0s Z=p [a1] (2; 3) Z=p [p]
Now we can consider the next differential in the spectral sequence. The subm*
*odule
complementary to the subalgebra
Z=p [a1] (2; 3) Z=p [p]
is a torsion module, and the subalgebra is torsion-free. Subsequent differentia*
*ls are the
dr for r > p, have bidegree (-1; r), and are derivations of Z=p [a1]-algebras. *
*Since
0s

of filtration degree greater than*
* p, and
since algebra demands that dr() , we find that the elements of ar*
*e cycles
for dr for every r > p. Algebraic reasons do not rule out the possibility that*
* the next
nonvanishing differential on an element of the complementary subalgebra
Z=p [a1] (2; 3) Z=p [p]
has a component in , but the filtration argument does in fact rule this out*
*. Thus
is a Z=p [a1]-submodule of E1 annihilated by ap1. We will show that the *
*higher
differentials introduce higher a1-torsion so that S1 is a set of Z=p [a1]-modul*
*e generators
36
for the summands of E1 isomorphic to Z=p [a1]=(ap1) as claimed in the theorem.*
* Also,
we have concluded that the next nonvanishing differential on
Z=p [a1] (2; 3) Z=p [p]
has only a torsion-free component, so it is detected in the localized spectral *
*sequence.
Putting this discussion together with the preceding theorem we obtain the rest *
*of P (1),
namely, that Ep+1 = Er(2), and that the formulas
dr(2)(x) = 0 for x 2 S1 [ {2; 3}
and
dr(2)(p) = ar(2)12
determine the next differential.
The argument for deducing P (n) from P (n - 1) is in outline exactly the sam*
*e as the
argument we have just finished, so we will just quickly repeat the steps. Accor*
*ding to
P (n - 1) we have the decomposition
n-1
E*;*r(n)= Z=p [a1] (n; n+1) Z=p [p ];
and we can calculate Er(n)+1from P (n - 1), first deriving the formula
nmn+pn-1m : fflnffln+1pnmn pn-1 m -1 r(n)
dr(n)(fflnnffln+1n+1p n-1) = n n+1 mn-1( ) n-1 (a1 n);
where ffli = 0 or 1, 0 mn-1 p - 1, and mn 0. One finds that the a1-torsion
submodule of E*;*r(n)+1is spanned by
Sn = Sn-1[
nmn ffln+1pn-1 pnmn ffln+1pn-1(p-2)pnmn
{ffln+1n+1np ; n+1 (n ) ; . .;.n+1 (n ) ; where mn 0 }
E*;0r(n)+1
and that each x 2 Sn - Sn-1 generates a submodule isomorphic to Z=p [a1]=(ar(n)*
*1). This
gives us
0s and the bide*
*gree
of dr that the next differential is determined in the localized spectral sequen*
*ce. By
reference to theorem 6.1 we obtain Er(n)+1= Er(n+1)and the desired formulas for*
* dr(n+1),
completing the proof of P (n).
Thus, by the stage E1 the torsion-free subalgebra has been reduced to Z=p [*
*a1] and
one has seen that the inductive steps from P (n - 1) to P (n) present E1 as a *
*sum of
cyclic modules over Z=p [a1] of the required a1-order. |
37
References
[1] Adams, J.F. Stable homotopy and generalized homology. University of Chicag*
*o Press,
1974.
[2] Araki, S. and Toda, H. Multiplicative structures in mod q cohomology theor*
*ies, I.
Osaka J. Math. 2(1965), 71-115.
[3] B"okstedt, M. Topological Hochschild homology. Preprint, 1987.
[4] B"okstedt, M. The topological Hochschild homology of Z and Z=p. Preprint, *
*1987.
[5] B"okstedt, M. The natural transformation from K(Z) to T HH(Z). Preprint, 1*
*987.
[6] B"okstedt, M. and Waldausen, F. The map BSG -! A(*) -! QS0.
[7] Bruner, R.R. The homotopy theory of H1 ring spectra. Lecture Notes in Math*
*emat-
ics, Vol 1176, 88-128. Springer, 1986.
[8] Cartan, H. and Eilenberg, S. Homological algebra. Princeton University Pre*
*ss, 1956.
[9] Dickson, L.E. History of the theory of numbers, Vol. I. Carnegie Insitute *
*of Wash-
ington, 1919.
[10] Elmendorf, A.D. Foundations of topological Hochschild homology. In prepara*
*tion.
[11] Goodwillie, T. Cyclic homology, derivations, and the free loopspace. To*
*pology
24(1985), 187-215.
[12] Kane, Richard. Memoir of the AMS.
[13] Kummer, D. U"ber die Erg"anzungss"atze zu den allgemeinen Reciprocit"atsge*
*setzen.
Journal f"ur die Reine und Angewandte Mathematik 44(1852), 93-146.
[14] Lang, S. Algebra. Addison-Wesley, 1965.
[15] May, J.P. Simplicial objects in algebraic topology. Van Nostrand, 1967.
[16] May, J.P. Geometry of iterated loop spaces. Lecture Notes in Mathematics, *
*Vol 271.
Springer, 1972.
[17] Miller, H.R. and Ravenel, D.C. Morava stabilizer algebras and the localiza*
*tion of
Novikov's E2-term. Duke Math.J.44 (1977), 433-447.
[18] Ravenel, D.C. Complex cobordism and stable homotopy groups of spheres. Aca*
*demic
Press, 1986.
38
[19] Robinson, A. Derived tensor products in stable homotopy theory. Topology 2*
*2(1983),
1-18.
[20] Singmaster, D. Divisibility of binomial and multinomial coefficients by pr*
*imes and
prime powers. A collection of manuscripts related to the Fibonacci sequenc*
*e, 18th
anniversary volume (V. Hoggatt, Jr. and M. Bicknell-Johnson, eds), The Fib*
*onacci
Association, Santa Clara, California, 1980, 98-113.
[21] Steinberger, M. Homology operations for H1 and Hn ring spectra. Lecture N*
*otes in
Mathematics, Vol 1176, 56-87. Springer, 1986.
[22] Waldhausen, F. Algebraic K-theory of spaces, a manifold approach.
[23] Waldhausen, F. Algebraic K-theory of spaces, localization, and the chromat*
*ic filtra-
tion of stable homotopy.
[24] Waldhausen, F. Algebraic K-theory of spaces, II.
[25] unexplained. AA
[26] unexplained. II
[27] unexplained. GG
[28] unexplained. WW
[29] unexplained. MM
[30] unexplained. SV
[31] unexplained. Jones
[32] unexplained. Wlecture
[33] unexplained. GoCal
[34] unexplained. GoLet
[35] unexplained. BHM
[36] unexplained. RavAmJ
[37] unexplained. Einfty
[38] unexplained. BrownPet
[39] unexplained. E2 term
39