. The Topological Hochschild Homology of the Gaussian Integers Ayelet Lindenstrauss Department of Mathematics The Technion Haifa 32000, Israel ayeletl@tx.technion.ac.il x0. Introduction This paper gives an explicit calculation of the 2-torsion in the topologica* *l Hochschild homology THH of rings of integers in quadratic extensions of the rationals wh* *ich are ramified at the prime 2 (Theorem (1.14)). The calculation of the p-torsion in * *THH of rings of integers in extensions unramified at p can be deduced from B"okstedt's* * calculation of this invariant for the integers (see (2.2)). This means that given the results * *of this paper, the 2-torsion of THH is known for any quadratic number ring. It also means t* *hat for rings of integerspin_extensionspwhich_ramify only at the prime 2, specifically,* * the Gaussian integers, Z[ 2], and Z[ -2 ], the homotopy type of the THH spectrum is comp* *letely known, and given in Theorem (2.3): 1Y0 THH (Z[i])' K(Z; 0)2 x K(Z=2jZ; 2j - 1)2 j=1 p ___ 1Y0 THH (Z[ 2 ])' K(Z; 0)2 x K(Z=2jZ; 2j - 1) x K(Z=4jZ; 2j - 1) : j=1 In comparison, B"okstedt's calculation in [4] 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=jZ int* *o Z=2jZ in higher dimensions. The standard inclusion of a number ring R into its completion at 2 induces * *an iso- morphism of stable homotopy groups with coefficients mod 2, ssS*(THH (R); Z=2* *Z) ~= ssS*(THH (Rb2); Z=2Z). The spectrum THH (Rb2) is interesting in light of the* * result of Hes- selholt and Madsen in [12], which tells us that for completions of number rings* * at a prime p, the topological cyclic homology spectrum and the algebraic K-theory spectrum be* *come the same after completion at p. Topological cyclic homology, introduced by B"oksted* *t, Hsiang, 1 and Madsen in [5], 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. Further results, by the author and Madsen ([15]), now calculate the topolog* *ical Hochschild homology of arbitrary rings of integers in algebraic extensions of Q. The topological Hochschild homology THH of an associative ring R with uni* *t was introduced by B"okstedt in [3]. 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* *et- ter target, from which more information about K*(R) could be pulled back. Goodw* *illie conjectured the existence of THH before it was defined, on the basis of his c* *alculus of functors, and he conjectured that the stable homotopy groups should be isomorph* *ic to the stable K-groups of R. After B"okstedt's definition of THH , Dundas and Mc* *Carthy confirmed the isomorphism of these invariants in [9]. Pirashvili and Waldhausen* * have fur- ther shown that these stable homotopy groups are isomorphic to the MacLane homo* *logy groups of R ([21]). Betley and Pirashvili in [2] have shown that for a fixed r* *ing R, the MacLane homology groups of R, with coefficients in functors from the category o* *f finitely generated free R-modules cross its opposite, can be thought of as derived funct* *ors of the 0'th MacLane homology group. Since regular MacLane homology is a special case o* *f this construction, their work gives a homological-algebra definition of MacLane homo* *logy, and hence also of THH . B"okstedt's definition of THH uses a formalism of functors with smash prod* *uct (FSPs); a different approach, of defining THH on a category of spectra with a strictl* *y associative smash product, has since been defined and used ([18], [10]). Work in progress i* *n Chicago and Osnabr"uck makes it clear that the two approaches are equivalent ([17]). In* * the case of Eilenberg-MacLane spectra of discrete rings, however, B"okstedt's FSPs give sma* *ller spaces, which are convenient to work with, and they shall be used in this paper. In [4] (see also Breen [7]), B"okstedt calculated THH (Z) and THH (Z=pZ* *) for p prime. Both of these proved much more interesting, and in the case of Z much m* *ore suggestive of torsion structure of K*(Z), than the respective Hochschild homolo* *gy groups (which vanish in positive dimensions). Results by Hesselholt on