... LODAY CONSTRUCTIONS ON TWISTED PRODUCTS AND ON TORI arXiv:2002.00715v1 [math.AT] 3 Feb 2020 ALICE HEDENLUND, SARAH KLANDERMAN, AYELET LINDENSTRAUSS, BIRGIT RICHTER, AND FOLING ZOU Abstract. We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology. Introduction When one studies commutative rings or ring spectra, important homology theories are topological Hochschild or its higher versions. These are specific examples of the Loday construction, whose definition relies on the fact that commutative ring spectra are enriched in simplicial sets: for a simplicial set X and a commutative ring spectrum R one can define the tensor X R as a simplicial spectrum whose n-simplices are ^ R. xXn By slight abuse of notation X R also denotes the commutative ring spectrum that is the geometric realization of this simplicial spectrum. This recovers topological Hochschild homology of R, THH(R), when X = S 1 , and higher topological Hochschild homology, THH[n] (R), for higher dimensional spheres S n . Tensoring satisfies several properties [8, VII, 2, 3], two of which are: If X is a homotopy pushout, X = X1 hX0 X2 , then the tensor product of R with X splits as a homotopy pushout in the category of commutative ring spectra which is the derived smash product: (X1 hX0 X2 ) R (X1 R) L (X0 R) (X2 R). A product of simplicial sets X Y gives rise to an iterated tensor product: (X Y ) R X (Y R). This last expression does not, however, imply that calculating the homotopy groups of (XY )R is easy. In particular, if one iterates the trace map from algebraic K-theory to topological Hochschild homology n times, one obtains a map K (n) (R) = K(K(. . . (K (R)) . . .)) (S 1 . . . S 1 ) R. | {z } | {z } n n Since iterated K-theory is of interest in the context of chromatic red-shift, one would like to know as much about (S 1 . . . S 1 ) R as possible. Date: February 4, 2020. 2000 Mathematics Subject Classification. Primary 18G60; Secondary 55P43. Key words and phrases. torus homology, (higher) Hochschild homology, (higher) topological Hochschild homology, stability, twisted Cartesian products. 1 In some good cases, the homotopy type of X R only depends on the suspension of X in the sense that if X Y , then one has X R Y R. This property is called stability. Stability for instance holds for Thom spectra R that arise from an infinite loop map to the classifying space BGL1 (S) (see Theorem 1.1 of [25]), or for R = KU or R = KO [14, 4]. One can also work relative to a fixed commutative ring spectrum R and consider commutative R-algebra spectra A and ask whether X R A only depends on the homotopy type of X. In this paper, we will often work with coefficients: we look at pointed simplicial sets X and place a commutative A-algebra spectrum C at the basepoint of X. In other words, when X is pointed then the inclusion of the basepoint makes X R A into a commutative A-algebra and we can look at LR X (A; C) = (X R A) A C, the Loday construction with respect to X of A over R with coefficients in C. We call the pair (A; C) stable if the homotopy type of LR X (A; C) only depends on the homotopy type of X. Note that the ring R is not part the notation when we say that (A; C) is stable although the question depends on the choice of R, so the context should specify the R we are working over. We call the commutative R-algebra A multiplicatively stable as in [14, R Definition 2.3] if X Y implies that LR X (A) LY (A) as commutative A-algebra spectra. If A is multiplicatively stable, then for any cofibrant commutative A-algebra C, the pair (A; C) is stable (see [14, Remark 2.5]). We investigate several algebraic examples, i.e., commutative ring spectra that are Eilenberg Mac Lane spectra of commutative rings. For instance we show that the pairs (HQ[t]/tm ; HQ) are not stable for all m > 2, extending a result by Dundas and Tenti [7]. We also prove integral and mod-p versions of this result. Work of Berest, Ramadoss and Yeung implies that the homotopy types of LHk X (HA; Hk) and Hk LX (HA) only depend on the homotopy type of X if k is a field and if A is a commutative Hopf algebra over k. We generalize this result to commutative Hopf algebra spectra. Moore introduced twisted cartesian products as simplicial models for fiber bundles. We develop a Serre type spectral sequence for Loday constructions of twisted cartesian products where the twisting is governed by a constant simplicial group. As a concrete example we compute the Loday construction with respect to the Klein bottle for a polynomial algebra over a field with characteristic not equal to 2. Content. In Section 1 we recall the definition of the Loday construction and fix notation. Section 2 contains the construction of a spectral sequence for the homotopy groups of Loday constructions with respect to twisted cartesian products. Our results on commutative Hopf algebra spectra can be found in Section 3. In Section 4 we prove that truncated polynomial algebras of the form Q[t]/tm and Z[t]/tm for m > 2 are not multiplicatively stable by comparing the Loday construction of tori to the Loday construction of a bouquet of spheres corresponding to the cells of the tori. We also show that for 2 6 m < p the Fp -algebra Fp [t]/tm is not stable. Acknowledgements. We thank the organizers of the third Women in Topology workshop, Julie Bergner, Angelica Osorno, and Sarah Whitehouse, and also the Hausdorff Institute of Mathematics for their hospitality during the week of the workshop. We thank the Hausdorff Research Institute for Mathematics, grants nsf-dms 1901795 and nsf-hrd 1500481AWM ADVANCE, and the Foundation Compositio Mathematica for their support of the workshop. We thank Maximilien Peroux for help with coalgebras in spectra, Inbar Klang for a helpful remark about norms, Mike Mandell for a helpful discussion on En -spaces, Jelena Grbic for pointing out [23] to us, and Thomas Nikolaus for -category support. AL was supported by Simons Collaboration Grant 359565. The last two authors thank the Department of Mathematics at Indiana University for its hospitality and BR thanks the Department of Mathematics at Indiana University for support as a short-term research visitor in 2019. 