. The Topological Hochschild Homology of the Gaussian Integers Ayelet Lindenstrauss Department of Mathematics The Technion Haifa 32000, Israel x0. Introduction The topological Hochschild homology THH of an associative ring R with uni* *t was introduced by B"okstedt in [2]. It is a `fattened' spectrum version of Hochschi* *ld homology, which was constructed in order to give the Dennis trace map Kn(R) ! HH n(R) a b* *etter target, from which more information about K*(R) could be pulled back. Goodwilli* *e con- jectured the existence of THH before it was defined on the basis of his calcu* *lus of functors, and he conjectured that the stable homotopy groups should be isomorphic to the * *stable K-theory of R. After B"okstedt's definition of THH , Dundas and McCarthy conf* *irmed the isomorphism of these invariants in [8]. Pirashvili and Waldhausen have furt* *her shown that these stable homotopy groups are isomorphic to the MacLane homology groups* * of R ([17]). Topological Hochschild homology can also be defined for more general spectr* *a than those arising from discrete rings ([15], [9]). In the case of Eilenberg-MacLan* *e spectra of discrete rings, however, B"okstedt's functor with smash product (FSP) setup * *is very concrete, and convenient to work with. In [3], B"okstedt calculated THH (Z) and THH (Z=pZ) for p prime. Both o* *f these proved much more interesting, and in the case of Z much more suggestive of tors* *ion structure of K*(Z), than the respective Hochschild homology groups (which vanis* *h in positive dimensions). Results by Hesselholt and Madsen in [11] (theorem 6.1), * *and the results of [13] extend the range of rings for which THH is known- in fact, b* *oth are special cases of a splitting phenomenon which occurs in THH (R) for R an A-al* *gebra spanned over A by linearly independent elements x1; : :x:nsuch that the set {0;* * x1; : :x:n} is closed under multiplication. For such R, one can see by a trick of Waldhause* *n or by direct construction that THH (R) is the smash product of THH (A) with a complex cons* *tructed from the xi's (if R = A[G], this complex is exactly ^BG ). It can be seen from the splitting above that in the case of group rings R =* * Z[G], a copy of THH (Z) sits inside THH (R) as a direct summand- from functoriality, * *it is clear that this should be the case simply because R is in this case augmented over Z.* * It was not, however, clear that in the case of unaugmented rings (such as number rings* *) there would not be similar behavior. B"okstedt constructs a spectral sequence for c* *alculating HS*(THH (R); Z=pZ) for R which is additively free over Z. The E2-term of this* * spectral sequence splits (0:0:1) E2*;*(THH (R); Z=pZ) ~=E2*;*(THH (Z); Z=pZ) HH *(R; Z=pZ) and there was no known case where this splitting did not last to the E1 term. * *For those rings where the splitting does last to the E1 term, topological Hochschild hom* *ology is 1 exactly as fine, as an invariant, as the Hochchild homology groups (taken over * *all choices of coefficients). This paper shows that such a splitting at the E1 level does not occur for * *p = 2 when R is the ring of integers in a quadratic extension of the rationals which ramif* *ies at 2. In fact, in this case the spectral sequence differentials on (0.0.1) are as non-* *trivial as they could possibly be, given the constraint that once we reduce modulo 2 we know by* * [13] that E1*;*(THH (R Z=2Z); Z=2Z) ~=E1*;*(THH (Z=2Z); Z=2Z) HH *(R; Z=2Z): A complete calculation is given for the 2-torsion in THH (R) for R the rin* *g of integers in any quadratic extension of the rationals (see theorem (2.10)).p_In_thepcase_* *of_rings of integers which ramify only at 2, the Gaussian integers, Z[ 2], and Z[ -2 ], t* *his concludes the calculation of THH , since for p 6= 2 the splitting in (0.0.1) must nece* *ssarily hold to the E1 term (the HH *(R; Z=pZ) sitting in the 0'th row of the spectral sequenc* *e vanishes in positive dimensions) and so the p-torsion can be read off directly. p __* *_ p ____ The basic result in the cases which ramify only at 2 (i.e. R = Z[i], Z[ 2]* *, Z[ -2 ]) is Theorem (3.3): 1Y0 THH (R) ' K(Z; 0)2 x (K(Z=2jZ; 2j - 1) x K(Z=4jZ; 2j - 1)): j=1 B"okstedt's calculation in [3] gives 1Y0 THH (Z) ' K(Z; 0) x K(Z=jZ; 2j - 1); j=0 and it turns out that the standard inclusion Z ,! R induces an isomorphism into* * one copy of Z on the 0-dimensional stable homotopy and multiplication by 2 from Z=j* *Z into Z=2jZ in higher dimensions. The homotopy groups in (3.3) are much smaller tha* *n the analogous ones for, say, the group ring R = Z[x]=(x2-1), though the Hochschild * *homology groups, with any choice of coefficients, coincide. Thus topological Hochschild * *homology is a fundamentally deeper invariant than Hochschild homology, not only in the base* * cases of Z and Z=pZ, but also when studying extensions. It is important to note that the stable homology calculations of THH (R) w* *ith co- efficients in Z=2Z for the rings R discussed here coincide with what we would g* *et for THH (Rb2). Thus the stable homotopy groups with coefficients in Z=2Z of the t* *wo spectra conicide. Knowing the latter spectrum is interesting in light of the result of* * Hesselholt and Madsen in [11], which tells us that for completions of number rings at a pr* *ime p, the topological cyclic homology spectrum and the algebraic K-theory spectrum become* * the same after completion at p. Topological cyclic homology, introduced by B"oksted* *t, Hsiang, and Madsen in [4], is a suitably defined fixedpoint set of the circle action on* * topological Hochschild homology. And thus, the calculation of the topological Hochschild ho* *mology of the extensions discussed in this paper is a step towards understanding the a* *lgebraic K-theory of their completions, in particular at the prime 2. 