group rings and* * results in [14] extended the range of rings for which THH is known- in fact, both are s* *pecial cases of a splitting phenomenon now described by Hesselholt and Madsen in section 6.1* * of [12]: when R is an A-algebra spanned over A by linearly independent elements x1; : :x* *:nsuch that the set {0; x1; : :x:n} is closed under multiplication, THH (R) is the sm* *ash product of THH (A) with a cyclic nerve complex constructed from the xi's (if R = A[G],* * this complex is homotopy equivalent to the free loopspace ^BG ). Pirashvili in [20] * * makes the same calculation in terms of MacLane homology, and his calculation generalizes * *to smooth algebras over Z, where the collapse at E2 of the spectral sequence he uses cont* *inues to hold. 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 number rings there would not be similar* * behavior. 2 B"okstedt constructs a spectral sequence for calculating 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 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* * [14] that E1*;*(THH (R Z=2Z); Z=2Z) ~=E1*;*(THH (Z=2Z); Z=2Z) HH *(R; Z=2Z): The calculation of these spectral sequence differentials forms the core of the * *calculation of the 2-torsion in THH (R) for these rings in theorem (1.14). As mentioned ea* *rlier, this concludes the calculation of the stable homotopy groups of THH (R) when R is r* *amified only at 2; other cases require further results which will appear in [15] to co* *nclude the calculation. The stable homotopy groups in the result (2.3) mentioned above are much sm* *aller than the analogous ones for, say, the group ring R = Z[x]=(x2- 1), though the H* *ochschild homology groups, with any choice of coefficients, coincide. Thus topological H* *ochschild 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. The first section of this paper describes the calculation of the 2-torsion * *in topologi- cal Hochschild homology of quadratic number rings in extensions of the rational* *s which ramify at 2, modulo some technical lemmas which are proved in the last section.* * The second section uses the results of the first to work out explicitly the topolog* *ical Hochschild homology of quadratic number rings which ramify only at 2. The last section de* *scribes in more detail the setup which is used for the calculations of the first sectio* *n, as well as proving the technical lemmas used there. 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, Winfried Dreckmann, Mi* *chael Larsen, and John Rognes for useful conversations I have had with them while wor* *king on it. x1. The Homology Calculation for Quadratic Rings Which Ramify at Two We will use B"okstedt's setup from [4], with the indexing of section 1.7 in* * [12] and the notations of [14]. These will be reviewed in the beginning of section (3.1)* *. B"okstedt constructs a spectral sequence converging to HS*(THH (R)), with (1:0:1) E2*;*~=HH*(HS*(K(R; 0); Z=pZ)) ~=HH *(HS*(K(Z; 0); Z=pZ)) HH *(R; Z=pZ) 3 if R is additively a free finitely spanned Z-module, and (1:0:2) E2*;*~=HH*(HS*(K(R; 0); Z=pZ)) ~=HH *(HS*(K(Z=pZ; 0); Z=pZ)) HH *(R) if R is additively a finite dimensional vector space over Z=pZ (all rings are a* *ssumed to have units, but are not necessarily commutative). As explained in the introduction, the Hochschild homology factor in (1.0.1* *) and (1.0.2) lives to the E1 term in the cases worked out in [12] and [14]. We wi* *ll see that this is not the case if R is the ring of integers in a quadratic extension of t* *he rationals which ramifies at 2 and p = 2. (1.1) Outline of the Calculation for the Ramified Case: Let R be the ring of integers in a quadratic extension of the rationals whi* *ch is ramified at p = 2. Consider the E2 term (1.0.1). By [4], the E2 term for THH (Z) is __ 2 HH *(HS*(K(Z; 0); Z=pZ)) = A[e3; e4; e8; : :]:=(ei ); dim(ei) = * *i; __ where A = HS*(K(Z; 0); Z=2Z) = Z=2Z[j; 2; 3; : :]:. By [13], HH *(R; Z=2Z) = HH *(R Z=2Z) ~=R=2R[ffl]=(ffl2) (a2); dim(ffl) = 1; dim* *(a2) = 2; where denotes the divided power