Two responses to Jim Lin's question about generators of H^*(K(Z/p,n);Z/p)...DMD
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From: Nick Kuhn <njk4x@adams.math.virginia.edu>
Date: Wed,  2 Jun 99 17:35:56 -0500
Subject: Re: question about generators

An answer to Jim Lin:

I assume by his question that he wants to know the degrees of the
algebra generators of the ring  H^*(K(Z/p,n);Z/p).  Here is an
answer when p=2. (odd would presumably be similar.)

Let F(n) be the free unstable module on an n dimensional class.  It
is well known that F(1) has basis x, x^2, x^4, ... with x one
dimensional, and that F(n) is the Sigma_n invariants of F(1)^otimes
n.  Let DF(n) denote the "double" of F(n). The description of
H^*(K(Z/2,n);Z/2) as the free unstable A-algebra on an n-dimensional
class makes it easy to see that the Poincare series of
QH^*(K(Z/2,n);Z/2) = Poincare series of F(n)/DF(n).
But F(n)/DF(n) = SF(n-1) (S= one suspension).

Thus we get:

PUNCHLINE  The number of algebra generators in H^d(K(Z/2,n);Z/2)
will equal the number of distinct ways of writing d-1 as a sum of
n-1 powers of 2.

EXAMPLE   Since 6 = 4 + 1 + 1 = 2 + 2 + 2, there are two algebra
generators in H^7(K(Z/2,4);Z/2).

REMARK To get explicit representatives for the indecomposables one
could use the Milnor basis, as Jim suspected.

Nick Kuhn

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Date: Wed, 2 Jun 1999 17:21:05 -0500 (EST)
From: Clarence Wilkerson <wilker@math.purdue.edu>
Subject: generators


Using the Milnor basis, I think this is basically a subset of the
following problem:

Let B = F_p[ x_1,........] be a polynomial algebra on generators
x_i of degree n_i, with
n_{i+1} > n_i for all i. Let J be the augmentation ideal.

Problem: Compute the Poincare series of J^r or B/J^r .

Clarence