ E ( σx ) ⊗ Γ( x n ) ⊗ P ( µ 0 ) → P ( µ 0 )  and let d 0 b e gener ate d mu ltiplic atively by d 0 ( γ j ( x n )) = nx n − 1 γ j − 1 ( x n ) for j ≥ 1 . Then we have a sp e ctr al se quenc e E ∗ , ∗ 1 = H ∗ ( E ∗ , ∗ 0 , d 0 ) = ⇒ T H H ∗ ( W n ( k ) , ( p )) . 13 As long as ν p ( i ) < n the fol lowing happ ens. The class µ i 0 was supp ose d to supp ort a diﬀer en t ial, but it is missing, so the tar get of the diﬀer ential survives. This gives an extr a class in T H H 2 i − 1 ( W n ( k ) , ( p )) . If ν p ( i ) ≥ n then µ i 0 survives t o give a class in T H H 2 i ( W n ( k )) ; running t he r elative sp e ctr al se qu enc e we then get one class less in T H H 2 i ( W n ( k ) , ( p )) . Henc e we ﬁn d the fol lo wing (c omp ar e The or em 2.13). Theorem 2.18. We have T H H 2 i ( W n ( k ) , ( p )) ∼ = W max( ν p ( i ) ,n − 1) ( k ) ⊕ M 0 ≤ j ≤ i − 1 W max( ν p ( j ) ,n ) ( k ) T H H 2 i − 1 ( W n ( k ) , ( p )) ∼ = W max( ν p ( i )+1 ,n ) ( k ) ⊕ M 1 ≤ j ≤ i − 1 W max( ν p ( j ) ,n ) ( k ) As in the DGA app earing in Theorem 2.13 w e will sometimes denote the class represented b y σ x in T H H ∗ ( W ( k ) , ( p )) o r T H H ∗ ( W n ( k ) , ( p )) b y λ 0 . 3. The trace method In this s ection we r eview the “tra ce metho d” for computing algebr aic K -theor y . Most of the materia l in this section is known, w e include it here for the reader’s conv enience and for ease of reference. In so me instanc e s we hav e generalized kno wn calculations from F p or Z p to k or W ( k ). W e also introduce some notation that will b e used in the later sections. 3.1. Fixed p oin ts and geometric ﬁxed p oin ts. F rom now on we will take G -sp ectrum to mean orthogona l G -sp ectrum in the sense of [28], se e also [2 4, Ap- pendix B]. Recall that an orthog onal G -sp ectrum is simply an orthogo nal sp ectrum with a G -action; the diﬀerence b etw een G - s pec tr a indexed on diﬀerent universes is taken car e o f b y v ar ying the mo del structure. In particula r, diﬀerent families of representations g ive diﬀerent notions of ﬁbr ant r eplacement. W e will use a v e r sion o f the p o sitive c omplete s table mo del structure from [24, Prop ositio n B.63] wher e the weak equiv alences a re deﬁned using only ﬁnite sub- groups of G . F or details see [1]. If we take the H -ﬁxe d po in ts for some H ≤ G w e get a W ( H )-spe c trum in the obvious wa y . It is impo rtant to note that taking ﬁxed p oints do es not commute with sp ectriﬁcation. In particular , if X is a p ointed G -s pace then (Σ ∞ G X ) H is very diﬀerent fro m Σ ∞ W ( H ) X H . Instead, the classic a l to m Dieck splitting gives a formula for (Σ ∞ G X ) H . Second, we hav e the geometric ﬁx e d point spec trum Φ H ( E ) , obtained as a left Kan extensio n as in [28, Deﬁnition V.4.3]. If we apply geometr ic ﬁxed points for a subgroup H ≤ G we again ge t a W ( H )-sp ectrum. T ak ing geometric ﬁxed p oints has the pr op erty that if X is a p ointed G -spac e then (Σ ∞ G X ) H ∼ = Σ ∞ W ( H ) X H . Now let G = S 1 and let H = C n . If E is an S 1 -sp ectrum then Φ C n ( E ) is an S 1 /C n -sp ectrum. The n ’th ro ot provides an isomorphism ρ n : S 1 → S 1 /C n , a nd we ca n use this to change Φ C n ( E ) back into an S 1 -sp ectrum ρ ∗ n Φ C n ( E ) . The functor Φ H preserves co ﬁbrations and weak eq uiv a le nc e s b etw een coﬁbrant ob jects in our mo del str ucture, a nd has a left adjoint we will denote by L Φ H . Deﬁnition 3.1 ([6]) . A cyclotomic stru ctur e on an S 1 -sp e ctrum E is a se quenc e of c omp atible maps ρ ∗ n Φ C n ( E ) → E 14 for al l n ≥ 2 such that t he induc e d map ρ ∗ n L Φ C n ( E ) → E is a we ak e quivalenc e. Her e c omp atible me ans that the two maps fr om ρ ∗ mn Φ C mn ( E ) = ρ ∗ m Φ C m ( ρ ∗ n Φ C n ( E ) ) to E agr e e. The canonical example o f a cyclotomic sp ectrum is Σ ∞ S 1 LX + , the equiv a r iant susp ension spec tr um of a free lo op space. In this cas e Φ C n (Σ ∞ S 1 LX + ) ≃ Σ ∞ S 1 /C n ( LX ) C n + , and we see that this is a cyclotomic sp ectrum beca us e ( LX ) C n ∼ = LX . W e a lso k now [1 1, 22] that T H H ( A ) as deﬁned by B¨ okstedt is a cyclotomic sp ectrum; the pr o of that T H H ( A ) a s deﬁned in Section 2.2 ab ov e is a cyclotomic sp ectrum is s imilar to the pro of found in [22, Sec tio n 2] in the class ical setting. This sho uld not b e surpr ising, since T H H (Σ ∞ Ω X + ) ≃ Σ ∞ LX + . Deﬁnition 3.2. L et A b e a ring or symmetric ring sp e ctru m. Then t he TR -gr oups of A ar e the homotopy gr ou ps of t he sp e ctr a TR m ( A ) = T H H ( A ) C p m − 1 . (This is often denote d TR m ( A ; p ) ; we le ave out the p to s implify the notation.) These sp ectra ar e r elated b y a num b er of maps , in a wa y that we now r ecall. There is a map F : TR m +1 ( A ) → TR m ( A ) called F ro b enius, which is giv en by inclusion of ﬁxed p oints. Deﬁnition 3.3. L et A b e a ring or symmetric ring sp e ctrum. Then TF( A ) is deﬁne d as TF( A ) = holim F TR m ( A ) . The F ro be nius ha s an asso cia ted transfer map V : TR m ( A ) → TR m +1 ( A ) c a lled the versc hiebung. There is a map d : TR m q ( A ) → TR m q +1 ( A ) deﬁned as m ultiplicatio n by the fundamental clas s of S 1 /C p m − 1 . Finally , there is a r estriction map R : TR m +1 ( A ) → TR m ( A ) , which is deﬁned us ing the cycloto mic structure on T H H ( A ). T o be precise, the map R : TR 2 ( A ) → TR 1 ( A ) = T H H ( A ) of non-equiv aria nt sp ectra is given b y the canonica l map fro m ﬁxed po in ts to geo - metric ﬁxed p oints, follow ed by the equiv alence of the geo metric ﬁxed p oints with T H H ( A ). Mor e genera lly R : TR m +1 ( A ) → TR m ( A ) is the C p m − 1 ﬁxed p oints of this ma p. If we b eef this up to include (vir tual) S 1 -representations the map R takes the form R : Σ α TR m +1 ( A ) → Σ α ′ TR m ( A ) , where α = [ β ] − [ γ ] ∈ RO ( S 1 ) and α ′ = ρ ∗ p ( α C p ), see [2 1, 1 9]. It is genera lly hard to understand ﬁxed point sp ectra directly , and it is sometimes useful to compar e the actual ﬁxed point sp ectr um TR m +1 ( A ) to the homotopy ﬁxed point s pectr um T H H ( A ) hC p m . Let T = T H H ( A ), let T hC p m denote the 15 homotopy orbit spectr um and let T tC p m denote the T a te sp ectrum. Then there is a fundamental diag ram [9, Theo rem 1.10 and Section 2], as follows. T hC p m N / / =   TR m +1 ( A ) R / / Γ m   TR m ( A ) ∂ / / b Γ m   Σ T hC p m =   T hC p m N h / / T hC p m R h / / T tC p m ∂ / / Σ T hC p m If we take the ho motopy inv er se limit over F we obtain a version o f the funda- men tal dia gram fea tur ing S 1 . Σ T hS 1 N / / =   TF( A ) Γ   R / / TF( A ) ∂ / / b Γ   Σ 2 T hS 1 =   Σ T hS 1 N h / / T hS 1 R h / / T tS 1 ∂ / / Σ 2 T hS 1 Now consider the sp ecial case A = Σ ∞ Ω X + , so T H H ( A ) ≃ Σ ∞ S 1 LX + . The tom Dieck s plitting s ays that (Σ ∞ S 1 LX + ) C p m ≃ _ 0 ≤ k ≤ m (Σ ∞ LX + ) hC p k . In this case the top r ow in the fundamen tal diagra m splits. In general, the ex istence of the top r ow in the fundamental diagram can b e thought of as a non-split version of the tom Dieck splitting for genera l A . Finally we get to topolo gical cyc lic homo logy . Deﬁnition 3.4. L et A b e a ring or symmetric ring sp e ctrum. The top olo gic al cyclic homolo gy TC( A ) of A is the homotopy e qu alizer TC( A ) → TF( A ) R ⇒ id TF( A ) . Alternativel y, it c an b e deﬁne d as the homotopy e qualizer TC( A ) → TR( A ) F ⇒ id TR( A ) , wher e TR( A ) = holim R TR m ( A ) , or as TC( A ) = holim R,F TR m ( A ) . There is a tr ace ma p trc : K ( A ) → TC( A ) which is an isomor phism o n homotopy g roups in degree ≥ 0 after p -completio n if A is e.g . a ﬁnite W ( k )-algebra [31]. These comparison results go thro ug h r elative TC a nd r elative K -theo r y . Given a functor F from rings (or s ymmetric ring sp ectr a ) to sp ectra and an ideal I ⊂ A , we deﬁne F ( A, I ) as the homotopy ﬁb er F ( A, I ) → F ( A ) → F ( A/I ) . This deﬁnes r elative K -theo ry and TC, and we hav e a rela tive tr ace map trc : K ( A, I ) → TC( A, I ) . 16 What McCa rthy [31] actually shows is that this relative trace ma p is an eq uiv a lence after p -completion when I is nilpo tent. (Actually the relative trace map is an equiv alence even b efore p -co mpleting, see [16] for details.) The calcula tion of TC( k ) r ecalled below plus Kr atzer’s calculation of K ( k ) [2 7] provides the bas e case which we use to conclude that the absolute trace ma p is an equiv alence in no n-negative degrees after p - completion for cer tain r ings. In particular this mea ns that up to p - completion we have K q ( W n ( k ) , ( p )) ∼ = TC q ( W n ( k ) , ( p )) for all q . In s o me c a ses we ca n use a result of Tsalidis to study TR m ( A ) in terms of the C p m T ate sp ectrum. Theorem 3.5 (Tsalidis, [37]) . L et A b e a c onne ctive symmetric ring sp e ctru m of ﬁnite typ e. Supp ose b Γ 1 : π q T H H ( A ) → π q T H H ( A ) tC p is an isomorphism for q ≥ q 0 . Then b Γ m : TR m q ( A ) → π q T H H ( A ) tC p m is an isomorphism for q ≥ q 0 for al l m . This allows for an induction argument, as follows. Recall [20, 9] that there is a T ate sp ectra l sequence conv e r ging to π ∗ T H H ( A ) tC p m , and that w e get sp ectral sequences conv er ging to π ∗ T H H ( A ) hC p m and π ∗ T H H ( A ) hC p m by (with a small mo diﬁcation in ﬁltra tion 0) r estricting to the ﬁrst or s e c ond quadrant, resp ectively . If the conditions of Tsalidis ’ Theorem hold and we unders ta nd TR m ∗ ( A ), we can often understand the spectr al sequence c o nv erging to π ∗ T H H ( A ) tC p m bec ause we know what it co n verges to in de g ree q ≥ q 0 . Then restr icting this sp ectra l sequence to the second q uadrant gives a spectr al sequence computing π ∗ T H H ( A ) hC p m , and this determines TR m +1 q ( A ) for q ≥ q 0 . By taking the homotopy in verse limit o ver F , we can also conclude that the maps Γ : TF q ( A ) → π q T H H ( A ) hS 1 and b Γ : TF q ( A ) → π q T H H ( A ) tS 1 are isomo r phisms for q ≥ q 0 + 1. 3.2. T op ologi cal cyclic hom ology o f k . Many computations r ely on the corr e- sp onding computations for k , so following [22] we sp ell this case o ut ﬁrst. Recall that T H H ∗ ( k ) = P ( µ 0 ) is a p olynomial a lgebra ov er the gr o und ﬁeld k on a degree 2 genera tor µ 0 . Then the T ate sp ectr al sequence co n verging to π ∗ T H H ( k ) tC p m takes the following fo r m: b E ∗ , ∗ 2 = P ( µ 0 ) ⊗ E ( u m ) ⊗ P ( t, t − 1 ) = ⇒ π ∗ T H H ( k ) tC p m . This is bigra ded by ﬁbe r degr ee a nd homo lo gical degree, with | µ 0 | = (2 , 0 ), | u m | = (0 , − 1) and | t | = (0 , − 2). The top ologica l deg ree is the sum of the tw o degree s . The class v 0 = tµ 0 represents multiplication by p a nd is a p ermanent cycle. W e hav e a diﬀerential d 2 m +1 ( u m ) . = t m +1 µ m 0 = tv m 0 , leaving b E ∗ , ∗ 2 m +2 = b E ∗ , ∗ ∞ = P m ( v 0 ) ⊗ P ( t, t − 1 ) . 17 This is the a sso ciated gr aded of π ∗ T H H ( k ) tC p m ∼ = W m ( k )[ t, t − 1 ] . When m = 1 the map b Γ 1 : T H H ∗ ( k ) → π ∗ T H H ( k ) tC p is given by b Γ( µ 0 ) . = t − 1 . This is a n iso morphism in non-neg ative degr ees and Tsalidis ’ Theore m applies. T o compute π ∗ T H H ( k ) hC p m we restrict the T ate spectr al sequence to the seco nd quadrant, a nd we ha ve E ∗ , ∗ 2 = P ( µ 0 ) ⊗ E ( u m ) ⊗ P ( t ) . W e hav e the same d 2 m +1 -diﬀerential, which leav es E ∗ , ∗ 2 m +2 = E ∗ , ∗ ∞ = P m ( v 0 ) { t i | i > 0 } ⊕ P m +1 ( v 0 ) { µ j 0 | j ≥ 0 } . This is the a sso ciated gr aded of π ∗ T H H ( k ) hC p m ∼ = W m ( k ) { t i | i > 0 } ⊕ W m +1 ( k ) { µ j 0 | j ≥ 0 } . Next we co mpute R : π ∗ T H H ( k ) C p m → π ∗ T H H ( k ) C p m − 1 , and here we need to be a little bit careful. F rom [22] we know that w e have an isomor phism ρ R m : π 0 T H H ( k ) C p m − 1 → W m ( k ) which is compatible with the restr ic tion map R . But we would like to take the inv erse limit o ver F r ather than R , b ecause we wan t to study TF( A ) rather than TR( A ). W e hav e a co mm utative dia gram . . . / / W 3 ( k ) F / / W 3 ( φ 3 )   W 2 ( k ) F / / W 2 ( φ 2 )   k φ   . . . / / W 3 ( k ) R / / W 2 ( k ) R / / k where W m ( φ m ) : W m ( k ) → W m ( k ) is an isomo r phism, and for our purp ose s it is better to use the isomorphism ρ F m : π 0 T H H ( k ) C p m − 1 → W m ( k ) given by ρ F m = W m ( φ m ) ◦ ρ R m . This extends to an isomorphism ρ F m : π ∗ T H H ( k ) C p m − 1 → W m ( k )[ µ 0 ] . Lemma 3.6. Supp ose we use the ab ove map ρ F m to identify π ∗ T H H ( k ) C p m − 1 with W m ( k )[ µ 0 ] . Then the F r ob enius map F : T H H ( k ) C p m → T H H ( k ) C p m − 1 is given by the usu al r estriction map on Witt ve ctors and by µ 0 7→ µ 0 . The r estriction map R : T H H ( k ) C p m → T H H ( k ) C p m − 1 is given by the usual r estriction map on Witt ve ctor fol lo we d by W m ( φ − 1 ) and by µ 0 7→ pλ m µ 0 for some unit λ m ∈ Z /p m . It follows that TF ∗ ( k ) ∼ = W ( k )[ µ 0 ] and that TC i ( k ) ∼ =      Z p if i = 0 cok er ( W ( φ ) − 1) if i = − 1 0 otherwise Here we have used that the (co)equalizer of W ( φ − 1 ) and 1 is isomor phic to the (co)kernel o f W ( φ ) − 1. 18 The trace ma p K ∗ ( k ) → TC ∗ ( k ) is, after p -completion, an isomorphism in degree 0 a nd trivia l in degr ee − 1, since K ( A ) is a co nnective s p ectr um fo r a n y r ing A . T ogether w ith Kratzer’s calculation [27, Coro llary 5.5] of K ( k ) this provides the base case wher e the trace map is an equiv alence on non-nega tive homotopy gro ups after p -adic completion. 