algebra. Thus (1.0.1) can be rewritten as __ 2 (2) (4) 2 * * (i)2 E2*;*= A[e3; e4; e8; : :]:=(ei ) R[ffl; a2; a2 ; a2 ; : :]:=(ffl ; * *(a2 ) ): __ The elements of A have filtration degree zero, the eiand ffl filtration degree * *one, so d2 and all higher spectral sequence differentials must vanish on them. It will be shown th* *at d2 must also vanish on all the generators a(i)2for dimension reasons, and the higher sp* *ectral sequence differentials all vanish on a2 since it is of filtration degree two. The most i* *mportant step in the calculation will be to demonstrate directly that for i 2, d3(a(i)2) = e* *3a(i-2)2; this shows that all monomials involving a(i)2, for i 2, but not involving e3, do no* *t survive to the E4 term, since they are not in kerd3, and on the other hand, neither do any* * terms involving e3, since they are in imd3. The result will be that __ 2 2 2 E4*;*~=A R[ffl; a2; e4; e8; : :]:=(ffl ; a2; ei); and because of the filtration degree of the generators of this term, E4 = E1 . * *Since the calculation is done over a field, the E1 term gives the correct linear structu* *re of the stable homology, and the multiplication of the homology of the associated graded objec* *t- we will show that the actual multiplicative structure on the stable homology is __ 2 HS*(THH (R); Z=2Z) ~=A R[ffl; a2]=(ffl); with_a22i-1representing e2iin the E1 term. Recall from [4] that HS*(THH (Z);* * Z=2Z) ~= A [e3; e4]=(e23); the inclusion Z ,! R sends e37! 0, e47! a22. Knowing the ring* * structure of stable homology makes it easier to calculate the Bockstein operators on its cla* *sses. Since THH (R) for finitely generated R is known a priori to be a product_of Eilenbe* *rg-MacLane spectra, its homotopy can be read off its stable homology (as a A -module, with* * action of the Bockstein operators). To begin to carry out this program, we need 4 Lemma (1.2) 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 (1:2:1) = 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. Note that by our construction, i satisfies a minimal polynomial over Z of t* *he form f(x) = x2+2ax+2b = 0; in the final formula, it will become important to know th* *e parity of a. (1.3) By [4], 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 first case we have __ i (1:3:1) HS*(K(Z; 0); Z=2Z) = A = Z=2Z[j; 2; 3; : :]:; degj = 2; degi = 2* * - 1 and so __ __ 2 (1:3:2) 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. 5 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 [19] and [8]. The reduction map Z 7! Z=2Z ind* *uces an inclusion A ,! A, and since corollary 2.5 in chapter III of [8] gives us the i* *mage of this inclusion map explicitly, we take the j of (1.3.1) 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 (1:3:3) M : THH (m)(R) x THH (n)(R) ! THH (m+n)(R) defined in section 1.7 of [12], which induces the shuffle product on the Ek ter* *ms for k 1 (see [14]). 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 i 7! i. So on the E2 terms we get a map __ 2 2 (1:3:4) 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 (1.3.4) on the first factor of S(K(R; 0); Z=2Z)) ~=HH (__A) HH (R; Z=2Z)-red! (1:3:5) 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 (1.3.5) is the E2 term of a spectral sequence which col* *lapses at E2 by [14]- 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 * * (1.3.5), 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 __ (1:3:6) im(dk|HH *(R;Z=2Z)) ker(red) = e3 . A[e4; e8; : :]: HH *(R; Z=2Z* *): We will now see that for k = 3 this inclusion is actually an equality, which me* *ans that the spectral sequence differentials are as `destructive' as they can be within the * *restriction of (1.3.6). If we study what is sitting in each coordinate of the spectral sequenc* *e, we will see 6 that d2|HH *(R;Z=2Z)has nothing low enough to hit, given (1.3.6). d3|HH *(R;Z=* *2Z)already has something it could hit. RZ=2Z is spanned over Z=2Z by i, or more precisely the image of the i chose* *n in (1.2) under the reduction mod 2 map. By the choice of i, its square vanishes under th* *e reduction mod 2. Thus, by the well-known description of the Hochschild homology of an ex* *terior algebra, the cycles 1ii. .i.and iii. .i.in HH k(R; Z=2Z) ~=HH k(RZ=2Z) span it over Z=2Z. For these generators we have Lemma (1.4) d3(1 i__i__._.