Let $U_\infty\Z_p$ be the group of infinite invertible upper triangular matrices with entries in the $p$-adic integers. Also let $\Aut_{\text{left-}\ell\text{-mod}}^0(\ell\wedge\ell)$ be the group of left $\ell$-module automorphisms of $\ell\wedge\ell$ which induce the identity on mod $p$ homology, where $\ell$ is the Adams summand of the $p$-adically complete connective $K$-Theory spectrum. In this thesis we construct and prove there is an isomorphism between these two groups. We will then determine a specific matrix (up to conjugacy) which corresponds to the automorphism $1\wedge\psi^q$ of $\ell\wedge\ell$ where $\psi^q$ is the Adams operation and $q$ is an integer which generates the $p$-adic units $\Z_p^\times$. We go on to look at the map $1\wedge\phi_n$ where $\phi_n=(\psi^q-1)(\psi^q-r)\cdots(\psi^q-r^{n-1})$ and $r=q^{p-1}$ under a generalisation of the map which gave us the isomorphism. Lastly we use some of the ideas presented to give us a new way of looking at the ring of degree zero operations on the connective $p$-local Adams summand via upper triangular matrices.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:555132 |
| Date | January 2011 |
| Creators | Stanley, Laura |
| Contributors | Whitehouse, Sarah |
| Publisher | University of Sheffield |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.whiterose.ac.uk/2015/ |
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