3.3. T op ologi cal cyclic homol ogy of W ( k ) . Next w e turn to the top ological cyclic ho mology o f W ( k ). In this cas e the T ate sp ectral sequence conv erging to π ∗ T H H ( W ( k )) tS 1 has E 2 -term b E ∗ , ∗ 2 =  W ( k ) { 1 } ⊕ M i ≥ 1 W ν p ( i )+1 ( k ) { λ 1 µ i − 1 1 }  ⊗ P ( t, t − 1 ) . F rom [9] we know the be havior of the corr esp o nding spectr a l sequence with mod p co eﬃcients, and this lets us say s ome things ab out this spec tr al seq ue nce . But rather tha n going int o details we will instead study the relative version, whic h has the form b E ∗ , ∗ 2 =  W ( k ) { p } ⊕ M i ≥ 1 W ν p ( i )+1 ( k ) { λ 0 µ i − 1 0 }  ⊗ P ( t, t − 1 ) . Prop ositio n 3. 7. The diﬀer entials in the ab ove sp e ctr al se quenc e ar e al l P ( t, t − 1 ) - line ar and given by d 2 r ( p r ) . = rt r λ 0 µ r − 1 0 . The E ∞ term is given by b E ∗ , ∗ ∞ = M i ≥ 1 W ν p ( i )+1 ( k ) { λ 1 µ i − 1 1 } ⊗ P ( t, t − 1 ) . Pr o of. W e hav e a long exact sequence of sp ectral sequences . . . → b E ∗ , ∗ r ( W ( k ) , ( p )) → b E ∗ , ∗ r ( W ( k )) → b E ∗ , ∗ r ( k ) → . . . for each r ≥ 2 . Consider the class p r ∈ b E 0 , 0 2 ( W ( k )). In the T ate sp ectra l sequence for k ther e is a hidden extensio n, with p r represented b y t r µ r 0 . The connecting map T H H ∗ ( k ) → T H H ∗− 1 ( W ( k ) , ( p )) maps µ r 0 to r λ 0 µ r − 1 0 , hence in the T ate s p ectr a l sequence it maps t r µ r 0 to rt r λ 0 µ r − 1 0 . If there was no suc h d 2 r diﬀerential this would break ex actness at the E r term, so the res ult fo llows.  Remark 3 .8. The ab ove r esult c an b e explaine d in a s imple but non-rigor ous way as fol lows. We have π ∗ T H H ( k ) tS 1 ∼ = W ( k )[ t, t − 1 ] , and by “c ompr essing” the ﬁltr ation in b E ∗ , ∗ 2 ( k ) the map fr om b E ∗ , ∗ 2 ( W ( k )) b e c omes surje ctive with kernel c onc entra t e d in o dd total de gr e e, isomorph ic to the given E ∞ term. This computes TF ∗ ( W ( k ) , ( p )) up to extensions in tw o wa ys , using either the map Γ : TF ∗ ( W ( k ) , ( p )) → π ∗ T H H ( W ( k ) , ( p )) hS 1 or the map b Γ : TF ∗ ( W ( k ) , ( p )) → π ∗ T H H ( W ( k ) , ( p )) tS 1 . One could now attempt to compute TC ∗ ( W ( k ) , ( p )) by un- derstanding the restr ic tion map R on TF ∗ ( W ( k ) , ( p )). This is complicated, and we do not now how to do this without pa s sing to mo d p co e ﬃcien ts and following [9, 1 0]. W e will omit a discussion of the mo d p calculation as w e will not need it. 19 3.4. T op ologi cal cyclic homolog y of k [ x ] / ( x n ) . W e als o need the computation of TC ∗ ( k [ x ] / ( x n )) from [21]. Supp ose Π is a pointed monoid, and let k (Π ) deno te the p ointed monoid algebra. Then T H H ( k (Π)) ≃ T H H ( k ) ∧ B cy ∧ (Π), and this is a n equiv alence of S 1 -equiv ariant spectra . In pa rticular, let Π n = { 0 , 1 , x, . . . , x n − 1 } so that k (Π n ) = k [ x ] / ( x n ). Then it is c lear that B cy ∧ (Π n ) splits as a wedge of homo - geneous summands, using the degree in x , and Hess elholt and Madsen calcula ted the S 1 -equiv ariant homotopy type o f as follows: Theorem 3.9 (Hesselholt-Madsen [21]) . The cyclic b ar c onst ruction B cy (Π n ) splits, S 1 -e quivariantly, as B cy (Π n ) ∼ = _ s ≥ 0 B cy (Π n ; s ) , wher e B cy (Π n ; 0 ) = S 0 , B cy (Π n ; s ) ≃ S 1 ( s ) + ∧ S λ d if n do es not divide s and B cy (Π n ; s ) sits as the mapping c one S 1 ( s/n ) + ∧ S λ d n → S 1 ( s ) + ∧ S λ d → B cy (Π n ; s ) if n divides s . Here d = ⌊ s − 1 n ⌋ , λ d = C (1) ⊕ . . . ⊕ C ( d ), and S 1 ( s ) denotes S 1 as an S 1 -space with an a ccelerated action. If p do e s no t divide n then this s impliﬁes after p -completion as B cy (Π n ) ∧ p ≃ ( S 0 ) ∧ p ∨ _ n ∤ s B cy (Π n ; s ) ∧ p . The homo topy g roups o f T H H ( k [ x ] / ( x n )) are given by the homo lo gy of the DGA P n ( x ) ⊗ E ( σx ) ⊗ Γ( x n ) ⊗ P ( µ 0 ) , with diﬀerential generated multiplicativ ely by d ( x n ) = nx n − 1 σ x . (Compare with Theorem 2 .1 3.) Because the ab ov e splitting is S 1 -equiv ariant it follows that TR m ( k [ x ] / ( x n )) ≃ _ s ≥ 0 TR m ( k [ x ] / ( x n ); s ) and that TF( k [ x ] / ( x n )) ≃ _ s ≥ 0 TF( k [ x ] / ( x n ); s ) . W e need to understand the T ate spec tr um and ho motopy ﬁxed p oint sp ectr um of T H H ( k [ x ] / ( x n ); s ). W e start with the T ate sp ectrum. It is convenien t to deﬁne z s = t ⌊ s/n ⌋ γ ⌊ s/n ⌋ ( x n ) x { s/n } , where { s/n } denotes the residue o f s mod n . This class lives in to tal deg ree 0 in the T ate sp ectral sequence. Then we can describ e the E 2 term o f the T ate sp ectral sequence conv erg ing to π ∗ T H H ( k [ x ] / ( x n ); s ) tS 1 as follows. W e start with b E ∗ , ∗ 0 ( s ) = P ( µ 0 ) ⊗ P ( t, t − 1 ) ⊗ k { z s , z s − 1 σ x } . If n | s and p ∤ n we deﬁne d 0 ( z s ) = tz s − 1 σ x , otherwise we deﬁne d 0 = 0. Then b E ∗ , ∗ 2 ( s ) is the homolo gy of ( b E ∗ , ∗ 0 ( s ) , d 0 ). If n | s and p ∤ n we hav e b E ∗ , ∗ 2 ( s ) = 0. The higher diﬀerentials are g iven by d 2 ν p ( s )+2 ( z s ) . = t ν p ( s )+1 µ ν p ( s ) 0 z s − 1 σ x 20 if n ∤ s and d 2 ν p ( n ) ( z s ) . = t ν p ( n )+1 µ ν p ( n ) 0 z s − 1 σ x if n | s . (This includes the a b ove d 0 diﬀerential in the case ν p ( n ) = 0.) This leav es the E ∞ term b E ∗ , ∗ ∞ ( s ) = P ν p ( s ) ( v 0 ) ⊗ P ( t, t − 1 ) ⊗ k { z s − 1 σ x } if n ∤ s and b E ∗ , ∗ ∞ ( s ) = P ν p ( n ) ( v 0 ) ⊗ P ( t, t − 1 ) ⊗ k { z s − 1 σ x } if n | s . This is the a sso ciated g raded of π ∗ T H H ( k [ x ] / ( x n ); s ) tS 1 = W ν p ( s ) ( k ) ⊗ P ( t, t − 1 ) ⊗ k { z s − 1 σ x } if n ∤ s and π ∗ T H H ( k [ x ] / ( x n ); s ) tS 1 = W ν p ( n ) ( k ) ⊗ P ( t, t − 1 ) ⊗ k { z s − 1 σ x } if n | s . Restricting the T ate sp e ctral seq uence to the s econd quadrant we obtain a cal- culation of π ∗ T H H ( k [ x ] / ( x n ); s ) hS 1 . It will b e co nv enient to in tro duce a no ther family o f e lemen ts. Let y s = µ −⌊ s/n ⌋ 0 γ ⌊ s/n ⌋ ( x n ) x { s/n } . Then the clas s µ j 0 y s is in T H H 2 j ( k [ x ] / ( x n ); s ) for j ≥ ⌊ s n ⌋ a nd the cla s s µ j 0 y s − 1 σ x is in T H H 2 j +1 ( k [ x ] / ( x n ); s ) for j ≥ d . (Recall that we deﬁned d = ⌊ s − 1 n ⌋ .) With this naming c onv ention we then get π ∗ T H H ( k [ x ] / ( x n ); s ) hS 1 = L i> 0 W ν p ( s ) ( k ) { t i µ d 0 y s − 1 σ x } ⊕ L j ≥ d W ν p ( s )+1 ( k ) { µ j 0 y s − 1 σ x } if n ∤ s and π ∗ T H H ( k [ x ] / ( x n ); s ) hS 1 = L i> 0 W ν p ( n ) ( k ) { t i µ d 0 y s − 1 σ x } ⊕ L j ≥ d W ν p ( n ) ( k ) { µ j 0 y s − 1 σ x } if n | s . The ab ov e calculation of π ∗ T H H ( k [ x ] / ( x n )) hS 1 and π ∗ T H H ( k [ x ] / ( x n )) tS 1 do es not, in itse lf, compute TF ∗ ( k [ x ] / ( x n )), b ecause Ts alidis’ Theor em do es not apply . But it is po ssible to co mpute TF ∗ ( k [ x ] / ( x n ); s ) fo r each s directly , identifying it with TR ν p ( s )+1 ∗− λ d − 1 ( k ) if n ∤ s a nd with the cokernel of V ν p ( n ) : TR ν p ( s/n )+1 ∗− λ d − 1 ( k ) → TR ν p ( s )+1 ∗− λ d − 1 ( k ) if n | s . And we hav e the following computation, see [21]. See also [19, 2] in the cas e k = F p . Theorem 3.10. L et λ b e an actual c omplex S 1 -r epr esentation. The n TR m ∗− λ ( k ) is c onc entra t e d in even de gr e e. If i ≥ dim C ( λ ) we have TR m 2 i − λ ( k ) = W m ( k ) . If dim C ( λ ( j − 1) ) > i ≥ dim C ( λ ( j ) ) t hen TR n 2 i − λ ( k ) = W m − j ( k ) . 