2 The first section of this paper describes the setup for the calculation. I* *n the sec- ond section, the actual calculation of the 2-torsion in topological Hochschild * *homology of quadratic number rings is described, modulo some technical lemmas which are pro* *ved in the fourth section. The third section uses the results of the second to work ou* *t explicitly the topological Hochschild homology of quadratic number rings which ramify only* * at 2. I would like to thank Wu-Chung Hsiang, and also Ib Madsen, for suggesting t* *his problem to me. I would also like to thank Eli Aljadeff, Michael Larsen, and Joh* *n Rognes for useful conversations I have had with them while working on it. x1. The Setup for the Calculation We briefly recall B"okstedt's setup from [3], with the indexing of section * *1.7 in [11] and the notations of [13]. Let R be an associative ring with unit. For a finite sim* *plicial object X:, we will use the notation R[|X:|] to denote the geometric realization of the* * simplicial object given in each degree by the free module RXn = R[Xn]=R*, with * the degen* *erate simplex at the basepoint. Let 1|X:|: |X:| ! R[|X:|] be the map induced by ff 7!* * 1 . ff, and for any two such objects |X:| and |Y:|, let |X:|;|Y:|: R [|X:|]^ R [|Y:|]! R [|X:| ^ |Y:|] be the map induced by R's multiplication. For m 0, we set (1:0:1) Tr(m)= hocolim (Sj0^ Sj1^ . .^.Sjr; R[Sj0] ^ R[Sj1] ^ . .^.R[Sjr] ^* * Sm ); (j0;j1;:::;jr)2(Im )r+1 where I is the category of integers n = {1; 2; : :;:n}, for n 0, and injective* * maps between them, and Im is the m-fold cartesian product of I with itself. S(;;;;:::;;)is * *defined to be S0, and for an element j = (n1; n2; : :n:m) 2 Im we define Sj as a smash product o* *f spheres S1 indexed by the elements of all the non-empty sets ni in j. Thus topologicall* *y each Sj is a sphere whose dimension is the sum of cardinalities of all the coordinates * *of j, and each R[Sj] is the Eilenberg-MacLane space of R (as an additive group) in that d* *imension. We will use integers n to denote objects of the form (n; ;; ;; : :;:;) 2 Im . T* *opologically, for m > 0 Tr(m)is homotopy equivalent to the space we would have gotten had we * *taken the homotopy colimit to run over I alone; the reason for the more complicated c* *ategory is to facilitate the definition of a product the topological Hochschild homolog* *y spectrum. For m = 0, the definition makes sense if we interpret I0 as the category consis* *ting of the empty set; this will be convenient for the simplicial calculations in the last * *section of this paper. We define cyclic simplicial boundary maps di : Tr(m)! Tr(m-1)for 0 i r wh* *ich `multiply' the ith and (i + 1)st(rth and 0th, in the case i = r) coordinates: w* *e identify Sji^ Sji+1with Sji+ji+1, and compose the maps in Tr(m)with Sji;Sji+1: R[Sji] ^ R[Sji+1] ! R[Sji^ Sji+1] = R[Sji+ji+1] 3 in the i'th coordinate. When i = r we do this treatment in the 0'th coordinate* *, using Sjr;Sj0. The + operation on Im used here is concatenation in each coordinate, + : ((n1; n2; : :;:nm ); (n01; n02; : :;:n0m)) 7! (n1 + n01; n2 + n02; :* * :;:nm + n0m): Degeneracy maps si : Tr(m)! Tr(m+1)for 0 i r are defined by inserting the maps 1S(;;;;:::;;): S(;;;;:::;;)! R[S(;;;;:::;;)] after the i'th coordinate. For each r, spectrum structure maps Tr(m)! Tr(m+1)are defined as in [13], e* *xcept that here we also use the embedding Im ,! Im+1 given by (n1; : :;:nm ) 7! (n1* *; : :;:nm ; ;) to induce a map of the homotopy colimits. These structure maps make {m 7! Tr(m)* *} for m 1 into an -spectrum. For each m, {r 7! Tr(m)} is a simplicial space whose realization is called * *THH (m)(R). Since the simplicial structure maps commute with the spectrum structure maps, t* *he lat- ter induce spectrum structure maps on the realizations, and the topological Hoc* *hschild homology spectrum THH (R) is the resulting spectrum {m 7! THH (m)(R)}. Note * *that B"okstedt's original indexing convention, with the category I as the fixed limi* *t category in (1.0.1), gives a very different THH (0)(R) than the one we get here (specifical* *ly, it gives the loopspace of THH (1)(R)); when looking at the spectrum THH (R) with our notati* *on, it is probably best to avoid the 0-space, and it will be used here only to simplify n* *otation in calculations. The filtration by simplicial `strata' on each THH (m)gives a filtration of* * the singu- lar chains on it, which can be used for a spectral sequence calculating its hom* *ology with coefficients in Z=pZ. For each THH (m)the E1 term of this spectral sequence w* *ill have E1r;*~=H*(Tr(m); Z=pZ). The spectrum suspension maps respect this filtration, a* *nd there- fore they induce maps of the associated spectral sequences. On the E1-term, the* * spectrum suspension maps send E1r;s(THH (m)(R); Z=pZ) ! E1r;s+1(THH (m+1)(R); Z=pZ); and for any fixed r and s these maps stabilize eventually, to become isomorphis* *ms from some point on. Let K(R; n) denote the Eilenberg-MacLane spectrum {m 7! K(R; n+m* *)}, where K(R; n) is the Eilenberg-MacLane space corresponding to R's additive grou* *p (of which R[Sn] is a model). Then as spectra, {m 7! Tr(m)} ' K(R; 0)^(r+1), and so mlim!1E1r;*+m(THH (m)(R); Z=pZ) ~=HS*({m 7! Tr(m)}; Z=2Z) ~=HS*(K(R; 0); Z=pZ* *)r+1 : The dimension-shifting maps on the spectral sequences above commute with bounda* *ry maps, and therefore they commute with the spectral sequence differentials. Beca* *use of this, the spectrum suspensions induce (dimension shifting) homomorphisms on the Ek te* *rms of the spectral sequence for all k > 1. So we can work with the limit spectral seq* *uence, which calculates HS*(THH (R); Z=pZ). The spectral sequence differential calculatio* *ns can be carried on in any THH (m) in which the class we want to work with is already re* *presented, and we will use this often. 