2 1. The Loday construction: basic features We recall some definitions concerning the Loday construction and we fix notation. For our work we can use any good symmetric monoidal category of spectra whose category of commutative monoids is Quillen equivalent to the category of E -ring spectra, such as symmetric spectra [12], orthogonal spectra [17] or S-modules [8]. As parts of the paper require us to work with a specific model category we chose to work with the category of S-modules. Let X be a finite pointed simplicial set and let R A C be a sequence of maps of commutative ring spectra. Definition 1.1. The Loday construction with respect to X of A over R with coefficients in C is the simplicial commutative augmented C-algebra spectrum LR X (A; C) given by ^ LR (A; C) = C A n X xXn \ where the smash products are taken over R. Here, denotes the basepoint of X and we place a copy of C at the basepoint. The simplicial structure of LR X (A; C) is straightforward: Face maps di on X induce multiplication in A or the A-action on C if the basepoint is involved. Degeneracies si on X correspond to the insertion of the unit maps A : R A over all n-simplices which are not hit by si : Xn1 Xn . As defined above, LR X (A; C) is a simplicial commutative augmented C-algebra spectrum. In the following we will always assume that R is a cofibrant commutative S-algebra, A is a cofibrant commutative R-algebra and C is a cofibrant commutative A-algebra. This ensures that the homotopy type of LR X (A; C) is well-defined and depends only on the homotopy type of X. Remark 1.2. When R A C is a sequence of maps of commutative rings, we can of course use the above definition for HR HA HC. The original construction by Loday [15, Proposition 6.4.4] used O C A xXn \ instead with the tensors taken over R as the n-simplices in LR X (A; C). This algebraic definition also makes sense if R is a commutative ring and A C is a map of commutative simplicial R-algebras. It continues to work if R is a commutative ring and A C is a map of graded-commutative R-algebras, with the n-simplices defined as above, but the maps between them require a sign correction as terms are pulled past each othersee [21, Equation (1.7.2)]. An important case is X = S n . In this case we write THH[n],R (A; C) for LR S n (A; C); this is the higher order topological Hochschild homology of order n of A over R with coefficients in C. Let k be a commutative ring, A be a commutative k-algebra, and M be an A-module. Then we define THH[n],k (A; M ) := LHk S n (HA; HM ). If A is flat over k, then THHk (A; M ) = HHk (A; M ) [8, Theorem IX.1.7] and this also holds for [n],k higher order Hochschild homology in the sense of Pirashvili [21]: THH[n],k (A; M ) = HH (A; M ) if A is k-flat [4, Proposition 7.2]. Given a commutative ring A and an element a A, we write A/a instead of A/(a). 3 2. A spectral sequence for twisted cartesian products We will start by letting R A be a map of commutative rings and we study Loday constructions LR B (A ) over a finite simplicial set B, where indicates a twisting by a discrete group G that acts on A via ring isomorphisms. This construction can be adapted as in Definition 1.1 and Remark 1.2 to allow coefficients in an A-algebra C if B is pointed, and to the case where R A is a map of commutative ring spectra, or R is a commutative ring and A is a graded-commutative R-algebra or a simplicial commutative R-algebra. If we have a twisted cartesian product (TCP) in the sense of [18, Chapter IV] E( ) = F B where the fiber F is a simplicial R-algebra and the simplicial structure group G acts on F by simplicial R-algebra isomorphisms, it is possible to generalize this definition of the Loday construction to allow twisting by a simplicial structure group, as expained in Definition 2.1 below. We show an example where such a TCP arises: if we start with a TCP of simplicial sets E( ) = F B with twisting in a simplicial structure group G acting on F simplicially on the left and with a map of commutative rings R A, we can use that twisting to construct a TCP with fiber equal R to the simplicial commutative R-algebra LR F (A) and with the structure group G acting on LF (A) by R-algebra isomorphisms. In that situation, we get that LR (A) = LR (LR (A) ), B E( ) F R R which generalizes the fact that for a product, LR F B (A) = LB (LF (A)). If the structure group G is discrete, i.e., if G is a constant simplicial group, LR E( ) (A) can be written as a bisimplicial set and we get a spectral sequence for calculating its homotopy groups. Definition 2.1. Let B be a finite simplicial set, R be a commutative ring, and A be a commutative R-algebra (or a graded-commutative R-algebra, or a simplicial commutative R-algebra). Let G be a discrete group acting on A from the left via isomorphisms of R-algebras, and let be a function from the positive-dimensional simplices of B to G so that (2.2) (b) (di b) (si b) (s0 b) = = = = [ (d0 b)]1 (d1 b) (b) (b) eG for for for for q > 1, b Bq , i > 2, q > 1, b Bq , i > 1, q > 0, b Bq , and q > 0, b Bq . The twisted Loday construction with respect to B of A over R twisted by is the simplicial com mutative (resp., graded-commutative, or bisimplicial commutative) R-algebra LR B (A ) given by O R LR A B (A )n = LBn (A) = bBn where the tensor products are taken over R, with O O Y d0 fb = gc with gc = (b)(fb ), cBn1 bBn di O bBn si O bBn fb = fb = O b:d0 b=c gc with gc = cBn1 O Y fb for 1 i n, and b:di b=c hd with hd = dBn+1 Y fb for 0 i n. b:si b=d We should think of the copy of A sitting over a simplex b Bn as sitting over its 0th vertex, and of (b) as translating between the A over bs 0th vertex and the A over bs 1st vertex. 4 Lemma 2.3. The definition above makes LR B (A ) into a simplicial set. Proof. To check this we need only check the relations involving d0 , since the ones that do not involve work in the same way that they do in the usual Loday construction. For j > 1, we get d0 dj = dj1 d0 because in both terms, for any c Bn2 we get the product over all b Bn with d0 dj b = dj1 d0 b = c of terms that are either (b)(fb ) or (dj b)(fb ). These are the same by the condition in Equation (2.2) above. For j = 1, we get the product over all b Bn with d0 d1 b = d0 d0 b = c of terms that are either (d1 b)(fb ) or (d0 b) (b)(fb ), which again agree by Equation (2.2). We get d0 s0 = id since (s0 b) = eG , and d0 si = si1 do for i > 0 since for those i, (si b) = (b). Following Moore, May considers the following simplicial version of a fiber bundle [18, Definition 18.3]: Definition 2.4. Let F and B be simplicial sets and let G be a simplicial group which acts on F from the left. Let : Bq Gq1 for all q > 0 be functions so that d0 (b) (di+1 b) (si+1 b) (s0 b) = = = = [ (d0 b)]1 (d1 b) di (b) si (b) eq for for for for q > 1, b Bq , i 1, q > 1, b Bq , i 0, q > 0, b Bq , and q > 0, b Bq . The twisted Cartesian product (TCP) E( ) = F B is the simplicial set whose n-simplices are given by E( )n = Fn Bn , with simplicial structure maps (i) d0 (f, b) = ( (b) d0 f, d0 b), (ii) di (f, b) = (di f, di b) i > 0, and (iii) si (f, b) = (si f, si b) i 0. These structure maps satisfy the necessary relations to be a simplicial set because of the conditions that satisfies. Definition 2.5. If R is a commutative ring and E( ) = C B is a TCP as in Definition 2.4 where C is a commutative simplicial R-algebra and the simplicial group G acts on C by R-algebra isomorphisms (that is, for every q 0, the group Gq acts on the commutative R-algebra Cq by R-algebra isomorphisms) then we can use the twisting to define the twisted Loday construction with respect to B of C over R, twisted by , O R Cn LR B (C )n = LBn (Cn ) = bBn N with twisted structure maps given on monomials bBn fb , with fb Cn for all b Bn , by O Y O (b)(d0 fb ), gc with gc = d0 