21 This is prov ed using an R O ( S 1 )-graded v er sion o f the fundamental dia g ram. F or any vir tua l S 1 -representation α we hav e a fundamen tal diagram T [ α ] hC p m N / / =   TR m +1 ( A )[ α ] R / / Γ m   TR m ( A )[ α ′ ] b Γ m   T [ α ] hC p m N h / / T [ α ] hC p m R h / / T [ α ] tC p m This dia gram ca n also b e used to compute R : TR m +1 ∗− λ ( k ) → TR m ∗− λ ′ ( k ). W e use Theorem 3.9 a b ove and ﬁnd (compare [22, Section 8 .2]) that if n ∤ s then TF( A [ x ] / ( x n ); s ) ≃ ( S 1 ( s ) + ∧ S λ d ∧ T H H ( A )) S 1 ≃ Σ F ( S 1 ( s ) + , T H H ( A ) ∧ S λ d ) S 1 ≃ Σ( T H H ( A ) ∧ S λ d ) C s up to p -completion. Similarly , if n | s then TF( A [ x ] / ( x n ); s ) sits in a co ﬁbration sequence Σ( T H H ( A ) ∧ S λ d ) C s/n V n − → Σ( T H H ( A ) ∧ S λ d ) C s → TF( A [ x ] / ( x n ); s ) . Hence TF ∗ ( k [ x ] / ( x n ); s ) ∼ = TR ν p ( s )+1 ∗− 1 − λ d ( k ) when n ∤ s and s imilarly for the ca se n | s . This is what Hess elholt and Madsen used to co mpute K ∗ ( k [ x ] / ( x n )). With this w e can describ e the maps Γ : TF ∗ ( k [ x ] / ( x n )) → T H H ( k [ x ] / ( x n )) hS 1 and b Γ : TF ∗ ( k [ x ] / ( x n )) → T H H ( k [ x ] / ( x n )) tS 1 . The map Γ sends TF( k [ x ] x/ ( x n ); s ) to T H H ( k [ x ] / ( x n ); s ) hS 1 and even though Ts alidis’ theorem do e s not apply we do hav e the following: Theorem 3.11. In de gr e e 2 i + 1 for i ≥ d the map Γ : TF 2 i +1 ( k [ x ] / ( x n ); s ) → π 2 i +1 T H H ( k [ x ] / ( x n ); s ) hS 1 is an isomorphism. In de gr e e 2 i + 1 for i < d the map Γ : TF 2 i +1 ( k [ x ] / ( x n ); s ) → π 2 i +1 T H H ( k [ x ] / ( x n ); s ) hS 1 is inje ctive. W e hav e a s imilar result for the map b Γ. In this case b Γ sends TF( k [ x ] / ( x n ); s ) to T H H ( k [ x ] / ( x n ); ps ). Theorem 3.12. In de gr e e 2 i + 1 for i ≥ d the map b Γ : TF 2 i +1 ( k [ x ] / ( x n ); s ) → π 2 i +1 T H H ( k [ x ] / ( x n ); ps ) tS 1 is an isomorphism. In de gr e e 2 i + 1 for i < d the map b Γ : TF 2 i +1 ( k [ x ] / ( x n ); s ) → π 2 i +1 T H H ( k [ x ] / ( x n ); ps ) tS 1 is inje ctive. If we use T he o rem 3.11 to na me elements in TF ∗ ( k [ x ] / ( x n )), the map b Γ is given by b Γ( µ a 0 y s − 1 σ x ) = t − a z ps − 1 σ x. F rom this we can read oﬀ the action of R : TF 2 i +1 ( k [ x ] / ( x n ); s ) → TF 2 i +1 ( k [ x ] / ( x n ); s/ p ) . 22 Theorem 3.13. Supp ose ν p ( s ) ≥ 1 . In de gr e e 2 i + 1 for i ≥ d the map R : TF 2 i +1 ( k [ x ] / ( x n ); s ) → TF 2 i +1 ( k [ x ] / ( x n ); s/ p ) is mu ltiplic ation by p i − d . In de gr e e 2 i + 1 for i < d the m ap R is an isomorphism. In particular this mea ns that there is a stable rang e. If i < d then R : TF 2 i +1 ( k [ x ] / ( x n ); s ) → TF 2 i +1 ( k [ x ] / ( x n ); s/ p ) is a n iso mo rphism, a nd if i = d it is surjective. 4. A spectral sequence on fixed points In this s e c tion we intro duce the ﬁlter ed r ing spectr al sequences on ﬁxed p oints and use it to ca lculate the asso ciated grade d of TF ∗ ( W n ( k ) , ( p )). Using this we prov e Theorem A. 4.1. A s p ectra l s equence for TR n ( A ) . The sp ectral s equence in Theor em 2 .9 comes fro m an S 1 -equiv ariant ﬁltration on T H H ( A ), so it is rea sonable to exp ect it to induce a ﬁltration on ﬁxed p oints as well. O nce we have this, we ge t an induced sp ectral sequence on ﬁxed p oints as well. Theorem 4.1. Supp ose A is a c omplete ﬁlter e d ring or symmetric ring sp e ct ru m with asso ciate d gr ade d GrA . Then t her e is a we akly c onver gent s p e ctr al se quenc e E s,t 1 = TR m s + t ( GrA ; s ) = ⇒ TR m s + t ( A ) . If A is c ommu tative then this is an algebr a sp e ctr al se quenc e. Pr o of. W e prove the case m = 2, the gener al case is s imila r. W e use the p - fold edgewise sub divisio n mo del of T H H , which is the S 1 -sp ectrum with V ’th space the geometric r ealization o f T H H [ p ] ( A ) q ( V ) = ho co lim I p ( q +1) Ω i 0 + ... + i p ( q +1) − 1  A i 0 ∧ . . . ∧ A i p ( q +1) − 1 ∧ S V  . The a dv antage of this mo del is that we have a simplicial a ction o f C p . While T H H ( A ) might not be ﬁbrant, the discussion in [22, Sectio n 2 a nd Ap- pendix A] implies that the ﬁxed points of T H H ( A ) (without ﬁbra n t r eplacement) calculates the co rrect homo topy type. W e hav e a ﬁltr a tion on ea ch T H H [ p ] ( A ) q ( V ) coming from the ﬁltration on each space A i in the sp ectrum A , and this induces a ﬁltration o n T H H [ p ] ( A ) whic h is equiv alent to the ﬁltration on T H H ( A ) co nsidered b efore. With this mo del it is clea r tha t ta king ﬁx e d p oints pre s erves the ﬁltratio n, s ince the re presentation spheres S V are all in ﬁltr a tion 0 .  There is of course a similar spectr al sequence conv e r ging to the homo to p y gr o ups of the re la tive s pectr um. Corollary 4.2. Supp ose A is a c omplete ﬁlter e d ring or symmetr ic ring sp e ctru m with asso ciate d gr ade d Gr A and let I = F 1 A ⊂ A . Then ther e is a sp e ctr al se qu enc es E s,t 1 = ( TR m s + t ( GrA ; s ) if s ≥ 1 0 if s = 0 = ⇒ TR m s + t ( A, I ) . A description o f the E 1 -term of this spectr al sequence for ( A, I ) = ( W n ( k ) , ( p )) follows from the calculations in [21], r ecalled in Sectio n 3.4 a bove. Because we will only need the corr esp onding sp ectra l sequenc e for TF we omit the details. 23 4.2. A sp ectral se quence for TF( A ) . The F rob enius F is simply the inclusio n of ﬁxed p oints, so it is compatible with the ﬁltration and we can take a ho motopy inv erse limit to g et a sp ectral sequence conv er g ing to TF ∗ ( A ). Theorem 4.3. Supp ose A is a c omplete ﬁlter e d ring or symmetric ring sp e ct ru m with asso ciate d gr ade d GrA . Then t her e is a we akly c onver gent s p e ctr al se quenc e E s,t 1 = TF s + t ( GrA ; s ) = ⇒ TF s + t ( A ) . As usual ther e is a rela tive version. Corollary 4.4. Supp ose A is a c omplete ﬁlter e d ring or symmetr ic ring sp e ctru m with asso ciate d gr ade d Gr A and let I = F 1 A ⊂ A . Then ther e is a sp e ctr al se qu enc es E s,t 1 = ( TF s + t ( GrA ; s ) if s ≥ 1 0 if s = 0 = ⇒ TF s + t ( A, I ) . F or A = W n ( k ) this E 1 -term is studied in [21] as reca lled in Sectio n 3 .4 ab ov e, and we ﬁnd the following. Prop ositio n 4.5. Supp ose p ∤ n . Then t he ab ove sp e ctr al se quenc e c onver ging to TF ∗ ( W n ( k ) , ( p )) has E 1 -term E