Periodicity of hermitian K-groups

PERIODICITY OF HERMITIAN K -GR OUPS A. J. BERRICK, M. KAR OUBI AND P . A. ØSTVÆR 0. Introduction and st a tements of main resul ts By the f undamental work of B ott [11] it is known that the homotopy gr oups o f classical Lie groups a re p er io dic, of per io d 2 or 8. F or instance, the general linear and symplectic groups satisfy the isomorphisms: π n (GL( R )) ∼ = π n +8 (GL( R )) π n (Sp( C )) ∼ = π n +8 (Sp( C )) π n (GL( C )) ∼ = π n +2 (GL( C )) These p erio dicity statements were in terpreted by Atiy ah, Hirzebr uch and others in the framework of top olog ical K -theor y o f a Banach algebra A : re c all that there a re isomorphisms K top n ( A ) ∼ = K top n + p ( A ), where K top n ( A ) = π n − 1 (GL( A )) if n > 0 and K top 0 ( A ) = K ( A ) is the usual Grothendieck group. Here p is the p erio d which is 2 or 8 accor ding as A is complex or real. W e refer to [37] and [50] for an o verview of the sub ject, b oth algebraically and top ologic a lly . A few years later , after higher algebr aic K -theory was introduced by Quillen, an analogo us p erio dicity statement was sought, o f the form K n ( A ) ∼ = K n + p ( A ), where A is now a discrete ring. The first co mputations show ed that a per io dicity isomorphism of this form is far from true in basic exa mples. How ever, if w e consider K -theory with finite c o efficien ts, a nd n is at least a certain b ound d , then some per io dicity conjectures app eared feasible, at least for certain r ings o f a g eometric nature. These co njectures were formulated for different prime pow er co efficient groups, and ar e essentially o f the fo llowing type ( n ≥ d ) K n ( A ; Z /m ) ∼ = K n + p ( A ; Z /m ). The r elationship b etw een the prime power m and the asso cia ted sma llest p erio d p is g iven by the following co nv ention, which we maintain throughout the paper . Con ven tion 0.1. F or Z /m co efficie nts, where m = ℓ ν with ℓ prime, the smallest per io d p is g iven by p =  sup  8 , ℓ ν − 1  if ℓ = 2 , 2( ℓ − 1) ℓ ν − 1 otherwise. Date : December 10, 2010. First and second authors partially supp orted by the National Universit y of Singap ore R-146- 000-097-112. Third author partially s upported by RCN 185335/V30. 1 2 BK O MA Y 26, 2022 Using techniques of algebraic geometry and a compariso n theorem with ´ etale K -theory , n umerous examples listed below sho wed that these conjectures hold. In the case of a 2- power, the first three a re particula r cases of Theorem 2 in [52], ba sed on the fundamental work of V o evodsk y [62]. In the cas e o f a n o dd prime power, the first four examples are consequences of the Bloch-Kato conjecture. Before giving these examples , we define the mo d 2 vir tual ´ etale co homologica l dimension vcd 2 ( A ) of a commutativ e ring A as the mo d 2 ´ etale cohomolo gical dimension of A ⊗ Z Z [ µ 4 ] obtained by adjoining a primitive fourth r o ot o f unity to A . F or conv enience, if ℓ is o dd, then vcd ℓ ( A ) denotes the mo d ℓ ´ etale cohomolo gical dimension cd ℓ ( A ) of A . F or ℓ fixed, here are the examples we consider. (1) Any field k of ch aracter istic char( k ) 6 = ℓ for which vcd ℓ ( k ) < ∞ . In this case, d = vcd ℓ ( k ) − 1 if vcd ℓ ( k ) 6 = 0 and d = 0 otherwis e. (2) The ring O F [1 /ℓ ] of ℓ -integers in any num b er field F . In this case, d = vcd ℓ ( F ) − 1 = 1 ( cf. [43 ] when ℓ = 2 ). (3) Any finitely generated and regular Z [1 /ℓ ]-algebr a A with finite mo d ℓ vir- tual ´ etale cohomolo gical dimension. In this case d = sup { vcd ℓ ( k ( s )) − 1 , 0 } , where k ( s ) is the r esidue field at any p oint s ∈ Spec ( A ). The same state- men t holds when replacing Z [1 /ℓ ] b y Q or by an y other field k of charac- teristic 6 = ℓ . The r egularity assumption on A can b e disp ensed with w hen w o r king with negative K -theo ry [4], [30], [31], [61]. As shown in [53, Theorem 4.5], this do es no t change the bo und d . (4) Gr o up ring s R [ G ] , where G is finite and R is a r ing o f ℓ -integers in a num b er field, as s hown in [66]. Her e d = 1 . F or some explicit computatio ns see [4 2]. (5) The ring C ( X ) o f real or complex contin uous functions on a co mpact spa ce X , as sho wn in [18], [46]. In this case d = 1 . In these ex amples, the per io dicity iso morphism b etw een the groups K n ( A ; Z /m ) and K n + p ( A ; Z /m ) is defined by taking cup-pro duct with a “Bott element” b K . F or p = 2 ν − 1 with ν ≥ 4 , one ca n construct this ele ment in the g r oup K p ( Z ; Z / 2 p ), such that its image in the top ologica l K -gro up 1 K p ( R ; Z / 2 p ) ∼ = Z / 2 p is the cla ss mo d 2 p of a gener a tor in K p ( R ) ∼ = Z . The cup-pro duct alluded to ab ov e is a pair ing ∪ : K n ( A ; Z /m ) × K p ( Z ; Z / 2 p ) − → K n + p ( A ; Z /m ). W e refer to Section 1 for precise definitions and the extension to o dd prime powers. As we ca n see in these examples, a key ro le is play ed by the infinite g e neral linear group GL( A ). How ever, it was already s hown in the works of Bo tt and Borel [10], and also in to po logical applications, that other infinite ser ie s asso ciated to clas sical Lie groups may be considere d a s well. More precisely , if we consider a r ing with inv olution A a nd a s ign of symmetry ε = ± 1 gener alizing the or thogonal ( ε = 1) or symplectic ( ε = − 1 ) case, one defines hig her hermitian K -g r oups, denoted in this pap er ε K Q n ( A ), in a parallel wa y to algebr aic K -groups K n ( A ). Thes e gro ups are asso ciated to the infinite ε -orthog onal g roup ε O ( A ). A typical example is when A is co mm utative and ε = − 1, in which case one recov ers the infinite symplectic group on A . W e r efer to the survey pap er [37] already mentioned ab ov e for pr ecise definitions. 1 W e shall wr ite K n instead of K top n when dealing with the field R of real n umbers or the field C of complex num b ers with their usual top ology , and l i ke wise f or sp ectra. Hermitian p eri od icity May 26, 2022 3 The main purpo se of this paper is to sho w tha t a p erio dicit y statement in a lg e- braic K -theory implies a simila r o ne in K Q -theory , when 1 / 2 ∈ A . Since K Q -theory with co efficien ts Z /m , with m = 2 ν , is the most imp ortant and difficult c a se, we state the main theore ms in this context, leaving the case of o dd prime p ow e r co- efficients to the end of this Intro ductio n and to Section 5 o f the main bo dy of the pap er. F or the first step in the ar gument, we intro duce a para meter q that is essentially p , apa r t from a s light mo dificatio n in the ca s e m = 16. Sp ecifically , we make the following conv ention. Con ven tion 0.2 . q =    8 if m ≤ 8 , 16 if m = 1 6 , m/ 2 otherwise. In other words, q = p except when m = 16, in which case q = 2 p . It is meaningful to sp eak o f per io dicity maps r aising dimension b y q , since q is a multiple o f p . As a conv enient notation, we w r ite K Q (resp. K ) for the K Q -theo ry (resp. K -theory) with co efficients in Z /m , the relationship b etw e en m a nd the p erio d p being as in Co n ven tion 0.1. One of our main theorems is the follo wing. Theorem 0.3. With the ab ove definitions, assu me that ther e exists an inte ger d such that the cu p-pr o duct map ∪ b K : K n ( A ) − → K n + p ( A ) with the Bott element in K p ( Z ; Z / 2 p ) is an isomorphism whenever n ≥ d . Then, for n ≥ d + q − 1 , ther e is also an isomorphism ε K Q n ( A ) ∼ = ε K Q n + p ( A ) . Surprisingly , the isomo rphism be t ween the K Q -groups is in g e neral no t given by cup-pro duct with a Bott element (see Remark 4.8 in Section 4). This rela tes to the fact that hermitian K - theory p oss esses more than one Bott element, as we now describ e. Wherea s in algebr aic K -theory universal Bott elements ar e to b e found in the K -gr oups o f the in tegers Z , here, b eca use we a r e working with rings containing 1 / 2, our Bott ele ments are to b e found in the hermitian K -gro ups of the ring of 2-integers Z ′ = Z [1 / 2]. As in a lg ebraic K -theor y , using the metho ds of [5], in this pap er we prov e the exis tence of a “p ositive Bott element” b + in 1 K Q p ( Z ′ ; Z / 2 p ) whose ima g e in K p ( Z ′ ; Z / 2 p ) ∼ = K p ( Z ; Z / 2 p ) is the Bott elemen t in K -theor y alluded to a b ov e. On the other ha nd, o ne of the ma in differences betw een algebraic and hermitian K -theory in our context is the existence of another element 2 u in − 1 K Q − 2 ( Z ′ ), which plays a n importa n t role in the fundamen tal theorem in her mitian K -theory [33]. W e now define the ne gative Bott element b − in hermitian K -theor y to b e the image of the elemen t u p/ 2 in the group 1 K Q − p ( Z ′ ; Z / 2 p ). T o make the statement of Theor em 0.3 more pre c ise, we note that the cup- pro duct with the p ositive Bo tt element in 1 K Q p ( Z ′ ; Z / 2 p ) deter mines a dir e ct system of abelian groups ε K Q n ( A ) − → ε K Q n + p ( A ) − → ε K Q n +2 p ( A ) − → · · · . 2 W e recall that the negativ e K -groups of a r egular no etherian r ing (for instance Z or Z ′ ) are trivial. 4 BK O MA Y 26, 2022 Symmetrically , cup-pro duct with the negative Bott ele ment in 1 K Q − p ( Z ′ ; Z / 2 p ) determines an inv erse system of abe lia n gr oups · · · − → ε K Q n +2 p ( A ) − → ε K Q n + p ( A ) − → ε K Q n ( A ). The theorem ab ov e can now b e restated in a more precise form. (Recall that the ov erba r denotes Z /m co efficients.) Theorem 0 .4. L et A b e any ring (with 1 / 2 ∈ A ), m , p and q b e 2 - p owers as in Conventions 0.1 and 0.2, and let d ∈ Z , su ch that the cup-pr o duct with the Bott element b K in K p ( Z ; Z / 2 p ) induc es an isomorph ism K n ( A ) ∼ = − → K n + p ( A ) whenever n ≥ d . Then, for n ≥ d , ther e is an exact se quenc e · · · θ − − → ε K Q n +1 ( A ) θ + − → lim − → ε K Q n +1+ ps ( A ) → lim ← − ε K Q n + ps ( A ) θ − − → ε K Q n ( A ) θ + − → lim − → ε K Q n + ps ( A ) wher e θ + (r esp e ctively θ − ) is induc e d fr om the cup-pr o duct with the p ositive Bott element b + (r esp. the ne gative Bott element b − ). Mor e over, for n ≥ d + q − 1 , ther e is a short split ex act se quenc e 0 → lim ← − ε K Q n + ps ( A ) θ − − → ε K Q n ( A ) θ + − → lim − → ε K Q n + ps ( A ) → 0 . It turns out that the inv er se limit is not alwa ys trivial. This p oint is discussed in Section 2 (where the in verse limit v anishes) a nd Section 4 (where it does not). How ever, for rings of geometr ic na ture and of finite mo d 2 virtual ´ etale cohomo- logical dimension, we conjecture that the inv erse limit is trivial. Definition 0.5. W e say that a ring A is hermitian r e gular if lim ← − ε K Q n + ps ( A ) and lim ← − 1 ε K Q n + ps ( A ) ar e trivial 3 . Remark 0 .6. It sho uld b e noted that subsequent to the origina l submission o f this pap er a t the beginning of F ebruary 2010, as a co ns equence o f a mor e re cent theo- rem of Hu, K riz and Ormsby [2 5] in c haracter istic 0, the author s a nd M. Schlic hting prov ed indep endently that a field of c ha r acteristic 0 that is of finite mo d 2 virtual ´ etale cohomolo g ical definition is hermitian r egular. F urthermore, Schlic hting ex- tended this theorem for fields o f characteristic p > 0 with the s ame co homologica l prop erties. This affirms Co njecture 6.6 of the pr e s ent pap e r , whic h implies in turn our Co njecture 0.14 and therefore considera bly extends the num b er of exa mples o f commutativ e rings (and schemes) that a re hermitian regula r . T he details of the pro ofs will app ea r in a forthcoming joint pap er of the authors a nd Sc hlich ting [7]. A particular example quoted b elow is giv en by suitable rings o f integers in a num- ber field. In Theorem 0.10, we state the p erio dicit y theo rem in this case with an independent pro of which will b e given in Section 2. A more general theorem is as follows. 3 As a matter of fact, with our hypothesis about p erio dicit y of the K -groups, we alwa ys ha v e lim 1 = 0 , since the inv er s e system satisfies the Mi ttag-Leffler condition as we shall see in Sections 3 and 4. Hermitian p eri od icity May 26, 2022 5 Theorem 0.7. L et A b e a ring which is hermitian r e gular and satisfies t he hyp oth- esis of the pr evious the or em for its K -gr oups. Then for n ≥ d , the cup-pr o duct with the p ositive Bott element induc es an isomorph ism ε K Q n ( A ) ∼ = − → ε K Q n + p ( A ) . More generally , in order to fully exploit the sp ectrum appro a ch and to improve the previous theorems, w e may consider a pointed CW-complex X and define the group K X ( A ) as the group of homotop y cla sses of pointed ma ps fro m X to K ( A ), where K denotes the K -theory s pe c trum. If X is a pointed s phere S n , we recover Quillen’s K -gr oup K n ( A ). F or brevity , we sha ll also wr ite K X + t ( A ) instead of K X ∧ S t ( A ), and K X − t ( A ) instead of K X ( S t A ), where S t A denotes the t -iter ated susp ension of A (see for instance [37] for the definition of the s usp e nsion and the basic definitions of v ario us K -theorie s ). W e adopt the same conven tions for hermit- ian K -theory , and also for alg ebraic or he r mitian K -theory with co efficients, a nd finally for s p ectr a. The prev ious theorem can now b e g eneralized as follows. Theorem 0 .8. L et A b e any ring (with 1 / 2 ∈ A ), m , p and q b e 2 - p owers as in Conventions 0.1 and 0.2, and let d ∈ Z , su ch that the cup-pr o duct with the Bott element b K in K p ( Z ; Z / 2 p ) induc es an isomorph ism K n ( A ) ∼ = − → K n + p ( A ) whenever n ≥ d . Then, if X is a ( d − 1) -c onne cte d sp ac e, ther e is an exact se quenc e ε K Q X +1 ( A ) θ + − → lim − → ε K Q X +1+ ps ( A ) − → lim ← − ε K Q X + ps ( A ) θ − − → ε K Q X ( A ) θ + − → lim − → ε K Q X + ps ( A ) → · · · . If X is ( d + q − 2) -c onne cte d, ther e is a split short exact se qu en c e 0 → lim ← − ε K Q X + ps ( A ) θ − − → ε K Q X ( A ) θ + − → lim − → ε K Q X + ps ( A ) → 0 . Final ly, if A is hermitian r e gu lar and if X is ( d − 1) -c onne cte d, the cu p-pr o duct with the p ositive Bott element induc es an isomorphi sm ε K Q X ( A ) ∼ = ε K Q X + p ( A ) . Corollary 0.9. F or any ( d + q − 2) -c onne ct e d sp ac e X and A as ab ove (not ne c- essarily hermitian r e gular), ther e is a p erio dicity isomorphism ε K Q X ( A ) ∼ = ε K Q X + p ( A ) . F or suitable subrings A S in a nu mber fie ld F , the previous results may be stated more precisely , by us ing the metho ds o f [6]. The rings A S , defined in Sec tion 2 below, g eneralize b oth the ring of S -integers (when S is finite) a nd the num b er field F itself. (More g e neral exa mples are consider ed in Section 6 and in [7].) Theorem 0.10. L et F b e a total ly r e al 2 -r e gu lar nu mb er fi eld as c onsider e d in [6] ; also, let m and p b e 2 -p owers as in Convention 0.1. Then, for al l int e gers n > 0 , the inverse limit lim ← − ε K Q n + ps ( A S ) is trivial ( i.e. A S is hermitian r e gu lar) and the “p ositive” Bott map β n = ∪ b + : ε K Q n ( A S ) − → ε K Q n + p ( A S ) 6 BK O MA Y 26, 2022 is an isomorphi sm. Mor e gener al ly, if X is any c onne cte d CW-c omplex, the Bott map β X : ε K Q X ( A S ) − → ε K Q X + p ( A S ) is an isomorph ism. F or completeness w e men tion the o dd-primar y analo g of Theorem 0 .4, whic h is prov ed in Section 5. Its applica tions a re rela ted to the Blo ch-Kato conjecture a s we men tioned at the b eginning. W e note that the hypo thesis 1 / 2 ∈ A may b e dr opp ed in this cas e. Theorem 0. 