Three quick postings to Lazarev's question............DMD
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Subject: Re: posting and proposal
Date: Mon, 24 Nov 2003 10:13:28 -0500 (EST)
From: Robert Bruner
Regarding Andrey's question:
Certinly Sq^0 has nonzero kernel. For example, Sq^0(h_0^4) =
h_1^4 = 0. For a positive stem example, Sq^0(Ph_2) = h_3 g = 0.
(Ext^{5,32} = 0, so there is no chance for it to be nonzero.)
In high dimensions there are a lot of elements in the kernel,
but I'd have to dig a bit to extract the exact data.
Bob
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Subject: Re: posting
Date: Mon, 24 Nov 2003 17:05:00 +0100
From: Christian Nassau
Well, since Sq^0(h0) = h1, and h1^k = 0 for k>=4, Sq^0 is certainly not
injective; a chart that shows the image of Sq^0 for p=2 can be found at
http://www.nullhomotopie.de/ext2/index.html
One can see there that the image is very small, so the kernel ought
to be big.
Christian
> Subject: question for the toplist
> Date: Wed, 19 Nov 2003 19:57:16 +0000
> From: A.Lazarev@bristol.ac.uk
>
> Let A be the Steenrod algebra when p=2 or the algebra of reduced
> powers for p odd. The operation of raising to the pth power in A
> induces an operation on Ext_A(Z/p,Z/p) usually denoted by P^0.
>
> My question is what is known about the kernel of P^0 in the Ext-
> algebra? Particularly, is it nontrivial?
>
> Best,
> andrey
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Subject: Re: posting and proposal
Date: Mon, 24 Nov 2003 13:12:18 -0800
From: John H Palmieri
Here is a response to Lazarev:
When p=2, it is nontrivial: the element h_0^4 in Ext^{4,4} is in the
kernel. (This is the "first" element in the kernel, in the sense that
P^0 is injective on the 1-, 2-, and 3-lines of Ext.) There are many
elements in the kernel that are not in the ideal (h_0^4), but I don't
know any explicit examples.
At odd primes, I hope that the kernel is nontrivial, but I don't
know how to prove it.
I have a relevant preprint:
What I know about P^0 acting on Ext is discussed mainly in the first
few pages of the introduction (see 1.2, 1.3, and 1.4 in particular).
I hope this helps. I'd be interested to hear anything anyone else
knows about this.
John
--
J. H. Palmieri
Dept of Mathematics, Box 354350 mailto:palmieri@math.washington.edu
University of Washington http://www.math.washington.edu/~palmieri/
Seattle, WA 98195-4350