s, ∗ 1 = TR ν p ( s )+1 ∗− λ d − 1 ( k ) if n ∤ s and E s, ∗ 1 = 0 if n | s for s ≥ 1 . Note that this is concentrated in o dd top ological deg r ee, and hence this sp ectra l sequence collapses at the E 1 -term. In particular , E s, ∗ 1 is a W ν p ( s )+1 ( k ) in suﬃcien tly high o dd tota l degr ee. Prop ositio n 4.6. Supp ose p | n . Then t he ab ove sp e ctr al se quenc e c onver ging to TF ∗ ( W n ( k ) , ( p )) has E 1 -term E s, ∗ 1 = ( TR ν p ( s )+1 ∗− λ d − 1 ( k ) if n ∤ s cok er  TR ν p ( s/n )+1 ∗− λ d − 1 ( k ) V ν p ( n ) − → TR ν p ( s )+1 ∗− λ d − 1 ( k )  if n | s for s ≥ 1 . In the cas e n | s the cokernel is isomorphic to W ν p ( n ) ( k ) in suﬃciently high o dd total degr e e, and again we see that the E 1 -term is conce n tr ated in odd top ologic al degree. Corollary 4.7. The sp e ctr al se quenc e c onver ging to TF ∗ ( W n ( k ) , ( p )) c ol lapses at the E 1 -term. W e compa re this to W ( k ), for which we ﬁnd the following. (See also P rop osition 3.7 ab ove.) Corollary 4.8. The sp e ctr al se quenc e c onver ging to TF ∗ ( W ( k ) , ( p )) has E 1 -term E s, ∗ 1 = TR ν p ( s )+1 ∗− 1 ( k ) for s ≥ 1 . This sp e ctr al se quen c e also c ol lapses at t he E 1 -term. 24 Pr o of of The or em A . Supp ose k → k ′ is a G -Galo is extension of p erfect ﬁelds of characteristic p for a ﬁnite gro up G . Then it follows from Cor ollary 4.7 a nd 4.8 that TF ∗ ( W n ( k ′ )) ∼ = TF ∗ ( W n ( k )) ⊗ W ( k ) W ( k ′ ) with the induced G -action. Hence the homotopy ﬁxed p oint sp ectral sequence H ∗ ( G ; TF ∗ ( W n ( k ′ ))) = ⇒ π ∗ (TF( W n ( k ′ )) hG ) collapses at the E 2 -term, and it fo llows that the canonica l map TF( W n ( k )) → TF( W n ( k ′ )) hG is a n equiv alence. The maps R and 1 a re G -equiv a riant, and homotopy equalize r s commute with homotopy ﬁxe d p oints. Hence the canonical map TC( W n ( k )) → TC( W n ( k ′ )) hG is an equiv alence as well. The statement of the theorem follows by taking connective cov ers a nd p -co mpleting.  5. A commuting square of spectral s equences In this se c tion w e compare tw o wa y s o f c a lculating the homotopy groups of the T ate sp ectrum T H H ( W n ( k ) , ( p )) tS 1 . The main purp ose is to better understand the map b Γ : TF ∗ ( W n ( k ) , ( p )) → π ∗ T H H ( W n ( k ) , ( p )) tS 1 and thereb y better under s tand the restriction ma p on TF ∗ ( W n ( k ) , ( p )). 5.1. A comm uting square. W e b egin with a ge neral observ ation abo ut what happ ens when we hav e a “commuting squar e” of sp ectra l sequences. W e claim no originality for this, but because w e could not ﬁnd exactly what we need in the literature we include this material her e . But see [4] for a rela ted discuss io n. Suppo se w e hav e a spectrum X with tw o compatible ﬁltrations on it. That means we have a sp ectrum F i,j X for each i and j , with ma ps F i,j X → F i − 1 ,j X and F i,j X → F i,j − 1 X which commute in the obvious sense, with X = F −∞ , −∞ X the homotopy colimit. By for getting one o f the ﬁltratio ns w e get a horizontal ﬁltration . . . → F i +1 , −∞ X → F i, −∞ X → F i − 1 , −∞ X → . . . and a corr esp onding hor izontal sp ectral se q uence, and by forgetting the o ther w e get a vertical ﬁltration . . . → F −∞ ,j +1 X → F −∞ ,j X → F −∞ ,j − 1 → . . . and a cor resp onding vertical sp ectra l seq uence. Similarly , for each j we hav e a horizontal ﬁltration . . . → F i +1 ,j X/F i +1 ,j +1 X → F i,j X/F i,j +1 X → F i − 1 ,j X/F i − 1 ,j +1 X → . . . and a c o rresp onding horizontal spe c tral sequence, and for ea ch i we have a vertical ﬁltration . . . → F i,j +1 X/F i +1 ,j +1 X → F i,j X/F i +1 ,j X → F i,j − 1 X/F i +1 ,j − 1 X → . . . and a cor resp onding vertical sp ectra l seq uence. 25 Putting all of these together we get a “commuting squar e ” of sp ectral sequences E i,j,k 1 = π i + j + k Gr i,j X + 3   ( E ′ 1 ) i + k,j = π i + j + k Gr j v X   ( E ′′ 1 ) i,j + k = π i + j + k Gr i h X + 3 π i + j + k X . Here Gr i,j X is the itera ted homo topy coﬁber F i,j X/F i +1 ,j X F i +1 ,j X/F i +1 ,j +1 X , while Gr i h X is the homotopy co ﬁber F i, −∞ /F i +1 , −∞ and s imila rly for Gr j v X . Question 5.1. Given a class in α ∈ π m X , how c an we ﬁn d al l the r epr esentatives of α in E ∗ , ∗ , ∗ 1 ? An element y ∈ E i,j,m − i − j 1 represents α if y lifts to an element ˜ y ∈ π m F i,j X which maps to α under the map F i,j X → X . Assuming the sp ectral s e quences conv erge w e can do the following: Fir st, let i 0 be the la rgest integer so that α is the image o f an element y ′ 0 ∈ π m F i 0 , −∞ X . Next, let j 0 be the la r gest integer so that y ′ 0 is the imag e of a n element y ′′ 0 ∈ π m F i 0 ,j 0 X . Then the image y 0 of y ′′ 0 in E i 0 ,j 0 ,m − i 0 − j 0 1 = π m Gr i 0 ,j 0 X represents α , and survives when we go counter-clo ckwise around the ab ove squar e o f sp ectral sequences. Next let i 1 < i 0 denote the largest in teger so that the image y ′ 1 of y ′′ 0 in π m F i 1 ,j 0 X is the image of some ele men t y ′′ 1 ∈ π m F i 1 ,j 1 X with j 1 > j 0 . W e can cho ose j 1 to b e maximal. Then the image y 1 of y ′′ 1 in π m Gr i 1 ,j 1 X also repr esents α . W e can con tinue like this until we ﬁnd a last elemen t y ′′ a ∈ π m F i a ,j a X . Its image y a ∈ π m Gr i a ,j a X represents α a nd surv ives when w e go c lo c kwise a round the ab ov e sq uare of sp ectra l seq ue nc e s . The fact that the ima g e of y ′′ b − 1 in π m F i b +1 ,j b − 1 X/F i b +1 ,j b − 1 +1 X is nonzero but maps to zer o in π m F i b ,j b − 1 X/F i b ,j b − 1 +1 X implies tha t ther e is some z b ∈ E i b ,j b − 1 ,m − i b − j b − 1 +1 1 = π m +1 Gr i b ,j b − 1 with d h i b − 1 − i b ( z b ) = y b − 1 . Similarly , there is so me z ′ b ∈ E i b ,j b − 1 ,m − i b − j b − 1 1 with d v j b − j b − 1 ( z ′ b ) = y b . I t fol- lows from a diagram chase that we can choose z ′ b = − z b . (The minus sig n is not impo rtant for our purp oses.) W e illus trate with the following zig-zag in the case a = 2 : y 0 z 1 ✳ d h 6 6 ♥ ♥ ♥ ♥ ♥ ♥ ✏ d v ( ( P P P P P P y 1 z 2 ✳ d h 6 6 ♥ ♥ ♥ ♥ ♥ ♥ ✏ d v ( ( P P P P P P y 2 A t one end of the zig- zag we ﬁnd the r epresentativ e for α that survives going counter-clockwise ar ound the sq ua re of sp ectral se q uences, at the other end we ﬁnd the representativ e that sur vives go ing clo c kwise. 