11. L et p and m b e o dd prime p owers as in Convention 0.1. L et b K b e the asso ciate d Bott element in K p ( Z ; Z /m ) (se e Se ct ion 1 for details). Now let A b e any ring and assume that, whenever n ≥ d , cu p-pr o duct with b K induc es an isomorphi sm K n ( A ) ∼ = K n + p ( A ) . Then ther e exists a “mixe d Bott element ” b in 1 K Q p ( Z ′ ) such that for n ≥ d , the cup-pr o duct with b induc es an isomorphi sm b etwe en the r elate d K Q -gr oups β n : ε K Q n ( A ) ∼ = − → ε K Q n + p ( A ) . Mor e gener al ly, if X is a ( d − 1 )-c onne cte d CW c omplex, then t he cup-pr o duct map with b indu c es an isomorphi sm β X : ε K Q X ( A ) ∼ = − → ε K Q X + p ( A ) . In Section 6 we note that work in prog ress by Schlich ting [56] allows us to extend our res ults from commutativ e rings to schemes S that are se pa rated, no etherian and of finite Kr ull dimension. More precisely , follo wing J ardine’s method for algebraic K -theory [29] we define an “´ etale” K Q -theor y , denoted by ε K Q ´ et n ( S ), where the co efficient groups ar e pr ime pow e r s. The ´ etale K Q -theory shares many prop erties with the ´ etale K -theory introduced by Dwyer and F riedlander [1 5]. F or ex a mple, there exists a comparison map σ : ε K Q n ( S ) − → ε K Q ´ et n ( S ). F or o dd pr ime p ow ers, there is an inv olution on the o dd torsion gro up ε K Q n ( S ). Let ε K Q ´ et n ( S ) + and ε K Q ´ et n ( S ) − denote the corr e sp onding eigenspa ces. O n the other hand, for a ny prime p ow er (o dd or even), the cup-pro duct map with the B ott element b + defined a bove induces a direct system of groups, w ho se colimit we sha ll denote b y ε K Q n ( S )  β − 1  , in the notatio n o f [60 ]. Next, we s tate t wo theor ems and a conjectur e in this co n text. Theorem 0 . 12. With the c o efficient gr oup Z /ℓ ν , wher e ℓ is an o dd prime, ther e is an isomo rphism ε K Q n ( S )  β − 1  ∼ = ε K Q ´ et n ( S ) for al l n if cd ℓ ( S ) < ∞ . Mor e over, the c omp arison map σ induc es an isomorphi sm ε K Q n ( S ) + ∼ = ε K Q ´ et n ( S ) for n ≥ sup { cd ℓ ( k ( s )) − 1 } s ∈ S . Hermitian p eri od icity May 26, 2022 7 Recall that S is uniformly ℓ -b ounded with b o und d if for all re sidue fields k ( s ) we have cd ℓ ( k ( s )) ≤ d . In the even t that S is uniformly ℓ -b ounded with bound d , then cd ℓ ( S ) ≤ n + d where n denotes the K rull dimension o f S ; an eleg ant pro of for this inequalit y is given in [39, Theorem 2.8]. A t the prime 2 w e prov e the following theorem, reminiscent of the main results in [16] and in [60]. Theorem 0. 13. With the c o efficient gr oup Z / 2 ν , ther e is an isomorph ism ε K Q n ( S )[ β − 1 ] ∼ = ε K Q ´ et n ( S ) for al l n if vcd 2 ( S ) < ∞ . Mor e over, the c omp arison map σ : ε K Q n ( S ) − → ε K Q ´ et n ( S ) is a split s u rje ction for n ≥ sup { vcd 2 ( k ( s )) − 1 } s ∈ S + q − 1 . More gener ally , we make the follo wing conjecture. Conjecture 0. 14. With the c o efficient gr oup Z / 2 ν the map σ is bije ctive whenever n ≥ sup { vcd 2 ( k ( s )) − 1 } s ∈ S . Using alg ebro-geo metric metho ds, in Theorem 6.5 b elow we s how how to reduce this conjecture to the case of fields. As men tioned above, the characteristic 0 ca s e was solved indep endently by the author s and M. Schlic hting, while the p ositive characteristic case was solved by Schlic hting. A pro of o f this conjecture in general will app ear in a joint paper with Schlic hting [7]. Let us no w briefly discus s the co nt ents of the paper . In Sectio n 1 , for 2-p ow er co efficients we carefully construct the Bott elements that play a n imp ortant role in this w ork, as referred to ab ov e. Section 2 is somewhat indep endent of the other sections. In particular, we prove a refined version of our theor ems in the case A is the ring of integers in a totally real 2- r egular num b er field. (This version is a particular ca se of the considera - tions in Section 6 for schemes. Assuming Conjecture 0.1 4, which will be prov en in [7], Theorem 2.1 may b e g iven an indep endent pro of in a muc h mo re general framework.) In Section 3, we in tro duce what w e call “higher K S C -theories”. These theorie s in some sense mea sure the devia tion of “ negative” p erio dicity of the K Q -groups. On the other hand, they ar e built by successive extensions of the K -gro ups. Therefore, they are p er io dic if the K -groups are p er io dic. Section 4 is devoted to the pro o f o f our main Theo rems 0.4 and 4.5 (for ar bitrary rings with 2 inv er tible and mod 2 ν co efficients). The pro of is roughly divided into t wo steps as follows. In the first one, we prov e a cruder p erio dicity statement for n ≥ d + q − 1. In the second, we use the K Q -sp e ctrum and an ar gument ab out coho mology theor ies to prov e the p er io dicity theorems in full generality . W e conclude this section with an upp er b ound of the K Q -gro ups in terms of the K - groups. In Section 5, w e study the ca se o f o dd prime p ow ers, which is par adoxically sim- pler in our framework. The main observ atio n is that the K Q -ring spectrum splits naturally as the pr o duct of tw o r ing sp ectra, the first one b eing the “ symmetric” part of the K -theory sp ectrum. Section 6 is more ge ometric in nature and generaliz e s the previous consideratio ns (when A is commut ative) to no ether ian sepa rated schemes of finite K rull dimension. 8 BK O MA Y 26, 2022 Here we rely heavily on the fundament al theorem in her mitian K -theory pr ov ed in the scheme framework by Schlic hting [5 6]. Finally , Sections 7 and 8 ar e devoted to selected applications: rings of integers in nu mber fields, smo o th complex algebraic v arieties, and rings of contin uous functions on compact space s . Another application, to hermitian K Q -theory of gr o up rings, is a cons e quence of an appendix to this pa per b y C. W eib el [66]. Ac knowledgemen ts . W e warmly thank the referee for v ery re lev ant commen ts on a previous version of this pap er. W e extend our thanks to Ma rco Schlic hting for discussions resulting in our joint work [7]. 1. Bott elements in K - and K Q -theories Let ℓ b e a prime num b er a nd S 0 /ℓ ν the mo d ℓ ν Mo ore sp ectrum. In [1, § 12 ], Adams constructed K O ∗ -equiv alences A ℓ ν : Σ p S 0 /ℓ ν − → S 0 /ℓ ν . The dimension shift p is sup { 8 , 2 ν − 1 } if ℓ = 2 and 2( ℓ − 1) ℓ ν − 1 if ℓ is o dd . As shown by Bousfield in [1 2, § 4], work of Mahowald and Miller implies that a sp ectrum E is KO - lo cal if and o nly if its mod ℓ homotopy g r oups are p erio dic via A ℓ for every prime ℓ . W e shall refer to the p erio dicity manifested in KO -local sp ectr a as Bott per io dicity . Note that KO - lo calizations ar e the same as K U -lo caliza tions [12, § 4]. In gener al there are sev eral c hoices of a n element A ℓ ν as ab ove if the o nly criterio n is that it induces a KO -isomorphism. W e are interested in pa rticular choices of elements pertaining to clas sical Bott p erio dicity . Let u denote a gener a tor of the infinite cyclic gr oup π 2 ( B U ). Then for r ≥ 1 the Bo tt elemen t u 2 r in π 4 r ( B U ) is independent of the c hoice of u . W e denote b y v the e lement o f π 8 r ( B O ) mapping to the Bott elemen t in π 8 r ( B U ) under the map induced b y co mplexification c : B O → B U . The mo d 2 ν Bott element in degre e 8 r > 0 is the generator v = id S 0 / 2 ν ∧ v ∈ K O 8 r ( S 0 / 2 ν ; Z / 2 ν ) = [ S 0 / 2 ν , K O ∧ S 0 / 2 ν ] 8 r . The element A 2 ν is ca lled a n Adams p erio dicity op e rator if it ma ps to the mo d 2 ν Bott element in degre e p under the na turally induced KO -Hurewicz ma p π ∗ ( S 0 / 2 ν ; Z / 2 ν ) → KO ∗ ( S 0 / 2 ν ; Z / 2 ν ) for S 0 / 2 ν . When ℓ 6 = 2, the definition of a mo d ℓ ν Bott element is the same as ab ov e, except that K O is replac e d by KU . Crabb and Knapp [14] hav e shown that there exist Adams per io dicity op era tors for a ll ℓ and ν ≥ 1. By smashing the unit ma p S 0 → E of a ring s pec trum E with S 0 /ℓ ν and pushing forward the class in π p ( S 0 /ℓ ν ; Z /ℓ ν ) represented by the ma p A ℓ ν , one obtains a class in the group π p ( E ; Z /ℓ ν ) that w e call a Bo tt ele ment. Next, fo r m = 2 p, where p = 2 υ − 1 is a 2-p ow er ≥ 8 , we study mo d m Bott elements in more detail for K - and K Q -theory in the exa mple of Z ′ . The case o f an o dd pr ime is de a lt with in Section 5. T o b egin, we shall consider “Bott elements” in K p ( Z ′ ; Z /m ) and 1 K Q p ( Z ′ ; Z /m ), whose images in K p ( R ; Z /m ) and 1 K Q p ( R ; Z /m ), respe ctively , ar e generators de- duced from classical Bott p erio dic ity for the real num b er s (as K Q -mo dules). This is well-known for the a lg ebraic K -gro ups; it is included her e for the sake of co m- pleteness. Hermitian p eri od icity May 26, 2022 9 B¨ okstedt’s squa r e of alg ebraic K -theory sp ectra in tr o duced in [9] K ( Z ′ ) # − → K ( R ) c # ↓ ↓ K ( F 3 ) # − → K ( C ) c # was verified to b e homo topy cartesian by Rog ne s -W eib el in [4 9], [65], a s a co nse- quence of V o evo dsky’s pro of of the Milnor conjecture. Here # means 2-adic com- pletions and c means co nnec tive cover. Smashing with S 0 / 2 ν yields a homo to py cartesian squar e (an ov erbar indicates reduction mod m ): K ( Z ′ ) − → K ( R ) c ↓ ↓ K ( F 3 ) − → K ( C ) c Denote b y K the corres po nding mo d m homo topy groups. By Bott p erio dicity and the isomo rphism K p − 1 ( Z ′ ) → K p − 1 ( F 3 ), there is a split s hort exac t sequence 0 − → K p ( Z ′ ) − → K p ( R ) ⊕ K p ( F 3 ) − → K p ( C ) − → 0. On the o ther hand, Quillen’s homotopy fibration Ω K ( C ) Ψ 3 − 1 − → Ω K ( C ) − → K ( F 3 ) − → K ( C ) Ψ 3 − 1 − → K ( C ) yields an exa ct sequence K p +1 ( C ) · m − → K p +1 ( C ) − → K p ( F 3 ) − → K p ( C ) · m − → K p ( C ), and hence the isomorphisms K p ( F 3 ) ∼ = m K p ( C ) ∼ = Z /m . Here n A denotes the kernel of the multiplication by n map o n an ab elian group A . Hence, diagra m chasing shows ther e ar e isomorphisms K p ( Z ′ ) ∼ = K p ( R ) and K p ( Z ′ ) ∼ = K p ( F 3 ). More prec isely , there exists a B o tt elemen t b K in K p ( Z ′ ) mapping at the same time to a gener ator of K p ( R ) and to a genera tor of K p ( F 3 ). W e pro ceed in the same manner in or de r to explicate Bott elements in he r mitian K -theory , having almost the ex a ct same prop er ties as their namesakes in algebraic K -theory . More precis ely , we shall prove the following theo rem: Theorem 1.1. L et p ≥ 8 a 2 -p ower and m = 2 p . Then the gr oup 1 K Q p ( Z ′ ; Z /m ) is isomorphic to Z /m ⊕ Z /m ⊕ Z / 2 . Ther e is a Bott element b + in 1 K Q p ( Z ′ ; Z /m ) that maps at the same time to a gener ator of Z /m in 1 K Q p ( F 3 ; Z /m ) ∼ = Z /m ⊕ Z / 2 and to a gener ator of 1 K Q p ( R ; Z /m ) , viewe d as a mo dule 4 over 1 K Q 0 ( R ; Z /m ) . Pro of. In the following pro of, we a re g oing to use the r esults of [5 , Theorem 6.1] and [6, Theorems 1.2, 1.5]. In [5], it is shown that the sq uare of hermitian K - theory completed connective spe ctra 1 KQ ( Z ′ ) c # − → 1 KQ ( R ) c # ↓ ↓ 1 KQ ( F 3 ) c # − → 1 KQ ( C ) c # 4 The r ing structure for K Q -theory with mo d 2 ν coefficients is w ell-defined i f ν ≥ 4. 10 BK O MA Y 26, 2022 is homoto py car tesian (Recall that 1 KQ ( C ) is just K O .) Reducing mo d m yields another homotopy cartesian squar e: 1 KQ ( Z ′ ) c − → 1 KQ ( R ) c ↓ ↓ 1 KQ ( F 3 ) c − → 1 KQ ( C ) c This in turn gives rise to a shor t exact sequence (1:1) 0 − → 1 K Q p ( Z ′ ) − → 1 K Q p ( R ) ⊕ 1 K Q p ( F 3 ) − → 1 K Q p ( C ) − → 0, which s plits since 1 K Q p ( R ) is a dir ect s um of t wo copies of 1 K Q p ( C ), s ay G ⊕ G . The first copy of G , say G 1 , is generated by the image of 1 under the Bott isomorphism 1 K Q 0 ( R ) ∼ = 1 K Q p ( R ) (see App endix B in [6]). The splitting is given by the isomorphism b etw een 1 K Q p ( C ) and the second co py of G , say G 2 . Therefore, we get an iso mo rphism 1 K Q p ( Z ′ ) ∼ = G 1 ⊕ 1 K Q p ( F 3 ) In order to finish the pro of of the theorem, we need to compute 1 K Q p ( F 3 ). By [19], there is a Bo ckstein ex act sequence (1:2) 0 − → Z / 2 − → 1 K Q p ( F 3 ) − → Z /m − → 0. In order to r esolve this extension problem, co ns ider the ma p 1 K Q 0 ( F 3 ) /m = Z /m ⊕ Z / 2 − → 1 K Q p ( F 3 ) given by cup-pro duct with an y elemen t that maps to the generator o f 1 K Q p ( C ) ∼ = Z /m under the Br auer lift 1 K Q p ( F 3 ) → 1 K Q p ( C ). This gives a splitting of the exact sequence (1:2), and therefor e 1 K Q p ( F 3 ) ∼ = Z /m ⊕ Z / 2. ✷ Remarks 1.2. By considering the for getful map from the hermitian K seq uence (1:1) to its algebraic K co unt erpart, o ne sees that the Bo tt element of 1 K Q p ( Z ′ ; Z /m ) maps to the Bott element in the corre sp o nding algebraic K -theory g roup under the map induce d b y the forgetful functor. Moreover, all the results for F 3 in the a bove also hold for any finite field F t with t elemen ts, pro vided t ≡ ± 3 (mo d 8). 2. Proof of the periodicity theorem for tot all y real 2 -regular number fields Let A b e the r ing of 2-integers in a totally real 2-r e gular num b er field F with r real embeddings. In [6 ], we proved that the squar e o f hermitian K -theory 2- completed connective spe ctra ε KQ ( A ) c # − → W r ε KQ ( R ) c # ↓ ↓ ε KQ ( F t ) c # − → W r ε KQ ( C ) c # is homoto py cartesian (with t a car efully chosen o dd prime and wher e # denotes 2-adic completion). There fo re, the mo d 2 ν reduction of this square, namely ε KQ ( A ) c − → W r ε KQ ( R ) c ↓ ↓ ε KQ ( F t ) c − → W r ε KQ ( C ) c is also homotopy cartes ian, since ε K Q − 1 ( A ) = 0 b y Lemmas 3.11 and 3.12 in [6]. Using this s quare, we deduce an enhanced v er sion of our p er io dicity theorem. Hermitian p eri od icity May 26, 2022 11 Theorem 2 . 1. F or n ≥ 0 and p = sup { 8 , 2 ν − 1 } for ν ≥ 1 , taking cup-pr o duct with the p ositive Bott element in 1 K Q p ( Z ′ ; Z / 2 ν ) induc es an isomorphi sm ε K Q n ( A ; Z / 2 ν ) ∼ = ε K Q n + p ( A ; Z / 2 ν ) . Pro of. Cup-pr o duct with the Bott element in 1 K Q p ( Z ′ ; Z / 2 ν ) induces a n isomor - phism of ε K Q -groups for the r ings F t , R and C , where F t is the finite field with t elements. This is due to KO -lo calness of the corres po nding hermitian K -theory sp ectra, and a n induction on the order of the co efficien t gr oup based o n the five lemma applied to the Bo ckstein exact seque nc e . There fore, the res ult follows from the five lemma together with the homotopy cartesian squar e ab ov e. ✷ Remark 2.2. The isomorphism fo r n = 0 reflects the fact that 2-r egularity implies that there is no nontrivial 2-torsion in the Picard gr o up o f A . W e no te that the num b er ν was related to the c hoice of t ≡ ± 3 (mod 8 ) in Theorem 1.1 (for A = Z ′ ). How ever, the num b er t that mak es the dia grams ab ove homotopy car tes ian (for A totally real 2-re g ular) is different in gener al. Therefore, we c a n improv e the prev io us result by repla cing m = 2 ν by M , which is the num b er m mult iplied by the 2-pr imary factor m ′ =  t 2 − 1 8  2 of ( t 2 − 1 ) / 8 (compare with Lemma 2.9 in [6]). More precisely , we have the following prop ositio n. Prop ositi o n 2.3. L et p = sup { 8 , 2 ν − 1 } and c onsider the c anonic al ly induc e d map 1 K Q p ( Z ′ ; Z / 2 ν ) − → 1 K Q p ( A ; Z / 2 ν ) . The image of the Bott element b in 1 K Q p ( A ; Z / 2 ν ) is the r e duction mo d 2 ν of a class b mod M , with M = m ·  t 2 − 1 8  2 as define d ab ove. Pro of. F or brev ity , we use the ab ov e notation, whereby m = 2 ν and M = mm ′ . As in the ca se of Z ′ considered in Sec tio n 1, we can wr ite the following diagram of exact sequences (where K Q = 1 K Q ): 0 ↓ K Q p ( R r ; Z /m ′ ) ⊕ K Q p ( F t ; Z /m ′ ) → K Q p ( C r ; Z /m ′ ) → 0 ↓ ↓ K Q p ( A ; Z / M ) → K Q p ( R r ; Z / M ) ⊕ K Q p ( F t ; Z / M ) → K Q p ( C r ; Z / M ) → 0 ↓ ↓ ↓ K Q p ( A ; Z /m ) → K Q p ( R r ; Z /m ) ⊕ K Q p ( F t ; Z /m ) → K Q p ( C r ; Z /m ) → 0 ↓ ↓ 0 0 Chasing in this diagram shows the reduction map K Q p ( A ; Z / M ) → K Q p ( A ; Z /m ) is surjective. Therefore, the Bo tt element b in K Q p ( A ; Z /m ) is the reductio n mo d m of a class b mo d M which we shall ca ll an exotic Bott element (we do not claim, how ever, that b is unique). ✷ Theorem 2.4. L et b b e an exotic Bott element in the gr oup 1 K Q p ( A ; Z / M ) define d ab ove. Then cup-pr o du ct with b induc es an isomorph ism β : ε K Q n ( A ; Z / M ′ ) ∼ = − → ε K Q n + p ( A ; Z / M ′ ) for every n ≥ 0 and diviso r M ′ of M . 