fb = cBn1 bBn (2.6) di O bBn si O bBn fb = fb = O b:d0 b=c gc with gc = cBn1 O Y di fb for 1 i n, and b:di b=c hd with hd = dBn+1 Y b:si b=d 5 si fb for 0 i n. Note that there are two sets of simplicial structure maps being used, those of C inside and those of B outside. This looks like the diagonal of a bisimplicial set, but since our twisting : Bq+1 Gq explains only how to twist elements in Cq , this is not the case unless the structure group G is a discrete group, viewed as a constant simplicial group. If the structure group G is discrete, there is overlap between Definition 2.1 and Definition 2.5. The simplicial commutative R-algebra case of Definition 2.1 actually gives a bisimplicial set: we use only the simplicial structure of B in the definition and if A also has simplicial structure, that remains untouched. The diagonal of that bisimplicial set agrees with the constant simplicial group case of Definition 2.5. Given any TCP of simplicial sets E( ) = F B as in Definition 2.4 and a map R A of commutative rings, we can construct LR F (A) B which is a TCP of commutative simplicial algebras R-algebras as in Definition 2.5 using the same structure group G and twisting function : Bq Gq1 . We use the simplicial left action of Gn on Fn which we denote by (g, f ) 7 gf to obtain a left action by simplicial R-algebra isomorphisms R Gn LR Fn (A) LFn (A) O O (g, af ) 7 ag1 f . (2.7) f Fn f Fn Since the original action of Gn on Fn was a left action, this is a left action. In the original monomial, the f th coordinate is af . After g Gn acts on it, the f th coordinate is bf = ag1 f . After h Gn acts on the result of the action of g, the f th coordinate is bh1 f = ag1 h1 f , which is the same as the result of acting by hg on the monomial. Proposition 2.8. If E( ) = F B is a TCP and R A is a map of commutative rings, and we use the simplicial set twisting function to construct a simplicial R-algebra twisting function to obtain a TCP LR F (A) B as above, we get that LR (A) = LR (LR (A) ). B E( ) F This uses the definition of the Loday construction of a simplicial algebra twisted by a simplicial group in Definition 2.5. Proposition 2.8 generalizes the well-known fact that for a product of simplicial sets, LR (A) = LR (LR (A)). F B B F R R Proof. Both LR E( ) (A) and LB (LF (A) ) have the same set of n-simplices for every n > 0: O O O O A= A ( A). = eE( )n bBn (f,b)Fn Bn f Fn We have to show that the simplicial structure maps agree with respect to this identification. For 1 6 i 6 n, for any choice of elements x(f,b) A, O O y(g,c) di x(f,b) = (g,c)Fn1 Bn1 (f,b)Fn Bn where y(g,c) = Y x(f,b) = Y b:di b=c (f,b):(di f,di b)=(g,c) Y f :di f =g x(f,b) . The internal product on the right-hand side is what we get from di on LR F (A) and the external product is what we get from di of LR , so this agrees with the definition in Equation (2.6). B 6 The proof that the si , 0 6 i 6 n agree is very similar. The interesting case is that of d0 . For any choice of elements x(f,b) A, the boundary d0 associated to LR E( ) (A) satisfies O O (2.9) d0 x(f,b) = y(g,c), (f,b)Fn Bn (g,c)Fn1 Bn1 where y(g,c) = Y x(f,b) = Y b:d0 b=c (f,b):d0 (f,b)=(g,c) Y f : (b)d0 f =g x(f,b) . R From the LR B (LF (A)) point of view, by Equation (2.6). O O O Y O d0 ( x(f,b) ) = (b)d0 x(f,b) bBn f Fn f Fn cBn1 b:d0 b=c = O Y O Y cBn1 b:d0 b=c = cBn1 b:d0 b=c = O (b) O Y gFn1 f :d0 f =g O Y x(f,b) gFn1 f :d0 f = (b)1 g Y (g,c)Fn1 Bn1 b:d0 b=c which is exactly what we got in (2.9). Y x(f,b) f :d0 f = (b)1 g x(f,b) , If G is a discrete group and E( ) is constructed using G, then for every q > 0 there is a function : Bq G satisfying the conditions listed in Equation (2.2) and G acts simplicially on F on the left. Theorem 2.10. If E( ) = F B is a TCP where the twisting is by a constant simplicial group G and if R A is a map of commutative rings so that (LR F (A)) is flat over R, then there is a spectral sequence (2.11) 2 R R Ep,q = p ((LR B ( LF (A) ))q ) p+q (LE( ) (A)). Here, LR F (A) is a graded commutative R-algebra. For any fixed p and q, we consider the degree R R R q part of LR Bp ( LF (A) ), (LBp ( LF (A) ))q . This forms a simplicial abelian group which in degree R p is (LR Bp ( LF (A)))q , with simplicial structure maps induced by those of B with the twisting by R , and p ((LR B ( LF (A) ))q ) denotes its pth homotopy group. The flatness assumption above is for instance satisfied if R is a field. Proof. Since the twisting is by a constant simplicial group G, we are able to form a bisimplicial R-algebra O O (2.12) (m, n) 7 A. bBm f Fn In the n-direction, the simplicial structure maps dFi and sFi will simply be the simplicial structure R maps of the Loday construction LR F (A) applied simultaneously to all the copies of LF (A) over all 7 B the b Bn . In the m direction, dB i and si are the simplicial structure maps of the twisted Loday construction, as in Equation (2.2) in Definition 2.1. These commute exactly because the simplicial structure maps in G are all equal to the identity. For any choice of xb LR F (A)n for all b Bm , O O O Y F xb = dB dFi (xb ) = (b)dFi (xb ) dB 0 di 0 bBm while dFi dB 0( O bBm cBm1 b:d0 b=c bBm xb ) = dFi which is the same since O Y cBm1 b:d0 b=c (b) xb = O Y dFi ( (b) xb ), cBm1 b:d0 b=c dFi ( (b) xb ) = di ( (b)) dFi (xb ) = (b) dFi (xb ). R R Note that since the twisting is by a constant simplicial group, LR E( ) (A) = LB (LF (A) ) is exactly the diagonal of the bisimplicial R-algebra in Equation (2.12). We use the standard result (see for instance [9, Theorem 2.4 of Section IV.2.2]) that the total complex of a bisimplicial abelian group with the alternating sums of the vertical and the horizontal face maps is chain homotopy equivalent to the usual chain complex associated to the diagonal of that bisimplicial abelian group. Since we know that the realization of the diagonal is homeomorphic to the double realization of the bisimplicial abelian group, in order to know the homotopy groups of the double realization of a bisimplicial abelian group, we can calculate the homology of its total complex with respect to the alternating sums of the vertical and the horizontal face maps. Filtering by columns gives an E 2 spectral sequence calculating the homology of the total complex associated to a bisimplicial abelian group consisting of what we get by first taking vertical homology and then taking horizontal homology. In the case of the bisimplicial abelian group we have in Equation (2.12), Pn the vertical qth homology of the columns will be the qth homology with respect to i=0 (1)i dFi of the complex O LR F (A) and this is isomorphic to q obtain N bBm q bBm R LF (A) . Since O bBm LR =( F (A) we assumed that (LR F (A)) is flat over R, we O (LR F (A)))q . bBm N Here, the subscript q denotes the degree q part of the graded abelian group bBm (LR F (A)). N R Moreover, the effect of the horizontal boundary map on bBm (LF (A)) is the boundary of the twisted Loday construction, with the action of G on the graded-commutative R-algebra (LR F (A)) R induced by that of G on the commutative simplicial R-algebra LF (A). As the boundary map preserves internal degree, we get the desired spectral sequence. 