4 The E1-term of the limit spectral sequence is E1r;*~=HS*(K(R; 0); Z=pZ)r+1 The spectral sequence d1 is exactly the Hochschild homology differential (for g* *raded alge- bras), and so we also have E2r;*~=HHr(HS*(K(R; 0); Z=pZ)) (in the graded sense). B"okstedt showed in [3] that this spectral sequence co* *llapses at Ep for R = Z and R = Z=pZ. We note that if R is additively a free Z- or Z=pZ-module, then HS*(K(R; 0); Z=pZ) ~=HS*(K(Z; 0); Z=pZ) R or HS*(K(R; 0); Z=pZ) ~=HS*(K(Z=pZ; 0); Z=pZ) R; respectively, and so, by a K"unneth formula argument using shuffle products wit* *h signs adjusted to respect the graded structure, (1:0:2) E2*;*~=HH*(HS*(K(R; 0); Z=pZ)) ~=HH *(HS*(K(Z; 0); Z=pZ)) HH *(R; Z=pZ) or (1:0:3) E2*;*~=HH*(HS*(K(R; 0); Z=pZ)) ~=HH *(HS*(K(Z=pZ; 0); Z=pZ)) HH *(R); respectively. The Hochschild homology factor in (1.0.2) and (1.0.3) is the one that l* *ives to the E1 term in the cases worked out in [11] and [13], as explained in the introdu* *ction, and we will see that this is not the case if R is the ring of integers in a quadrat* *ic extension of the rationals which ramifies at 2 and p = 2. x2. The 2-Torsion Calculation We now specialize our homology calculation to the case p = 2. By B"okstedt'* *s argument in [3] (see also [13]), the topological Hochschild homology spectrum of any ri* *ng which is additively finitely generated over Z is equivalent to the restricted product of* * Eilenberg- MacLane spectra corresponding to cyclic groups of various orders. These can be * *read off from the stable homology of THH (R)- the p-torsion in homotopy is, for each p,* * determined by the homology with mod p coefficients and the action of the Bockstein operato* *rs on it. Thus this section will effectively calculate the 2-torsion in THH (R) for R t* *he ring of integers in any quadratic extension of the rationals. There are two cases to consider- the first is the case where R comes from a* * quadratic extension that does not ramify at 2. But then, by (3.1), HH *(R) ~=0 for * > 0* * and we apply the easy lemma (3.2) to find that the 2-torsion in THH (R) consists of * *exactly two copies of that in THH (Z), namely two copies of K(Z=2aZ; 2m - 1) for each m = * *2ab, with b a positive odd integer. The case where there is ramification at 2 is more interesting. 5 Lemma (2.1) If R is the ring of integers in a quadratic extension of the ratio* *nals which ramifies at 2, there is an element i 2 R such that i and 1 span R linearly over* * Z, i2 is divisible by 2 in R, and i2=2 is invertible in bR2. Proof. Note that to satisfy the second and third requirements of the lemma, it is * *enough to know that the 2-adic valuation of the i we will pick is 1=2 (all elements of a * *fixed valuation are products by Rb2-units of each other, and for any element of R which is divi* *sible by 2 in bR2, the quotient of that element by 2 must be in R). So we start with an* *y "i2 R with valuation 1=2- these must exist since R is dense in the completion bR2, an* *d bR2is a principal ideal domain in which the ideal 2R, of elements whose valuation is 1* *, satisfies 2R = I2 for some ideal I. The valuation of "i+ 2a will of course be 1=2 as well* *, for any a 2 R, and the claim is now that we can pick an a so that i = "i+ 2a will satis* *fy the first condition. We know there is some x 2 R which, together with 1, spans R linearl* *y over Z and so we look at R=2R, which consists of the four classes 2R, 1 + 2R, x + 2R* *, and 1 + x + 2R. The elements of the first class have valuations 1, those of the se* *cond class valuations of 0. Thus "ibelongs to the third or fourth class, and so there som* *e element a 2 R for which "i+ 2a = x or "i+ 2a = 1 + x. * * tu We define (2:1:1) = i2=2; which exists in R by our choice of i. In the case of the Gaussian integers, we* * can for example think of i = 1 + i, and then = i. From [3] we know that B"okstedt's spectral sequences calculating HS*(THH * *(Z); Z=2Z) and HS*(THH (Z=2Z); Z=2Z) collapse at the E2 term- in the former case we have ___ i (2:1:2) HS*(K(Z; 0); Z=2Z) = A = Z=2Z[j; 2; 3; : :]:; degj = 2; degi = 2* * - 1 and so ___ ___ 2 (2:1:3) E1*;*= E2*;*= HH*(A ) ~=A [e3; e4; e8; : :]:=(ei) where e3 is represented by 1 j and e2iby 1 i. In the second case we have HS*(K(Z=2Z; 0); Z=2Z) = A = Z=2Z[1; 2; 3; : :]:; degi= 2i- 1 and so E1*;*= E2*;*= HH*(A) ~=A[1; 2; : :]:=(2i) where i is represented by 1 i. For calculations later in the paper, it will be important to specify our ex* *act choice of i and i . We do not take i to be the standard Milnor generators, but rather* * their images under the canonical anti-automorphism of the dual of the Steenrod algebr* *a, which are called Oi in the notation of [16] and [7]. The reduction map Z 7! Z=2Z ind* *uces an 6 ___ inclusion A ,! A, and since corollary 2.5 in chapter III of [7] gives us the i* *mage of this inclusion map explicitly, we take the j of (2.1.2) to be the pre-image of 1 2* *, and the i there (for i > 1) to be the pre-images of the respective i. Note that the above formulae refer to a multiplicative structure on B"okste* *dt's spectral sequence- this is in fact induced by a multiplicative structure (2:1:4) M : THH (m)(R) x THH (n)(R) ! THH (m+n)(R) defined in section 1.7 of [11], which induces the shuffle product on the Ek ter* *ms for k 1 (see [13]). For such k, we also know that this product commutes with the * *maps Ek*;*(THH (m)(R); Z=2Z) ! Ek*;*+`(THH (m+`)(R); Z=2Z) induced by the spectrum* * suspen- sion maps, and so when we want to calculate a product of two stable filtered ho* *mology classes we can pick representatives for them on any THH (m), THH (n) that we pl* *ease. By our choice of generators, the reduction map Z 7! Z=2Z induces a map send* *ing j 7! 21and i7! i. So on the E2 terms we get a map ___ 2 2 (2:1:5) A[e3; e4; e8; : :]:=(ei) ! A[1; 2; : :]:=(i) ___ with A ,! A, e3 7! 