5.2. The commuting square for the T ate sp ectrum. Given a ﬁltered ring A , we now hav e t wo co mpa tible ﬁltra tions o n T H H ( A ) tS 1 , so b y a minor reindexing of the setup in the previo us section we g et a co mm uting sq ua re of sp ectral sequences 26 as follows: b E i,j, ∗ 1 = π i + j T H H ( Gr A ; j ) ⊗ P ( t ± 1 ) + 3   ( b E ′ 1 ) i + ∗ ,j = π i + j + ∗ T H H ( Gr A ; j ) tS 1   ( b E ′′ 2 ) i + j, ∗ = π i + j T H H ( A ) ⊗ P ( t ± 1 ) + 3 π i + j + ∗ T H H ( A ) tS 1 W e have a similar comm uting square of spectr a l sequence s with relative groups everywhere co mputing π ∗ T H H ( A, I ) tS 1 . Now co nsider the above squa re for ( A, I ) = ( W n ( k ) , ( p )). Then b E ∗ , ∗ , ∗ 1 = T H H ∗ ( k [ x ] / ( x n ) , ( x )) ⊗ P ( t, t − 1 ) , and we unders tand three o f the four sp ectra l seque nc e s completely . Recall that we deﬁned z s = t ⌊ s/n ⌋ γ ⌊ s/n ⌋ ( x n ) x { s/n } , a nd that the top horizontal sp ectral sequence has diﬀerentials z s 7→ t ν p ( s )+1 µ ν p ( s ) 0 z s − 1 σ x for n ∤ s and z s 7→ t ν p ( n )+1 µ ν p ( n ) 0 z s − 1 σ x for n | s . If we start with b E ∗ , ∗ , ∗ 0 = M s ≥ 1 k { z s , z s − 1 σ x } ⊗ P ( t, t − 1 ) and interpret some of the diﬀerentials as d 0 diﬀerentials then this makes sense even when p ∤ n and n | s . The right hand side vertical sp ectral s equence c o llapses, b ecaus e the E 1 term is concentrated in o dd tota l degr ee. Finally , the left hand side v ertical sp ectral s equence has diﬀer en tials which w e think o f as generated mu ltiplicatively by d 1 ( µ 0 ) = σ x , where we use the Leibniz rule d r +1 ( y p ) = y p − 1 xd r ( y ). This is only mo r ally true, b eca use T H H ( W n ( k ) , ( p )) is only a non-unital ring spectr um and there is no class µ 0 in T H H 2 ( W n ( k ) , ( p )). If we were considering the non-r elative version of the left ha nd side vertical s p ectr al sequence then this would b e ac c ur ate. In any case, by Examples 2.5 and 2.17 we know all the diﬀerentials in the left hand side vertical sp ectral sequence as well. Example 5 . 2. F or a typic al example of how su ch a zig-zag works, let us c onsider the c ase p = 3 and n = 4 , and let us start with the class x 4 xσ x in total de gr e e 3 , which survives going clo ckwise ar ound the squar e of sp e ct ra l se quenc es. First we use the zig-zag x 4 xσ x µ 0 x 4 x ✭ d v 4 4 ❤ ❤ ❤ ❤ ❤ ❤ ✕ d h * * ❯ ❯ ❯ ❯ ❯ ❯ tµ 0 x 4 σ x to r eplac e x 4 xσ x by the r epr esentative tµ 0 x 4 σ x . Next we use t he zig-zag tµ 0 x 4 σ x tµ 2 0 x 4 ✮ d v 4 4 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✔ d h * * ❚ ❚ ❚ ❚ ❚ ❚ t 3 µ 4 0 x 2 σ xσ x to r eplac e it by t he r epr esentative t 3 µ 4 0 x 2 σ x . Her e d h c an b e c alculate d using yet another zig-zag. 27 Final ly we use the zig-zag t 3 µ 4 0 x 2 σ x t 3 µ 5 0 x 2 ✯ d v 5 5 ❥ ❥ ❥ ❥ ❥ ❥ ✔ d h ) ) ❚ ❚ ❚ ❚ ❚ ❚ t 4 µ 5 0 xσ x to ﬁn d t he r epr esentative t 4 µ 4 0 xσ x that su rvives going c oun ter-clo ckwise ar ound the squar e of sp e ctr al se quenc es. (If we wer e c onsidering T H H ( W 4 ( k )) r ather than T H H ( W 4 ( k ) , (3 )) we would have had a diﬀer ential d v 1 ( t 4 µ 6 0 ) = t 4 µ 5 0 xσ x and our class would r epr esent zer o is π 3 T H H ( W 4 ( k )) tS 1 .) Remark 5. 3. It is p ossible to describ e the b ottom horizontal sp e ctr al se quenc e as wel l. We omit the details as we wil l not ne e d them. 6. A common spectral sequence f or homotopy orbits and topological cyclic homol ogy Suppo se as usual that A is a complete ﬁltere d r ing or symmetric ring sp ectrum and let I = F 1 A . In addition to the usual homotopy orbit spectra l sequence, we get a spectra l sequence conv er ging to π ∗ Σ T H H ( A, I ) hS 1 coming from Σ T H H ( A, I ) hS 1 being the homotopy ﬁb er of R : TF( A, I ) → TF( A, I ). The amazing thing is that we hav e another sp ectral sequence with the same E 1 term conv erg ing to TC ∗ ( A, I ). The sta rting p oint is the following r esult. Theorem 6.1. Supp ose A is a c omplete ﬁleter e d ring or symmetric ring sp e ctru m. Then R : TR m +1 ( A ) → TR m ( A ) sends F s TR m +1 ( A ) to F ⌈ s/p ⌉ TR m ( A ) and R : TF( A ) → TF( A ) sends F s TF( A ) → F ⌈ s/p ⌉ TF( A ) . Pr o of. W e prove the case m = 1, the gener al case is s imila r. W e use the p - fold edgewise sub division mo del of T H H considered in the pro of o f Theorem 4.1 ab ov e. Fixed p oints by the action of C p are ta ken spa cewise, and a ﬁxed p oint of a term in the colimit deﬁning T H H [ p ] ( A ; V ) q lo oks like ( a 0 ∧ . . . ∧ a q ) ∧ p ∧ v where v ∈ ( S V ) C p . Now, if a i is homog eneous of ﬁltra tion | a i | , this is in ﬁltration degree p ( | a 0 | + . . . + | a q | ). Applying R r eplaces this by ( a 0 ∧ . . . ∧ a q ) ∧ v , which has ﬁltration de g ree | a 0 | + . . . + | a q | .  Deﬁnition 6.2. Supp ose A is a c omplete ﬁ lter e d ring or symmetric ring sp e ctrum. L et F s Σ T H H ( A ) hS 1 denote the homotopy ﬁb er F s Σ T H H ( A ) hS 1 → F s TF( A ) R − → F ⌈ s/p ⌉ TF( A ) and let F s TC( A ) denote the homotopy e qualizer F s TC( A ) → F s TF( A ) R ⇒ I F ⌈ s/p ⌉ TF( A ) . Theorem 6. 3. Supp ose A is a c omplete ﬁlter e d ring or symmetric ring sp e ctrum. Then ther e is a sp e ctr al se quenc e with E s,t 1 = ker  TF s + t ( GrA ; s ) R → TF s + t ( GrA ; s/p )  ⊕ cok er  TF s + t +1 ( GrA ; s ) R → TF s + t +1 ( GrA ; s/p )  28 for s ≥ 1 and E 0 ,t 1 = 0 , c onver ging t o π ∗ Σ T H H ( A, I ) hS 1 . Ther e is another sp e ctr al se quenc e with the same E 1 term c onver ging to TC s + t ( A, I ) . Mor e over, al l diﬀer entials t hat multiply the ﬁltr ation by a factor of less than p ar e isomorphic and al l extens ions that multiply the ﬁltra t ion by a factor of less t han p ar e isomorphi c. Pr o of. It is clea r that there is a spectra l sequence asso ciated to the ﬁltration of Σ T H H ( A, I ) hS 1 and ano ther sp ectral sequence asso ciated to the ﬁltration of TC( A, I ), a nd we can compute the E 1 term o f the ﬁrst sp ectral s equence using the diagram F s +1 Σ T H H ( A, I ) hS 1 / /   F s +1 TF( A ) R / /   F ⌈ ( s +1) /p ⌉ TF( A )   F s Σ T H H ( A, I ) hS 1 / /   F s TF( A ) R / /   F ⌈ s/p ⌉ TF( A )   Gr s Σ T H H ( A, I ) hS 1 / / Gr s TF( A ) R / / Gr s/p TF( A ) A simila r diag r am, with R replaced b y R − I on the top tw o rows, calculated the E 1 term o f the s e c ond sp ectra l seq uence. The las t part, comparing shor t diﬀerentials and shor t extensions in the tw o sp ectral sequences, follows by Lemma 6.4 b elow.  Lemma 6.4 (Brun [14, Lemma 5.3]) . Supp ose s < t ≤ ps . Th en F s TC( A ) /F t TC( A ) ≃ F s Σ T H H ( A ) hS 1 /F t Σ T H H ( A ) hS 1 . This is esp ecially useful b eca us e we can compute π ∗ T H H ( W n ( k )) hS 1 through a range of degr ees (co mpare [14, Pr op osition 6.4 and 7.2]). Prop ositio n 6. 