2.1. Norms and finite coverings of S 1 . The connected n-fold cover of S 1 given by the degree n map can be made into a TCP as follows. Let B = S 1 be the standard simplicial circle and Cn = hi be the cyclic group of order n with generator . The twisting function : Sq1 Cn sends the non-degenerate simplex in S11 to and is then determined by Equation (2.2). Let F = Cn , viewed as a constant simplicial set, and let Cn act on F from the left. Then E( ) = F B is in fact another simplicial model of S 1 with n non-degenerate 1-simplices. Therefore, LR (A) LR1 (A) and (LR (A)) = HHR (A) E( ) E( ) S 8 R n is the constant commutative for every commutative R-algebra A. In this case, LR FA = A simplicial R-algebra, with the Cn -action given by (a1 an ) = an a1 an1 . As LR F (A) is a constant simplicial object, we obtain that ( AR n , = 0, L R (A) = F 0, > 0. If A is flat over R, the spectral sequence of Equation (2.11) is 2 R n R Ep,q = p (LR ) )q p+q LR E( ) (A) = HHp+q (A). S 1 (A But here, the spectral sequence is concentrated in q-degree zero, and hence it collapses, yielding p (LR1 (AR n ) ) = HHR (A). p S LSE( ) (A) With Proposition 2.8 we can identify if A is a commutative ring spectrum and we recover the known result (see for instance [2, p. 2150]) that THHCn (NeCn A) THH(A). (2.13) 1 Here, THHCn (A) = NCSn (A) is the Cn -relative THH defined in [2, Definition 8.2], where NeCn A is the Hill-Hopkins-Ravanel norm. See also [1, Definition 2.0.1]. The identification in (2.13) is an 1 1 instance of the transitivity of the norm: NCSn NeCn A NeS A. 2.2. The case of the Klein bottle. For the Klein bottle we compute the homotopy groups of the Loday construction of the polynomial algebra k[x] for a field k using our TCP spectral sequence and we confirm our answer using the following pushout argument. We assume that the characteristic of k is not 2, so 2 is invertible in k. Note that the Klein bottle can be represented as a homotopy pushout K (S 1 S 1 ) hS 1 D 2 . Since the Loday construction converts homotopy pushouts of simplicial sets into homotopy pushouts of commutative algebra spectra, we obtain LkK (k[x]) LkS 1 S 1 (k[x]) L Lk S1 (k[x]) LkD2 (k[x]). Homotopy invariance of the Loday construction yields that LkD2 (k[x]) = k[x], and as LkS 1 (k[x]) = HHk (k[x]) is well known to be isomorphic to k[x] (x) as a graded commutative k-algebra, we get that LkS 1 S 1 (k[x]) = LkS 1 (k[x]) k[x] LkS 1 (k[x]) = k[x] (xa , xb ) where the indices a and b allow us to distinguish between the generators emerging from each of the circles Sa1 Sb1 . Let Sc1 represent the circle along which we glue the disk, and call the corresponding generator in dimension one for the Loday construction over it xc . Let Sa1 denote the circle that Sc1 will go twice around in the same direction and Sb1 denote the circle that it will go around in opposite directions. So we have a projection K Sb1 . We can calculate LkK (k[x]) with a Tor spectral sequence whose E 2 -page is (2.14) 2 E, = Lk (k[x]) Tor, S 1 LkS 1 S 1 (k[x]), LkD2 (k[x]) k[x](xc ) (k[x] (xa , xb ), k[x]). = Tor, We need to understand the LkS 1 (k[x])-module structure on k[x] (xa , xb ), so we need to understand the map k[x] (xc ) k[x] (xa , xb ). 9 Since k[x] in both cases is the image of the Loday construction on a point, we know that x on the left maps to x on the right. If we map Sc1 to Sa1 Sb1 and then collapse Sa1 to a point, we end up with a map Sc1 Sb1 that is contractible, so if we only look at the xb part of the image of xc in (xa , xb ) (that is, if we augment xa to zero) we get zero. We deduce that k[x](xc ) Tor, (k[x] (xa , xb ), k[x]) (x ) k[x] = Tor, (k[x], k[x]) Tor, c ((xa ), k) Tork, ((xb ), k) (x ) = k[x] Tor, c ((xa ), k) (xb ). (x ) In order to calculate Tor, c ((xa ), k), we map Sc1 to Sa1 Sb1 and then collapse Sb1 to a point. This gives a map Sc1 Sa1 that is homotopic to the double cover of the circle as depicted below. We consider elements of LSc1 (k[x]), which we think of as built on the top circle, and of LSa1 (k[x]), which we think of as built on the bottom circle, and write them as sums of tensor monomials of ring elements with subscripts indicating the simplex each ring element lies over. Under this map, we have 1 vr1 x0 := 1s0 v0 1s0 v1 x0 11 7 1s0 v x x1 := 1s0 v0 1s0 v1 10 x1 7 1s0 v x Then d0 d1 maps these elements to the following: 0 r x0 7 1v0 xv1 xv0 1v1 x1 7 xv0 1v1 1v0 xv1 v0 Note that the sum of the images under d0 d1 is zero, and so x0 + x1 is a cycle with one copy of x in simplicial degree 1, which is what xc should be. Monomials that put the copy of x over s0 vi are the image under d0 d1 +d2 of monomials that put one copy of x over s20 vi , so do not contribute to the homology, and all cycles not involving those and involving only one copy of x are multiples of x0 + x1 , and so x0 + x1 represents xc . But we r v know that xa is represented by 1s0 v x , so we get that that xc 7 2xa . We take the standard resolution of k as a (xc )-module: ... xc // (xc ) xc // (xc ) Since we saw above that xc 7 2xa , tensoring () (xc ) (xa ) yields 2xa ... // (xa ) 2xa // (xa ) (x ) Since we assume that 2 is invertible in k, we get that Tor c ((xa ), k) = k, and so when 2 is invertible in k, the spectral sequence in Equation (2.14) has the form E2 = k[x] k (xb ) = k[x] (xb ), , and therefore also collapses for degree reasons, yielding Lk (k[x]) = k[x] (x). K Remark 2.15. In fact we have shown that LkK (k[x]) = LkS 1 (k[x]) and that the projection K Sb1 induces this isomorphism. 10 Now we want to get the same result using our TCP spectral sequence for A = k[x]. We will use the following simplicial model for the Klein bottle: K = (I S 1 )/(0, t) (1, flip(t)) where flip is the reflection of the circle about the y-axis. If we use the same model of the circle with two vertices and two edges that we used in the double cover picture above but we reverse the orientation on 0 so that both edges go top to bottom, this is a simplicial map preserving v0 and v1 and exchanging the i . The flip map induces a map on (LkS 1 (k[x]) = k[x] (x) sending x 7 x and x 7 x. The fact that x 7 x comes from the fact that it is the image of the Loday construction over a point. Using the same notation and argument as before, with the different orientation on 0 , x can be represented by x0 x1 , so exchanging the i sends x to x. The nontrivial twist : S 1 C2 = hi maps the non-degenerate 1-cell S11 to and is then determined by Equation (2.2), yielding (2.16) d0 (a0 a1 . . . an ) = a0 a1 a2 . . . an . The TCP spectral sequence (2.11) in this case takes the form ! 