0, and e2i 7! i. The standard inclusion of Z=2Z in R Z=2Z* * is obtained from the standard inclusion of Z in R by tensoring mod 2, and thus the* * reduction map R ! R Z=2Z induces the map of (2.1.5) on the first factor of S(K(R; 0); Z=2Z)) ~=HH (___A) HH (R; Z=2Z)-red! (2:1:6) HH *(H* * * HH *(HS*(K(R Z=2Z; 0); Z=2Z)) ~=HH *(A) HH *(R Z=2Z): On the second factor, red induces the obvious isomorphism HH *(R; Z=2Z) ~= HH ** *(R Z=2Z). The right side of (2.1.6) is the E2 term of a spectral sequence which col* *lapses at E2 by [13]- in particular, we know that all the higher differentials of the spectr* *al sequence vanish on the HH *(R Z=2Z) factor. Thus when studying the spectral sequence differentials on the left side of * * (2.1.6), we know by naturality that their values on the HH *(R; Z=2Z) factor must be in the* * kernel of red, i.e. for k 2 ___ (2:1:7) im(dk|HH *(R;Z=2Z)) ker(red) = e3 . A[e4; e8; : :]: HH *(R; Z=2Z* *): We will now see that this inclusion is actually an equality, which means that t* *he spectral sequence differentials are as `destructive' as they can be within the restricti* *on of (2.1.7). If we study what is sitting in each coordinate of the spectral sequence, we wil* *l see that d2|HH *(R;Z=2Z)has nothing low enough to hit, given (2.1.7). d3|HH *(R;Z=2Z)a* *lready has something it could hit. We recall from formula (1.8.4) of [12] that the cycles * *1ii. .i. and i i i . .i.in HH k(R; Z=2Z) span it over Z=2Z (the crucial aspects to remem* *ber are that R = Z[i]=(i2 + a . i + b) for some choice of a; b 2 Z since i and 1 sp* *an R, and that i2 0 if we tensor the Hochschild homology complex with Z=2Z coefficients)* *. For these generators we have 7 Lemma (2.2) d3(1 i__i__._.-.iz_____") = e3 . (1 i__i__._.-.iz_____") k times k-4 times Since the spectral sequence differentials satisfy the Leibniz rule with res* *pect to the multiplication of the spectral sequence (this is simply because the differentia* *ls are the composition of the geometric boundary map with a quotient map, and the multipli* *cation is induced by a continuous multiplication map of spaces), they commute in parti* *cular with the HH 0(R; Z=2Z)-module structure, and so (2.2) implies immediately that d3(i__i__._.-.iz_____") = e3 . (i__i__._.-.iz_____"): k+1 times k-3 times Thus, ker d3|HH *(R;Z=2Z)= HH 0(R; Z=2Z) HH 1(R; Z=2Z) HH 2(R; Z=2Z) HH 3(R; Z=2Z* *): We recall the multiplicative structure of HH *(R; Z=2Z), as given by propositio* *n (1.15) in [12]: HH *(R; Z=2Z) ~=R=2R[ffl]=(ffl2) (a2); where (a2) is the divided power algebra spanned linearly over Z=2Z by elements * *a(i)2, i 0 (with a(0)2= 1 and a(1)2= a2) which multiply by the rule a(i)2. a(j)2= i+* *jia(i+j)2. Following the proof there we see that in our case 1 i represents the 1-dimensi* *onal class ffl, and 1 i i the 2-dimensional class a2 (and in general 1 i2k the 2k-dime* *nsional class a(k)2). Using this notation, and the Leibniz rule again, we write ___ 2 2 2 kerd3 = A R[ffl; a2; e3; e4; e8; : :]:=(ffl ; a2; ei) ___ (2) (4) * * 2 (i)2 2 e3 . (a(2)2; a(4)2; a(8)2; : :):. AR[ffl; a2; a2 ; a2 ; : :;:e3; e4; e8* *; : :]:=(ffl ; (a2 ) ; ei); ___ (2) (4) 2 (i)2 2 im d3 = e3 . A R[ffl; a2; a2 ; a2 ; : :;:e3; e4; e8; : :]:=(ffl ; (a2 ) ; e* *i) and so ___ E4*;*= kerd3__imd~=A R[ffl; a2; e4; e8; : :]:=(ffl2; a22; e2i* *): 3 Now in fact all the multiplicative generators_of_this last algebra have filtrat* *ion degree smaller than four: 0 in the case of A R, 1 in the case of ffl and the ei, and * *2 in the case of a2. We deduce that all the dk for k 4 must vanish, and so ___ 2 2 2 (2:2:1) E1*;*~=A R[ffl; a2; e4; e8; : :]:=(ffl ; a2; ei): Since we are working over a field, this is isomorphic_to_HS*(THH (R); Z=2Z) a* *s_far_as linear structure goes; they are also isomorphic as A modules, since multiplication by * *A involves only smashing the 00th coordinate with some K(Z; n) and adjusting dimensions- a* *n oper- ation which preserves all filtration degrees. We still need to study HS*(THH * *(R); Z=2Z)'s multiplicative structure in order to determine the action of the Bockstein oper* *ators on it, which is necessary for reading off the Eilenberg-MacLane spectra in THH (R). 8 Terminology (2.3) We will call a chain on some THH (m)(R) an unfiltered cycle* * if its boundary is zero in the singular chain complex of THH (m)(R) (as opposed to the* * associated filtered quotients). We will call a cycle with mod 2 coefficients absolute if i* *t is the reduction of an integral cycle. Claim (2.4) (Summary of facts from [3]) We can_choose_unfiltered_homology cla* *sses e4, e3 on THH (Z) such that HS*(THH (Z); Z=2Z) ~=A [e3; e4]=(e23), with the mod * *2 Bockstein operator sending fi1(e4) = e3. Moreover, we can pick them to be of a specific f* *orm which will be explained in section (4.3). Finally, reduction modulo 2, red : THH (R) ! TH* *H (R Z=2Z) satisfies red*(e4) = 21, where 1 is the unfiltered class B"okstedt chos* *e to represent 1 in the E1 term, for which HS*(THH (Z=2Z; Z=2Z) ~=A[1 ]. Since the inclusion homomorphism Z ,! R sends B"okstedt's E1 classes e2i (* *which are represented by combinations of his e4 and its powers ) into classes we have* * used for R with the same names, the image of his e4 (which we will call by the same name* *) in HS*(THH (R); Z=2Z) and its powers will span representatives of all the e2i in* * (2.2.1). Since reduction mod 2 maps e4to the class 21, which is algebraically_independen* *t_of A in HS*(THH (RZ=2Z); Z=2Z), e4is algebraically independent of A in HS*(THH (R);* * Z=2Z). We get that ___ S (2:4:1) A [e4] H* (THH (R); Z=2Z): The elements of R Z=2Z in E10;0can be represented by the 0-chains on THH (0)(* *R) corresponding to r : S0 7! R[S0] sending the non-basepoint to r times itself- t* *his really forms a 0-dimensional subcomplex closed under the multiplication M and so its h* *omology sits as_a_subalgebra of HS*(THH (R); Z=2Z); this subalgebra satisfies no nont* *rivial relations with A[e4] because no such relations hold in the E1 term. We can therefore say* *, in addition to (2.4.1), that ___ S (2:4:2) A R[e4] H* (THH (R); Z=2Z): Lemma (2.5) The class ffl in (2.2.1) can be represented by an