12 BK O MA Y 26, 2022 Pro of. W e just co py the pr o of of Theor em 2.1, using the five lemma, since this per io dicity statement holds fo r the ring s R , C a nd F t (see the independent lemma below for the field F t ). ✷ Lemma 2.5. L et F t b e a fi n ite field with t elements and let p = sup { 8 , 2 ν − 1 } and m = 2 ν . Then the image of the Bott element by the c anonic al map 1 K Q p ( Z ′ ; Z /m ) − → 1 K Q p ( F t ; Z /m ) is the r e duction mod m of a class mo d M ′ (with m | M ′ ) if and only if ( M ′ ) 2 ≤ m · (( t 2 − 1) / 8) 2 , wher e ( i ) 2 is the 2 -primary p art of i . Pro of. W e look at the following commutativ e dia g ram, with exa ct rows: − → 1 K Q p ( F t ; Z / M ′ ) − → 1 K Q p − 1 ( F t ) · M ′ − → 1 K Q p − 1 ( F t ) ↓ α M ′ ↓ · M ′ 2 − ν ↓ id − → 1 K Q p ( F t ; Z / 2 ν ) − → 1 K Q p − 1 ( F t ) · 2 ν − → 1 K Q p − 1 ( F t ) The Bott element in the gro up 1 K Q p ( F t ; Z / 2 ν ) maps nontrivially into the g r oup 1 K Q p − 1 ( F t ) and its imag e is divisible by M ′ 2 − ν . On the other hand, we know by [19] that 1 K Q p − 1 ( F t ) is cy c lic o f order M , where M is the 2-pr imary pa rt of ( t p/ 2 − 1), which is also the 2 -primary part o f ( t 2 − 1) · p/ 4 by Lemma 2.7 in [6]. This nu mber is a lso the 2-prima ry part of 2 ν · ( t 2 − 1 ) / 8. The r efore, a simple dia g ram chase shows that α M ′ is sur jective if a nd only if M ′ | M . ✷ Now let us c o nsider a nonzer o prime ideal p in A , and the q uotient field A/ p . There is a comm uta tive diagr am Z ′ − → A/ p ↓ ↓ A − → A/ p where the right v e r tical arr ow is the identit y map. Since the Bo tt element in the K Q -gro up 1 K Q p ( A/ p ; Z / 2 ν ) is the reduction mo d 2 ν of a class mo d M , where M is a p ow er of 2, we have an isomor phism 1 K Q p ( A/ p ; Z / M ) ∼ = Z / M ⊕ Z / 2 according to the computations of the K Q -theo ry of finite fields in [1 9] and Sectio n 1. It follows that ther e is a p erio dicity is o morphism ε K Q n ( A/ p ; Z / M ′ ) ∼ = ε K Q n + p ( A/ p ; Z / M ′ ) for n ≥ 0 a nd an y M ′ | M , given by the cup-pro duct with an e x otic Bo tt e le men t. F or the next tw o results, we recall from [6, P rop osition 2.1] that F contains a unique dyadic pr ime (that is, pr ime ideal lying over the rational pr ime (2 )). F o r a ny set S of v aluatio ns in F including the dyadic v alua tion and the infinite ones , we define A S to co nsist o f the elements in F whos e v aluations not in S ar e non-nega tive. Thu s, when S is finite, A S is just the ring of S -integers. When S comprises only the dyadic v alua tio n a nd the infinite ones, A S = A ; while, when S c omprises all v aluations, A S = F . Hermitian p eri od icity May 26, 2022 13 Theorem 2.6. L et p = sup { 8 , 2 ν − 1 } for ν ≥ 1 . Then, for n > 0 , cup-pr o duct with an exotic Bott element in 1 K Q p ( A ; Z / M ) induc es an isomorphism β : ε K Q n ( A S ; Z / M ′ ) ∼ = − → ε K Q n + p ( A S ; Z / M ′ ) for any M ′ such that 2 | M ′ | M . Pro of. W e use the homotopy fibr ation _ ε U ( A/ p ) − → ε KQ ( A ) − → ε KQ ( A S ) noted in [24], where p r uns through all nonzer o pr ime idea ls in S . F or the cor - resp onding mo d M ′ reductions (indicated as usual by an ov er bar) where M ′ | M , there is a homotopy fibration _ ε U ( A/ p ) − → ε K Q ( A ) − → ε K Q ( A S ) . The maps in this fibratio n ar e c ompatible with cup-pro ducts with elements of K Q ∗ ( A ). The U - theo ry sp ectra of finite fields are KO -local as we show ed more pre- cisely ab ove (this is a consequence of the same prop er ty fo r the K and K Q -theo ries). F rom these facts, the five lemma implies the Bott p erio dicit y isomo rphism ε K Q n ( A S ; Z / M ′ ) ∼ = ε K Q n + p ( A S ; Z / M ′ ) for n > 0, given by the cup-pro du ct with an exotic Bott element. ✷ Theorem 2.7. L et A S b e as b efor e, and let b b e an exotic Bott element in the gr oup 1 K Q p ( A ; Z / M ) . Then, for any c onne cte d CW-c omplex X , cup-pr o duct with b induc es an isomorph ism β : ε K Q X ( A S ; Z / M ′ ) ∼ = ε K Q X + p ( A S ; Z / M ′ ) for any M ′ such that 2 | M ′ | M . Mor e over, when A S = A , the pr evious isomorph ism holds for any CW-c omplex, not ne c essarily c onn e cte d . Pro of. By Theore m 2.6, the Bott map β is an isomorphism when X is a sphere S n for n ≥ 1 . According to general facts a b o ut repr e s entable cohomolo g y theories [8], it follows tha t β is also an isomor phism if X is a connected CW-c o mplex, (finite or infinite, thanks to Milnor’s lim 1 exact sequence). If A S = A , then the Bo tt map β is also an isomorphism when X = S 0 . There fo re, the previous isomorphism holds also for not necessarily connected CW- c o mplexes. ✷ 3. Higher KSC-theories The useful co ncept of top olog ical K -theor y based upo n s elf c onjugate vector bundles K S C was intro duced by Ander son [2] and Green [2 0]. In [3 3, p. 281], for a ring A with inv olution, the sp ectrum KS C ( A ) was defined as the ho motopy fiber of 1 − τ , whe r e τ is the dualit y functor in a lgebraic K -theor y τ : K ( A ) − → K ( A ). The imp or tance of K S C - theo ry b eco mes e v ident from the ho motopy fibration [33, p. 282] KS C ( A ) − → Ω ε KQ ( A ) σ (2) − → Ω − 1 − ε KQ ( A ), 14 BK O MA Y 26, 2022 which implies a lo ng exact sequence (for legibilit y we omit the ring A in the no ta- tion) · · · → ε K Q n +2 s (2) → − ε K Q n → K S C n → ε K Q n +1 s (2) → − ε K Q n − 1 → · · · . The morphism s (2) betw een the K Q -g r oups is the p erio dicity ma p made explicit in [33]. It is defined by taking cup-pro duct with a g e ne r ator of the free par t of the group − 1 K Q − 2 ( Z ′ ) ∼ = 1 W 0 ( Z ′ ) ∼ = Z ⊕ Z / 2. (Recall our assumption that 1 / 2 ∈ A .) W e should note that this cup-pro duct induces a mor phism b etw e en cohomolo g y theor ies, and thence the asso ciated K Q - sp ectra and K Q -sp e ctra, acco rding to Brown’s representabilit y theor em. It turns out that the K S C -groups mea sure the failure of negative Bott p erio dic ity for the K Q -g roups. T o k eep track o f the degr ee shift we let K S C (2) (resp. KS C (2) ) denote the K S C -groups (resp. K S C -sp ectr um). There exist higher analo gs o f this sp ectrum corr esp onding to degr ee shifts by 4, 8 and higher 2-powers. The nex t version, denoted 5 by ε KS C (4) ( A ), is the homotopy fib er of the com- po site map σ (4) : Ω ε KQ ( A ) σ (2) − → Ω − 1 − ε KQ ( A ) Ω ( − 2) σ (2) − → Ω − 3 ε KQ ( A ). Prop ositi o n 3.1. Ther e exists a homotopy fibr ation of sp e ctr a ε KS C (2) ( A ) − → ε KS C (4) ( A ) − → Ω − 2 ( ε KS C (2) ( A )) and a long ex act se quenc e (we again omit the ring A for c onvenienc e) · · · → ε K Q n +2 s (4) → ε K Q n − 2 → ε K S C (4) n → ε K Q n +1 s (4) → ε K Q n − 3 → · · · . Pro of. This is just the observ ation that for t wo composa ble maps u and v , there is a homotopy fibration F ( u ) − → F ( v ◦ u ) − → F ( v ), wher e F ( f ) denotes the homotopy fib er of some map f . ✷ Iterating, for r > 4 a 2-p ow er , we pro cee d similarly a nd define ε KS C ( r ) ( A ) as the homoto py fib er of the ma p σ ( r ) : Ω ε KQ ( A ) − → Ω − r +1 ε KQ ( A ) where σ ( r ) = Ω − r / 2 σ ( r / 2) ◦ σ ( r / 2) . In the o ther direction, if we a llow the conv ent ion ε KS C (1) ( A ) = Ω K ( A ), then from the orig inal definition of K S C ab ov e the fo llowing also holds for r = 2. Prop ositi o n 3.2. F or a 2 -p ower r ≥ 2 , ther e is a homotopy fibr ation of sp e ctr a ε KS C ( r / 2) ( A ) − → ε KS C ( r ) ( A ) − → Ω − r / 2 ε KS C ( r / 2) ( A ) and an asso ciate d long exact se quenc e · · · → ε K Q n +2 s ( r ) → ε ′ K Q n +2 − r → ε K S C ( r ) n → ε K Q n +1 s ( r ) → ε ′ K Q n +1 − r → · · · wher e ε ′ = − ε if r = 2 and ε ′ = ε if r > 2 . ✷ Finally , we show that the higher K S C - theories dep ends o n the sign of symmetry ε . 5 A pri ori , this theory depends on ε . A pro of of this statemen t may be found in Lemma 3.3 below. F or K S C - theory with coefficients, we can als o argue by con tr adiction as i n Lemma 3.13. Hermitian p eri od icity May 26, 2022 15 Lemma 3.3. L et F b e a finite field of char acteristic 6 = 2 . Then the gr oup 1 K S C (4) 1 ( F ) is isomorph ic t o Z / 2 , while − 1 K S C (4) 1 ( F ) = 0 . Pro of. Let us dr op the field F for no tational conv enience . Then the g roup − 1 K S C (4) 1 fits into the exact s equence − 1 K Q 3 − → − 1 K Q − 1 − → − 1 K S C (4) 1 − → − 1 K Q 2 − → − 1 K Q − 2 . W e hav e − 1 K Q − 1 = 0 b y the same argument use d in the pr o of of Lemma 3.11 in [6], where we sho uld replace R F by F . W e also have − 1 K Q 2 = 0 by a r esult o f F riedlander [19]. Ther efore, − 1 K S C (4) 1 ( F ) = 0. On the other hand, the group 1 K S C (4) 1 fits into the exact s equence 1 K Q 3 − → 1 K Q − 1 − → 1 K S C (4) 1 − → 1 K Q 2 α − → 1 K Q − 2 . F or the same reaso n as ab ov e, we hav e 1 K Q − 1 = 0. W e also hav e 1 K Q 2 = Z / 2 by a n ana logous result of F r iedlander [19]. The per io dicity map α can be factored though the group − 1 K Q 0 which is isomorphic to Z . Therefore , Ker( α ) = Z / 2 and the gr o up 1 K S C (4) is isomor phic to Z / 2. ✷ W e can mimic the previous de finitio ns by taking spe c tra o r groups mo d m , wher e m is related to p accor ding to our conv ention 0 .1. In that case, w e sha ll write KQ instead of KQ , K S C instead of K S C , etc. Prop ositi o n 3.4. L et d b e the nu mb er define d in the Intr o duction ( i.e. the starting p oint of p erio dicity for t he K -gr oups). Then for any 2 -p ower r ≥ 2 , t he p ositive Bott map σ : ε K S C ( r ) n ( A ) → ε K S C ( r ) n + p ( A ) is an isomorph ism if n ≥ d + r − 2 . Pro of. W e argue b y iteratio n o n the 2-p ow er r , using the following diagr am of exact sequences from (3.2): (3:3) K S C ( r / 2) n +1 − r / 2 → K S C ( r / 2) n → K S C ( r ) n → K S C ( r / 2) n − r / 2 → K S C ( r / 2) n − 1 ↓ ↓ ↓ ↓ ↓ K S C ( r / 2) n ++ p +1 − r / 2 → K S C ( r / 2) n + p → K S C ( r ) n + p → K S C ( r / 2) n + p − r / 2 → K S C ( r / 2) n + p − 1 Commutativit y of this diagram follows from the fact that the vertical maps are induced by cup-pr o duct with the p o sitive Bott ele ment in K Q - theo ry a s constructed in Section 1, a nd that a ll maps are K Q -mo dule maps. By induction, we know that the vertical ma ps , with the po ssible exception of the middle one, are is omorphisms if n − r / 2 ≥ d + r/ 2 − 2, that is n ≥ d + r − 2. W e conclude thanks to the five lemma. ✷ Remark 3.5. A v ariant of this prop ositio n is to c o nsider a parameter space X instead of a spher e S n . Mor e precisely , by the metho d o f pro of of Theor em 2.7, the Bott map σ : ε K S C ( r ) X ( A ) → ε K S C ( r ) X + p ( A ) is an isomo rphism if the space X is ( d + r − 3)-connected. 16 BK O MA Y 26, 2022 Next w e consider the failure of po sitive p -p erio dic ity in hermitian K -theory . This is enco ded in the homotopy fib er of the p erio dicity map given by the cup-pro duct with the positive Bott element ε KQ ( A ) − → Ω p ε KQ ( A ), which we shall denote by P ε KQ ( A ). In the same way , we denote by P K ( A ) the homotopy fib er of the p erio dicity map in K -theory K ( A ) − → Ω p K ( A ). According to o ur genera l a ssumptions, the homo topy groups P K n ( A ) of P K ( A ) v anish if n ≥ d . On the other hand, we can introduce , cf. [33], the homotopy fibe r s P ε U ( A ) and P − ε V ( A ) of the hyper b o lic and forgetful maps P K ( A ) → P ε KQ ( A ) and P − ε KQ ( A ) → P K ( A ), respec tively . Prop ositi o n 3.6. Ther e is a homotopy e quivalenc e P − ε V ( A ) ≃ Ω( P ε U ( A )) . Mor e over, the c omp osition Ω 2 ( P ε KQ ( A )) − → Ω( P ε U ( A )) ≃ P − ε V ( A ) − → P − ε KQ ( A ) is induc e d by cup-pr o duct with the ne gative Bott element in the gr oup − 1 K Q − 2 ( Z ′ ) . Pro of. The fundamental theor em of hermitian K -theor y [33] exhibits an explicit homotopy equiv alence (given b oth ways) b etw een the sp ectra − ε V ( A ) and Ω ε U ( A ). Moreov er, with the no tation KQ ( Z ′ ) = 1 KQ ( Z ′ ) ∨ − 1 KQ ( Z ′ ), these maps are KQ ( Z ′ )-mo dule maps. Therefore, the reduction mo d m o f these sp ectra is a lso a homotopy equiv alence. Since the comp osition Ω 2 ( P ε KQ ( A )) − → Ω( P ε U ( A )) ≃ P − ε V ( A ) − → P − ε KQ ( A ) is a KQ ( Z ′ )-mo dule map, it is defined by the cup-pr o duct with the negative Bott element, as pr ov ed in [3 3]. ✷ Lemma 3. 7 . If P K n ( A ) = 0 for n ≥ d , then the ne gative Bott map P ε K Q n +2 ( A ) → P − ε K Q n ( A ) is an isomorph ism for n ≥ d and is a monomorphism for n = d − 1 . Pro of. T his follo ws from the diagram ( A omitted) with exact ro ws P K n +2 − → P ε K Q n +2 − → P ε ¯ U n +1 − → P K n +1 ↓ ∼ = P K n +1 − → P − ε ¯ V n − → P − ε K Q n − → P K n the vertical isomorphis m b eing a consequence of the fundamen tal theor em in her- mitian K -theo ry . ✷ Iteration of this Bott map induces a further isomorphism P ε K Q n +4 ( A ) ∼ = − → P ε K Q n ( A ). The classica l induction method [5, (3 .5)], adapted to this case, ena bles us to prove the following theor em. W e recall that the overbar over the K Q indicates reduc- tion mo d m , where m was defined in the Introductio n. Strictly sp eaking , one has to take m ≥ 16, in the theorem, so that K Q ∗ ( Z ′ ) is an a sso ciative ring — see F o otnote 4. Ho wev er, if m < 16, w e can co nsider a ll these groups a s mo dules over K Q ∗ ( Z ′ ; Z / 16 ) a nd the Bott ma p still mak es sense. Hermitian p eri od icity May 26, 2022 17 Theorem 3. 