2 k k = p+q LkK (k[x]) Ep,q = p LS 1 (LS 1 (k[x])) q and since LkS 1 (k[x]) = k[x] (x), 2 Ep,q = p LkS 1 k[x] (x) ! , q which is the pth homotopy group of the simplicial k-vector space whose p-simplices are LkSp1 (k[x] (x) ) . q For each p, LkS 1 (k[x](x)) LkS 1 (k[x])k LkS 1 ((x)), and so LkS 1 (k[x](x) ) LkS 1 (k[x])k p p p LkS 1 ((x) ). We can think of this tensor product of simplicial k-algebras as the diagonal of a bisimplicial abelian group, and again by [9, Theorem 2.4 of Section IV.2.2] the total complex of a bisimplicial abelian group with the alternating sums of the vertical and the horizontal face maps is chain homotopy equivalent to the usual chain complex associated to the diagonal of that bisimplicial abelian group. But in this case of a tensor product, the total complex was obtained by tensoring together two complexes, and since we are working over a field its homology is the tensor product of the homology of the two complexes, so k LS 1 k[x] (x) LkS 1 ((x) ) . = LkS 1 (k[x]) The first factor is just the Hochschild homology of k[x]. It sits in the 0th row of the E 2 term since x has internal degree zero, and gives us (LkS 1 (k[x]) = k[x] (x)) concentrated in positions (0, 0) and (1, 0). All spectral sequence differentials vanish on it for degree reasons, and so it will just contribute k[x] (x) to the E term. The second factor in the E 2 term is the twisted Hochschild homology for (x). To calculate it, we can use the normalized chain complex and therefore we only have to consider non-degenerate elements, which means that we only have two elements to take into account in any given simplicial degree: 11 p-degree 0 1 2 ... 1 1 x 1 x x . . . x x x x x x . . . Elements of the form x . . . x will map to zero under the Hochschild boundary map. We need to consider the odd and even cases of differentials on elements of the form 1 x . . . x. The di maps in the twisted and untwisted Hochschild complex are all the same except d0 , which incorporates the twisting action of . Therefore we have d(1 (x)2k ) = x2k + (1)2k (1)x2k = 2x2k d(1 (x)2k+1 ) = x2k+1 + (1)2k+1 (1)x2k+1 = 2x2k+1 . Here, the first 1 comes from the action on x as in (2.16) and the extra 1 in brackets come from passing the one-dimensional x past an odd or an even number of copies of itself. Since we are assuming that 2 is invertible in k, we get that the second part of the E 2 term has only k left in degree 0. So, if 2 is invertible in k, then the entire E 2 term is just k[x] (x) in the 0th row, and the TCP spectral sequence collapses and confirms that Lk (k[x]) = k[x] (x). K 3. Hopf algebras in spectra We start by describing what we mean by the notion of a commutative Hopf algebra in the -category of spectra, Sp. We consider the -category CAlg of E -ring spectra. Definition 3.1. A commutative Hopf algebra spectrum is a cogroup object in CAlg. Hopf algebra spectra are fairly rare, so let us list some important examples. Example 3.2. If G is a topological abelian group, then the spherical group ring S[G] = +G equipped with the product induced by the product in G, the coproduct induced by the diagonal map G G G, and the antipodal map induced by the inverse map from G to G is a commutative Hopf algebra spectrum. This follows from the fact that the suspension spectrum functor + : S Sp is a strong symmetric monoidal functor. Here S denotes the -category of spaces. Example 3.3. If A is an ordinary commutative Hopf algebra over a commutative ring k and A is flat as a k-module then the Eilenberg-Mac Lane spectrum HA is a commutative Hopf algebra spectrum over Hk because the canonical map HA Hk HA H(A k A) is an equivalence. We use the fact that the category of commutative ring spectra is tensored over unpointed topological spaces and simplicial sets in a compatible way [8, VII, 2, 3]. If U denotes the category of unbased (compactly generated weak Hausdorff) spaces and X U , then for every pair of commutative ring spectra A and B there is a homeomorphism of mapping spaces ([8, VII, Theorem 2.9]) (3.4) CS (X A, B) = U (X, CS (A, B)). Here, CS denotes the (ordinary) category of commutative ring spectra in the sense of [8]. By [16, Corollary 4.4.4.9], (3.4) corresponds to an equivalence of mapping spaces of -categories (3.5) CAlg(X A, B) S(X, CAlg(A, B)). See also [22, 2] for a detailed account on tensors in -categories. 12 If we consider a commutative Hopf algebra spectrum H, then the space of maps CS (H, B) has a basepoint: the composition of the counit map H S followed by the unit map S B is a map of commutative ring spectra. The functor that takes an unbased space X to the topological sum of X with a point + is left adjoint to the forgetful functor the category of pointed spaces, Top , to spaces, so we obtain a homeomorphism (3.6) U (X, CS (H, B)) = Top (X+ , CS (H, B)) and correspondingly, an equivalence in the context of -categories (3.7) S(X, CAlg(H, B)) S (X+ , CAlg(H, B)). For path-connected spaces Z, May showed that the free En -space on Z, Cn (Z), is equivalent to n n Z [19, Theorem 6.1]. Segal extended this result to spaces that are not necessarily connected. He showed that for well-based spaces Z there is a model of the free E1 -space, C1 (Z), as follows: The spaces C1 (Z) and C1 (Z) are homotopy equivalent, C1 (Z) is a monoid, its classifying space BC1 (Z) is equivalent to (Z) [24, Theorem 2], and thus, C1 (Z) BC1 (Z) is a group completion. We can apply this result to Z = X+ because X+ is well-based, thus BC1 (X+ ) (X+ ). Note that BC1 (X+ ) (X+ ). Nikolaus gives an overview about group completions in the context of -categories [20]. He shows that for every E1 -monoid M , the map M BM gives rise to a localization functor of cateories in the sense of [16, Definition 5.2.7.2], such that the local objects are grouplike E1 -spaces. In particular, there is a homotopy equivalence of mapping spaces [16, Proposition 5.2.7.4] MapE1 (BC1 (X+ ), Y ) MapE1 (C1 (X+ ), Y ) if Y is a grouplike E1 -space. Here, E1 denotes the -category of E1 -spaces. If H is a commutative Hopf-algebra, then the space CAlg(H, B) is a grouplike E1 -space. Therefore, by using Equations (3.5) and (3.7), we obtain a chain of homotopy equivalences CAlg(X H, B) S(X, CAlg(H, B)) S (X+ , CAlg(H, B)) MapE1 (C1 (X+ ), CAlg(H, B)) MapE1 (BC1 (X+ ), CAlg(H, B)) MapE1 ((X+ ), CAlg(H, B)). If (X+ ) (Y+ ) is an equivalence of pointed spaces, then (X+ ) (Y+ ) as grouplike E1 -spaces and therefore we get a homotopy equivalence CAlg(X H, B) CAlg(Y H, B). Applying the Yoneda Embedding to the above equivalence yields the following result: Theorem 3.8. If H is a commutative Hopf algebra spectrum and if (X+ ) (Y+ ) is an equivalence of pointed spaces, then there is an equivalence X H Y H in CAlg. Remark 3.9. If X is a pointed simplicial set, then the suspension (X+ ) is equivalent to (X) S 1 . Therefore, if X and Y are pointed simplicial sets, such that (X) (Y ) as pointed simplicial sets, then we also obtain an equivalence between (X+ ) and (Y+ ). Segals result also works for larger n than 1. If two spaces are equivalent after an n-fold suspension, then an En -coalgebra structure on a Hopf algebra is needed for the Loday construction to be equivalent on these two spaces. There are indeed interesting spaces that are not equivalent after just one suspension, but that need iterated suspensions to become equivalent: 13 Christoph Schaper [23, Theorem 3] shows that for affine arrangements A one needs at least a (A + 2)-fold suspension in order to get a homotopy type that only depends on the poset structure of the arrangement. Here, A is a number that depends on the poset data of the arrangement, namely the intersection poset and the dimension function. For homology spheres, the double suspension theorem of James W. Cannon and Robert D. Edwards [6, Theorem in 11] states that the double suspension 2 M of any n-dimensional homology sphere M is homeomorphic to S n+2 . Here, a single suspension does not suffice unless M is an actual sphere. 