unfiltered cla* *ss fflwhich is absolute, i.e. fik(ffl) = 0 for all the Bockstein operators fik, k 1, and * *which satisfies ffl2= 0. Lemma (2.6) The class a2 in (2.2.1) can be represented by an unfiltered clas* *s a2 with fi1(a2) = iffl(see (2.1)), for which e4= a22. Now we know by (2.5) that in addition to__(2.4.2), HS*(THH (R); Z=2Z) co* *ntains a class fflwhich is_linearly_independent_of A R[e4] by the E1 term and whose * *square__ is identified into A R[e4] as zero. And (2.6) tells us that beyond this copy* * of A ___ R[ffl; e4]=(ffl2) HS*(THH (R); Z=2Z), we have a class a2, linearly independe* *nt of A R[ffl; e4]=(ffl2) by the E1 term, whose square is identified to e4. The algebr* *a formed by all of these accounts for all of HS*(THH (R); Z=2Z), by dimension_counting_in the* * E1 term. Since a2 satisfies_no_nontrivial linear relation with A R[ffl; e4]=(ffl2)_and_* *e4is algebraically independent of A R[ffl]=(ffl2), a2 is algebraically independent of A R[ffl]=(* *ffl2). We thus deduce 9 ___ Corollary (2.7) HS*(THH (R); Z=2Z) ~=A R[ffl; a2]=(ffl2). * * ___ If we look at the result (2.7) to find generators of HS*(THH (R); Z=2Z) * *over A , we find two absolute classes, which can be taken to be and i, in dimension 0 (the* *se two are linearly independent since 's invertibility in R Z2 means that it is equal to * *1 or 1 + i). Alternatively, these can be taken to be 1 and i. In addition to this, for any 1* * ` we have the products of (a2)`and (a2)`-1fflwith the 0-dimensional generators. To determine the 2-torsion Eilenberg-MacLane spectra which appear in THH (* *R), we need to look at how the Bockstein operators act on these generators. We know fr* *om (2.6) that fi1(a2) = iffl fi1(ia2) = i2ffl= 2ffl 0: Moreover, the calculation used in (2.6)for showing this (formula (4.5.1)) works* * also with coefficeints in Z=8Z, from which we can deduce fi2(ia2) = ffl: Lemma (2.8) fi1(a22) = 0 but fi2(a22) = ia2ffl. As before, this implies that for i = 1; 2, fii(ia22) = 0, but fi3(ia22) = a* *2ffl. For higher powers, we have part (vi) of proposition 1.5 in [14], which tells us tha* *t for an even- dimensional mod 2 homology class y on an infinite loopspace, if fii(y) = 0 for * *1 i < k (with k > 1) we have fii(y2) = 0 for 1 i k and fik+1(y2) = yfik(y). This will inductivelyagive us the behavior of the higher Bockstein operators defined on t* *he classes a22, and from them we can as before deduce the behavior on the multiples of the* *se classes a+1* *b-1__2a by i. For powers (a2)`, with ` = 2ab and b odd, we observe that (a2)`= (a2)2 * * 2(a2) , and applying the Leibniz rule to products of a lifting of (a2)to a chain with a* *ppropriate coefficients, we get finally that ___ Claim (2.9) HS*(THH (R); Z=2Z) is generated over A by two 0-dimensional gene* *rators, in addition to generators (a2)`, i(a2)`, (a2)`-1ffl, and (a2)`-1fflfor any 1 `* *. If ` = 2ab for b odd, these are related by fi2a+1((a2)`) = i(a2)`-1ffland fi2a+2(i(a2)`) =* * (a2)`-1ffl. We get Theorem (2.10) Let R be the ring of integers in a quadratic extension of the r* *ationals which ramifies at 2; then THH (R) consists of Y1 0 K(Z; 0)2 x K(Z=2a+1Z; 2` - 1) x K(Z=2a+2Z; 2` - 1) `=1;`=2ab; (b;2)=1 and p-torsion spectra for p 6= 2. x3. Concluding the Calculation for R Which Ramifies Only at 2 We need two lemmas- 10 Lemma (3.1) Let R be the ring of integers in an extension of the rationals whic* *h does not ramify at a prime p. Then HH 0(R; Z=pZ) ~=R Z=pZ and HH *(R; Z=pZ) ~=0 for * >* * 0. Proof. The fact that HH 0(R; Z=pZ) ~=R Z=pZ follow from R's commutativity, since * *that implies that the first differential in the standard bar complex calculating Hoc* *hschild ho- mology is the zero map. For the second fact, we know by the flatness of bZpthat HH *(Rbp=bZp) ~=HH * **(R) bZp. Now by assumption, the quotient field of bRpis an unramified extension of bQp, * *and so by Lemma (1.4) in [12], HH *(Rbp=bZp) = 0 for * > 0. This tells us that HH *(R) co* *nsists, in positive dimensions, entirely of q-torsion for q 6= p, from which we get by the* * universal coefficient theorem (and the fact that HH 0(R) ~=R is torsion-free) that HH *(R* *; Z=pZ) ~=0 for * > 0. * * tu Lemma (3.2) If R is an extension of the integers which is additively a free Z-* *module of rank k, and if HH 0(R; Z=pZ) ~= R Z=pZ and HH *(R; Z=pZ) ~= 0 for * > 0, th* *en HS*(THH (R); Z=pZ) ~=HS*(THH (Z); Z=pZ)k as a ring with action of the Bocks* *tein op- erator and higher Bockstein operators when they are defined (and as a result of* * this, the p-torsion Eilenberg-MacLane spectra in the decomposition of THH (R) consists o* *f exactly k copies of what we get for THH (Z)). Proof. By (1.0.2), E2*;*(THH (R); Z=pZ)~=HH*(R; Z=pZ) E2*;*(THH (Z); Z=pZ) ~=(R Z=pZ) E2*;*(THH (Z); Z=pZ) and the copy of R Z=pZ sits in the coordinate (0; 0) where no nontrivial diffe* *rentials can map from it or into it. Therefore this R Z=pZ lives on to the E1 term. The sp* *litting E1*;*(THH (R); Z=pZ) ~=(R Z=pZ) E1*;*(THH (Z); Z=pZ) implies a similar sp* *litting of stable homology since it tells us that the subalgebra R Z=pZ satisfies no rela* *tions with the rest of the homology, beyond the obvious relations in a tensor product. Si* *nce the classes in this R Z=pZ can be represented by absolute cycles, multiplication b* *y them commutes with any Bockstein operators that are defined. * * tu In our case we know from [3] that the p-torsion in THH (Z) consists of a * *copy of K(Z=paZ; 2pab - 1) for each b coprime to p and a 1. Using the last lemma, and* * the result of (2.10), we conclude Theorem (3.3) Let R be the ring of integers in a quadratic extension of the rat* *ionals which ramifies only at 2; then 1Y0 THH (R) ' K(Z; 0)2 x (K(Z=2jZ; 2j - 1) x K(Z=4jZ; 2j - 1)): j=1 11 x4. Proofs of the Technical Lemmas The proofs of the technical lemmas are done by simplicial calculations. Si* *nce these are much easier to write when we look at the smash products R[Sj0] ^ R[Sj1] ^ .