-.iz_____") = e3 . (1 i__i__._.-.iz_____") k times k-4 times (1.5) Since the spectral sequence differentials satisfy the Leibniz rule with* * respect to the multiplication of the spectral sequence (this is simply because the differe* *ntials 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 (1.4) 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 [13]: 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 (1:5:1) E1*;*~=A R[ffl; a2; e4; e8; : :]:=(ffl ; a2; ei): 7 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 0'th 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). Terminology (1.6) 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 (1.7) (Summary of facts from [4]) 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 (3.4). 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 ]. (1.8) Since the inclusion homomorphism Z ,! R sends B"okstedt's E1 classes e2* *i(which are represented by powers of e4) 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 (1.5.1). Since reduction mo* *d 2 maps e4 to the class 21, which is algebraically_independent of A in HS*(THH (R Z=2Z)* *; Z=2Z), e4 is algebraically independent of A in HS*(THH (R); Z=2Z). We get that __ S (1:8:1) 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 forms a 0-dimensional subcomplex closed under the multiplication M and so its homology * *sits as a_subalgebra of HS*(THH (R); Z=2Z); this subalgebra satisfies no nontrivial r* *elations with A [e4] because no such relations hold in the E1 term. We can therefore say, in* * addition to (1.8.1), that __ S (1:8:2) A R[e4] H* (THH (R); Z=2Z): Lemma (1.9) The class ffl in (1.5.1) 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. 8 Lemma (1.10) The class a2 in (1.5.1) can be represented by an unfiltered cla* *ss a2 with fi1(a2) = (i + a)ffl(see (1.2)- i satisfies R = Z[i]=(i2 + 2ai + 2b)), for whic* *h e4= a22. Now we know by (1.9) that in addition to_ (1.8.2), 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 (1.10) tells us that beyond this cop* *y 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 __ Corollary (1.11) HS*(THH (R); Z=2Z) ~=A R[ffl; a2]=(ffl2). * * __ If we look at the result (1.11) 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 Z=2Z means that it is equal e* *ither to 1 or to 1 + i). Alternatively, these can be taken to be 1 and i. In addition to t* *his, for any 1 ` we have the products of (a2)`and (a2)`-1fflwith the 0-dimensional generato* *rs. To determine the 2-torsion Eilenberg-MacLane spectra which appear in THH (* *R), we need to look at the way the Bockstein operators act on these generators. We kno* *w from (1.10) that fi1(a2) = (i + a)ffl fi1(ia2) = (i2 + ai)ffl= -(2b + ai)ffl 0 in the case of a even, but fi1(ia2) = (i2 + ai)ffl= -(2b + ai)ffl iffl in the case of a odd. For even a, the calculation used in the proof of (1.10) for showing the ca* *lculation above (formula (3.6.1)) works also with coefficeints in Z=8Z, from which we can* * deduce fi2(ia2) = ffl: Lemma (1.12) fi1(a22) = 0 but fi2(a22) = (i + a)a2ffl. As before, this implies in the case of odd a that fi2(ia22) ia2ffl, and in* * the case of even a that for i = 1; 2, fii(ia22) = 0, but fi3(ia22) = a2ffl. For higher powe* *rs, we have part (vi) of proposition 1.5 in [16], which tells us that for an even-dimensional mo* *d 2 homology class y on an infinite loopspace, if fii(y) = 0 for 1 i < k (with k > 1) we ha* *ve fii(y2) = 0 for 1 i k and fik+1(y2) = yfik(y). This will inductivelyfgivefus the behavio* *r of the higher Bockstein operators defined on the classes (a2)2, and from them we can a* *s before deduce the behavior on the multiples of these classes by i. For powers (a2)`, w* *ith ` = 2fffi i fi-1_j2ff+1 ff and fi odd, we observe that (a2)`= (a2) 2 (a2)2 , and applying the Leibni* *z