8. Assume that P K n ( A ) = 0 for n ≥ d . Assume mor e over that P ε K Q n ( A ) = 0 for ε = ± 1 and for n = d and d + 1 . Then the cup-pr o duct with the Bott element in 1 K Q p ( Z ′ ) induc es a morphism β n : ε K Q n ( A ) ∼ = − → ε K Q n + p ( A ) which is an isomorp hism for n ≥ d + 1 and a monomorphi sm for n = d . Pro of. The 2- pe r io dicity of the P K Q -groups shown ab ov e (with a change of symmetry) implies that P ε K Q n ( A ) = 0 for n ≥ d and ε = ± 1. F ro m the exac t sequence P ε K Q n ( A ) − → ε K Q n ( A ) β n − → ε K Q n + p ( A ) − → P ε K Q n − 1 ( A ) , we deduce the required isomorphism (sta r ting fro m n = d + 1) and a monomorphism for n = d . ✷ Unfortunately , this strateg y is not efficient to establish Bott per io dicity b ecause the star ting po int o f the induction is no t always v alid (see the end of Section 4 fo r counterexamples a nd Section 2 for e x amples). Therefore, we are go ing to take a no ther appr oach tow a rds Bott p er io dicity . F r o m now on, w e often assume implicitly that the K -groups ar e p erio dic s tarting in degree d . More prec is ely , the Bott map K n ( A ) − → K n + p ( A ) is an iso morphism for n ≥ d , which implies that P K n ( A ) = 0 in the same r ange. Our aim is to prove a s imilar p erio dicity a s sertion for K Q -theory , as we a nnounced in the Introductio n. Let us inv es tig ate in detail the comp osition of the tw o “opp osite” per io dicity maps ε KQ ( A ) u − → Ω q ε KQ ( A ) v − → ε KQ ( A ). Prop ositi o n 3 .9. L et m and q b e 2 -p owers as in Convention 0.2 . Then the cu p- pr o duct b etwe en the images of the ne gative and p ositive Bott elements in 1 K Q ± q ( Z ′ ; Z /m ) is r e duc e d to 0 . Ther efor e, the c omp ositions v ◦ u and u ◦ v ar e nul lh omotopic. Pro of. Since , by e.g. [5, pp. 797 , 799], 1 K Q 0 ( Z ′ ; Z /m ) embeds in 1 K Q 0 ( R ; Z /m ) ⊕ 1 K Q 0 ( F 3 ; Z /m ), we co nsider separately the pro jections of the comp os ites to each summand. In each cas e, the key p oint is that the neg ative B ott element is a p ow er of an element in degr ee − 2. Pro jection to 1 K Q 0 ( R ; Z /m ). W e co mpute the cup-pr o duct of the tw o Bo tt elements, using the fir st step in [36, Lemma 1 .1] 6 , that is, the t welv e-ter m ex act sequences of [33] for b oth Z ′ and the to p o logical ring R . As in [36, Lemma 1.1], they show tha t the map Z ⊕ Z / 2 ∼ = 1 W − 4 ( Z ′ ) − → 1 W − 4 ( R ) ∼ = Z is given b y ( w , α ) 7→ 2 z wher e z generates 1 W − 4 ( R ). W e now use so me s tandard facts: 6 Erratum: in the statemen t of Lemma 1.1 of [36], one should replace 8 y by 16 y b ecause i n the proof the inclusion − 1 W top − 6 ֒ → 1 W ′ top − 8 is s tr i ct. 18 BK O MA Y 26, 2022 (i) there is a multiplicativ e isomorphism b etw e en K n ( R ) and 1 W n ( R ) for all n ∈ Z [3 2, Th´ eor` eme 2 .3], and when n is a multiple o f 8, each group is Z , generator y n say; (ii) the cup-s quare z 2 of the generato r z of K − 4 ( R ) ∼ = 1 W − 4 ( R ) is 4 times a generator y − 8 of K − 8 ( R ) ∼ = Z ; (iii) the c up-square o f an y generator o f the free pa r t of 1 W − 4 ( Z ′ ) pr o jects to a generator o f the free part of 1 W − 8 ( Z ′ ) ∼ = Z ⊕ Z / 2. Let us write q = 2 i +3 where i ≥ 0. F ro m these fac ts , under the ma p Z ⊕ Z / 2 ∼ = 1 W − q ( Z ′ ) − → 1 W − q ( R ) ∼ = Z , ( w, α ) 2 i +1 is sent to (2 z ) 2 i +1 = 2 2 i +1 · (4) 2 i y − q = 2 q/ 2 y − q . Now co nsider the com- m uting diagra m 1 W − q ( Z ′ ) ∼ = Z ⊕ Z / 2 ∼ = ← − 1 K Q − q ( Z ′ ) ∼ = Z ⊕ Z / 2 − → K − q ( Z ′ ) = 0 ↓ ↓ ↓ 1 W − q ( R ) ∼ = Z ← − 1 K Q − q ( R ) ∼ = Z ⊕ Z − → K − q ( R ) ∼ = Z Since the tw o lower horiz o ntal maps corres po nd to the c o kernel of the hyperb olic map (on the left) a nd to the sig nature map (on the rig ht ), they send a pa ir ( u, v ) ∈ Z ⊕ Z to ( u − v ) y − q ∈ 1 W − q ( R ) a nd ( u + v ) y − q ∈ K − q ( R ) resp ectively . Now, by (iii) ab ov e, the element 1 ∈ Z ⊂ 1 K Q − q ( Z ′ ) ma y b e ta ken as (up to sign) the pro jection of ( w , α ) 2 i +1 , and therefo re ma ps b oth on the left to ± 2 q/ 2 y − q ∈ 1 W − q ( R ) a nd on the rig ht to 0 ∈ K − q ( R ). Hence, the image ( u, v ) ∈ 1 K Q − q ( R ) of 1 must hav e the form ± (2 q/ 2 − 1 , − 2 q/ 2 − 1 ). Therefore, since a lwa ys m ≤ 2 q/ 2 − 1 (the r e ason for the change from p to q ), if we now take K Q -theor y with co efficients in Z /m , then the cup-pro duct of the tw o Bott elements may b e written σ = (0 , 0 , η ) ∈ 1 K Q 0 ( Z ′ ; Z /m ) ∼ = Z /m ⊕ Z / m ⊕ Z / 2 , as calcula ted from the B o ckstein exa ct sequence 1 K Q 0 ( Z ′ ) − → 1 K Q 0 ( Z ′ ) − → 1 K Q 0 ( Z ′ ; Z /m ) − → 1 K Q − 1 ( Z ′ ) = 0 and Lemma 3.11 of [6]. Its image in 1 K Q 0 ( R ; Z /m ) ∼ = Z /m ⊕ Z /m is thus ± (2 q/ 2 − 1 , − 2 q/ 2 − 1 ) = (0 , 0). Pro jection to 1 K Q 0 ( F 3 ; Z /m ). T o compute the cup-pr o duct γ of the ima ges of the t wo Bott elements in 1 K Q ± q ( F 3 ; Z /m ), we exploit the definition of the neg a tive Bott element as the iterated power of an element in − 1 K Q − 2 ( Z ′ ). Therefore, γ is the ima g e of the p o s itive Bott element in 1 K Q q ( F 3 ; Z /m ) under the following comp osition: 1 K Q q ( F 3 ; Z /m ) → − 1 K Q q − 2 ( F 3 ; Z /m ) → · · · → 1 K Q 4 ( F 3 ; Z /m ) → → − 1 K Q 2 ( F 3 ; Z /m ) → 1 K Q 0 ( F 3 ; Z /m ) According to F riedlander [19], we have − 1 K Q 2 ( F 3 ) = − 1 K Q 1 ( F 3 ) = 0. Therefore, from another Bockstein e xact sequence, − 1 K Q 2 ( F 3 ; Z /m ) = 0 , and hence γ = 0. Finally , for the last par t of the prop osition, we use well-known facts in cohomol- ogy theories mod 2 k [3, I. p. 7 5] to prov e that the comp os ite maps v ◦ u and u ◦ v are nu llhomotopic. Mo re s pe cifically , the multip lication by 2 s on cohomology theor ie s mo d 2 k is null-homotopic if s ≥ k and s ≥ 2. ✷ Hermitian p eri od icity May 26, 2022 19 F or the next step, we need the following well-kno wn Lemma ( cf. [3, I. p. 75] again) which is a consequence of the splitting of the mu ltiplication by m ′ on the sp ectrum S 0 /m , where m and m ′ are 2- powers defined b elow. Lemma 3.10. L et h ∗ b e a c ohomolo gy the ory re pr esente d by a s p e ctrum S and m b e a 2 - p ower. L et h ∗ ( − ; Z /m ) b e the asso ciate d c ohomolo gy the ory r epr esente d by the sp e ctrum S /m = S ∧ S 0 /m . Final ly , let T m ′ b e the homotopy fib er of the m ap S /m − → S /m define d by the multiplic ation by a 2 -p ower m ′ , wher e m ′ ≥ s up { 4 , m } . Then we have a c anonic al splitting T m ′ ∼ S /m × Ω( S /m ) . Let us denote by F the generic homoto py fiber of the maps descr ibed b e fo re the lemma. Ther e is a homotopy fibra tion F ( u ) − → F ( v ◦ u ) − → F ( v ). According to the ab ov e cons ide r ations, F ( u ) is the sp ectrum P ε KQ ( A ), while F ( v ) is the sp ectrum Ω q − 1 ε KS C ( q ) ( A ). On the o ther hand, as a conseque nc e of Pro po sition 3 .9 and the previous lemma applied to the sp ectrum of her mit- ian K -theo ry , F ( v ◦ u ) may be canonically ide ntified with the pr o duct of sp ectra ε KQ ( A ) × Ω ε KQ ( A ). Ther efore, by taking homotopy gr oups of the pr evious fibra- tion, we get the ex a ct sequence K S C ( q ) n + q → P K Q n → K Q n ⊕ K Q n +1 → K S C ( q ) n + q − 1 → P K Q n − 1 . As a piece of conv enient notation, set K Q n,n +1 = K Q n ⊕ K Q n +1 . More g enerally , we shall a lso use the nota tion K Q X,X +1 for the direct sum K Q X ⊕ K Q X +1 . Prop ositi o n 3.11 . We have t he fol lowing t wo diagr ams of exact se quenc es wher e the vertic al maps ar e induc e d re sp e ctively by the cup-pr o duct with the ne gative or p ositive Bott element: P K Q n → K Q n,n +1 → K S C ( q ) n + q − 1 ↑ P β ′ n ↑ ↑ σ ′ n P K Q n + q → K Q n + q,n + q +1 → K S C ( q ) n +2 q − 1 and P K Q n → K Q n,n +1 → K S C ( q ) n + q − 1 ↓ P β n ↓ ↓ σ n P K Q n + q → K Q n + q,n + q +1 → K S C ( q ) n +2 q − 1 In these diagr ams, P β ′ n is an isomorphism if n ≥ d and a monomorphism if n = d − 1 . We also have P β n = 0 if n ≥ d − 1 . Final ly, σ n (r esp. σ ′ n ) is an isomorphism (r esp. the zer o map) if n ≥ d − 1 . 20 BK O MA Y 26, 2022 Pro of. The second diag ram is included in a bigger o ne with ho r izontal exa ct se- quences of K Q ( Z ′ )-mo dules, where we r ecall that K Q ( Z ′ ) = 1 K Q ( Z ′ ) ⊕ − 1 K Q ( Z ′ ): K S C ( q ) n + q → P K Q n → K Q n,n +1 → K S C ( q ) n + q − 1 → P K Q n − 1 ↓ ↓ P β n ↓ ↓ ↓ K S C ( q ) n +2 q → P K Q n + q → K Q n + q,n + q +1 → K S C ( q ) n +2 q − 1 → P K Q n + q − 1 W e claim that the second vertical ma p P β n is reduced to 0 if n ≥ d − 1. In order to see this, w e consider the r everse map P β ′ n : P K Q n + q − → P K Q n given by the cup-pro duct with the negative Bo tt element. This is a mono morphism for n ≥ d − 1 according to Lemma 3.7. Since the c up-pro duct betw een the po sitive and the negative Bott elements is equal to 0 acco rding to Pr op osition 3.9, we hav e necessarily P β ′ n = 0 . On the other hand, as we hav e seen in P rop osition 3.4, the po sitive Bo tt ma p σ n is an isomorphism K S C ( q ) n + q − 1 ∼ = K S C ( q ) n +2 q − 1 for n + q − 1 ≥ d + q − 2, i.e. for n ≥ d − 1. The reverse map σ ′ n is reduced to 0 since its comp osite with σ n is trivial (note that all ma ps in these diagrams are K Q ( Z ′ )-mo dule maps).. ✷ Prop ositi o n 3.12. If n ≥ d + q − 1 , we have a split short exact se quenc e 0 → P ε K Q n ( A ) → ε K Q n,n +1 ( A ) → ε K S C ( q ) n + q − 1 ( A ) → 0 . Pro of. Let us c onsider a bigger diagram, where we now choose n ≥ d + q − 1: K S C ( q ) n → P K Q n − q → K Q n − q,n − q +1 → K S C ( q ) n − 1 σ n − q +1 ↓ ∼ = ↓ 0 ↓ α n − q ւ γ σ n − q ↓ ∼ = K S C ( q ) n + q → P K Q n → K Q n,n +1 → K S C ( q ) n + q − 1 σ n +1 ↓ ∼ = ↓ 0 ↓ α n σ n ↓ ∼ = K S C ( q ) n +2 q → P K Q n + q → K Q n + q,n + q +1 → K S C ( q ) n +2 q − 1 W e would like to insert a ma p γ : K S C ( q ) n − 1 − → K Q n,n +1 that render s this diagr a m commutativ e. F or this, we cons ider the other c omp osition Ω q ε KQ ( A ) v − → ε KQ ( A ) u − → Ω q ε KQ ( A ). W e get a ma p fr o m Ω − q ( F ( v ) ) to Ω − q ( F ( u ◦ v )) that induces the requir ed map γ since u ◦ v is nullhomotopic. The commutativit y of the a bove diagr a m with γ inserted is a consequence of the homotopy commutativ e square Ω q ε KQ ( A ) v − → ε KQ ( A ) ↓ Ω q ( u ) ↓ u Ω 2 q ε KQ ( A ) Ω q ( v ) − → Ω q ε KQ ( A ) where the vertical (resp. horizo ntal) maps are defined by the cup-pro duct with the positive (r esp. negative) Bott element . Therefore , we g e t the s ho rt split ex a ct sequence 0 − → P K Q n − → K Q n,n +1 − → K S C ( q ) n + q − 1 − → 0 , which is an abbreviated formulation of our P rop osition. ✷ Hermitian p eri od icity May 26, 2022 21 Remark 3.13. The gr oup ε K Q n,n +1 therefore deco mpo ses in tw o dis tinct wa ys as the direct s um of tw o groups, that is ε K Q n,n +1 = ε K Q n ⊕ ε K Q n +1 and ε K Q n,n +1 ∼ = P ε K Q n ⊕ ε K S C ( q ) n + q − 1 . In the app endix to [6] and also in the theor y of stabilized Witt gro ups [3 8], we give many examples of rings A such that K n ( A ) = 0 for all n ∈ Z and therefor e ε K S C ( q ) n + q − 1 ( A ) = 0. This implies that our tw o dir ect sum deco mpo sitions are not the same in gener al, since we may choose A s uch that the tw o groups ε K Q n ( A ) a nd ε K Q n +1 ( A ) ar e not 0. Moreover, this r emark may b e used to show that the higher ε K S C -theory dep ends a priori on the sign of sy mmetry ε . An exa mple o f this fact is A = Z ′ , where we know that P ε K Q n ( Z ′ ) = 0. On the other hand, from the table of the K Q - g roups of Z ′ [5], it is easy to see that 1 K Q n,n +1 ( Z ′ ) 6 = − 1 K Q n,n +1 ( Z ′ ) in general. F or a b etter under standing o f the p erio dicity s tatement s we shall prov e in full generality in Section 4, let us co ns ider the case where the 2-pr imary ab elian gro ups ε K Q n ( A ) ar e finite. The ca tegory of finite 2-primary ab elian gr oups is o f cours e well understo o d: its Grothendieck group (with resp ect to direct sums) is freely generated by the gro ups Z / 2 k . On the other hand, it follows fro m P rop osition 3.1 2 that the groups ε K Q n,n +1 ( A ) = ε K Q n ( A ) ⊕ ε K Q n +1 ( A ) are p erio dic of pe r io d q with resp ect to n for n ≥ d + q − 1. Mo re pr ecisely , P ε K Q n is perio dic for n ≥ d according to Lemma 3 .7 a nd ε K S C ( q ) n + q − 1 is p erio dic for n + q − 1 ≥ d + q − 2, i.e . for n ≥ d − 1, acc o rding to Pro p osition 3.4. Let us wr ite α r for the class of the gro up ε K Q r + d + q − 1 in the Gro thendieck group and put τ = α q − α 0 . W e hav e the identities α q = α 0 + τ , α q +1 = α 1 − τ , α q +2 = α 2 + τ , etc. In general, we may prove b y induction on s the formula α r + qs = α r + ( − 1) r sτ when r ≥ 0. Prop ositi o n 3 .14. L et us assume that the cu p-pr o duct with t he Bott element in- duc es an isomorph ism K n ( A ; Z / m ) → K n + p ( A ; Z / m ) for n ≥ d , and that t he hermitian K -gr oups ε K Q n ( A ; Z /m ) ar e finite for n ≥ d + q − 1 . Then these gr oups ar e p erio dic with r esp e ct to n when n ≥ d + q − 1 , of p erio d q . In p articular, we have τ = 0 in the formulas ab ove. Pro of. In the previous computation, let us write τ = k P i =1 u i − k ′ P j =1 v j where u i and v i are clas ses of nonze r o irreducible mo dules and where k and k ′ are chosen minimal. W e hav e the t wo identit ies, v a lid for all s ≥ 0 : α qs = α 0 + sτ α qs +1 = α 1 − sτ 22 BK O MA Y 26, 2022 F rom the former, for all s uch s , the mo dule s k ′ P j =1 v j is alwa ys a summand o f ε K Q d + q − 1 ( A ; Z /m ). Thu s, k ′ = 0 . Likewise, from the latter identit y , k = 0. Hence, τ = 0 . ✷ Another ex ample o f a p erio dicity statement in hermitian K - theo ry is to consider the case where A is the ring with inv olution B × B op . Here B op is the opp osite algebra of B , the involution on A b eing defined by ( b, b ′ ) 7→ ( b ′ , b ). It is easy to see that ε K Q n ( A ) ∼ = K n ( B ) and that the negative p erio dicity ma p ε K Q n ( A ) → − ε K Q n − 2 ( A ) is reduced to 0. Therefor e, we have the isomorphisms ε K S C n ( A ) = ε K S C (2) n ( A ) ∼ = K n +1 ( B ) ⊕ K n ( B ), and, more g enerally ε K S C ( p ) n ( A ) ∼ = K n +1 ( B ) ⊕ K n − p +2 ( B ) for p a 2-p ow er . If we no w reduce these theories mo d m a nd ass ume the p ositive p -p erio dicity of the asso ciated K -groups (for n ≥ d ), then the last identit y can als o be written as ε K S C ( p ) n + p − 1 ( A ) ∼ = K n + p ( B ) ⊕ K n +1 ( B ) ∼ = K n ( B ) ⊕ K n +1 ( B ) ∼ = ε K Q n ( A ) ⊕ ε K Q n +1 ( A ), which is a particular case of (3.12), since P ε K Q n ( A ) = 0. Note that the p os itive Bott map ε K Q n ( A ) → ε K Q n + p ( A ) is here an isomorphism for n ≥ d . In the spir it of Tho mason, one may consider the p erio dized K Q -theor y , which we shall deno te by ε K Q n ( A )  β − 1  = lim − → ε K Q n + ps ( A ). W e have the following theorem, q uite similar to Connes’ exact seq uence r elating cyclic and Ho chsc hild homologies [13] . Prop ositi o n 3.15. L et u s take n ≥ d + r − 2 . Then (with the ring A omitte d fr om notation) we have the exact se quenc e · · · → ε K Q n +2  β − 1  → ε ′ K Q n +2 − r  β − 1  → ε K S C ( r ) n → ε K Q n +1  β − 1  → ε ′ K Q n +1 − r  β − 1  → · · · → ε K Q d − 1+ r  β − 1  → ε ′ K Q d − 1  β − 1  , wher e ε ′ = − ε if r = 2 and ε ′ = ε if r > 2 . Mor e over, if also r ≥ q , we have a splitting ε K S C ( r ) n ( A ) ∼ = ε K Q n +1 ( A )  β − 1  ⊕ ε K Q n +2 ( A )  β − 1  . Note that for r = q , this splitting is a lso prov e d in the b eg inning of the pro of of Theorem 4.2 . Pro of. T he first part of the prop osition is a direct consequence of Prop osition 3.4 showing that the K S C ( r ) -groups ar e p er io dic for n ≥ d + r − 2. F or the second part, we notice that the map b etw een the K Q -groups is 0, since the cup-pro duct b etw een the pos itive and negative Bott element s is 0, a ccording to Prop ositio n 3 .9. Thus, the seq uence decomp oses in to s hort exact sequences 0 → ε ′ K Q n +2 − r ( A )  β − 1  − → ε K S C ( r ) n ( A ) − → ε K Q n +1 ( A )  β − 1  → 0. Now, the in version of β yields an isomorphism ε ′ K Q n +2 − r ( A )  β − 1  ∼ = ε K Q n +2 ( A )  β − 1  Hermitian p eri od icity May 26, 2022 23 since q | r . Finally , the splitting of these sequences is as before a co ns equence o f a general statement on cohomo logy theor ie s . One has to replace n by a parameter space X , as w e shall also do in the next section. ✷ Remarks 3.16 . The most interesting cases of this prop ositio n a re when o ne has r = 2 or r = q . The pro p o sition a lso shows that the groups ε K S C ( r ) n ( A ) are isomorphic for r ≥ q . 