4. Truncated polynomial algebras One way of showing that a commutative R-algebra spectrum A is not multiplicatively or linearly stable is to prove that the homotopy groups of the Loday construction LR T n (A) differ from those of W n R k W W L n (A), as in [7]. Here, we write (n) S for the k -fold -sum of S k . Indeed, there is a Sk k=1 k (nk) homotopy equivalence n _ _ (T n ) ( S k ). k=1 (n) k If A is augmented over R, then for proving that R A is not multiplicatively or additively stable, it suffices to show that R W W LR k (A; R). T n (A; R) 6 L n k=1 (n) S k See [14, 2] for details and background on different notions of stability. In the following we restrict our attention to Eilenberg-Mac Lane spectra of commutative rings and we will use this strategy to show that none of the commutative Q-algebras Q[t]/tm for m > 2 can be multiplicatively stable. We later generalize this to quotients of the form Q[t]/q(t) where q(t) is a polynomial without constant term and to integral and mod-p results. Pirashvili determined higher order Hochschild homology of truncated polynomial algebras of the form k[x]/xr+1 additively when k is a field of characteristic zero [21, Section 5.4] in the case of odd spheres. A direct adaptation of the methods of [4, Theorem 8.8] together with the flowchart from [5, Proposition 2.1] yields the higher order Hochschild homology with reduced coefficients for all spheres. See also [7, Lemma 3.4]. Proposition 4.1. For all m > 2 and n > 1 ( Q (xn ) Q[yn+1 ], [n],Q HH (Q[t]/tm ; Q) = Q[xn ] Q (yn+1 ), if n is odd, if n is even. In both cases Hochschild homology of order n is a free graded commutative Q-algebra on two generators in degrees n and n + 1, respectively, and the result does not depend on m. We will determine for which m and n we get a decomposition of the form Q m m W W (4.2) L Q k (Q[t]/t ; Q). T n (Q[t]/t ; Q) = L n k=1 (nk) S Note that the right-hand side is isomorphic to n O O L Q (Q[t]/tm ; Q) Sk k=1 (n) k where all unadorned tensor products are formed over Q. Thus, if we have a decomposition as in m (4.2), then we can read off the homotopy groups of LQ T n (Q[t]/t ; Q) with the help of Proposition 4.1. 14 Expressing Q[t]/tm as the pushout of the diagram t7tm Q[t] // Q[t] t70 Q allows us to express the Loday construction for Q[t]/tm , now viewed as a commutative HQ-algebra spectrum, as the homotopy pushout of the diagram LHQ T n (HQ[t]; HQ) t7tm // LHQ T n (HQ[t]; HQ) t70 HQ and so HQ m L LHQ T n (HQ[t]/t ; HQ) LT n (HQ[t]; HQ) LHQ (HQ[t];HQ) HQ. Tn As Q[t] is smooth over Q, LHQ T n (HQ[t]; HQ) is stable [7, Example 2.6]. So we can write HQ W LHQ T n (HQ[t]; HQ) L n k=1 W (nk) Sk (HQ[t]; HQ). Again, we obtain an isomorphism Wn L Q k=1 W n k ( ) (Q[t]; Q) = Sk n O O k=1 [k],Q HH (Q[t]; Q) n k ( ) and with the help of [5, Proposition 2.1] we can identify the terms as follows: ( Q[xk ], if k is even , [k],Q HH (Q[t]; Q) = Q (xk ), if k is odd. Lemma 4.3. There is an isomorphism of graded commutative Q-algebras Wn L Q k=1 W LQ T n (Q[t];Q) (nk) Sk Q (Q[t]/tm ; Q) = LT n (Q[t]; Q) Tor (Q, Q). Proof. We already know that (4.4) Wn L Q k=1 W (nk) (Q[t]/tm ; Q) = Sk n O O k=1 [k],Q HH (Q[t]/tm ; Q) = n k n O O k=1 ( ) gFQ (xk , yk+1 ) n k ( ) where gFQ (xk ) denotes the free graded commutative Q-algebra generated by an element xk in degree k and gFQ (xk , yk+1 ) denotes the free graded commutative Q-algebra generated by an element xk in degree k and an element yk+1 in degree k + 1. Nn N n gFQ (xk ), we obtain that As LQ k=1 T n (Q[t]; Q) = ( ) k LQ (Q[t];Q) (Q, Q) Tor T n = n+1 O O =2 ( gFQ (y ) n 1 ) and hence the tensor product of the two gives a graded commutative Q-algebra isomorphic to (4.4) 15 Q Let A denote the graded commutative Q-algebra LQ T n (Q[t]; Q) and B denote LT n (Q[t]; Q) viewed as an A -module via a morphism of graded commutative Q-algebras f : A B . Lemma 4.5. Let f1 : A B be the morphism f1 = B A where A : A Q is the augmentation that sends all elements of positive degree to zero and where B : Q B is the unit map of B . Let f2 : A B be any map of graded commutative algebras such that there is ,fi an element x An with n > 0 such that f2 (x) = w 6= 0. Let TorA , (B , Q) denote the graded Tor-groups calculated with respect to the A -module structure on B given by fi . Then A ,f2 ,f1 dimQ (Tor, (B , Q))n < dimQ (TorA , (B , Q))n L ,fi A ,fi where (TorA , (B , Q))n = r+s=n Torr,s (B , Q). Proof. Let P Q be an A -free resolution of Q. We want to choose P efficiently, in the following sense: since QLis concentrated in degree zero and A0 = Q, we can choose P0 to be A . Then we nj choose P1 = jI1 A with the minimal possible number of copies of A in each suspension degree, beginning from the bottom (that is, the only reason we add a new n A is if there is a class in P1 that has not yet been L hit bynjthe suspensions of A in lower dimensions that we already have) to guarantee that d1 : jI A0 P0 is injective, and moreover M M ker(d1 : nj A0 P0 ) nj ker(A ). jI1 jI1 And of course we need Im(d1 : P1 P0 ) = ker(A : A Q) and similarly for higher . For every > 0 we choose P with M P = nj A jI so that d : L nj jI A0 P1 is injective and moreover M M ker(d : nj A0 P1 ) nj ker(A ). jI Then we get jI M Im(d : P P1 ) = ker(d1 : P1 P2 ) nj ker(A ). jI1 The Tor groups we want are the homology groups of M M B A P = B A nj A n j B = jI jI with respect to the differential id d for either A -module structure. As f1 : A B factors through the augmentation, we claim that the differentials in the chain complex B A P with the A -module structure given by f1 L are trivial: they are of the form L id d where d is the differential of P . As d sends every nj 1 jI nj A to something in jI1 nj ker(A ), M (id d)(b A nj 1) Q{b} A nj ker(A ) = 0 jI1 for all b B . Hence A ,f1 Tor,s (B , Q) = ( M nj B )s = jI M jI 16 nj Bs . ,f1 In particular, we have of course TorA 0,s (B , Q) = Bs for all s. For the A -module structure on B given by f2 we obtain that ,f2 TorA 0, (B , Q) = B A Q but here, the tensor product results in a nontrivial quotient of B . Recall that we assumed that f2 (x) = w 6= 0. The element w 1 B A Q is trivial because the degree of x is positive and hence A (x) = 0: w 1 = f2 (x) 1 = 1 A (x) = 