rule to products of a lifting of a2 to a chain with appropriate coefficients, we get 9 __ Claim (1.13) HS*(THH (R); Z=2Z) is generated over A by two 0-dimensional gen* *erators, in addition to generators (a2)`, i(a2)`, (a2)`-1ffl, and i(a2)`-1fflfor any 1 * *`. In the case that a in i's minimal polynomial (see (1.2)) is even, then if ` = 2fffi for fi * *odd, these are related by fiff+1((a2)`) = i(a2)`-1ffland fiff+2(i(a2)`) = (a2)`-1ffl. In the * *case that a is odd, they are related by fiff+1((a2)`) = (i + 1)(a2)`-1ffland fiff+1(i(a2)`) = * *i(a2)`-1ffland We get Theorem (1.14) Let R be the ring of integers in a quadratic extension of the r* *ationals which ramifies at 2; for a suitable i 2 R, R = Z[i]=(i2 + 2ai + 2b), a; b 2 Z. * *Then if a is even, THH (R) consists of Y1 0 K(Z; 0)2 x K(Z=2ff+1Z; 2` - 1) x K(Z=2ff+2Z; 2` - 1) `=2f`=1ffi; (fi;2)=1 and p-torsion spectra for p 6= 2, while if a is odd, it consists of Y1 0 K(Z; 0)2 x K(Z=2ff+1Z; 2` - 1) x K(Z=2ff+1Z; 2` - 1) `=2f`=1ffi; (fi;2)=1 and p-torsion spectra for p 6= 2. x2. Concluding the Calculation- Unramified Extensions In the previous section, while looking at rings of integers in quadratic ex* *tensions of the rationals which are ramified at the prime 2, we have not discussed those wh* *ich do not ramify at 2, for which the calculation of THH is much simpler. This calculati* *on is given in the following two lemmas. For rings of integers in quadratic extensions whi* *ch ramify only at p = 2, these lemmas will conclude the calculation of the stable homotop* *y type of the spectrum THH (R). Lemma (2.1) 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 follows from R's commutativity, since* * com- mutativity makes the first differential in the standard bar complex calculating* * Hochschild homology into 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 [13], 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 10 Lemma (2.2) 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) ~=R HS*(THH (Z); Z=pZ) as a A -algebra with action of t* *he Bock- stein operator, 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 of exactly k copies of what we get for THH (Z)). Proof. By (1.0.1), 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 By [4], 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 (1.14), we concl* *ude Theorem (2.3) Let R be the ring of integerspin_a_quadratic extension of the rat* *ionals which ramifies only at 2, i.e. R = Z[i] or R = Z[ 2 ]. 1Y0 THH (Z[i]) ' K(Z; 0)2 x K(Z=2jZ; 2j - 1)2 j=1 p ___ 1Y0 THH (Z[ 2 ]) ' K(Z; 0)2 x K(Z=2jZ; 2j - 1) x K(Z=4jZ; 2j - 1) : j=1 x3. Proofs of the Technical Lemmas (3.1) The Definition of THH and B"okstedt's Spectral Sequence We will start the presentation of the technical proofs by reviewing B"okste* *dt's definition of THH from [3] and [4] (with the indexing of section 1.7 of [12], and the * *notations of [14]). Let R be an associative ring with unit. For a finite simplicial object X:, * *we will use the notation R[|X:|] to denote the geometric realization of the simplicial obje* *ct given in each degree by the free module RXn = R[Xn]=R*, with * the degenerate simplex at* * the 11 basepoint. Let 1|X:|: |X:| ! R[|X:|] be the map induced by ff 7! 1 . ff, and fo* *r 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 (3:1:1) 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 easier of the simplicial calculation* *s which follow. We define cyclic simplicial boundary maps di : Tr(m)! Tr(m)-1for 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] 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)+1for 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 [14], 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 12 B"okstedt's original indexing convention, with the category I as the fixed limi* *t category in (3.1.1), gives a very different THH (0)(R) than the one we get here (specifical* *ly, it gives the loopspace of THH (1)(R)). If we want to think of THH (R) as an -spectrum, we * *should look at {m 7! THH (m)(R)} for m 1, but we will use THH (0)(R) in those calcu* *lations which already work over it, since it is a much simpler space- the geometric rea* *lization of the standard bar complex for Hochschild homology. 