* * .^. R[Sjk] themselves, rather than on the looped-down versions, we will work on the* * delooped THH (m)'s for different values of m as necessary. As explained when setting up* * B"okstedt's spectral sequence, the spectrum suspension maps induce dimension-lifting homomo* *rphisms of the E1-terms of the respective spectral sequences, which commute with the sp* *ectral sequence differentials di, i 1. The maps induced by multiplication commute wi* *th the spectral sequence suspension maps too, since the multiplication itself homotopy* *-commutes with the spectrum suspension maps. So we can carry out our calculations of diff* *erentials and products on whichever THH (m)we please, as long as the classes we are inte* *rested in are defined on it. Since in our case we are interested in the Hochschild homolo* *gy classes in (1.0.2) which are already images of classes in the spectral sequence for THH * *(0)(R), this is not a problem. More specifically, elements representing Hochschild homology classes in HH * *q(R; Z=2Z) can be given as sums of monomials a0 a1 . . .aq with ai 2 R. In [13], it was sufficient to represent each such monomial by the q-simplex corresponding to th* *e map (S0)^(q+1) ! R[S0]^(q+1) which sends the non-basepoint to the point (a0; a1; : * *:;:aq), because if the monomials were chosen wisely, the sum of the q-simplices corresp* *onding to them was an unfiltered cycle. So we were looking at q-dimensional cycles on THH* * (R)(0) which came from 0-dimensional cycles on Tq(0). Here the situation is more comp* *licated: we will, in general, want to represent each a0 a1 . . .aq by the (q + 1)-dime* *nsional chain on R[S1]^(q+1)which will, in the notation defined below, be called a0__^ * *a1__^ . .^.aq__ (or sometimes also by chains which will be called a0^ a1__^ . .^.aq__in the not* *ation below, on R[S0] ^ R[S1]^q). These will not, in general, be unfiltered cycles, but we w* *ill explicitly complete them with lower filtration elements in order to calculate the spectral* * sequence differentials on them. In effect what we are doing is building a subcomplex of* * some THH (m)(R) where the homology class in question is represented in the E2 term,* * and doing the calculation on the spectral sequence associated to that subcomplex. The fundamental mechanism will be to identify r-dimensional chains on R[Sj0* *] ^ R[Sj1] ^ . .^.R[Sjk] as (r + k)-dimensional chains on THH (m)(R), where m is th* *e sum of cardinalities of all the coordinates of all the ja's. This will work by identif* *ying R[Sj0] ^ R[Sj1] ^ . .^.R[Sjk] into Tk(m)via OE:R[Sj0] ^. .^.R[Sjk]!(Sj0^. .^.Sjk; R[Sj0] ^. .^.R[Sjk] ^ Sj0^ . .^.S* *jk); (4:0:1) x7! 7! (x; ) 8 2 Sj0^. .^.Sjk : We are, of course, using the fact that Sj0^ . .^.Sjk is topologically an Sm . W* *e regard this sphere as a smash product of circles S1 indexed by the disjoint union of t* *he Sja. Once we have identified R[Sj0] ^ R[Sj1] ^ . .^.R[Sjk] into Tk(m)via OE, it * *is clear how any r-dimensional chain C on it corresponds to an (r + k)-dimensional chain OE** *(C) x k on THH (m)(R). For brevity's sake we will, however, omit writing OE from now on* *, and refer to this (r + k)-chain on THH (m)(R) as C x k. 12 Thus our indexing of Sm is an ad-hoc one created for a particular chain, bu* *t if we make the identification for a particular chain on Tk(m)we can make a compatible iden* *tification for the images of the chain under the simplicial space face maps. This Sm shoul* *d be viewed as a fixed suspension coordinate, enabling us to call the chains on the loopspa* *ces by more convenient names. The simplicial space boundary maps can, with this understanding, simply be * *viewed as applying the smash maps on the appropriate coordinates for chains on R[Sj0]^R[S* *j1]^. .^. R[Sjk]. When we look at the last simplicial boundary map, the wraparound term, * *there can be a correction of sign needed- for most of the calculations here, this does no* *t matter, since the coefficients are in Z=2Z, but since the suspension coordinate Sm is not inv* *olved and the chains discussed are images of chains from (Sj0; R[Sj0]) ^ (Sj1; R[Sj1]) ^ . .^* *.(Sjk; R[Sjk]) (appropriately suspended), the sign correction is the `stable dimension' of the* * chain in the last coordinate times the sum of the `stable dimensions' of the chains it is pu* *shed in front of. (4.1) Notations. We will abbreviate the notation for B"okstedt's basic FSP multiplication ma* *ps |X:|;|Y:|, and say that for any j; j0 2 Im , denotes Sj;Sj0: R[Sj] ^ R[Sj0] ! R[Sj+j0] an* *d also the map this induces on simplicial chains. Moreover, since the |X:|;|Y:|are strictl* *y associative, we will use also to denote iterated applications of from R[Sj0] ^ R[Sj1] ^ . * *.^.R[Sja] to R[Sj0+j1+...+ja] and the map they induce on simplicial chains. For any (r + k)-dimensional chain on THH (m)(R) which is of the form C x k * *with C an r-dimensional chain on Tk(m), its boundary has, by the Leibniz formula, two * *components; since we are working with coefficients in Z=2Z, signs are irrelevant and we wri* *te that the boundary @(C x k) = b(C x k) + d(C x k) with b(C x k) = (@C) xPk corresponding to the boundary map of simplicial chains* * on Tk(m), and d(C xk) = ki=0di(C)xk-1 corresponding to the simplicial space stru* *cture maps. For a 2 R, we let a denote the 0-dimensional chain 1 . (a . (the non-basep* *oint)) on R[S0] = R[S;;:::;;]. If we take a simplicial model of S1 with one non-degenerat* *e 1-simplex ff, we let a__denote the 1-dimensional chain 1 . aff on R[S1] = R[S(1;;;:::;;)]* *. Note that with this notation, we have for any a1; a2 2 R, (a1 ^ a2__) = (a1__^ a2) = a1a2__. * *We also set a____= (a__^ 1__) = (1__^ a__), and deduce (a1 ^ a2____) = (a1____^ a2) = a1a2_* *___for any a1; a2 2 R. More generally, for any two chains C1; C2 on some R[Sj1], R[Sj2] and any a * *2 R we have ((C1 ^ a) ^ C2) = (C1 ^ (a ^ C2)); and since R is commutative we also have (C ^ a) = (a ^ C) for any chain C on R[Sj]. 