1 0 = 0. Therefore, A ,f1 ,f2 dimQ TorA 0,n (B , Q) < dimQ Tor0,n (B , Q). ,f2 The other Tor-terms in total degree n of the form TorA r,s (B , Q) with r + s = n are subquotients of M nj Bs jIr and hence for all (r, s) with r + s = n and r > 0 we obtain A ,f1 ,f2 dimQ TorA r,s (B , Q) 6 dimQ Torr,s (B , Q). Note that if f : A B factors through the augmentation A Q then A TorA (B , Q) = B Tor (Q, Q). We use Lemma 4.5 to prove the following result. Theorem 4.6. Let n > 2. Then Q n W dimQ n LQ T n (Q[t]/t ; Q) < dimQ n L n k=1 W n k ( ) Sk (Q[t]/tn ; Q). In particular, for all n > 2 the pair (Q[t]/tn ; Q) is not stable and Q Q[t]/tn is not multiplicatively stable. The n = 2 case of Theorem 4.6 was obtained earlier by Dundas and Tenti [7]. Before we prove the theorem, we state the following integral version of it: Corollary 4.7. For all n > 2 the pair (Z[t]/tn ; Z) is not stable and Z Z[t]/tn is not multiplicatively stable. Proof of Corollary 4.7. If for some n > 2 the pair (Z[t]/tn ; Z) were stable, then in particular LZT n (Z[t]/tn ; Z) = LZWn k=1 W (nk) S k (Z[t]/tn ; Z). Localizing at Z \ {0} would then imply Q n W L Q T n (Q[t]/t ; Q) = L n k=1 in contradiction to Theorem 4.6. W (nk) S k (Q[t]/tn ; Q) 17 4.1. Proof of Theorem 4.6. We prove Theorem 4.6 by identifying an element in A of positive degree that is sent to a nontrivial element of B . More precisely, we will show that the map that sends t to tn sends the indecomposable element in n LQ S n (Q[t]; Q) up to a unit to the element dt1 . . . dtn n LQ (Q[t]; Q). S 1 ...S 1 We consider both elements as elements of n LQ T n (Q[t]; Q) via the inclusions of summands Q n L Q (Q[t]; Q) n LQ T n (Q[t]; Q) n LS n (Q[t]; Q). S 1 ...S 1 In the following we consider T n as the diagonal of an n-fold simplicial set where every ([p1 ], . . . , [pn ]) ()n is mapped to Sp11 . . . Sp1n . Then LQ T n (Q[t]; Q) can also be interpreted as the diagonal of an n-fold simplicial Q-vector space with an associated n-chain complex. By abuse of notation we still denote this n-chain complex by LQ T n (Q[t]; Q). We use the following notation concerning the n-chain complex LQ T n (Q[t]; Q): 0m = (0, 0, . . . , 0) and 1m = (1, 1, . . . , 1) are the vectors containing only 0 or 1, respectively, repeated m times. A vector V Nn is viewed as a multi-degree of an element in the n-chain complex. A vector v Nn for which 0n 6 v 6 V in every entry can be thought of as specifying a coordinate in the multi-matrix of an element in multi-degree V. We call the ith entry of a vector v Nn the ith place in v. It is always assumed that V = 1n if not otherwise specified. Each element of LQ T n (Q[t]; Q) in degree V = (v1 , . . . , vn ) is a multi-matrix of dimension (v1 + 1, . . . , vn + 1) with entries in Q[t] at coordinates v 6= 0n and an entry in Q at coordinate 0n . xv for x Q[t] and v Nn is the multi-matrix with term x at coordinate v and 1 at other coordinates. We say a term is trivial if it is 1 in all its coordinates. Therefore xv yw for x, y Q[t] and v, w Nn is the product of xv and yw in degree V of LQ T n (Q[t], Q) regarded as an n-simplicial ring. Explicitly, if v 6= w, it is the multi-matrix with x at coordinate v, y at coordinate w, and 1 elsewhere; if v = w, it is the multi-matrix with xy at coordiante v and 1 elsewhere. Suppose that C is an n-chain complex with differentials d1 , . . . , dn in the n different directions, then the total chain complex Tot(C ) has differential in component (v1 , . . . , vn ) given by d= n X (1)v1 +...+vi1 di . i=1 Pvi In our case we will have each di = j=0 (1)j di,j where di,j : Cv1 ,...,vn Cv1 ,...,vi 1,...vn is the face map. We are interested in low degrees, especially in 1n . Any vi = 1 will imply di = 0 since the di are cyclic differentials and Q[t] is commutative. This allows us to eliminate the di from d. We have the following three lemmas about homologous classes and tori of different dimensions: Lemma 4.8 (Split Moving Lemma). Let a, b be coordinates in degree 1n1 (that is, in 22. . .2dimensional matrices). Then x(a,1) y(b,1) x(a,0) y(b,1) + x(a,1) y(b,0) . Proof. Their difference is a boundary of an element of degree (1n1 , 2): d(x(a,1) y(b,2) ) = (1)n1 dn (x(a,1) y(b,2) ) = x(a,0) y(b,1) x(a,1) y(b,1) + x(a,1) y(b,0) . 18 For example, when n = 2, a = 0, b = 1, the 1 x 1 x d = 1 1 y 1 difference is 1 1 x 1 x + . y 1 y y 1 Let b be a coordinate of a multi-matrix of an element in degree 1nm such that b 6= 0nm . For any multi-matrix c in degree W Nm , we can form the following multi-matrix in degree (W, 1nm ) Nn : at coordinate (a, 0nm ); ca c(,0) y(0,b) has terms yb at coordinate (0m , b); 1 elsewhere. Lemma 4.9. The following is a chain map: Q Tot(LQ T m (Q[t], Q)) Tot(LT n (Q[t], Q)); c 7 c(,0) y(0,b) . Proof. Clearly di (c(,0) y(0,b) ) = di c(,0) y(0,b) for 0 6 i 6 m. But since the multi-degree of c(,0) y(0,b) is V = (W, 1nm ) Nn and whenever vi = 1, di = di,0 di,1 = 0, we also get di (c(,0) y(0,b) ) = 0, for m < i 6 n. This lemma also applies when y(0,b) is replaced by another multi-matrix that has more than one nontrivial term, as long as the nontrivial terms are all in coordinates of the form (0m , b) for b in degree 1nm and b 6= 0nm . It has the following immediate corollary: Lemma 4.10 (Orthogonal Moving Lemma). Let b be a coordinate in degree 1nm such that b 6= 0nm . Let c, c be elements in multi-degree W Nm . If c c in multi-degree W, then c(,0nm ) y(0m ,b) c(,0nm ) y(0m ,b) in multi-degree (W, 1nm ) Conceptually, the moving lemmas tell us how to move the nontrivial elements x, y in certain multi-matrices to lower coordinates. They are stated for a special case for simplicity, but of course they work for any permulation of copies of Nn in the statement. The split moving lemma says that if we have xv and yw where the coordinates share a 1 in a particular place, the 1s can be moved to coordinate 0 separately. The orthogonal moving lemma says that the x in xv and the y in yw can be moved separately if they are supported in orthogonal tori (that is, have their nontrivial entries in different coordinates). Proposition 4.11. Let v and w be two coordinates of degree 1n . (1) If v and w are both 0 in the ith place for some 1 6 i 6 n, then xv yw 0. In particular, if v 6= 1n , then xv 0. (2) In general, xv y w X xv y w , v 6v,w 6w, v +w =1n where the sum is taken over all coordinates v and w such that They are place-wise no greater than v and w respectively; They take 1 in complementary places. 19 (3) For k > 1 and n > 1, we have the following homologous relation: X k (t )1n k Y tw i w1 ,...,wk 6=0n , i=1 w1 +...