4. Proof of the periodicity theorems Our aim in this Section is essentially to prov e Theor ems 0 .4 and 0.7. W e b egin with a lemma showing that it suffices to consider only the par ameter q of Conv ention 0.2, r ather than the desired per io d p of Conv ent ion 0.1 when dealing with direct or inv erse limits. Lemma 4.1 . L et m b e a 2 -p ower and p, q b e as in Conventions 0.1 and 0.2; that is, m p q ≤ 8 8 8 16 8 16 ≥ 32 m/ 2 m/ 2 Then lim − → ε K Q n + ps ( A ; Z /m ) = lim − → ε K Q n + qs ( A ; Z /m ) and lim ← − ε K Q n + ps ( A ; Z /m ) = lim ← − ε K Q n + qs ( A ; Z /m ) . Pro of. W e may fo cus on the exceptional case where m = q = 16 and p = 8. Here , by Theorem 1.1 there is a p os itive Bott element b + ∈ 1 K Q 8 ( Z ′ ), multiplication b y which gives rise to the direct system of abe lia n gr oups ε K Q n ( A ) − → ε K Q n +8 ( A ) − → ε K Q n +16 ( A ) − → ε K Q n +24 ( A ) − → · · · . The direct limit of its subsy stem ε K Q n ( A ) − → ε K Q n +16 ( A ) − → ε K Q n +32 ( A ) − → · · · app ears in the exa ct sequence of Theorem 4.2 b elow. Howev er, since this subsystem is cofina l, its dir ect limit is pr e c isely that of the or ig inal system. In other words, we may replace the term lim − → ε K Q n +16 s ( A ) by lim − → ε K Q n +8 s ( A ). Since the negative Bott elemen t orig ina tes in − 1 K Q − 2 ( Z ′ ), a similar argument shows that the term lim ← − ε K Q n +16 s ( A ) may b e replaced by lim ← − ε K Q n +8 s ( A ). ✷ W e now star t the pr o of of the p e r io dicity theor ems whic h will be a conseq uence of our co nsiderations in Section 3. Theorem 4.2. L et A b e a ring with involution such that 1 / 2 ∈ A and let m and p b e 2 -p owers ac c or ding to Convention 0.1. We assume t he existenc e of an inte ger d , su ch that the cup-pr o duct with the Bott element in K p ( Z ; Z /m ) induc es an isomo rphism K n ( A ; Z /m ) ∼ = − → K n + p ( A ; Z /m ) . for n ≥ d . F or such n , ther e is an exact se quenc e · · · θ + n − → lim − → ε K Q n +1+ ps ( A ; Z /m ) → lim ← − ε K Q n + ps ( A ; Z /m ) 24 BK O MA Y 26, 2022 θ − n − → ε K Q n ( A ; Z /m ) θ + n − → lim − → ε K Q n + ps ( A ; Z /m ) , which for n ≥ d + q − 1 gives a split short exact se quenc e 0 → lim ← − ε K Q n + ps ( A ; Z /m ) θ − n − → ε K Q n ( A ; Z /m ) θ + n − → lim − → ε K Q n + ps ( A ; Z /m ) → 0 . Pro of. F or n ≥ d , we co nsider the diagra m of exact sequences of Pr op ositions 3.11 and 3.12 (for co nv enience, we a gain dr op the ring A a nd the sig n o f sy mmetry ε in the notation): K S C ( q ) n + q → P K Q n → K Q n,n +1 → K S C ( q ) n + q − 1 → P K Q n − 1 P β n ↓ 0 ↓ α n σ n ↓ ∼ = 0 → P K Q n + q → K Q n + q,n + q +1 → K S C ( q ) n +2 q − 1 → 0 Since the dire c t limit of the P K Q n + qs is equa l to 0, w e see that lim − → K Q n + qs,n +1+ q s ∼ = K S C ( q ) n + q − 1 ∼ = Im( α n + q ) which has a lready b een pr ov en in P rop osition 3.1 5 . W e a lso hav e a reverse diagram of exact sequences (4:4) P K Q n → K Q n,n +1 → K S C ( q ) n + q − 1 → P β ′ n ↑ ∼ = ↑ α ′ n + q σ ′ n ↑ 0 0 → P K Q n + q → K Q n + q,n + q +1 → K S C ( q ) n +2 q − 1 → 0 where the vertical maps are now induce d b y the cup-pro duct with the negative Bo tt element. The first vertical map is an isomo rphism, while the la st o ne is reduced to 0, by P rop osition 3.11. F rom the splitting of exact sequences a fforded by P rop osition 3.12, we hav e lim ← − K Q n + qs,n +1+ q s ∼ = lim ← − P K Q n + qs ∼ = Im( α ′ n + q ) ∼ = P K Q n by Lemma 3 .7. The first exact sequence in the first diag ram ab ove implies the exactness of the middle row o f the diagram (4:5) lim ← − K Q n +1+ qs → K Q n +1 θ + n +1 → lim − → K Q n +1+ qs ↓ ↓ ↓ lim ← − K Q n + qs,n +1+ q s χ − → K Q n,n +1 χ + → lim − → K Q n + qs,n +1+ q s ↓ ↓ ↓ lim ← − K Q n + qs θ − n → K Q n θ + n → lim − → K Q n + qs Since the middle row is the direct sum of first and third rows, the exactness of the middle r ow implies the exa ctness o f the third row. W e apply the same ar gument for the le ft part o f the r equired exact seq uence. Now supp ose that n ≥ d + q − 1 . Then Prop o sition 3.12 implies that in the diagram ab ove χ + is a split epimorphism. The result follows. ✷ Recall from Definition 0.5 in the In tro duction that a r ing A is hermitian r e gular if the inverse limits lim ← − K Q n + ps ( A ) a nd lim ← − 1 K Q n + ps ( A ) are reduced to 0 for all n . Hermitian p eri od icity May 26, 2022 25 Examples 4.3. W e r emark that the seco nd c ondition (with lim 1 ) is a lwa ys fulfilled if A has a p e rio dic K -group after a certain rang e , since in Section 3 we have shown that the inv erse system  K Q n + ps ( A )  satisfies the Mittag- Leffler pro per ty . W e hav e a lso seen in Section 2 that suitable r ing s o f int egers in a num b er field a r e hermitian r egular. O n the other hand, acco rding to Hu, Kriz and Ormsby [25] (r esp. Schlic hting), if k is a field of finite mo d 2 vir tual ´ eta le cohomolog ical dimension a nd of characteristic 0 (resp. p ), we hav e a ho motopy eq uiv alence 1 KQ ( k ) ≃ K ( k ) h Z / 2 . Since K n ( k ) ∼ = K n + p ( k ) for n lar ge enough b y [52] and [62], the p os itive Bott ma p 1 K Q n ( k ) − → 1 K Q n + p ( k ) is also an isomor phism fo r n lar ge enough. The s ame statement is true for the gr oups − 1 K Q n , as we see b y relating the groups 1 K Q, − 1 K Q and K S C (see Section 3). As a conclusio n, the nega tive Bott ma p ε K Q n + p ( k ) − → ε K Q n ( k ) is tr ivial for n larg e enough, which implies that k is hermitia n reg ular by Theo rem 4.2. Mor e g enerally , as a spec ial case of the r esults in Section 6 and a planned joint pap er with Schlic hting [7], any commutativ e algebra A whose residual fields a re all of finite mo d 2 virtual ´ etale c ohomolog ic al dimensio n is hermitian regula r. In [7], these results will be generalized in the scheme framework. Remark 4.4. It is easy to see that the in verse systems of hermitian K - groups  ε K Q n + ps ( A )  and Witt g roups  ε W n + ps ( A )  are e q uiv alent since we hav e the following factoriz ation of the negative Bott map: ε K Q n +8 ( A ) → ε W n +4 ( A ) → ε K Q n ( A ) → ε W n − 4 ( A ). Therefore, the lim and lim 1 groups may a s well b e computed with hig her Witt groups. Theorem 4. 5 . With the same hyp otheses as in the pr evious the or em, let u s assume mor e over that the ring A is hermitian r e gular. Then, for n ≥ d , the p ositive Bott map ε K Q n ( A ) − → ε K Q n + p ( A ) is an isomorph ism. Pro of. Acco r ding to Theorem 4.2, it is enough to show that θ + d is sur jective. F r om the long exa c t sequence of P rop osition 3.2, w e obtain the map of e x act se q uences (4:6) K Q d + q s d + q → K Q d → K S C ( q ) d + q − 2 → K Q d + q − 1 ↓ ↓ θ + d ↓ ∼ = ↓ γ lim − → K Q d + q s u → lim − → K Q d + q s → K S C ( q ) d +2 q − 2 → lim − → K Q d − 1+ q s Observe fr om Theo rem 4 .2 that γ is injective beca use the inv erse limits ar e reduced to 0. Since u is defined b y the cup-pro duct with the negative Bott element, it is reduced to 0. An elementary diag ram chase now shows that θ + d is sur jective. ✷ Although fo r simplicity we have presented the arguments only in the case w he r e X is a sphere, we observe that the groups o btained in the ex a ct sequences of the previous theo r ems can be considered a s co homology theor ies with r esp ect to po inted 26 BK O MA Y 26, 2022 spaces X , if w e replace the v ario us s pe c tra inv o lved by their s ufficiently connected asso ciated sp ectra (so that the lo w - dimensional cohomolo gy gr oups are triv ial). In order to pass fro m exactness of the s equence to s plit exactness as we did a t the end o f Section 3, w e ma y app ea l to a g eneral fact abo ut c o homology theories, that any sur jective morphis m lik e ε θ + X : ε K Q X ( A ) − → lim − → ε K Q X + qs ( A ) alwa ys admits a section. The fo llowing theorems are the analog s of the previo us ones w ith a pa rameter space X . Theorem 4.6 . L et A b e a ring with involution s u ch that 1 / 2 ∈ A , m , p and q b e 2 -p owers ac c or ding to Convention 0.1 and 0.2. We assume t he existenc e of an inte ger d such that for n ≥ d the cup-pr o duct with the Bott element in K p ( Z ; Z /m ) induc es an isomorp hism K n ( A ; Z /m ) ∼ = − → K n + p ( A ; Z /m ) . (a) If X is ( d + q − 2) -c onne cte d, we have a split short exact se quenc e 0 → lim ← − ε K Q X + ps ( A ; Z /m ) θ − n − → ε K Q X ( A ; Z /m ) θ + n − → lim − → ε K Q X + ps ( A ; Z /m ) → 0 . As a c onse quenc e, the gr oups ε K Q X ( A ; Z /m ) ar e “p erio dic” with r esp e ct to X of p erio d p , mor e pr e cisely ε K Q X ( A ; Z /m ) ∼ = ε K Q X + p ( A ; Z /m ) . In p articular, if n ≥ d + q − 1 , ther e is an isomorphism ε K Q n ( A ; Z /m ) ∼ = ε K Q n + p ( A ; Z /m ) . (b) Mor e over, if A is hermitian r e gular, the pr evious statements ar e stil l tru e if we r eplac e the numb er q by 1 . Corollary 4. 7. F or A, X as in The or em 4.6(a), and n ≥ d + q − 1 , the p ositive Bott map β n : ε K Q n ( A ; Z /m ) − → ε K Q n + p ( A ; Z /m ) has (i) its image natur al ly isomorphic to the p erio dize d K Q -the ory, that is Im( β n ) ∼ = lim − → ε K Q n + ps ( A ; Z /m ) , and (ii) its kernel and c okernel natur al ly isomorphic t o lim ← − ε K Q n + ps ( A ; Z /m ) . Con- se quently, if β n is either inje ct ive or surje ctive then it is an isomorph ism. Mor e over, if A is hermitian re gular, the same statement s r emain true on r eplac- ing q by 1 . Pro of. Here, we chase the following co mm utative diag ram, where the vertical maps are given by the cup-pr o duct with the p o sitive Bo tt element : 0 → lim ← − ε K Q n + ps ( A ) θ − − → ε K Q n ( A ) θ + − → lim − → ε K Q n + ps ( A ) → 0 ↓ 0 ↓ β n ↓ ∼ = 0 → lim ← − ε K Q n + p + ps ( A ) θ − − → ε K Q n + p ( A ) θ + − → lim − → ε K Q n + p + ps ( A ) → 0 ✷ Hermitian p eri od icity May 26, 2022 27 Remark 4.8. One may a sk if the ma p θ + n in Theorem 4 .2 is an isomorphis m in general for n sufficiently large. This is not the case how ever, as is shown in the Appendix C to [6] and in [38], where we giv e man y examples of ring s with trivial K -theory and nontrivial K Q -theory . It follows from the 12 - term exa ct se q uence of [3 3, p. 278 ], tha t for such r ings θ − n is an is omorphism. F rom the short exact sequence of Theorem 4 .2, θ + n m ust therefore v anish, a lthough the K Q -theory is nontrivial. Thus, θ + n fails to be an iso morphism. Other examples may be found in the pap er of Hu, Kriz and Ormsby [25] for comm uta tive rings and schemes. How ever, one may ho p e it is so for the examples o f commutativ e rings A consid- ered in the Introductio n, which a re o f “geo metr ic nature”. See also Section 6 for an analog ous co njecture in the category of schemes. W e finish this section with a n application to the computation of the K Q -groups in terms o f the K -g roups w he n these gr oups are finite. (This result formally da tes from the Decem b er 2010 resubmission of the pap er.) F or reading conv enience, we again suppress the index ε ∈ { ± 1 } . Theorem 4.9 . L et us assum e the hyp otheses of The or em 4.6 and that for n ≥ d the K n -gr oups ar e fin ite, of or der k n . Then for n ≥ d t he K Q n -gr oups ar e finite, of or der kq n subje ct to the ine quality kq n + kq n +1 ≤ k d + k d +1 + · · · + k d + q − 1 and e quality kq n = kq n + q . Pro of. According to (3 :3 ), there is an exact sequence · · · − → K S C ( r / 2) n − → K S C ( r ) n − → K S C ( r / 2) n − r / 2 − → · · · . In the case r = 2, the outer tw o gro ups are r e s pe ctively K n +1 and K n , and s o assumed finite when n ≥ d . In general, if we denote b y s r n the order of the gr oup K S C ( r ) n when it is finite, we therefore have the inequa lity s r n ≤ s r / 2 n − r / 2 + s r / 2 n . F or ins ta nce, with n ≥ d , s 2 n +1 ≤ k n +1 + k n +2 s 4 n +3 ≤ s 2 n +1 + s 2 n +3 ≤ k n +1 + k n +2 + k n +3 + k n +4 . . . s q n + q − 1 ≤ k n +1 + k n +2 + · · · + k n + q = k d + · · · + k d + q − 1 , where the equality follows fro m the assumption of K -p erio dicity . On the other hand, since the r ing A is hermitian re gular, acco rding to Theorem 4.2 and its pro of we hav e s q n + q − 1 = kq n + kq n +1 for n ≥ d . This g ives the required inequality . Finally , the equality comes fro m Theorem 4.5. ✷ Example 4.10. If m = 8, we hav e q = 8. Therefore, for n ≥ d , we hav e the inequality kq n + kq n +1 ≤ k d + k d +1 + · · · + k d +7 . In particula r, if we ass ume that the gr oups ε K Q n ( A ) and ε K Q n +1 ( A ) ar e finitely generated, and that for some r and ε the Witt gr oup ε W r ( A ) has an infinite free 28 BK O MA Y 26, 2022 summand, then b y p erio dicity o f Witt groups a fter tensor ing with Q , one of the four gr oups ε K Q n , ε K Q n +1 ( ε ∈ { ± 1 } ) must b e nonzero . It now follows that a t least one o f the g roups K d ( A ) , . . . , K d +7 ( A ) must also b e nonzero. 5. The case of o dd prime power coefficients F or the sa ke of completeness, we should a lso study K Q -theory with o dd prime power co efficients, which is m uch easier to handle, starting fr o m known results in K -theory . As is well known, if ℓ is an o dd prime, there is a rema rk able Bott element b K in the g r oup K 2( ℓ − 1) ℓ ν − 1 ( Z ; Z /ℓ ν ) (see Se c tio n 1 of this pap er). In par ticular, its image in the top ologic a l K -group K 2( ℓ − 1) ℓ ν − 1 ( R ; Z /ℓ ν ) ∼ = K 2( ℓ − 1) ℓ ν − 1 ( C ; Z /ℓ ν ) ∼ = Z /ℓ ν is the image of an integral B ott genera tor in K 2( ℓ − 1) ℓ ν − 1 ( C ) ∼ = Z . According to Conv ention 0 .1, we shall write p = 2( ℓ − 1) ℓ ν − 1 (for the pe rio d) and m = ℓ ν (for the or de r o f the co efficient gro up). Let us a ssume now tha t A is one of the e x amples of algebr a s descr ib ed in the Int ro duction. F or o dd primes, we use the Blo ch-Kato co njectur e , which is now a theorem prov e n by Rost and V oevo dsky , cf. [54], [5 5], [6 3], [21], [58], [59], and [64]. This implies that the cup-pr o duct with the B ott element b induces an isomorphism K n ( A ; Z /m ) ∼ = − → K n + p ( A ; Z /m ) whenever n ≥ d for the type o f rings A considered in the Int ro duction. In o rder to extend our previo us results to hermitian K -theor y with o d d prime power co efficients, it is conv enient to describ e more geometrically elements o f the K -groups a nd K Q -groups. This descr iption in terms o f “v irtual” fla t bundles is given in detail in App e ndix 1 of [35]. F or ins ta nce, an element of ε K Q n ( A ) can b e describ ed a s a flat A -bundle E ov er an homology sphere of dimension n , provided with a nondegenera te q uadratic form q . With this la nguage, we can eas ily define an inv olutio n on the K Q -groups: it is induced by the co r resp ondence ( E , q ) 7→ ( E , − q ) . Let us no w consider the groups ε K Q n ( A ) ′ = ε K Q n ( A ) ⊗ Z Z ′ . The tensor pr o duct of virtual A -bundles (when A is commutativ e) induces a ring str uc tur e on the direct sum o f all these groups (with ε = ± 1). On the o ther hand, the previous inv olution enables us to split e a ch g roup ε K Q n ( A ) ′ as a dir ect sum ε K Q n ( A ) ′ + ⊕ ε K Q n ( A ) ′ − . Lemma 5.1. The su m de c omp osition of the ε K Q n ( A ) ′ describ e d ab ove is a ring pr o duct de c omp osition. In other wor ds, the cup-pr o duct map b etwe en elements of K Q ′ + and K Q ′ − is r e duc e d to 0 . Pro of. Since we make 2 inv ertible in the K Q ′ -groups inv o lved, one may think o f an element z o f K Q ′ + as a sum ( E , q ) + ( E , − q ) and an element z ′ of K Q ′ − as a difference ( E ′ , q ′ ) − ( E ′ , − q ′ ), wher e q and q ′ are ε - and ε ′ -quadratic forms. The pro duct z · z ′ is therefore (with q ⊗ q ′ an εε ′ -quadratic form) ( E ⊗ E ′ , q ⊗ q ′ ) + ( E ⊗ E ′ , − q ⊗ q ′ ) − ( E ⊗ E ′ , q ⊗ − q ′ ) − ( E ⊗ E ′ , − q ⊗ − q ′ ) , which is of c ourse zer o. ✷ Corollary 5. 