5. F or 2 i ≤ 2 p − 2 we have π 2 i T H H ( W n ( k )) hS 1 ∼ = W n ( i +1) ( k ) and for 2 i − 1 ≤ 2 p − 3 we have π 2 i − 1 T H H (( W n ( k )) hS 1 = 0 . Pr o of. Recall that the ho motopy or bit sp ectral sequence lo o ks like π ∗ T H H ( A )[ t − 1 ] = ⇒ π ∗ T H H ( A ) hS 1 , and recall that through degree 2 p − 2 we have π 2 i T H H ( W n ( k )) ∼ = W n ( k ) and π 2 i − 1 T H H ( W n ( k )) = 0. Hence the homotopy orbit spectra l sequence collapses at the E 2 -term throug h the ra nge of degr ees we conside r . This then shows that π ∗ T H H ( W n ( k )) hS 1 has the r equired length ov er k . T o show that the ex tensions are maxima lly nontrivial, w e consider the co rre- sp onding ho motopy or bit spec tr al s equence with mo d p co eﬃcients: V (0) ∗ T H H ( W n ( k ))[ t − 1 ] = ⇒ V (0) ∗ T H H ( W n ( k )) hS 1 . Let β n denote the elemen t in V (0) 1 T H H ( W n ( k )) which is co ming fro m p n − 1 ∈ T H H 0 ( W n ( k )) ∼ = W n ( k ). Then we hav e an immediate diﬀerential d 2 ( t − 1 ) . = β n 29 and it follows tha t w e hav e a diﬀer e n tia l d 2 ( t − i ) . = t − i +1 β n for all i ≤ p − 1. This implies that V (0) 2 i T H H ( W n ( k )) hS 1 ∼ = k for 2 i ≤ 2 p − 2, and it follows tha t the ex tensions ar e maxima lly nontrivial. The result no w follo ws be c ause the only maxima lly nontrivial extension of W ni ( k ) by W n ( k ) is W n ( i +1) ( k ). (Recall that all the iden tiﬁca tions with Witt vectors are additive only; no multiplicativ e s tructure is implied.)  Corollary 6.6. F or 2 i − 1 ≤ 2 p − 3 we have π 2 i − 1 F 1 Σ T H H ( W n ( k )) hS 1 ∼ = W ( n − 1) i ( k ) and for 2 i ≤ 2 p − 2 we have π 2 i F 1 Σ T H H ( W n ( k )) hS 1 = 0 . Pr o of of The or em B. Bec a use TF ∗ ( W n ( k ) , ( p )) is concentrated in o dd to ta l degr ee it follows that in this case all the diﬀeren tials in the ab ov e sp ectra l sequence co n- verging to TC ∗ ( W n ( k ) , ( p )) go from o dd to even total degree. The result now follows by a co un ting a rgument.  If instead we use the non-rela tiv e K - theory sp ectrum K ( W n ( k )), w e pick up an extra K ( k ) a nd we get the following. Corollary 6.7. S u pp ose k = F q is a ﬁnite ﬁeld with q elements. Then | K 2 i − 1 ( W n ( k )) | | K 2 i − 2 ( W n ( k )) | = q ( n − 1) i ( q i − 1) for al l i ≥ 2 . Finally we can prov e the las t main res ult. Pr o of of The or em C. W e co mpare the diﬀerentials a nd ex tens io ns in the s p ectr al sequence co n verging to π ∗ Σ T H H ( W n ( k ) , ( p )) hS 1 to the diﬀerentials and extensio ns in the sp ectral sequence co n verging to TC ∗ ( W n ( k ) , ( p )), using Theo rem 6 .3. T o compute diﬀerentials we need to calcula te R . T o do this, we use the commu- tative diagram TF ∗ ( W n ( k ) , ( p )) R / / Γ   TF ∗ ( W n ( k ) , ( p )) b Γ   π ∗ T H H ( W n ( k ) , ( p )) hS 1 R h / / π ∗ T H H ( W n ( k ) , ( p )) tS 1 and the observ ation (Theor e ms 3.11 and 3.12) tha t Γ and b Γ a re injective. Given an element in TF ∗ ( W n ( k ) , ( p )), we need to be able to ﬁnd a repres ent ative for its image under b Γ in the T ate s pectr al sequence; fo r this we us e the “co mm utative square” b E ∗ , ∗ , ∗ 1 = T H H ∗ ( k [ x ] / ( x n ) , ( x )) ⊗ P ( t, t − 1 ) + 3   π ∗ T H H ( k [ x ] / ( x n ) , ( x )) tS 1   T H H ∗ ( W n ( k ) , ( p )) ⊗ P ( t, t − 1 ) + 3 π ∗ T H H ( W n ( k ) , ( p )) tS 1 30 of sp ectra l seq uences dis cussed in Sectio n 5 a b ove. W e use the collapsing s p ectr a l seq uence E ∗ , ∗ 1 = TF ∗ ( k [ x ] / ( x n ) , ( x )) hS 1 = ⇒ TF ∗ ( W n ( k ) , ( p )) and the injective map Γ : TF ∗ ( k [ x ] / ( x n ) , ( x )) → π ∗ T H H ( k [ x ] / ( x n ) , ( x )) hS 1 to name ele men ts in TF ∗ ( W n ( k ) , ( p )). Recall that we deﬁned y s = µ −⌊ s/n ⌋ 0 γ ⌊ s/n ⌋ ( x n ) x { s/n } . W e are interested in the class µ a 0 y s − 1 σ x , whic h lives in TF 2 a +1 ( k [ x ] / ( x n ); s ) for a ≥ ⌊ s − 1 n ⌋ . (If p ∤ n and n | s then this class is zero.) Through the ra nge of deg rees we a re interested in w e hav e a ≤ p − 1. Also r ecall that w e deﬁned z s = t ⌊ s/n ⌋ γ ⌊ s/n ⌋ ( x n ) x { s/n } . Then t b z s − 1 σ x lives in the E 2 term o f the T ate sp ectra l sequence converging to π ∗ T H H ( k [ x ] / ( x n ); s ) tS 1 for all b . The hea rt of the argument is the following. Even though the cla ss t b z s − 1 σ x , in- terpreted as lying in b E ∗ , ∗ , ∗ 1 , is killed by a diﬀer e n tia l in the vertical sp e ctral sequence conv erg ing to T H H ∗ ( W n ( k ) , ( p )) ⊗ P ( t, t − 1 ), we can use the ab ov e commu tative square of sp ectra l sequences to ﬁnd a diﬀerent representativ e for this class in b E ∗ , ∗ , ∗ 1 . Start with a non-zero class µ a 0 y s − 1 σ x ∈ TF 2 a +1 ( k [ x ] / ( x n ); s ) with ⌊ s − 1 n ⌋ ≤ a ≤ p − 1. Then b Γ( µ a 0 y s − 1 σ x ) = t − a z ps − 1 σ x. If the p ower of t in t − a z ps − 1 σ x = t ⌊ ( ps − 1) /n ⌋− a γ ⌊ ( ps − 1) /n ⌋ ( x n ) x { ( ps − 1) /n } σ x is non-negative then ther e is nothing to do: t − a z ps − 1 σ x lies in the res triction of the T ate spectr al sequence for ( W n ( k ) , ( p )) to the second quadrant and hence t − a z ps − 1 σ x is in the imag e of the map R h . W e conclude that µ a 0 y s − 1 σ x is R of a class in ﬁltration ps . (W e alrea dy knew that, b ecause in this case R : TF 2 a +1 ( k [ x ] / ( x n ); ps ) → TF 2 a +1 ( k [ x ] / ( x n ); s ) is a n isomo rphism.) If the p ower of t in t − a z ps − 1 σ x is negative we can use the zig- zag t − a z ps − 1 σ x t − a µ 0 z ps − 1 ✬ d v 3 3 ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ✗ d h + + ❲ ❲ ❲ ❲ ❲ ❲ t − a +1 µ 0 z ps − 2 σ x to r eplace t − a z ps − 1 σ x with the representativ e t − a +1 µ 0 z ps − 2 σ x . If p ∤ n a nd n | ps − 1 then the horizontal diﬀeren tial on t − 1 µ 0 z ps − 1 is not given b y this formula but rather by a pplying a nother zig-zag t − a +1 µ 0 z ps − 2 σ x t − a +1 µ 2 0 z ps − 2 ✬ d v 3 3 ❣ ❣ ❣ ❣ ❣ ❣ ✗ d h + + ❲ ❲ ❲ ❲ ❲ ❲ t − a +2 µ 2 0 z ps − 3 σ x but the end r esult is the same. 31 W e can co nt inu e lik e this un til w e obtain a representativ e t − a + b µ b 0 z ps − b − 1 σ x with a non- negative p ow er of t . By o ur a ssumptions this is always p ossible, a nd it happ ens for some b ≤ p − 1. (Thr ough this rang e of degrees w e never encounter µ b 0 for p | b , s o the diﬀerential o n µ b 0 is a lwa ys a s simple a s p oss ible. W e also never encounter z c for p | c , so the diﬀere ntial on z c is also as simple as p ossible.) W e conclude that this class is in the image of R h , and that µ a 0 y s − 1 σ x is R of a class in ﬁltration ps − b . A similar discussion shows that mu ltiples o f µ a 0 y s − 1 σ x ar e in the ima g e of R , starting with the fact that b Γ( p a ′ µ a 0 y s − 1 σ x ) is repres ent ed by t − a + a ′ µ a ′ 0 z ps − 1 σ x in the T ate sp ectra l seq uence co n verging to π ∗ T H H ( k [ x ] /x n , ( x )) tS 1 . F rom the ab ove dis c ussion we see that through degr e e 2 p − 1 the only “lo ng” diﬀerential in the spe c tr al seque nce conv erg ing to π ∗ Σ T H H ( W n ( k ) , ( p )) hS 1 is the one coming from R ( µ p − 1 0 σ x ) = µ p − 1 0 σ x . With our conv entions, R is given by the inv erse F rob e nius (see Sectio n 3.2) and is an is o morphism k ∼ = E 1 , 2 p − 2 1 = E 1 , 2 p − 2 p d p = R − − − − → E p,p − 2 p = E p,p − 2 1 ∼ = k . In the spectr al s equence fo r top olo gical cyclic ho mology we have k ∼ = E 1 , 2 p − 2 1 = E 1 , 2 p − 2 p d p = R − 1 − − − − − → E p,p − 2 p = E p,p − 2 1 ∼ = k . The kernel of d p in this particula r bidegree is Z /p and the cokernel is coker ( φ − 1 ) : k → k . This proves Theorem C up to extens io ns. It follows from Theorem 6.3 and Prop ositio n 6.5 that a ll the extensio ns except o ne are maximally nontrivial. The one extension we ca nno t calculate in this w ay g o es from ﬁltration 1 to ﬁltration 3 (or , in some cases, to ﬁltration 4 or 5 ) in degree 2 p − 3, b eca use s uc h an extension o nly makes sense after the d p − 1 diﬀerential happ ens, and F 1 TC( W n ( k )) /F p TC( W n ( k )) is just outside the ra nge where Lemma 6.4 applies. The picture b elow shows what the spec tral sequence lo o ks lik e in this degr ee in the cas e p = 5, n = 5; the dashed line is the extensio ns we ca nno t calcula te using Theor e m 6.3. 