+wk =1n In particular, if k = n and we let ei denote the coordinate that has 1 at the ith place and 0 at other places, we get n (4.12) (t )1n n! n Y tei . i=1 Also, if k > n, this gives us (tk )1n 0 Proof. The class in (1) is a cycle because everything is in multi-degree 1n is a cycle; it is nullhomologous because it is in the image of the degeneracy si,0 in the ith place. For (2) we write |v| for the sum of the places of the vector v. We induct on |v| + |w|. Notice that a coordinate v of degree 1n is just a sequence of length n of 0s and 1s and |v| is just the number of 1s in it. For |v| + |w| 6 n, there are two cases: One is that v and w are both 0 in one place. Then the claim holds because the right-hand side is the empty sum and the left-hand side is 0 by part (1). The other case is that v + w = 1n . Then the claim also holds because the right-hand side has only one copy that is exactly the left-hand side. Assume that the claim is true for |v| + |w| 6 m where m > n and suppose now |v| + |w| = m + 1. Since m + 1 > n + 1, v and w have to be both 1 in some place. Without loss of generality, we assume that v = (v0 , 1), w = (w0 , 1) where v0 , w0 6 1n1 . By the Split Moving Lemma (Lemma 4.8), xv yw x(v0 ,0) yw + xv y(w0 ,0) . Since |(v0 , 0)| + |w| = |v| + |(w0 , 0)| = m, by inductive hypothesis we have that X X xv y w x(v0 ,0) yw + xv y(w0 ,0) v0 6v0 ,w 6w, (v0 ,0)+w =1n = X v 6v,w0 6w0 , v +(w0 ,0)=1n xv y w . v 6v,w 6w, v +w =1n For (3) we order the pair (k, n) by the lexicographical ordering. We induct on (k, n). When k = 1, the claim is trivially true. Suppose the claim is true for all pairs less than (k, n) where k > 2. Taking v = w = 1n , x = t and y = tk1 in part (2), we get that X X (4.13) (tk )1n tw1 (tk1 )v = tw1 (tk1 )v . w1 6=0n w1 +v =1n w1 +v =1n 20 The second step above uses that t0n = 0 because t is 0 in the Q[t]-module Q. Let m = |v |. By the inductive hypothesis, we have (4.14) (t k1 )1m k Y X twi w2 ,...,wk 6=0m , i=2 w2 +...+wk =1m For each wi which is a coordinate of degree 1m , we add in 0 in places where v is 0 to make it a coordinate of degree 1n . Denote it by wi . Then the Orthogonal Moving Lemma (Lemma 4.10), (4.13) and (4.14) combine to k X Y k (t )1n tw i . w1 ,...,wk 6=0n , i=1 w1 +...+wk =1n Qn For any n > 2, we call t1n the diagonal class and denote it by n . We call i=1 tei the volume form and denote it by voln . If we include S 1 T n as the ith coordinate and identify the first Hochschild homology group with the Kahler differentials, the generator dt of HHQ 1 (Q[t]; Q) maps to the generator we call dti in the Loday construction of the torus. In this sense voln corresponds to the degree-n class dt1 . . . dtn . Proof of Theorem 4.6. By Equation (4.12) we know that the map t 7 tn induces a map on L Q T n (Q[t]; Q), that sends the diagonal class, n , to n!voln . Hence, by Lemma 4.5 we know that Q n n W W dimQ n (LQ k (Q[t]/t ; Q)). T n (Q[t]/t ; Q)) < dimQ n (L n n S k=1 (k) In particular, Q n n W W (LQ k (Q[t]/t ; Q)). T n (Q[t]/t ; Q)) (L n n S k=1 (k) Remark 4.15. For the non-reduced Loday construction LQ T n (Q[t]), parts (1) and (2) of Proposition 4.11 are still true. Part (3) will become k (t )1n k Y X tw i w1 +...+wk =1n i=1 and Equation (4.12) is no longer true. 4.2. Q[t]/tm on T n for 2 6 m < n. We know that for Q[t]/tn we get a discrepancy between n of the Loday construction on the n-torus and that of the bouquet of spheres that correspond to the cells of the n-torus. We use this to first show that Q[t]/tm causes a similar discrepancy for 2 6 m < n. Proposition 4.16. Let 2 6 m 6 n. Then Q m W m L Q T n (Q[t]/t ; Q) m L n k=1 W n k ( ) Sk (Q[t]/tm ; Q). Proof. We consider the Tor-spectral sequence LQ (Q[t];Q) Tn Tor, Q m ( LQ T n (Q[t]; Q), Q) LT n (Q[t]/t ; Q) 21 Q m where the LQ T n (Q[t]; Q)-module structure on LT n (Q[t]; Q) is induced by t 7 t . The m-chain (m) complex C := LQ T m (Q[t]; Q) can be considered as an n-chain complex whose m + 1, . . . , ncoordinates are trivial. Then (m) C (n) = LQ T m (Q[t]; Q) C := LQ T n (Q[t]; Q) (n) is a sub-n-complex of C . We know that m 7 m!volm in the homology of the total complex of (m) (n) C and hence the same is true in C . Therefore the map Q m L Q T n (Q[t]; Q) m LT n (Q[t]; Q) m that is induced by t 7 tm is nontrivial and by Lemma 4.5 the dimension of m LQ T n (Q[t]/t ; Q) is strictly smaller than the dimension of Wn m L Q k=1 W n k ( ) Sk (Q[t]/tm ; Q). 4.3. Quotients by polynomials without constant term. Let q(t) = am tm + . . . + a1 t Q[t]. Then we can still write Q[t]/q(t) as a pushout Q[t] t7q(t) // Q[t] t70 // Q[t]/q(t) Q hence the above methods carry over. Proposition 4.17. Let m0 be the smallest natural number with 1 6 m0 6 m with am0 6= 0. Then Q W m0 LQ T m0 (Q[t]/q(t); Q) m0 L m0 k=1 W (mk0 ) S k (Q[t]/q(t); Q). Q Proof. If m0 = 1, then t HHQ 1 (Q[t]; Q) maps to (q(t)) HH1 (Q[t]; Q) under the map t 7 q(t). In the module of Kahler differentials this element corresponds to a1 dt + 2a2 tdt + . . . + mam tm1 dt but all these summands are null-homologous except for the first one. So t 7 a1 t 6= 0 and this, along with Lemma 4.5, proves the claim. We denote by m0 (q(t)) the element (q(t))1m0 . If m0 > 1, then the diagonal element m0 (t) maps to m X m0 (q(t)) = ai m0 (ti ) i=m0 and this is homologous to (m0 )!am0 volm0 + terms of higher t-degree by (4.12). Hence m0 (t) maps to a nontrivial element and again Lemma 4.5 gives the claim. 22 4.4. Truncated polynomial algebras in prime characteristic. We know that for commutative Hopf algebras A over k the Loday construction is stable, so Loday constructions of truncated polynomial algebras of the form Fp [t]/tp have the same homotopy groups when evaluated on an n-torus and on the corresponding bouquet of spheres. However, we show that there is a discrepancy for truncated polynomial algebras Fp [t]/tn for 2 6 n < p. Theorem 4.18. Assume that 2 6 n < p and n 6 m, then F F (LTpm (Fp [t]/tn ; Fp )) (LWpm k=1 In particular for all 2 6 n < p the pair (Fp [t]/tn ; F p) W m k ( ) Sk (Fp [t]/tn ; Fp )). is not stable. Proof. We consider the case m = n. The cases n < m follow by an argument similar to that for Proposition 4.16. As Fp [t] is smooth over Fp , we know that Fp Fp [t] is stable, so that F F (LTpn (Fp [t]; Fp )) = (LWpn k=1 [k],Fp and HH W n k ( ) (Fp [t]; Fp )) = Sk n O O k=1 [k],Fp HH (Fp [t]; Fp ) n k ( ) (Fp [t]; Fp ) is calculated in [4, 8] so that we obtain [k],Fp HH (Fp [t]; Fp ) = Bk+1 B B where B1 = Fp [t] and Bk+1 = Tor,k (Fp , Fp ) where the grading on Tor,k (Fp , Fp ) is the total F [2],F grading. Thus in low degrees this gives HH p (Fp [t]; Fp ) = Fp (t) with |t| = 1, HH p (Fp [t]; Fp ) = N F (0 t) with |0 t| = 2. 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Department of Mathematics, University of Oslo, Box 1053, Blindern, NO - 0316 Oslo, Norway E-mail address: aliceph@math.uio.no Department of Mathematics, Michigan State University, 619 Red Cedar Rd, East Lansing, MI 48840, USA E-mail address: klander2@msu.edu Department of Mathematics, Indiana University, 831 East 3rd Street, Bloomington, IN 47405, USA E-mail address: alindens@indiana.edu Fachbereich Mathematik der Universitat Hamburg, Bundesstrae 55, 20146 Hamburg, Germany E-mail address: birgit.richter@uni-hamburg.de Department of Mathmatics, University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA E-mail address: foling@math.uchicago.edu 24 ...