2. The ring pr o duct de c omp osition of the dir e ct sum of the gr oups ε K Q n ( A ) ′ induc es a ring pr o duct de c omp osition of the dir e ct sum of the gr oups K Q n ( A ; Z /m ) when m is an o dd prime p ower > 3 . Hermitian p eri od icity May 26, 2022 29 Pro of. The coro llary follows from g e neral a rguments about co homology theories mo d m [3]. ✷ Remark 5.3 . One also has an inv o lution o n the K -groups induced by the duality functor, as was noticed alrea dy in Section 3. If we perfor m the tensor pro du ct by Z [1 / 2] or we take co efficients in the group Z /m with m o dd, the symmetric part is in bijectiv e corre s po ndence with the symmetric part of the corr esp onding K Q - group. This cor r esp ondence is induced by the for getful functor or the h yp erb olic functor [33]. Remark 5.4. Before int ro ducing the Bott elements in this s itua tion, it is worth men tioning that the fundamental theorem in hermitian K -theo r y holds for arbitrar y rings (we no longer assume that 1 / 2 ∈ A ) when we lo calize aw ay from 2. The details may be found in [3 6, Le mma 1.1 ]. Mo re precisely , in this case the symmetric part KQ ( A ) ′ + of the sp ectrum of K Q ( A ) ′ is the sy mmetric pa rt of the spectrum K ( A ) ′ , whereas the a ntisymmetric part K Q ( A ) ′ − has p er io dic homotopy gro ups of p erio d 4, which ar e the higher Witt g roups. This r e mark also applies when w e take mo d m co efficients with m o dd. Let b + denote the Bott e le men t in 1 K Q p ( Z ; Z /m ) + = 1 K Q p ( Z ) + corres p o nding to the usua l Bott ele ment b K in K p ( Z ; Z /m ) = K p ( Z ; Z /m ) + = K p ( Z ) + . The following Theorem is now obvious, since K + ∼ = K Q + . Theorem 5.5. L et A b e any ring su ch that the cup-pr o duct map with t he K -the ory Bott element b K induc es an isomorphism K n ( A ) ∼ = K n + p ( A ) for n ≥ d . Then, taking cup-pr o duct with b + also induc es an isomorphism ε K Q n ( A ) + ∼ = ε K Q n + p ( A ) + for the same values of n . ✷ On the other hand, there is another “B ott ele ment” b ′ that lies in 1 K Q p ( Z ) − = 1 W p ( Z ; Z /m ), which is the higher Witt group mo d m . It is the image mo d m of a suitable p ow er o f u ∈ − 1 W 2 ( Z ) constructed in [33] and [36, T heo rem 1.4]. Theorem 5 .6. F or an o dd prime ℓ , let p = 2( ℓ − 1) ℓ ν − 1 and m = ℓ ν as in (0.1). L et A b e any ring s u ch that its K - t he ory mo d m is p erio dic for n ≥ d , the p erio dicity b eing given by the cup-pr o duct with t he Bott element b K . As usual, denote by K Q the K Q -gr oups mod m . Then, for n ≥ d , ther e is an isomorphi sm ε K Q n ( A ) θ + − − → lim − → ε K Q n + ps ( A ) wher e θ + − is induc e d by the cup-pr o duct with the sum b + + b ′ of the two pr eviously define d Bott elements in the gr oup 1 K Q p ( Z ) = 1 K Q p ( Z ) + ⊕ 1 K Q p ( Z ) − . Pro of. If we consider the direct sum ε K Q n ( A ) = ε K Q n ( A ) + ⊕ ε K Q n ( A ) − , then the cup-pro duct with this element b ′ is trivial on the summand ε K Q n ( A ) + , and induces an iso morphism from ε K Q n ( A ) − to ε K Q n + p ( A ) − after [32, Th´ eo r` eme 3.9 ]. 30 BK O MA Y 26, 2022 In a par allel way , the cup-pr o duct with b + is trivia l o n the ter m ε K Q n ( A ) − and induces an iso morphism from ε K Q n ( A ) + to ε K Q n + p ( A ) + , as shown in Cor ollary 5.2. The theorem follows. ✷ Remark 5. 7. By cho osing a “negative Bott element” in 1 K Q − p ( Z ) − , we obtain an equiv alent v ersion of the pr evious theorem, in analo gy with Theorem 0.4, as the following shor t split exact sequence: 0 → lim ← − ε K Q n + ps ( A ) θ − − → ε K Q n ( A ) θ + − → lim − → ε K Q n + ps ( A ) → 0. In this exact seq ue nc e the inv er se system (r esp. direct system) is g iven by taking cup-pro duct with the neg a tive (resp. positive) Bott elemen t in 1 K Q − p ( Z ) − (resp. 1 K Q p ( Z ) + ). Remark 5. 8. F or simplicity , w e hav e assumed the ring A commutativ e in order to define in ternal cup-pro ducts. How ever, a closer lo ok at the ar guments shows that we hav e in fact used “ external” cup-pro ducts of the type K Q ( A ) × K Q ( Z ) − → K Q ( A ) Therefore, the previous theorem extends ea sily to nonco mmut ative r ings, such as group rings. 6. Generaliza tion to schemes and ´ et ale theories The pro of and the statement of Theorem 0.4 apply v erbatim to schemes for whic h 2 is in vertible. This follows since the fundamental theor em in hermitian K -theory has been g eneralized by Schlic hting – see his work in progress o n exact catego ries with weak equiv alences and duality [56]. (Of cours e, in the context of CW-sp ectra , weak equiv alences are in fact homotopy equiv alence s.) In this sec tion we view this generaliza tion in the context of the ´ etale descent problem for hermitian K -theory . Let S b e a reg ular and separ a ted no etherian s cheme of finite Krull dimensio n (in the int erest of g eneralizing these assumptions the inclined re ader may compar e with [53] and [56]). Throug hout, for a fixed prime ℓ , w e a ssume that O S is a shea f of Z [1 /ℓ ]- mo dules. In pa r ticular, fo r the imp or tant ca se ℓ = 2, 2 is in vertible in S . F or the definition of the hermitian K -theor y sp ectrum ε KQ ( S ), a nd the forgetful and hyperb olic ma ps b etw een the algebraic a nd hermitia n K -theory of S , we refer to Schlic hting’s work [56]. In particula r , as for rings, we may form the fib ers ε V ( S ) and ε U ( S ) of the forgetful and hyper b o lic maps, resp ectively . The genera lization of the fundamen tal theor em to schemes in [56] shows that there is a homo topy equiv alence (6:7) − ε V ( S ) ≃ Ω ε U ( S ) such that the comp osite map Ω 2 ε KQ ( S ) → Ω ε U ( S ) → − ε V ( S ) → − ε KQ ( S ) is given b y the cup-pr o duct with the negative Bott elemen t in − 1 K Q − 2 ( Z ′ ). With these results in hand, the pro of o f Theor e m 0.4 carries ov er to the setting of sc hemes. That is, if cup-pro duct with the Bo tt ele ment in K p ( Z ; Z /m ), m and p b eing 2 -p ow er s linked by our Con ven tion 0.1, induces an isomor phis m K n ( S ; Z /m ) ∼ = − → K n + p ( S ; Z /m ) Hermitian p eri od icity May 26, 2022 31 for n ≥ d , then, for n ≥ d + q − 1 there is a split s hort exac t s equence 0 → lim ← − ε K Q n + ps ( S ) θ − − → ε K Q n ( S ) θ + − → lim − → ε K Q n + ps ( S ) → 0 . W e exp ect that the map θ + is an isomorphism in many cases of geometric interest. (F or n ≥ d , and not just ≥ d + q − 1: compare with Theorem 4.5.) This question is closely rela ted to the so-ca lled ´ eta le descent pro blem for hermitian K -theor y and explicit computations, whic h are o ur main concerns in this section. Jardine introduced in [28] the Bott p er io dic ´ eta le hermitian K -theo ry spec tr a of S w ith mod ℓ ν -co efficients ε KQ ´ et /ℓ ν ( S ). More precisely , ε KQ ´ et /ℓ ν ( S ) is the KO - o r equiv alently KU - lo calization of Jardine’s ´ etale hermitian K -theory . By cons truction of the Bott perio dic ´ etale theory , there exists an induced mod ℓ ν compariso n map Γ S : ε KQ /ℓ ν ( S ) − → ε KQ ´ et /ℓ ν ( S ) obtained by taking glo bal sections of a globally fibrant mo del for the presheaf ε KQ /ℓ ν ( ) on s ome sufficiently large ´ etale site of S . Similarly , the mo d ℓ ν ´ etale self-conjugate K -theory KS C ´ et /ℓ ν ( S ) of S is defined by taking a globally fibrant mo del of the presheaf KS C /ℓ ν ( ). Later in this section we shall ma ke use o f sp ecific fibrant mo dels. Recall that the notation vcd ℓ stands for the mo d ℓ virtual co ho mologica l di- mension. When ℓ is odd, then vcd ℓ coincides with the usual mod ℓ cohomolo gical dimension cd ℓ . Although cd 2 is infinite in the examples of Z [1 / 2 ] a nd R , the num- ber vcd 2 is finite in b oth ca ses. F or a pr o of of the next r esult we refer to [40, Prop ositio n 6 .1]. Lemma 6. 1 . If vcd ℓ ( S ) < ∞ then the ´ etale hyp er c ohomolo gy pr eshe af H • ´ et ( − , ε KQ /ℓ ν ( )) is a glob al ly fi br ant mo del for ε KQ /ℓ ν ( − ) . The same fibr ancy r esult holds for the pr eshe af KS C /ℓ ν ( − ) . ✷ The ´ etale desce n t problem fo r s e lf-conjugate K -theory ca n b e solved ea s ily using the solution for a lgebraic K -theory [52]. F or a p oint s ∈ S , le t k ( s ) deno te the corres p o nding residue field. Theorem 6. 2. The c omp arison m ap KS C /ℓ ν ( S ) − → K S C ´ et /ℓ ν ( S ) is a we ak e quivalenc e on sup { vcd ℓ ( k ( s )) − 2 } s ∈ S -c onne cte d c overs. Henc e, if vcd ℓ ( S ) < ∞ then ther e is a we ak e quivalenc e L KU KS C /ℓ ν ( S ) − → K S C ´ et /ℓ ν ( S ) ≃ H • ´ et ( S, L KU KS C /ℓ ν ( )) . Pro of. There exists a natura lly induced commutativ e diagram of fib er sequence s of presheaves of sp ectra KS C /ℓ ν ( − ) − → K / ℓ ν ( − ) − → K /ℓ ν ( − ) ↓ ↓ ↓ KS C ´ et /ℓ ν ( − ) − → K ´ et /ℓ ν ( − ) − → K ´ et /ℓ ν ( − ) ✷ 32 BK O MA Y 26, 2022 Similarly to Theorem 6 .2, we do not exp ect that the compar is on map Γ S is a weak equiv a le nce in general, but in many cases of int erest it sho uld b e a w eak equiv alence on s ome connected cov er. Let ℓ b e an o dd prime, and let the subscript + as in K + denote the symmetr ic part of K -theory . Then the fo r getful and hyperb olic functors induce isomor phisms betw een the s ymmetric par ts of the a lgebraic and her mitian K -gr oups o f S a t ℓ . This a llows us to infer the following res ult b y referring to [52] and to Cor ollary 6 .8 for the s ymmetry of ´ etale hermitian K - theory at o dd primes. Theorem 6. 3. L et ℓ 6 = 2 . The c omp arison map ε KQ /ℓ ν ( S ) + − → ε KQ ´ et /ℓ ν ( S ) + ≃ ε KQ ´ et /ℓ ν ( S ) is a we ak e quivalenc e on sup { vcd ℓ ( k ( s )) − 2 } s ∈ S -c onne cte d c overs. Henc e, if vcd ℓ ( S ) < ∞ then ther e is a we ak e quivalenc e L KU ε KQ / ℓ ν ( S ) + − → ε KQ ´ et /ℓ ν ( S ) ≃ H • ´ et ( S, L KU ε KQ / ℓ ν ( )) . ✷ A t ℓ = 2, we prove Theorem 0.13 stated in the Introductio n. In the following, let n ≥ sup { vcd 2 ( k ( s )) − 1 } s ∈ S + q − 1. Inv erting the p ositive Bott element in the direct sum decompositio n ε K Q n,n +1 ( S ) ∼ = P ε K Q n ( S ) ⊕ ε K S C ( p ) n + p − 1 ( S ) yields ε K Q n,n +1 ( S )[ β − 1 ] ∼ = ε K S C ( p ) n + p − 1 ( S )[ β − 1 ] . In order to simplify the right ha nd side of this isomorphism, we first use the fact that if v cd 2 ( S ) < ∞ then there is a w eak equiv alence KS C ( r ) ( S )[ β − 1 ] − → KS C ( r ) ´ et ( S ). Then, by ´ etale des cent for s elf-conjugate K - theory shown in Theor em 6.2, the induced compa r ison ma p KS C ( r ) ( S ) − → K S C ( r ) ´ et ( S ) is a weak equiv alence on sup { vcd 2 ( k ( s )) + r − 4 } s ∈ S -connected cov ers . Hence, there is an isomo rphism ε K S C ( p ) n + p − 1 ( S )[ β − 1 ] ∼ = ε K S C ( p ) n + p − 1 ( S ). As a result, ε K Q n,n +1 ( S ) = ε K Q n ( S ) ⊕ ε K Q n +1 ( S ) maps by a split surjection on to its Bott loca lization ε K Q n,n +1 ( S )[ β − 1 ] ∼ = ε K Q ´ et n ( S ) ⊕ ε K Q ´ et n +1 ( S ). By lo oking at one comp onent a t a time, we deduce that there is a split surjection ε K Q n ( S ) − → ε K Q n ( S )[ β − 1 ] and an isomo rphism ε K Q n ( S )[ β − 1 ] ∼ = ε K Q ´ et n ( S ) for all n . ✷ Hermitian p eri od icity May 26, 2022 33 Our next r esult is a lo cal-global comparison theorem. It will b e a conseq ue nc e of the homo topical setup due to Jardine, see e.g. [29], and the r igidity theorem for hermitian K -theor y of henselian pa irs proven by the seco nd author in [3 4, Theor em 4], cf. the unpublished work [27] for a n approach using homotopy theory of simplicial presheav es. The sp ecific result we use is as follows. Theorem 6.4. ( [34] ) L et ( A, I ) b e a henselian p air with q ∈ A × ∩ Z such that I is invariant by the involution on A , and λ + λ = 1 for some λ ∈ A if q is even. Then the map of rings with involutions A → A/I induc es an isomorphism ε K Q n ( A ; Z /q ) ∼ = − → ε K Q n ( A/I ; Z /q ) for al l ε and n ≥ 0 . W e note that the shar p er b ound for the connected co vers in the theore m below (relative to that in Theorem 0.1 3) equals the o ne shown for algebraic K -theory in [52]. Theorem 6.5. Supp ose that Γ k ( s ) is a we ak e quivalenc e on (v cd 2 ( k ( s )) − 2) - c onne cte d c overs for every r esidue field k ( s ) of S . Then the c omp arison map Γ S : ε KQ / 2 ν ( S ) − → ε KQ ´ et / 2 ν ( S ) is a we ak e quivalenc e on sup { vcd 2 ( k ( s )) − 2 } s ∈ S -c onne cte d c overs. ✷ Pro of. There is a functoria lly induced commutativ e diagram with the mo d 2 compariso n map displa yed on top: ε KQ / 2( S ) − → H • ´ et ( S, L KU ε KQ / 2( )) ↓ ↓ H • Nis ( S, ε KQ / 2( )) − → H • Nis ( S, H • ´ et ( S, L KU ε KQ / 2( ))) W e claim that the vertical maps are weak equiv a le nces. By the Nisnevich descent theorem in [5 6], this holds for the le ft hand side. F or the rig ht hand side, the ´ etale top ology is finer than the Nisnevich one; so, the direct image functor maps H • ´ et ( S, L KU ε KQ / 2( )) to a globa lly fibrant ob ject on the Nisnevich site of S . W e claim that the mo d 2 compar is on ma p is a stalkwise weak equiv alence on the given connected cov e r for the Nisnevich top ology . In fact, let A be a Hensel lo cal ring with residue field k and co nsider the functorially induced commutativ e diagram ε KQ / 2( A ) − → ε KQ / 2( k )) ↓ ↓ H • ´ et ( A, L KU ε KQ / 2( )) − → H • ´ et ( k , L KU ε KQ / 2( )) Combining the pr evious theo rem concerning rigidity for hermitian K -theory and the equiv a lence betw een the ´ etale sites of A a nd k , this reduces the stalkwise weak equiv alence to the assumed ca se of fields. It fo llows that the low er horizontal map in Diagram (6) is a stalkwise weak equiv alence on the same connec ted cov ers betw een glo bally fibr ant ob j ects, and hence it is a p oint wise weak equiv alence on sup { v cd ( k ( s ) − 2 } s ∈ S -connected covers. ✷ Motiv ated by the lo cal hypotheses of The o rem 6.5, we make the following for ecast of the outco me of the ´ etale descent pr oblem for hermitian K -theory . Conjecture 6.6. Supp ose that k is a field of char acteristic 6 = 2 . Then the c om- p arison map Γ k : ε KQ / 2 ν ( k ) − → ε KQ ´ et / 2 ν ( k ) 34 BK O MA Y 26, 2022 is a we ak e quivalenc e on ( vcd 2 ( k ) − 2 )-c onne cte d c overs. Conjecture 6.6, in co njunction with Theorem 6.5, predicts that, in many cases o f int erest, her mitian K -theor y is Bott p erio dic on so me connected cover. O ur earlier results on Bott p erio dicity can b e taken a s oblique evidence for this co njecture. F or n ≥ vcd 2 ( k ) + q − 1 , rec all the ex act s equence of K Q -gr o ups with 2-p ow er co efficients: 0 → lim ← − ε K Q n + ps ( k ) θ − − → ε K Q n ( k ) θ + − → lim − → ε K Q n + ps ( k ) → 0 . By Bott p erio dicity , Co njecture 6.6 implies that the in verse limit is trivial, i.e. lim ← − ε K Q n + ps ( k ) = 0 . In other words, the field k should b e hermitian regular (Definition 0.5). Co nv ersely , if the inverse limit is trivial, then there is a n isomorphism θ + : ε K Q n ( k ) ∼ = − → lim − → ε K Q n + ps ( k ) for n ≥ vcd 2 ( k ) − 1, a ccording to our Theorem 4.5. As noted in the Introductio n, a pro of o f the ab ove conjecture is to app ear in a join t pap e r with Schlic hting [7]. Lemma 6.1 can be motiv ated b y the conditionally conv ergent r ight half-pla ne cohomolog ical desce nt sp ectral seque nce established by Thomason [60]: (6:8) ε E p,q 2 = H p ´ et ( S, e π q L KU ε KQ / ℓ ν ( )) = ⇒ π q − p H • ´ et ( S, L KU ε KQ /ℓ ν ( )) . Here the co efficient sheaf indicated by e π ∗ is the ´ etale shea fification of the presheaf of stable homotopy groups π ∗ L KU ε KQ /ℓ ν ( ). The concept of “ conditional con- vergence” for sp ectral seque nce s w as in tro duced by Boar dman in [8 ]. A useful consequence is that the descent sp ectral sequence (6:8) conv erg es s trongly provided that ther e exists o nly a finite num b er o f nontrivial different ials. Thus, the sp ectral sequence (6:8) is strongly con vergen t if S has finite mo d ℓ ´ eta le cohomolog ical di- mension. (This will b e the case in all the exa mples w e consider.) The d r -differential in (6:8) ha s bidegree ( r , 1 − r ). In o rder to identif y the ´ etale stalk s of ε KQ / ℓ ν ( ), and consequently the