Subject: Re: PC and P series
Date: Fri, 24 May 2002 12:18:22 -0400 (EDT)
From: "Douglas C. Ravenel"
There is unlikely to be any more information of this sort. Let
V1_*, V2_* and V3_* be the three graded vector spaces, so we have
a long exact sequence
a_i b_i c_i a_{i-1}
V1_i -------> V2_i -------> V3_i -------> V1_{i-1} ---------->...
Let A(t), B(t) and C(t) be Poincare series for the ranks of the
graded maps a, b and c. Then we have
P1(t) = C(t)/t + A(t)
P2(t) = A(t) +B(t)
P3(t) = B(t) +C(t)
which leads to
(1+t)A(t) = -t*P1(t) - P2(t) + P3(t)
(1+t)B(t) = t*P1(t) - t*P2(t) - P3(t)
(1+t)C(t) = t*P1(t) - t*P2(t) + t*P3(t)
Setting t equal to -1 reduces each of these to your equation.
The other possibility is to set t equal to a zero of either A, B,
or C, but this requires knowledge of the morphisms. If C(t)=0
for all t, ie if the long exact sequence is a collection of short
exact sequence, then the third equation above reduces to
P2(t) = P1(t) + P3(t).
as expected.
Doug
On Fri, 24 May 2002, Don Davis wrote:
> __________________________________________________
>
> Subject: Question on Poincare series for the Topology list
> Date: Tue, 14 May 2002 16:38:37 +0100
> From: Martin Crossley
>
> In trying to find a painless way of introducing
> British undergraduates to homology, I hit the
> following problem:
>
> Suppose you have three graded vector spaces (with
> Poincare series P_1, P_2, P_3) which are linked by
> a long exact sequence (like the homology exact
> sequence of a pair). Then you have an equation like
> P_2(-1) = P_3(-1) + P_1(-1).
> But this does not contain as much information as
> the exact sequence, even if you suppose no knowledge
> of the morphisms.
> Is there a formula in the Poincare series that
> catches more of the information ?
>
> Martin Crossley
>
>
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