2p-4 2p-3 The fact that this par ticular extensions is as cla imed follows by co mparing with TC 2 p − 3 ( W ( k ) , ( p )).  References [1] V i gleik Angeltv eit, Andrew Blumberg, T eena Gerhardt, Michael A. Hill, Tyler La w s on, and Michae l Mandell. Relative cyclotomic sp ectra and topological cyclic homology via the norm. arXiv:1401.500 1 . [2] V i gleik Angeltveit and T eena Gerhardt. RO ( S 1 )-graded TR-groups of F p , Z and ℓ . J. Pur e Appl. Algebr a , 215(6):1405–1419 , 2011. [3] V i gleik Angeltv eit, M i c hael A. Hil l, and Tyler Lawson. T op ological Hochsc hild homology of ℓ and k o. Amer. J. Math. , 132(2):297–33 0, 2010. [4] V i gleik Angeltve it and John A. Lind. Uniqueness of B P h n i . Pr eprint, arXiv:1501 .01448 , 2015. 32 [5] V i gleik Angeltv eit and John Rognes. Hopf algebra structure on top ological Ho chsc hild ho- mology . Algebr. Geo m. T op ol. , 5:1223–1290 (electronic), 2005. [6] A ndr ew Blumberg and M ic hael Mandell. The homotop y theory of cyclotomic s pectra. Pr eprint, arXiv:1303.1694 , 2013. [7] J. Michae l Boardman. Conditionally con vergen t s p ectral sequences. In Homotopy invariant algebr aic structur es (Baltimor e, MD, 1998) , volume 239 of Conte mp. Math. , pages 49–84. Amer. Math. Soc., Providence , RI, 1999. [8] M . B¨ okstedt, W. C. Hsiang, and I. M adsen. The cyclotomic tr ace and algebraic K -theory of spaces. Invent. Math. , 111(3):465– 539, 1993. [9] M . B¨ okstedt and I. Madsen. T op ological cyclic homology of the integ ers. Ast´ erisque , (226):7– 8, 57–143, 1994. K -theory (Strasb ourg, 1992). [10] M . B¨ okstedt and I. M adsen. Algebraic K -theory of lo cal n umber ﬁelds: the unramiﬁed case. In Pr osp e cts in top olo gy (Princeton, NJ, 1994) , volume 138 of Ann. of Math. St ud. , pages 28–57. Princeton Univ. Press, Princeton, NJ, 1995. [11] M ar cel B¨ okstedt. T op ological ho c hsc hi ld homology . Unpublishe d . [12] M ar cel B¨ okstedt. The topological ho ch sc hild homology of Z and Z /p . Unpublishe d . [13] M . Brun. T op ological Hochsc hild homology of Z /p n . J. Pur e Appl. A lgebr a , 148(1):29–76, 2000. [14] M or ten Brun. Filtered top ological cyclic homology and relative K -theory of nilp oten t ideals. Algebr. Ge om. T op ol. , 1:201–230 (electronic), 2001. [15] M or ten Brun, Zbigniew Fiedorowicz, and Rainer M. V ogt. On the multiplicativ e structure of topological Ho chsc hild homology . Algebr. Ge om. T op ol. , 7:1633–1650, 2007. [16] Bj ørn Ian Dundas, Thomas G. Go odwil lie, and Randy McCarthy . The lo c al structur e of algebr aic K -the ory , vo l ume 18 of Algebr a and Applic ations . Spri nger, Berl in, 2012. [17] Leonard Ev ens and Eric M. F riedlander. On K ∗ ( Z /p 2 Z ) and related homology groups. T r ans. Am er. Math. So c. , 270(1):1–46, 1982. [18] Thomas Geisser. On K 3 of Witt v ectors of l ength t wo o ver ﬁnite ﬁelds. K - The ory , 12(3):193– 226, 1997. [19] T eena Gerhardt. The RS 1 -graded equiv ariant homotop y of T H H ( F p ). Algebr aic and Geo - metric T op olo gy , 8(4):1961–19 87, 2008. [20] J. P . C . Greenlees and J. P . May . Generalized Tate cohomology . Mem. Amer. Math. So c. , 113(543) : viii+178, 1995. [21] Lars Hesselholt and Ib Madsen. Cyclic p olytopes and the K -theory of truncat ed p olynomial algebras. Invent. M ath. , 130(1):73–97 , 1997. [22] Lars Hesselholt and Ib M adsen. On the K -theory of ﬁnite algebras ov er Witt vectors of p erfect ﬁelds. T op olo gy , 36(1):29–101, 1997. [23] Lars Hesselholt and Ib M adsen. On the K -theory of l ocal ﬁelds. A nn. of Math. (2) , 158(1):1– 113, 2003. [24] M i c hael A. Hill, Michael J. Hopkins, and Douglas C. Rav enel. On the non-existence of ele- men ts of Kerv aire inv ariant one. [25] M ar k Hov ey , Bro oke Shipley , and Jeﬀ Smith. Symmetric sp ectra. J. Amer. Math. So c. , 13(1):149– 208, 2000. [26] J. F. Jardine. A closed mo del structure f or diﬀeren tial graded algebras. In Cy clic c ohomo lo gy and nonc ommutative geo met ry (Waterlo o, ON, 1995) , volume 17 of Fields Inst. Commun. , pages 55–58. Amer. Math. So c., Providence, RI, 1997. [27] Ch. Kratzer. λ -structure en K -th´ eorie alg ´ ebrique. Comment. Math. Helv. , 55(2):233–2 54, 1980. [28] M . A. Mandell and J. P . May . Equiv ariant orthogonal sp ectra and S -modules. Me m. Am e r. Math. So c. , 159(755):x+108 , 2002. [29] M . A. Mandell, J. P . May , S. Sc hw ede, and B. Shipley . Mo del categories of diagram s p ectra. Pr o c. L ondon Math. So c . (3) , 82(2):441–512 , 2001. [30] J. Pet er May . A general algebraic approac h to Steenrod op erations. In The Steenr o d Algebr a and its Applic ations (Pr o c. Conf. to Celeb ra t e N. E. Ste e nro d’s Six tieth Birthday, Batt el le Memorial Inst., Columbus, Ohio, 1970) , Lecture Notes in Mathematics, V ol. 168, pages 153–231. Springer, Berlin, 1970. [31] Randy McCarthy . Relativ e algebraic K -theo ry and topological cyclic homology . A cta Math. , 179(2):197 –222, 1997. 33 [32] J. E. McClur e and R. E. Staﬀeldt. On the topological Ho chsc hil d homology of b u. I. Amer. J. Math. , 115(1):1–45 , 1993. [33] Stephen A. M itc hell. The algebraic K -theory spectrum of a 2-adic lo cal ﬁeld. K -The ory , 25(1):1–37 , 2002. [34] John Rognes. Al gebraic K -theory of the tw o-adic in tegers. J. Pur e Appl. Algebr a , 134(3) : 287– 326, 1999. [35] John Rognes. The pro duct on topological Hochsc hild homology of the inte gers wi th mo d 4 coeﬃcients. J. Pur e Appl. Algebr a , 134(3) : 211–218, 1999. [36] John Rognes. T opological cyclic homology of the integers at tw o. J. Pur e Appl. Algebr a , 134(3):219 –286, 1999. [37] Stavros Tsal idis. T opological Hoc hsc hild homology and the homot opy descen t problem. T op ol- o gy , 37(4):913–93 4, 1998. Ma thema tical Sciences In stitute, Australian Na tiona l University, Australia E-mail addr ess : vigleik.a ngeltveit@anu.edu.au 34

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