E 2 -page of (6:8), cf. [2 8], [5 7, Theor e m 2.6], we inv oke the Rigidity The o rem 6.4 together with the homotop y equiv alences (6:9) ε KQ /ℓ ν ( A ) ≃ ε KQ /ℓ ν ( C ) ≃  K /ℓ ν ( R ) ε = 1 Ω 4 K /ℓ ν ( R ) ε = − 1 for a strict Hensel lo cal ring A . The ab ove is very similar to the case of a lgebraic K -theory , where the ´ etale sheaf a s so ciated to the presheaf U 7→ π n K /ℓ ν ( U ) is the T ate twisted sheaf of ro ots of unity µ ⊗ k ℓ ν when n = 2 k is even, and trivial when n is odd. F or KS C and ε KQ at ℓ we have: Corollary 6.7 . ( [57] ) The ´ etale she af asso ciate d to the pr eshe af U 7→ π n ( KS C /ℓ ν ( U )) is given by: Hermitian p eri od icity May 26, 2022 35 n mo d 4 ℓ = 2 ℓ 6 = 2 4 k µ ⊗ 2 k 2 ν µ ⊗ 2 k ℓ ν 4 k + 1 µ ⊗ 2 k +1 2 0 4 k + 2 µ ⊗ 2 k +1 2 0 4 k + 3 µ ⊗ 2 k +2 2 ν µ ⊗ 2 k +2 ℓ ν Corollary 6.8 . ( [28] , [57] ) The ´ et ale she af asso ciate d to the pr eshe af U 7→ π n ( ε KQ / ℓ ν ( U )) is given as fol low s. (1) F or ℓ = 2 by: n mo d 8 ε = 1 , ν = 1 ε = − 1 , ν = 1 ε = 1 , ν > 1 ε = − 1 , ν > 1 8 k µ ⊗ 4 k 2 µ ⊗ 4 k 2 µ ⊗ 4 k 2 ν µ ⊗ 4 k 2 ν 8 k + 1 µ ⊗ 4 k +1 2 0 µ ⊗ 4 k +1 2 0 8 k + 2 µ ⊗ 4 k +1 4 0 ( µ ⊗ 4 k +1 2 ) ⊕ 2 0 8 k + 3 µ ⊗ 4 k +1 2 0 µ ⊗ 4 k +1 2 0 8 k + 4 µ ⊗ 4 k +2 2 µ ⊗ 4 k +2 2 µ ⊗ 4 k +2 2 ν µ ⊗ 4 k +2 2 ν 8 k + 5 0 µ ⊗ 4 k +3 2 0 µ ⊗ 4 k +3 2 8 k + 6 0 µ ⊗ 4 k +3 4 0 ( µ ⊗ 4 k +3 2 ) ⊕ 2 8 k + 7 0 µ ⊗ 4 k +3 2 0 µ ⊗ 4 k +3 2 (2) F or ℓ 6 = 2 and ε = ± 1 by µ ⊗ 2 k ℓ ν if n = 4 k , and trivial otherwise. Remark 6.9. In Corollar y 6.8, the (4 , 2)- p er io dicity in the change of symmetry betw een the ε = 1 and ε = − 1 cases in the table for ℓ = 2 is giv en by cup-pro duct with a generator of − 1 K Q 4 ( C ; Z / 2 ν ). The case ℓ 6 = 2 is similar. In degrees 8 k + 2, recall that R P 2 is a mo d 2 Mo ore space and g K O ( R P 2 ) ∼ = Z / 4 ge ne r ated by the tangent bundle, while the universal co efficient seq uence splits fo r ν > 1. As a conse quence of Lemma 6.1 and C o rollar y 6.8, we co nclude that if vcd 2 ( S ) < ∞ then there exist conditionally con vergent cohomolog ical sp ectr al se q uences 1 E p,q 2 =                H p ´ et ( S, µ ⊗ 4 k 2 ) q = 8 k H p ´ et ( S, µ ⊗ 4 k +1 2 ) q = 8 k + 1 H p ´ et ( S, µ ⊗ 4 k +1 4 ) q = 8 k + 2 H p ´ et ( S, µ ⊗ 4 k +1 2 ) q = 8 k + 3 H p ´ et ( S, µ ⊗ 4 k +2 2 ) q = 8 k + 4 0 q ≡ 5 , 6 , 7 (mo d 8 )                = ⇒ 1 K Q ´ et q − p / 2( S ) , and − 1 E p,q 2 =                H p ´ et ( S, µ ⊗ 4 k 2 ) q = 8 k H p ´ et ( S, µ ⊗ 4 k +2 2 ) q = 8 k + 4 H p ´ et ( S, µ ⊗ 4 k +3 2 ) q = 8 k + 5 H p ´ et ( S, µ ⊗ 4 k +3 4 ) q = 8 k + 6 H p ´ et ( S, µ ⊗ 4 k +3 2 ) q = 8 k + 7 0 q ≡ 1 , 2 , 3 (mo d 8 )                = ⇒ − 1 K Q ´ et q − p / 2( S ) . 36 BK O MA Y 26, 2022 And for ν ≥ 2 , the descent spec tral sequences tak e the for ms 1 E p,q 2 =                H p ´ et ( S, µ ⊗ 4 k 2 ν ) q = 8 k H p ´ et ( S, µ ⊗ 4 k +1 2 ) q = 8 k + 1 H p ´ et ( S, µ ⊗ 4 k +1 2 ) ⊕ 2 q = 8 k + 2 H p ´ et ( S, µ ⊗ 4 k +1 2 ) q = 8 k + 3 H p ´ et ( S, µ ⊗ 4 k +2 2 ν ) q = 8 k + 4 0 q ≡ 5 , 6 , 7 (mo d 8)                = ⇒ 1 K Q ´ et q − p / 2 ν ( S ) , and − 1 E p,q 2 =                H p ´ et ( S, µ ⊗ 4 k 2 ν ) q = 8 k H p ´ et ( S, µ ⊗ 4 k +2 2 ν ) q = 8 k + 4 H p ´ et ( S, µ ⊗ 4 k +3 2 ) q = 8 k + 5 H p ´ et ( S, µ ⊗ 4 k +3 2 ) ⊕ 2 q = 8 k + 6 H p ´ et ( S, µ ⊗ 4 k +3 2 ) q = 8 k + 7 0 q ≡ 1 , 2 , 3 (mo d 8)                = ⇒ − 1 K Q ´ et q − p / 2 ν ( S ) . F or ℓ 6 = 2 and cd ℓ ( S ) < ∞ , the descent sp ectr al sequence ta kes the form ε E p,q 2 = ( H p ´ et ( S, µ ⊗ q 2 ℓ ν ) q ≡ 0 (mo d 4) 0 q 6≡ 0 (mo d 4 ) ) = ⇒ ε K Q ´ et q − p /ℓ ν ( S ) . Note that the E 2 -pages are independent of the symmetry ε . This is not surprising since on symmetric parts K - theory mo d ℓ ν maps by a weak equiv alence to hermitian K -theory mo d ℓ ν . The following results are concerned with Bousfield ℓ -adic c o mpletions (denoted by #) of the se lf-conjugate and hermitian K -theory sp ectra . Bousfield in tr o duced this notion in [12]. First we shall tabulate the corr e s po nding ´ etale sheav es. Let Z ⊗ k ℓ denote the k th T a te twist of the ℓ -a dic integers. In the example of self-c o njugate K -theory the ´ etale s heav es are pe r io dic in the following sense. Corollary 6.1 0. The ´ etale she af asso ciate d to the pr eshe af U 7→ π n ( KS C ( U ) # ) of ℓ -adic al ly c omplete d self-c onjugate K -the ory is given by: n mo d 4 ℓ = 2 ℓ 6 = 2 4 k Z ⊗ 2 k 2 Z ⊗ 2 k ℓ 4 k + 1 µ ⊗ 2 k +1 2 0 4 k + 2 0 0 4 k + 3 Z ⊗ 2 k +2 2 Z ⊗ 2 k +2 ℓ F or her mitian K - theo ry the ´ etale s he aves are p erio dic in the follo wing sense. Corollary 6.1 1. The ´ etale she af asso ciate d to the pr eshe af U 7→ π n ( ε KQ ( U ) # ) of ℓ -adic al ly c omplete d hermitian K -the ory is given as fol lows. Hermitian p eri od icity May 26, 2022 37 n mo d 8 ε = 1 ε = − 1 8 k Z ⊗ 4 k 2 Z ⊗ 4 k 2 8 k + 1 µ ⊗ 4 k +1 2 0 8 k + 2 µ ⊗ 4 k +1 2 0 8 k + 3 0 0 8 k + 4 Z ⊗ 4 k +2 2 Z ⊗ 4 k +2 2 8 k + 5 0 µ ⊗ 4 k +3 2 8 k + 6 0 µ ⊗ 4 k +3 2 8 k + 7 0 0 (1) F or ℓ = 2 by: (2) F or ℓ 6 = 2 and ε = ± 1 by Z ⊗ 2 k ℓ if n = 4 k , and trivial otherwise. In the follo wing, ´ etale coho mology is co ntin uous ´ etale co ho mology [15], [26]. As a result of the pr evious co rollar ies, the descent sp ectra l sequences for the 2-completed ´ etale self-conjuga te ´ etale K -theory and her mitian K - theo ry of S take the forms 1 E p,q 2 =        H p ´ et ( S, Z ⊗ 2 k 2 ) q = 4 k H p ´ et ( S, µ ⊗ 2 k +1 2 ) q = 4 k + 1 H p ´ et ( S, Z ⊗ 2 k +2 2 ) q = 4 k + 3 0 q = 4 k + 2        = ⇒ K S C ´ et q − p ( S ) # , 1 E p,q 2 =            H p ´ et ( S, Z ⊗ 4 k 2 ) q = 8 k H p ´ et ( S, µ ⊗ 4 k +1 2 ) q = 8 k + 1 H p ´ et ( S, µ ⊗ 4 k +1 2 ) q = 8 k + 2 H p ´ et ( S, Z ⊗ 4 k +2 2 ) q = 8 k + 4 0 q ≡ 3 , 5 , 6 , 7 (mo d 8)            = ⇒ 1 K Q ´ et q − p ( S ) # , and − 1 E p,q 2 =            H p ´ et ( S, Z ⊗ 4 k 2 ) q = 8 k H p ´ et ( S, Z ⊗ 4 k +2 2 ) q = 8 k + 4 H p ´ et ( S, µ ⊗ 4 k +3 2 ) q = 8 k + 5 H p ´ et ( S, µ ⊗ 4 k +3 2 ) q = 8 k + 6 0 q ≡ 1 , 2 , 3 , 7 (mo d 8)            = ⇒ − 1 K Q ´ et q − p ( S ) # . F or ℓ 6 = 2 the desce n t sp ectral se q uences take the forms 1 E p,q 2 =    H p ´ et ( S, Z ⊗ 2 k ℓ ) q = 4 k H p ´ et ( S, Z ⊗ 2 k +2 ℓ ) q = 4 k + 3 0 q ≡ 1 , 2 (mo d 4)    = ⇒ K S C ´ et q − p ( S ) # , and ε E p,q 2 = ( H p ´ et ( S, Z ⊗ q 2 ℓ ) q ≡ 0 (mo d 4) 0 q 6≡ 0 (mo d 4) ) = ⇒ ε K Q ´ et q − p ( S ) # . Our nex t ob jectiv e is to compute ℓ - adically completed ´ etale self-co njugate a nd hermitian K -groups in terms of ´ etale cohomolog y gro ups. T o this end w e need s ome more notation. 38 BK O MA Y 26, 2022 Let A • B denote an ab elian g r oup extension of B b y A , s o that there ex ists a short exact sequence 0 → A → A • B → B → 0 . Lemma 6.12 . I f cd 2 ( S ) = 2 and H 0 ´ et ( S, Z ⊗ i 2 ) = 0 for i > 0 then the 2 -c omplete d ´ etale hermitian K -gr oups of S ar e c ompute d up to extensions in the fol lo wing table. n mo d 8 1 K Q ´ et n ( S ) # − 1 K Q ´ et n ( S ) # 8 k > 0 H 2 ´ et ( S, µ ⊗ 4 k +1 2 ) • H 1 ´ et ( S, µ ⊗ 4 k +1 2 ) 0 8 k + 1 H 1 ´ et ( S, µ ⊗ 4 k +1 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +1 2 ) 0 8 k + 2 H 2 ´ et ( S, Z ⊗ 4 k +2 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +1 2 ) H 2 ´ et ( S, Z ⊗ 4 k +2 2 ) 8 k + 3 H 1 ´ et ( S, Z ⊗ 4 k +2 2 ) H 2 ´ et ( S, µ ⊗ 4 k +3 2 ) • H 1 ´ et ( S, Z ⊗ 4 k +2 2 ) 8 k + 4 0 H 2 ´ et ( S, µ ⊗ 4 k +3 2 ) • H 1 ´ et ( S, µ ⊗ 4 k +3 2 ) 8 k + 5 0 H 1 ´ et ( S, µ ⊗ 4 k +3 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +3 2 ) 8 k + 6 H 2 ´ et ( S, Z ⊗ 4 k +4 2 ) H 2 ´ et ( S, Z ⊗ 4 k +4 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +3 2 ) 8 k + 7 H 2 ´ et ( S, µ ⊗ 4 k +5 2 ) • H 1 ´ et ( S, Z ⊗ 4 k +4 2 ) H 1 ´ et ( S, Z ⊗ 4 k +4 2 ) Remark 6. 13. The a ssumption o n the v anishing of H 0 ´ et ( S, Z ⊗ i 2 ) fo r i > 0 in Lemma 6.12 is a commonplace and holds for the examples considered in Section 7. W e note, how ever, that the assumptions in Lemma 6.12 are not satisfied for the field of r eal nu mbers, and for n umber fields with a t leas t one real embedding. Lemma 6.14. If ℓ is an o dd prime and cd ℓ ( S ) ≤ 7 the ℓ -c omplete d ´ etale hermitian K -gr oups of S ar e c ompute d up to extensions in the fol low ing t able. n mo d 4 ε K Q ´ et n ( S ) # 4 k > 0 H 4 ´ et ( S, Z ⊗ 2 k +2 ℓ ) • H 0 ´ et ( S, Z ⊗ 2 k ℓ ) 4 k + 1 H 7 ´ et ( S, Z ⊗ 2 k +4 ℓ ) • H 3 ´ et ( S, Z ⊗ 2 k +2 ℓ ) 4 k + 2 H 6 ´ et ( S, Z ⊗ 2 k +4 ℓ ) • H 2 ´ et ( S, Z ⊗ 2 k +2 ℓ ) 4 k + 3 H 5 ´ et ( S, Z ⊗ 2 k +4 ℓ ) • H 1 ´ et ( S, Z ⊗ 2 k +2 ℓ ) Corollar y 6.11 a nd the co rresp onding re s ult for alg e br aic K -theory imply the next result b y ins pe c tio n. Corollary 6.1 5. The ´ etale she af asso ciate d to the pr eshe af U 7→ π n ( ε V ( U ) # ) of ℓ -adic al ly c omplete d hermitian V - the ory is given as fol lows. (1) F or ℓ = 2 by: (2) F or ℓ 6 = 2 by Z ⊗ 2 k +1 ℓ if n = 4 k + 1 , and trivial otherwise. The prev ious cor o llary allows us to immediately iden tify the E 2 -page o f the de- scent sp ectral sequence for ℓ -adica lly completed ´ etale hermitia n V -gr oups in ter ms of ´ etale cohomolog y . As a consequence, imp osing a co mmonplace restriction on the ´ etale cohomolog ical dimensio n yields the following computation. Lemma 6.16. If c d 2 ( S ) = 2 and H 0 ´ et ( S, Z ⊗ i 2 ) = 0 for i > 0 , t hen the 2 -c omplete d ´ etale V -gr oups of S ar e c ompute d up to extensions in the fol lowing table. Hermitian p eri od icity May 26, 2022 39 n mo d 8 ε = 1 ε = − 1 8 k 0 0 8 k + 1 Z ⊗ 4 k +1 2 Z ⊗ 4 k +1 2 8 k + 2 µ ⊗ 4 k +1 2 0 8 k + 3 µ ⊗ 4 k +2 2 0 8 k + 4 0 0 8 k + 5 Z ⊗ 4 k +3 2 Z ⊗ 4 k +3 2 8 k + 6 0 µ ⊗ 4 k +3 2 8 k + 7 0 µ ⊗ 4 k +4 2 n mo d 8 1 V ´ et n ( S ) # − 1 V ´ et n ( S ) # 8 k ≥ 0 H 2 ´ et ( S, µ ⊗ 4 k +1 2 ) • H 1 ´ et ( S, Z ⊗ 4 k +1 2 ) H 1 ´ et ( S, Z ⊗ 4 k +1 2 ) 8 k + 1 H 2 ´ et ( S, µ ⊗ 4 k +2 2 ) • H 1 ´ et ( S, µ ⊗ 4 k +1 2 ) 0 8 k + 2 H 1 ´ et ( S, µ ⊗ 4 k +2 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +1 2 ) 0 8 k + 3 H 2 ´ et ( S, Z ⊗ 4 k +3 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +2 2 ) H 2 ´ et ( S, Z ⊗ 4 k +3 2 ) 8 k + 4 H 1 ´ et ( S, Z ⊗ 4 k +3 2 ) H 2 ´ et ( S, µ ⊗ 4 k +3 2 ) • H 1 ´ et ( S, Z ⊗ 4 k +3 2 ) 8 k + 5 0 H 2 ´ et ( S, µ ⊗ 4 k +4 2 ) • H 1 ´ et ( S, µ ⊗ 4 k +3 2 ) 8 k + 6 0 H 1 ´ et ( S, µ ⊗ 4 k +4 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +3 2 ) 8 k + 7 H 2 ´ et ( S, Z ⊗ 4 k +5 2 ) H 2 ´ et ( S, Z ⊗ 4 k +5 2 ) • H 0 ´ et ( S, µ ⊗ 4 k +4 2 ) The previo us computations are supplemented by mo r e sp ecialized examples in the next section. F o r the earliest ´ eta le K -theory computations we refer the reader to [1 5] and [60]. It is worth while to p oint out tha t the difference betw een the ´ etale K -theory , in lo c. cit. , and the ´ etale her mitian K -theory computations in this pap er is r e minis c ent of the situa tio n for the classica l Atiy a h-Hirzebruch spectr al s equences based on complex and rea l to p o logical K -theo ry . This ana logy is evident on the level of ´ etale stalks b y co mparison with the complex and real K -theories of a p oint. 7. Applica tio ns to finite fields, lo cal and global fields In this section we po int out some computational co nsequences of the ab ov e re- sults. The examples are geometric in nature a nd re late to finite fields, and to lo cal and global n umber fields. Our main interest a nd fo cus a re on the 2-primary computations. Example 7.1 . In what follows, we apply Lemma 6.14 to some classes of examples. Throughout, ℓ is an o dd pr ime num b er. (1) If S is a d -dimensiona l smo o th complex v ariety then cd ℓ ( S ) ≤ 2 d . Lemma 6.14 co mputes the group ε K Q ´ et n ( S ) # up to extensions if S is of dimension at most 3. F or curves and sur faces there are no undetermined extensions. Detailed computations of the algebraic K -theor y o f S were w o rked out in [44] and [45]. (2) If S is a smo oth curve ov er a nu mber field then cd ℓ ( S ) = 4. In this ca s e, for n > 0, there are no undetermined extensions in the computatio n of ε K Q ´ et n ( S ) # given in Lemma 6 .14. F or detailed computations o f the alg e- braic K -theo ry of S we refer to [51]. (3) The ring O F [1 /ℓ ] of ℓ - int egers in a ny num b er field F has co homologica l dimension cd ℓ ( O F [1 /ℓ ]) = cd ℓ ( F ) = 2. In particular, ε K Q ´ et n ( O F [1 /ℓ ]) # is 40 BK O MA Y 26, 2022 trivial when 0 < n ≡ 0 , 1 (mo d 4 ) a nd finite when n ≡ 2 (mo d 4). The same cohomo logical dimension b ound holds fo r lo cal num b er fields and their v aluation ring s, e.g. the field of ℓ -adic num b ers. Let F b e a field of characteristic 6 = 2 and ζ r be a primitive r th ro ot of unity . F or i ∈ Z , let w i ( F ) b e the maxima l 2-p ow er 2 n such that the exp o ne nt of the Galois g roup of F ( ζ 2 n ) /F divides i . If F contains ζ 4 and i = 2 λ k with k o dd , then w i ( F ) = 2 r + λ where r is max ima l such that F contains a primitive 2 r -ro ot o f unity . If i is o dd, w i ( Q ( √ − 1)) = 4, while if √ − 1 6∈ F then w i ( F ) = 2. In all our e xamples the num b er w i ( F ) is finite. Using Lemma 6.1 2 and the ´ etale coho mo logy groups of finite fields , we tabulate the 2-co mpleted ´ etale hermitian K - g roups of F t for t o dd. Our findings a re in agreement with F rie dlander’s c o mputation of the hermitian K -gr oups o f F t in [1 9]. Example 7. 2. L e t F t be a finite field with an o dd num b er of elements t . The 2-completed ´ etale her mitian K -gr oups of F t are computed in the following table. n mo d 8 1 K Q ´ et n ( F t ) # − 1 K Q ´ et n ( F t ) # 8 k > 0 Z / 2 0 8 k + 1 ( Z / 2) 2 0 8 k + 2 Z / 2 0 8 k + 3 Z /w 4 k +2 ( F t ) Z /w 4 k +2 ( F t ) 8 k + 4 0 Z / 2 8 k + 5 0 ( Z / 2) 2 8 k + 6 0 Z / 2 8 k + 7 Z /w 4 k +4 ( F t ) Z /w 4 k +4 ( F t ) The extension pr oblem for 1 K Q ´ et 8 k +1 ( F t ) # can b e res olved using the homotopy fibration [33] KS C ( F t ) − → Ω ε KQ ( F t ) σ (2) − → Ω − 1 − ε KQ ( F t ) . The gro up K S C 8 k ( F t ) # has order 2 , c f. Example 7.1 2, and is a direct summand of 1 K Q 8 k +1 ( F t ) # . This a lso r esolves the extension pro blem in degree 8 k + 5 for ε = − 1. Example 7. 