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. 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 [4] 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 13 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, we get formulae (1.0.1) and (1.0.2* *), E2*;*~=HH*(HS*(K(R; 0); Z=pZ)) ~=HH *(HS*(K(Z; 0); Z=pZ)) HH *(R; Z=pZ) and E2*;*~=HH*(HS*(K(R; 0); Z=pZ)) ~=HH *(HS*(K(Z=pZ; 0); Z=pZ)) HH *(R); respectively. (3.2) Notations. 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 (1.0.1) which are already images of classes in the spectral sequence for T* *HH (0)(R) or THH (1)(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 [14], 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_^ a* *1_^ . .^.aq_ (or sometimes also by chains which will be called a0^ a1_^ . .^.aq_in the notat* *ion 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 14 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); (3:2:1) 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. Our indexing of Sm is an ad-hoc one created for a particular chain, but 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. Essentially w* *hat we are doing is embedding a complex, representing the homology class in question, into* * a fixed THH (R)(m), and this complex is constructed step by step from liftings of the * *spectral sequence differentials. 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. 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 15 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 a* *1; 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]. (3.3) Proof of Lemma (1.4). We represent 1 i__i__._.-.iz_____"by the chain 1_^ i_^ . .^.i_x k, as expl* *ained 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 (3:3:1) 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 : 16 We consider the 3-chain (1_^M1)+(M1^1_) on R[S2]. It is a cycle, even when we l* *ook 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 reduce 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-j x k-2: j=0 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 [14]. 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-j x 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 (3:3:2) s(4;;;;;:::;;)= (i_^ M2 ^ i_) + (M2 ^ i2__) + (M1 ^ _^ M1 ^ _) 17 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 sphere. * *The homotopy colimit will make both of these cycles homologous to cycles on R[S(4;;;:::;;;4;* *;;;;;)], which are their suspensions in the respective complementary four coordinates. And since t* *hese cycles will represent the same homology class, their images must be homologous and the* *refore we see that there is a chain h on Tk(k+1)-4such that b(h) = s(4;;;;;:::;;)+ s(;* *;:::;;;4;;;;;;). We set k-4 X Lh = (1_^ i_j^ h ^ i_^k-4-j) x k-3 j=0 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 the same as s(4;;;;;:::;;)except for having fewer extra indi* *ces of ;, and M is our multiplication map of (1.3.3), 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 shuffle product, concatenating indexing 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 degene* *rate chain at the basepoint. Thus i j red*(s(4;;;;;;))= red*(M1 ^ _^ M1 ^ _) = red* (M1 ^ 1_) ^ (M1 ^ 1_) ^ 2 : For the 2-simplex M1 that we have chosen, [red*(M1)] = 1 2 H2((R Z=2Z)[S1]; Z=* *2Z), and since smashing with the unit sphere 1_is our suspension map R[Sk] ! R[Sk+1]* * which sends stable classes to their suspensions which have the same name, [red*((M1 ^* * 1_))] = 1 2 H3((R Z=2Z)[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 (3:3:3) [s(4;;;;;;)] = j 2 H6(R[S4]; Z=2Z): 18 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 the* * E3-term for a times any a. And by (3.3.3), (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 (3.4) Addendum to (1.7). The following details from B"okstedt's calculation are used in the calculat* *ions; they