Three quick postings to Lazarev's question............DMD ________________________________________________________ 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 _____________________________________________________ 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 ________________________________________________________ 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