3 . Lemma 6.1 2 applies to every dy adic lo cal field F , i.e. every finite extension of the 2-adic num b ers Q 2 . If the field extension degree [ F : Q 2 ] = d , then H 0 ´ et ( F, Z ⊗ i 2 ) = δ i 0 Z 2 , H 1 ´ et ( F, Z ⊗ 1 2 ) = Z d +1 2 ⊕ Z / 2, H 2 ´ et ( F, Z ⊗ 1 2 ) = Z 2 , H 1 ´ et ( F, Z ⊗ i 2 ) =  Z d 2 ⊕ Z / 2 i > 1 o dd, Z d 2 ⊕ Z /w i ( F ) i even, and H 2 ´ et ( F, Z ⊗ i 2 ) =  Z /w i − 1 ( F ) i > 1 o dd, Z / 2 i even. The 2- completed ´ etale hermitian K -groups of F a re computed up to extensions in the following table. F or a no n-dyadic lo cal field, i.e. a finite extension o f the p - a dic num b ers Q p for some o dd prime p , the 2-co mpleted ´ etale her mitian K -g roups are c o mprised of finite gr o ups in p o sitive deg rees. The ´ etale coho mology computation le a ding to this conclusion is giv en in [41, P rop osition 7.3.1 0]. Hermitian p eri od icity May 26, 2022 41 n mo d 8 1 K Q ´ et n ( F ) # − 1 K Q ´ et n ( F ) # 8 k > 0 Z / 2 • ( Z / 2) d +2 0 8 k + 1 ( Z / 2) d +2 • Z / 2 0 8 k + 2 Z / 2 • Z / 2 Z / 2 8 k + 3 Z d 2 ⊕ Z /w 4 k +2 ( F ) Z / 2 • ( Z d 2 ⊕ Z / w 4 k +2 ( F )) 8 k + 4 0 Z / 2 • ( Z / 2) d +2 8 k + 5 0 ( Z / 2) d +2 • Z / 2 8 k + 6 Z / 2 Z / 2 • Z / 2 8 k + 7 Z / 2 • ( Z d 2 ⊕ Z / w 4 k +4 ( F )) Z d 2 ⊕ Z /w 4 k +4 ( F ) Example 7.4. The 2-completed ´ etale hermitian K -gr oups of a finite extension F of Q p for p o dd are computed up to extensions in the following ta ble. n mo d 8 1 K Q ´ et n ( F ) # − 1 K Q ´ et n ( F ) # 8 k > 0 Z / 2 • ( Z / 2) 2 0 8 k + 1 ( Z / 2) 2 • Z / 2 0 8 k + 2 Z / 2 • Z / 2 Z / 2 8 k + 3 Z /w 4 k +2 ( F ) Z / 2 • Z /w 4 k +2 ( F ) 8 k + 4 0 Z / 2 • ( Z / 2) 2 8 k + 5 0 ( Z / 2) 2 • Z / 2 8 k + 6 Z / 2 Z / 2 • Z / 2 8 k + 7 Z / 2 • Z /w 4 k +4 ( F ) Z /w 4 k +4 ( F ) A totally imag inary num b er field F is called 2 -r e gular if the 2- Sy low subgro up of K 2 ( O F ) is tr iv ial. The Gaussian num b ers Q ( √ − 1) is a n example of such a nu mber field. F o r the ´ etale c o homology of O F [1 / 2] the 2-r egular assumption im- plies that H 2 ´ et ( O F [1 / 2] , Z ⊗ i 2 ) is triv ial for i 6 = 0 , 1 [48, Prop osition 2 .2]. Moreov er, H 1 ´ et ( O F [1 / 2] , Z ⊗ i 2 ) identifies with Z c 2 ⊕ Z /w i ( F ) for i 6 = 0, a nd H 1 ´ et ( O F [1 / 2] , µ ⊗ i 2 ) ∼ = ( Z / 2) c +1 where c denotes the num b er of pair s of complex embeddings of the num- ber field F . By way of example, the num b er w i ( Q ( √ − 1)) = 2 2+( i ) 2 for all i , wher e ( i ) 2 is the 2 - adic v aluation of i . With these preliminar ies in hand, w e are r eady to state the following computa- tion. Example 7.5. Let F be a totally imag inary 2-regula r num b er field with c pair s of complex embeddings. The 2-co mpleted ´ etale hermitian K -groups of its ring of 2-integers O F [1 / 2] a re computed up to extensions in the following ta ble. Remark 7.6. W e exp ect that 1 K Q ´ et 8 k +1 ( O F [1 / 2]) # and − 1 K Q ´ et 8 k +5 ( O F [1 / 2]) # are elementary ab elian 2-g roups o f ra nk equal to c + 2 . In the following discussio n of ´ etale V - theory we shall sp ecialize Lemma 6.16 to the previous examples of finite fields, lo ca l fields and tota lly imagina ry 2-r e gular nu mber fields. As for hermitian ´ etale K -theory , it turns out that the ´ etale V -g roups of dyadic and non-dyadic lo cal n umber fields are co mpletely differen t; although in some deg rees we ar e only able to compute these gr oups up to extensions, we ca n conclude that the former allow free summands in some degr ees while the latter are alwa ys finite ab elia n gr oups. 42 BK O MA Y 26, 2022 n mo d 8 1 K Q ´ et n ( O F [1 / 2]) # − 1 K Q ´ et n ( O F [1 / 2]) # 8 k > 0 ( Z / 2 ) c +1 0 8 k + 1 ( Z / 2) c +1 • Z / 2 0 8 k + 2 Z / 2 0 8 k + 3 Z c 2 ⊕ Z /w 4 k +2 ( F ) Z c 2 ⊕ Z /w 4 k +2 ( F ) 8 k + 4 0 ( Z / 2) c +1 8 k + 5 0 ( Z / 2) c +1 • Z / 2 8 k + 6 0 Z / 2 8 k + 7 Z c 2 ⊕ Z /w 4 k +4 ( F ) Z c 2 ⊕ Z /w 4 k +4 ( F ) Our computation of the 2-completed ´ etale V -gr oups of finite fields is in agreement with Hiller’s r esults for V - g roups in [2 2]. Example 7. 7. L e t F t be a finite field with an o dd num b er of elements t . The 2-completed ´ etale V - groups of F t are computed in the following table. n mo d 8 1 V ´ et n ( F t ) # − 1 V ´ et n ( F t ) # 8 k ≥ 0 Z /w 4 k +1 ( F t ) Z /w 4 k +1 ( F t ) 8 k + 1 Z / 2 0 8 k + 2 ( Z / 2) 2 0 8 k + 3 Z / 2 0 8 k + 4 Z /w 4 k +3 ( F t ) Z /w 4 k +3 ( F t ) 8 k + 5 0 Z / 2 8 k + 6 0 ( Z / 2) 2 8 k + 7 0 Z / 2 The extension pr o blem in degree 8 k + 2 c a n b e resolved using that 1 K Q 8 k +2 ( F t ) # has o r der 2 and is a direct summand of 1 V 8 k +2 ( F t ) # . Likewise, this also res olves the extensio n problem in degr ee 8 k + 6 for ε = − 1. Next w e turn to lo cal n um b er fields. W e find it co nv enient to distinguish betw een dyadic and non-dy adic lo cal fields. Example 7.8. The 2-c o mpleted ´ etale V - groups of a dyadic lo cal num be r field F of degree d are computed up to extensions in the following table. n mo d 8 1 V ´ et n ( F ) # − 1 V ´ et n ( F ) # 8 k ≥ 0 Z / 2 • ( Z d 2 ⊕ Z / 2) Z d 2 ⊕ Z / 2 8 k + 1 Z / 2 • ( Z / 2 ) d +2 0 8 k + 2 ( Z / 2) d +2 • Z / 2 0 8 k + 3 Z /w 4 k +2 ( F ) • Z / 2 Z /w 4 k +2 ( F ) 8 k + 4 Z d 2 ⊕ Z / 2 Z / 2 • ( Z d 2 ⊕ Z / 2) 8 k + 5 0 Z / 2 • ( Z / 2) d +2 8 k + 6 0 ( Z / 2) d +2 • Z / 2 8 k + 7 Z /w 4 k +4 ( F ) Z /w 4 k +4 ( F ) • Z / 2 If i is even, the n umber w i ( Q 2 ) = 2 2+( i ) 2 . F or non-dyadic lo cal num b er fields the ´ eta le V -g roups turn out to be torsion ab elian gro ups. Hermitian p eri od icity May 26, 2022 43 Example 7.9. The 2- completed ´ etale V -groups of a non-dyadic lo ca l n um b er field F a re computed up to extensions in the following ta ble. n mo d 8 1 V ´ et n ( F ) # − 1 V ´ et n ( F ) # 8 k ≥ 0 Z / 2 • Z / 2 Z / 2 8 k + 1 Z / 2 • ( Z / 2 ) 2 0 8 k + 2 ( Z / 2) 2 • Z / 2 0 8 k + 3 Z /w 4 k +2 ( F ) • Z / 2 Z /w 4 k +2 ( F ) 8 k + 4 Z / 2 Z / 2 • Z / 2 8 k + 5 0 Z / 2 • ( Z / 2) 2 8 k + 6 0 ( Z / 2) 2 • Z / 2 8 k + 7 Z /w 4 k +4 ( F ) Z /w 4 k +4 ( F ) • Z / 2 Our las t example concerning ´ etale V -theo r y deals with totally imaginary 2- regular num b er fields. W e refer to the discussion prior to Example 7.5 for some of the salient features of these n umber fields. Example 7.10. Le t F b e a totally imaginary 2-r egular num b er field with c pairs of complex embeddings. The 2-co mpleted ´ etale V -g roups o f its r ing of 2-integers O F [1 / 2] are computed up to extensio ns in the following table. n mo d 8 1 V ´ et n ( O F [1 / 2]) # − 1 V ´ et n ( O F [1 / 2]) # 8 k ≥ 0 Z c 2 ⊕ Z /w 4 k +1 ( F ) Z c 2 ⊕ Z /w 4 k +1 ( F ) 8 k + 1 ( Z / 2) c +1 0 8 k + 2 ( Z / 2) c +1 • Z / 2 0 8 k + 3 Z / 2 0 8 k + 4 Z c 2 ⊕ Z /w 4 k +3 ( F ) Z c 2 ⊕ Z /w 4 k +3 ( F ) 8 k + 5 0 ( Z / 2) c +1 8 k + 6 0 ( Z / 2) c +1 • Z / 2 8 k + 7 0 Z / 2 Remark 7. 11. W e exp ect that 1 V ´ et 8 k +2 ( O F [1 / 2]) # and − 1 V ´ et 8 k +6 ( O F [1 / 2]) # are elementary ab elian 2 -gro ups of r ank equal to c + 2. The next examples concer n self-co njugate algebr aic K -theory . Example 7. 1 2. The 2-c o mpleted K S C -groups of a finite field F t of o dd charac- teristic are giv en in the following table. n mo d 4 K S C n ( F t ) # 4 k > 0 Z / 2 4 k + 1 Z / 2 4 k + 2 Z /w 2 k +2 ( F t ) 4 k + 3 Z /w 2 k +2 ( F t ) Recall that τ de no tes the dua lity functor in algebraic K -theory . The map π n (1 − τ ) : K n ( F t ) # − → K n ( F t ) # is multiplication by 2 if n ≡ 1 (mod 4) and triv ial otherwise. 44 BK O MA Y 26, 2022 Example 7.13. Le t F b e a totally imaginary 2-r egular num b er field with c pairs of complex embeddings. The 2-c ompleted K S C -groups of its ring of 2 - int egers O F [1 / 2] are giv en in the following table. n mo d 4 K S C n ( O F [1 / 2]) # 4 k > 0 ( Z / 2) c +1 4 k + 1 Z / 2 4 k + 2 Z c 2 ⊕ Z /w 2 k +2 ( F ) 4 k + 3 Z c 2 ⊕ Z /w 2 k +2 ( F ) The map π n (1 − τ ) : K n ( O F [1 / 2]) # − → K n ( O F [1 / 2]) # is multiplication by 2 if n ≡ 1 (mo d 4 ) and trivial otherwise. There is an exact sequence (with A = O F [1 / 2]) 0 → K S C 2 n +1 ( A ) # → K 2 n +1 ( A ) # → K 2 n +1 ( A ) # → K S C 2 n ( A ) # → 0 . In the examples a bove, the assertio ns concerning π n (1 − τ ) follow by inspec tio n, using the c o mputations of K n ( F t ) # [47] and K n ( O F [1 / 2]) # [48, Theor em 3 .1]. A systema tic approach to the K S C -computations is to first compute the descent sp ectral sequence for ´ etale K S C -theory o btained fro m Corolla r y 6.10, and then inv oke Theorem 6.2. In gener al, if cd 2 ( S ) < ∞ , this a pproach gives “in sufficiently high degrees ” a strongly co nvergen t co ho mologica l sp ectral seq uence 1 E p,q 2 =        H p ´ et ( S, Z ⊗ 2 k 2 ) q = 4 k H p ´ et ( S, µ ⊗ 2 k +1 2 ) q = 4 k + 1 H p ´ et ( S, Z ⊗ 2 k +2 2 ) q = 4 k + 3 0 q = 4 k + 2        “ = ⇒ ” K S C q − p ( S ) # . 8. Applica tio ns to group rings, complex v arieties and co mmut a tive Banach algebras Let G b e a finite group and R b e the ring of S -integers in a num b er field. Let m b e a prime p ow er which is prime to the order of G . In [66], W eib el pr ov es a per io dicity theo rem for the higher algebr aic K -theory o f the gro up ring A = R [ G ], with co efficients in Z /m . More precis e ly , the cup-pr o duct with the B ott element in K -theory induces an isomorphism K i ( A ; Z /m ) ∼ = K i + p ( A ; Z /m ) for i > 0. Here, the in tegers m and p are linked ac cording to our Conv ention 0.1. W e c an a pply our p erio dicity theo r ems of Section 4 in order to show that, fo r 1 / 2 ∈ A and any inv olutio n o n A , for instance the one induced by g 7→ g − 1 , we hav e a split short exact se quence for m a p ow er of 2 and i > q − 1 . 0 − → lim ← − ε K Q i + ps ( A ; Z /m ) θ − n − → ε K Q i ( A ; Z /m ) θ + n − → lim − → ε K Q i + ps ( A ; Z /m ) − → 0 . Here the num be r q is g iven by our conv ention 0.2. W e note that W eibe l’s theor em is also tr ue for i ≥ 0 if we r eplace the n umber ring R by a lo ca l field. In par ticula r, the K Q -g roups are also p erio dic, i.e . ε K Q i ( A ; Z /m ) ∼ = ε K Q i + p ( A ; Z /m ) Hermitian p eri od icity May 26, 2022 45 at least for i > q − 1 . In fact, a more careful ana lysis forces us to distinguish tw o cases accor ding to the par it y of m . If m is even, the condition that the o rder to G is prime to m implies that G is of o dd order. Acco rding to the famous theorem o f F e it and Thompson [17], this implies that G is solv a ble . Therefore, for 2 -primary co efficients, we hav e a per io dicity statement only for a sp ecial clas s of solv able groups. If m is o dd, w e a lready know, without the hypo thes is 1 / 2 ∈ A , that the gr o up ε K Q i ( A ; Z /m ) splits into the direct sum ε K Q i ( A ; Z /m ) ∼ = ε K Q i ( A ; Z /m ) + ⊕ ε K Q i ( A ; Z /m ) − . In this direct sum deco mpo sition, the g r oup ε K Q i ( A ; Z /m ) − is the higher Witt group with Z /m co efficients and we hav e the per io dicity iso mo rphism ε K Q i ( A ; Z /m ) − ∼ = − e K Q i +2 ( A ; Z /m ) − for all v alues i ∈ Z . On the other hand, the gro up ε K Q i ( A ; Z /m ) + may be iden- tified with K i ( A ; Z /m ) + , the symmetric par t o f K i ( A ; Z /m ) with res pe c t to the inv olution given b y the duality . Since this inv olution is compatible with the B o tt map, W eib el’s theorem [66] implies another p erio dicity isomorphism ε K Q i ( A ; Z /m ) + ∼ = ε K Q i + p ( A ; Z /m ) + but only if i > 0 . Summarizing, we hav e pr ov ed the fo llowing theorem. Theorem 8.1 . L et G b e a finite gr oup and let R b e the ring of S -inte gers in a numb er field. If m is a 2 -p ower and if G is of o dd or der, then we have a p erio dicity isomorphi sm ε K Q i ( R [ G ] ; Z /m ) ∼ = ε K Q i + p ( R [ G ] ; Z /m ) if 1 / 2 ∈ R and if i > q − 1 . On the other hand, if m is an o dd prime p ower and if G is an arbitr ary fi nite gr oup whose or der is prime to m , we have the same p erio dicity isomorphi sm, with only the r estriction that i > 0 . Remark 8.2. As in Section 6, we may co njecture that, in the case whe r e m is a 2-p ow er , the inverse limit lim ← − ε K Q i + ps ( A ; Z /m ) is r educed to 0. In other words, we conjecture that the ring A is hermitian re gular according to Definition 0.5. This will imply that the p o sitive Bott map ε K Q i ( R [ G ] ; Z /m ) − → ε K Q i + p ( R [ G ] ; Z /m ) is an isomo rphism for i > 0 according to Theorem 4 .5. Let us now turn our a tten tion to a smo oth complex v ariety S of dimens io n n . As we briefly ment ioned in Section 6, the ´ etale dimension of S is 2 n . As a consequence of Artin-Gr othendieck theory , it is well-known that the B etti c o homology of S with co efficients Z /m is iso morphic to the mo d m ´ etale cohomolo g y . The same re s ult is v alid for any cohomolog y theo r y by the metho d initiated by Dwyer and F r iedlander [15]. F or instance, the mo d m ´ etale K -theory of S coincides with the mo d m complex top olog ic al K -theory o f Atiy ah a nd Hirzebruch. By the sa me argument, the mod m ´ etale 1 K Q -theory coincides with the mo d m K -theor y of complex vector bundles provided with a nondegenerate s ymmetric bilinear form. This theory is w ell understo o d and is detailed for instance in App endix B to [6]: it is the us ua l mo d m top olo gical rea l K -theory . In the same wa y , the mo d m ´ etale − 1 K Q -theory 46 BK O MA Y 26, 2022 coincides with the mo d m K - theory o f complex vector bundles provided with a nondegenera te a n tisymmetric bilinea r form . This theor y is also well under sto o d: it is the usual mo d m topo logical symplectic K -theory . In bo th cases , we shall write ε K Q top n ( S ), with ε = 1 , − 1 if we consider symmetric or antisymmetric bilinear forms resp ectively . Theorem 8 .3. Le t S b e a smo oth c omplex variety of dimension n . Then the mo d m ´ etale ε K Q -the ory of S c oincides with the mo d m top olo gic al K -the ory of its c omplex p oints, r e al or symple ctic ac c or ding to ε . Mor e over, the c anonic al map ε K Q i ( S ) − → ε K Q ´ et i ( S ) ∼ = ε K Q top i ( S ( C )) is split surje ctive 7 when i ≥ 2 n + q − 1 . Mor e over, for o dd prime p ower c o efficients, it is an isomorphism for i ≥ 2 n − 1 , with an identific ation of ε K Q top i ( S ) with K top i ( S ) + , the symmetric p art of K top i ( S ) with r esp e ct t o t he involution induc e d by the duality functor. Pro of. This theo rem is mostly a conse q uence of the general results in Section 6. What rema ins to b e shown is that ε K Q i ( S ) − is zero for o dd prime p ow er co efficients: this is a conse q uence of the fact that − 1 has a squar e ro ot in C . Therefore, the classical Witt gr oup and also the higher Witt groups hav e only 2-torsio n. ✷ Let us now consider a rea l or c o mplex commutative Bana ch alg ebra A . It is a theorem of Fisher [18] and Prasolov [46] that the natural map K alg i ( A ; Z / m ) − → K top i ( A ; Z / m ) is an iso morphism for i ≥ 1 . In par ticular, the g r oups K alg i ( A ; Z / m ) are p erio dic of p erio d 2 if A is complex and of p erio d 8 if A is real. In this context, it is natural to state the follo wing conjecture. Conjecture 8.4. L et A b e a r e al or c omplex c ommutative Banach algebr a with involution. Then the natur al map ε K Q alg i ( A ; Z / m ) − → ε K Q top i ( A ; Z / m ) is an isomorph ism for i ≥ 1 . Applying the general arguments in this pa per , we can prove a theor em that would also b e a cons e q uence of this conjecture, namely the p erio dicity of the groups ε K Q alg i ( A ; Z / m ), which we simply write ε K Q i ( A ; Z / m ). More pr ecisely , the theo- rem of Fisher and P rasolov implies tha t the Bott map K i ( A ; Z / m ) → K i + p ( A ; Z / m ) is an iso mo rphism for i ≥ 1 , with m and p b eing r elated by our Conv ention 0.1. F rom Theorem 0.13, we therefore deduce the following p er io dicit y pattern for the groups ε K Q i ( A ; Z / m ). Theorem 8.5. L et A b e a re al or c omplex c ommutative Banach algeb r a with invo- lution. Then we have an isomorph ism ε K Q i ( A ; Z / m ) ∼ = ε K Q i + p ( A ; Z / m ) for i ≥ q , wher e m , p and q ar e 2 -p owers r elate d by our Conventions 0.1 and 0.2. ✷ 7 In [ 7], we show that in fact this canonical map is an is omor phism when i ≥ 2 n − 1. Hermitian p eri od icity May 26, 2022 47 As a matter of fact, if m is an odd prime p ow er, we can prove a muc h b etter result. F or, w e know alr eady by o ur genera l theor y that the subgroup ε K Q i ( A ; Z / m ) − is per io dic of p erio d 4 for all v alue s o f i . Moreov er, ε K Q i ( A ; Z / m ) + is isomorphic to K i ( A ; Z / m ) + , the symmetric part o f K - theory which (as a dir ect conse q uence of the theorem of Fisher and Praso lov) is p er io dic of p erio d 4 if A is co mplex or real. Summar izing, we g et the following more precise theor em for m a n odd prime power. Theorem 8.6. L et A b e a re al or c omplex c ommutative Banach algeb r a with invo- lution and let m b e an o dd prime p ower. Then, for i ≥ 1 , we have an isomorphism given by the cu p-pr o duct with a Bott element ε K Q i ( A ; Z / m ) ∼ = − → ε K Q i +4 ( A ; Z / m ) . References [1] J. F. Adams. On the group J ( X ) IV. T op ology 5 : 21 − 71 , 1966 . [2] D. W. Anderson. 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