13 (4.2) Proof of Lemma (2.2). We represent 1 i__i__._.-.iz_____"by the chain 1__^ i__^ . .^.i__x k, as e* *xplained above, k times and proceed to trace its differentials. We have k-2X d(1__^ i__^kx k) = 1__^ i__^j^ 2____^ i__^k-2-jx k-1: j=0 We define (4:2:1) M1 = (s0(ff) + s1(ff)); a 2-simplex in the model of R[S1] defined from the simplicial structure we have* * chosen for S1. M1's boundary is precisely 1__+ 1__+ 2__. Letting k-2X L1 = 1__^ i__^j^ (M1 ^ __) ^ i__^k-2-jx k-1; j=0 we observe that b(L1) = d(1__^ i__^kx k); while i d(L1) = (1__^ M1 ^ __) ^ i__^k-2+ (M1 ^ ____) ^ i__^k-2 k-4XjX + 1__^ i__^`^ 2____^ i__^j-`^ (M1 ^ __) ^ i__^k-4-j j=0`=0 + 1__^ i__^`^ (M1 ^ __) ^ i__^j-`^ 2____^ i__^k-4-j k-3X + 1__^ i__^j^ (i__^ M1 ^ __) ^ i__^k-3-j j=0 j k-2 + 1__^ i__^j^ (M1 ^ i____) ^ i__^k-3-j x : We consider the 3-chain (1__^M1)+(M1^1__) on R[S2]. It is a cycle, even when we* * look at it as a chain with integral coefficients. But we know that H3(Z[S2]; Z) ~=H3(CP* *1 ; Z) ~=0, and of course our R is additively just a direct sum of copies of Z, so there ex* *ists a 4- dimensional chain M2 on R[S2] whose boundary is (1__^ M1) + (M1 ^ 1__). We red* *uce coefficients mod 2 and obtain a chain which we will also call M2. Setting i L2 = (M2 ^ __) ^ i__^k-2 k-4XjX + 1__^ i__^`^ (M1 ^ __) ^ i__^j-`^ (M1 ^ __) ^ i__^k-4-j j=0`=0 k-3X j + 1__^ i__^j^ (M2 ^ i__) ^ i__^k-3-jx k-2: j=0 14 We see that b(L2) = d(L1); in particular this shows that the spectral sequence differential d2[1__^i__^kxk* *] = [d(L1)] = 0 in the E2-term, a fact which we already know for dimension reasons and the R Z* *=2Z calculation in [13]. We let ik-6XXj X` L = 1__^ i__^m^ (M1 ^ __) ^ i__^`-m^ (M1 ^ __) ^ i__^j-`^ (M1 ^ __)* * ^ i__^k-6-j j=0 `=0m=0 k-5XjX + 1__^ i__^`^ (M2 ^ i__) ^ i__^j-`^ (M1 ^ __) ^ i__^k-5-j j=0`=0 k-5XjX + 1__^ i__^`^ (M1 ^ __) ^ i__^j-`^ (M2 ^ i__) ^ i__^k-5-j j=0`=0 k-4X j + (M2 ^ __) ^ i__^j^ (M1 ^ __) ^ i__^k-4-jx k-3: j=0 Comparing, k-4X d(L2) + b(L) = 1__^ i__j^ (i__^ M2 ^ i__) ^ i__^k-4-j j=0 + 1__^ i__j^ (M2 ^ i2____) ^ i__^k-4-j k-3 + 1__^ i__j^ (M1 ^ __^ M1 ^ __) ^ i__^k-4-jx : Now the sum (4:2:2) s(4;;;;;:::;;)= (i__^ M2 ^ i__) + (M2 ^ i2____) + (M1 ^ __^ M1 ^ * *__) is a 6-dimensional cycle on R[S4] = R[S(4;;;;;:::;;)] as it sits in Tk(k+1)-4in* * the formula above. We want to relate it to the 6-dimensional cycle s(;;:::;;;4;;;;;;)on R[S(;;:::;* *;;4;;;;;;)] which has the same structure as s(4;;;;;:::;;), but over the differently indexed sphe* *re. The ho- motopy colimit will make both of these cycles homologous to cycles on R[S(4;;;:* *::;;;4;;;;;;)], which are their suspensions in the complementary four coordinates, respectively* *. And since these cycles will represent the same homology class, their images must be homol* *ogous and therefore we see that there is a chain h such that b(h) = s(4;;;;;:::;;)+ s(;;:* *::;;;4;;;;;;). We set k-4X Lh = (1__^ i__j^ h ^ i__^k-4-j) x k-3 j=0 15 and get k-4X d(L2) + b(L + Lh)= 1__^ i__j^ s(;;:::;;;4;;;;;;)^ i__^k-4-jx k-3) j=0i k-4 k-4 1 j = M* (1__^ i__ ) x x (1 ^ s(4;;;;;;)) x where s(4;;;;;;)is essentially s(4;;;;;:::;;)with the extra indices not used, a* *nd M is our multiplication map of (2.1.4), mapping THH (k-3)x THH (4)! THH (k+1)in this * *case. This is because as far as non-degenerate simplices are concerned, M is a shuffl* *e product, concatenating coordinates Ik-3 x I4 ! Ik+1. What remains is to study the cycle s(4;;;;;;)on R[S4]. Recall that reductio* *n modulo 2 induces a map K(Z; n) ! K(Z=2Z; n) which is injective on homology. Since R i* *s, additively, just a free Z-module, we deduce that reduction modulo 2 map, red : * *R[S4] ! (R Z=2Z)[S4] induces a homology injection as well. If we apply the reduction m* *ap redto a chain, any smash product involving a simplex 2a, 2a__, or 2a____becomes a deg* *enerate chain at the basepoint. Thus i * * j red*(s(4;;;;;;))= red*(M1 ^ __^ M1 ^ __) = red* (M1 ^ 1__) ^ (M1 ^ 1__) ^ * *2 : Now for the 2-simplex M1 that we have chosen, [red*(M1)] = 1 2 H2(R[S1]; Z=2Z),* * and since smashing with the unit sphere 1__is our suspension map R[Sk] ! R[Sk+1] wh* *ich sends stable classes to their suspensions which have the same name, [red*((M1 ^ 1__))* *] = 1 2 H3(R[S2]; Z=2Z). And so [red*(((M1 ^ 1__) ^ (M1 ^ 1__)))] = 21= red*(j). Recall that we chose our i so that would be invertible in R Z=2Z ~=Z=2Z[i* *]=(i2). We know that the invertible elements in Z=2Z[i]=(i2) are 1 and 1+i. Both of the* *ir squares are equal to 1. So for any C 2 Hn(R[Sk]; Z=2Z), [2C] = [C] as long as k > 0. Thus, since [red*(s(4;;;;;;))] = [red*(2j)] = [red*(j)] we get by injectivity (4:2:3) [s(4;;;;;;)] = j 2 H6(R[S4]; Z=2Z): Now we have found a chain on THH (k+1)(R) with @ L1 + L2 + L + Lh) = i j 1__^ i__^kx k + M* (1__^ i__k-4) x k-4 x (1 ^ s(4;;;;;;)) x 1 ; where the first summand in the boundary is of filtration degree k, and the seco* *nd of filtration degree k - 3. 1__^ i__^ax a represents (1 i__i__._.-.iz_____") in t* *he E3-term for a times any a. And by (4.2.3), (1 ^ s(4;;;;;;)) x 1 represents e3 = 1 j in the E3-ter* *m. So we get that d3(1 i__i__._.-.iz_____") = (1 i__i__._.-.iz_____") . e3: k times k-4 times * *tu 16 (4.3) Addendum to (2.4). The following details from B"okstedt's calculation will be used in the calc* *ulations to follow; they are listed and explained in the notation established in the beginn* *ing of this section. Recall however that B"okstedt's calculation used Milnor's original cla* *sses i, and we are using their images under the canonical anti-automorphism; his calculatio* *ns work with this new choice, and in fact are simplified (see section 7 of [16] and se* *ction 4.2 of [11]). B"okstedt's map : S1 x hocolimj2Im(Sj; R[Sj] ^ Sm ) ! THH (m)(R) of [2] s* *ends, in our notation, *(ffxC) = (1^C)x1 for ff the 1-simplex in our standard (one 0-sim* *plex, one non-degenerate 1-simplex) model of S1, m the total cardinality of j, and C * *a chain on R[Sj]. In his proof of lemma (1.2) in [3], B"okstedt represents the iby i = i1, an* *d calculates that i = [*(ffxai)] where the aiare cycles for which [a1] = 1 2 Hm+1 (Z=2Z[Sj];* * Z=2Z), and [ai] = (unit) . i+ (decomposables) for i > 1. By section 4.2 of [11], with * *our choice of i we get [ai] = i for all i. So if we pick e4 = [(1 ^ "2) x 1] for some cyc* *le "2 representing 2 2 Hm+2 (Z[Sj]; Z=2Z), this clearly represents e4 in the E1 ter* *m of the spectral sequence calculating the stable homology of THH (Z), and the reductio* *n modulo 2 will send e47! 