are listed and explained in the notation established in the section (3.2). Rec* *all however that B"okstedt's calculation used Milnor's original classes i, and we are using* * their images under the canonical anti-automorphism; his calculations work with this new choi* *ce, and in fact are simplified (see section 7 of [19] and section 4.2 of [12]). B"okstedt's map : S1 x hocolimj2Im(Sj; R[Sj] ^ Sm ) ! THH (m)(R) of [3] 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 [4], 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 [12], 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 th* *eir duals Sq3u and Sq2u (where u is the fundamental class) and recalling that Sq1Sq2 = Sq* *3. This means that e3represents 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. (3.5) Proof of Lemma (1.9). 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 (1.3.3) gives M*(fflx ffl) = 2 . [1 ^ i ^ i x 2] 0: * *tu 19 (3.6) Proof of Lemma (1.10). 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 (3.3.1). 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 co* *nsidering the first summand in the representative we have chosen for a2, we see that it r* *epresents the class a2 in the filtered complex. Considering (1 ^ i_^ i_) x 2+ (1 ^ (M1^ _) x 1 as a chain with coefficients* * in Z=4Z, we see that its total boundary 2 1 @ (1 ^ i_^ i_) x + (1 ^ (M1 ^ _) x (3:6:1) = 2 . (i_^ i_x1) - 1 ^ 2__x 1 + 1 ^ 2__x1 - 2 . (1 ^ __x1) = 2 . (i_^ i_x 1 - 1 ^ __x1) and thus fi1(a2) = [i_^i_x1-1^__x1], a 3-dimensional homology class on on THH (* *2)which is the sum of double suspensions- of the 1-dimensional class [i ^ i x 1] which * *we now call iffl(once in each coordinate), and of the 1-dimensional class [1 ^ x1] (twice i* *n the second coordinate). To analyze the second term, recall from (1.2) that i2 = 2 = -2(a* *i + b), or in other words = -ai - b. Thus [1 ^ x 1] = a[1 ^ i x 1] + b[1 ^ 1 x 1] but 1 ^ 1x1 = @(1 ^ 1 ^ 1x2), so we are left with fi1(a2) = (i + a)ffl. Since we use our names for the designated classes and their suspensions, th* *is shows that fi1 applied to an appropriately stabilized a2gives an appropriately stabil* *ized (i + a)ffl. 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 (1.5.1)* * for the former, and lemma (6.3) in [14] along with proposition (1.15) in [13] 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 [4] 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 [14] to be HS*(THH (R Z=2Z); Z=2Z) ~=A[1 ] (R Z=2Z)["ffl]=("ffl2) ("a2) (Recall that proposition (1.15) in [13] gives the correct multiplication on HS* **(THH (R Z=2Z); Z=2Z) since the calculation in [14] 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). 20 The representative "a2used in [14] 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 (3* *.4), 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. So red*(a22) 21, whereas we know from (3.4) 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 (3.7) Proof of Lemma (1.12). Proposition 1.5 in [16] tells us that fi1((a2)2) vanishes. Moreover, part* * (vi) of this proposition tells us that for an even-dimensional homology class y with mod 2 c* *oefficients on an infinite loopspace, fi2(y2) = yfi1(y) + Qdimy(fi1(y)): We want to apply this to y = a2. The desired result would follow if we knew tha* *t Q1(ffl) = 0, but that follows from Lemma 2.9 in [4], which specializes in our case to the cl* *aim that Q1([1 ^ i x 1]) = 1 ^ Q1(i) x 1 = 1 ^ iQ1(1) = 0 where the equality before last follows from the fact that the multiplication of* * R's Eilenberg- MacLane spectrum commutes with multiplication by R, and the last equality follo* *ws from the fact that Q0 is the only Q-operation which does not vanish on 1 (see for ex* *ample part (4) of theorem 1.1 in chapter III of [8]). For a multiple of fflby an element r* * 2 R, we observe that in THH (R), too, all the structure we have defined commutes with multipli* *cation by elements of R, and thus Q1(rffl) = r2Q1(ffl) = 0. * * tu 21 REFERENCES [1]J. F. 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