2 = 21. Once e4 is thus picked, one defines e3= fi1(e4), which gives e3= [(1 ^ "j) * *x 1] for a cycle "jrepresenting j. We know that fi1(2 ) = j for example by looking at thei* *r duals Sq3u and Sq2u (where u is the fundamental class) and recalling that Sq1Sq2 = Sq3. Cl* *early this e3 represents e3. ___ To show e4's algebraic independence over A , B"okstedt uses the fact that_* *it_reduces to 21, where 1 is algebraically independent of A, and so in particular of A A. (4.4) Proof of Lemma (2.5). We represent ffl by the homology class fflof the 1-dimensional chain 1 ^ i * *x 1 on THH (0). This is already a cycle in the unfiltered complex- moreover, it is a* *n absolute cycle so all Bockstein operators vanish on it. Finally, since our representativ* *e of fflcomes from a chain on 0-dimensional spheres, the multiplication of (2.1.4) gives M*(fflx ffl) = 2 . [1 ^ i ^ i x 2] 0: * *tu (4.5) Proof of Lemma (2.6). We represent a2 by the homology class a2 of the 4-dimensional chain (1 ^ i_* *_^ i__) x 2 + (1 ^ (M1 ^ __)) x 1 on THH (2)(R), for M1 of (4.2.1). Since d((1 ^ i__^ i__) x 2) = (1 ^ 2____) x 1 = b((1 ^ (M1 ^ __) x 1) and d((1 ^ (M1 ^ __)) x 1) = 0, this is a cycle in the unfiltered complex. By c* *onsidering the first summand in the representative we have chosen for a2, we see that it r* *epresents the class a2 in the filtered complex. 17 Considering (1 ^ i__^ i__) x 2+ (1 ^ (M1^ __) x 1 as a chain with coefficie* *nts in Z=4Z, we see that its total boundary 2 1 @ (1 ^ i__^ i__) x + (1 ^ (M1 ^ __) x (4:5:1) 1 1 1 * * 1 = 2 . (i__^ i__x ) - 1 ^ 2____x + 1 ^ 2____x = 2 . (i__^ * *i__x ) and thus fi1(a2) = [i__^ i__x 1], a 3-dimensional homology class on on THH (2)w* *hich is the double suspension (once in each coordinate) of the 1-dimensional class [i ^i x1* *] which we now call iffl. Since we use our names for the designated classes and their susp* *ensions, this shows that fi1 applied to an appropriately stabilized a2 gives an appropriately* * stabilized iffl. In order to calculate a22, we compare the E1 terms of B"okstedt's spectral* * sequences for THH (R) and THH (R Z=2Z). Reduction modulo 2 induces a map (see (2.2.1)* * for the former, and lemma (6.3) in [13] along with proposition (1.15) in [12] for* * the latter), ___ 2 2 2 red * * 2 A R[ffl; a2; e4; e8; : :]:=(ffl ; a2; ei)-! A[1; 2; 3; : :]: (R Z=2Z)[ffl* *]=(ffl ) (a2): ___ B"okstedt's work in [3] shows that red embeds A ,! A and sends e2i 7! i. Clear* *ly the R Z=2Z in the (0; 0) coordinate goes to itself by the identity, and the filter* *ed classes ffl, a2 to the classes of the same name in the right hand side. Thus red induces an * *injection on the E1 terms, and so it must induce an injection HS*(THH (R); Z=2Z) ,! HS*(THH (R Z=2Z); Z=2Z): The latter is known from [13] to be HS*(THH (R Z=2Z); Z=2Z) ~=A[1 ] (R Z=2Z)["ffl]=("ffl2) ("a2) (Recall that proposition (1.15) in [12] gives the correct multiplication on HS* **(THH (R Z=2Z); Z=2Z) since the calculation in [13] uses sums of simplices of the form r* *0^ r1^ . .^. rk x k, with ri 0-chains on R[S0]'s, so M simply induces a shuffle product on t* *hem). The representative "a2used in [13] is homologous to 1 ^ i ^ i x 2 (actuall* *y it is the sum of this `monomial' with @((1 ^ i ^ 1 ^ i) x 3)). From the discussion in (4* *.3), we know that red* (1 ^ (M1 ^ 1__)) x 1 represents the class 1 . We, however, have* *, a copy of 1 ^ (M1^ __) x 1, and while we know that the reduction of is invertible in * *R Z=2Z, it may not be equal to 1. We write "1= red*(M1 ^ 1__), and note that @(1 ^ ^ "1x 2) = ( ^ "1+ 1 ^ ( ^ "1) + "1^ ) x 1: This gives us red*(a2) = "a2+ 1 + [("1 ^ ) x 1]; red*(a22) "a22+ 221 + [("1 ^ ) x 1]2 221 21; the equality next to last being true since ("1 ^ ) x 1 = M*("1 x (1 ^ x 1)) and M*((1 ^ x 1) x (1 ^ x 1)) = 2 . (1 ^ ^ x 2) 0, and the last equality being* * true since the two invertible elements in R Z=2Z both give 1 when squared. 18 So red*(a22) 21, whereas we know from (4.3) that if we pick e4 = 1 ^ "2x * *1 for some chain "2representing 2 2 Hk+3(Z[Sk]; Z=2Z) Hk+3(R[Sk]; Z=2Z) we will have red*(e4) = 21: Since we know the reduction modulo 2 induces an inclusion on stable homology, t* *his shows that a22= e4. * * tu (4.6) Proof of Lemma (2.8). fi1((a2)2) vanishes since iffland a2 commute. To show that the higher Bock* *stein operator fi2 does not vanish, we refer again to part (vi) of proposition 1.5 in* * [14]. Madsen uses the Pontrjagin squaring operation to show that for an even-dimensional hom* *ology class y with mod 2 coefficients on an infinite loopspace, fi2(y2) = yfi1(y) + Qdimy(fi1(y)): We want to apply this to y = a2, and so the lemma would follow if we demonstrat* *e that a Q operation vanishes on i__^ i__. Recall, however, that in the case of R com* *mutative, multiplying the zero'th coordinate in any Tr(m)by an element of r commutes with* * all the structure we have defined- and in particular, it commutes with the multiplicati* *on. Thus for any homology class x on any THH (m)(R) (m > 0), any r 2 R, and any i, Qi(rx) = * *r2Qi(x). So in our case, Q4(i__^ i__) = i2Q4(1__^ i__) = 2Q4(1__^ i__) 0: * *tu REFERENCES [1]J. F. Adams, Stable homotopy and generalised homology, Universtity of Chica* *go Press 1974. [2]M. 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