Talk:Divided power structure

Formatting

Somebody more experienced than I am please check the formatting for the references section. Do I need to put a link to the reference somewhere near the beginning?

Schepler 22:48, 18 November 2006 (UTC)

Be more specific on the dual-to-symmetric-algebra example?

I'm wondering whether it would be worth it to indicate exactly what the PD structure on $(S^{\cdot }M){\check {~}}$ is, or whether it is a bit too complex and would obscure things.

If I included it, it would go something like:

Addition is just the normal pointwise addition of functions. For multiplication, given $\phi ,\psi :S^{\cdot }M\to A$, their product $\phi \psi :S^{\cdot }M\to A$ is defined so that for $x_{1},\ldots ,x_{n}\in M$,

$(\phi \psi )(x_{1}\cdots x_{n})=\sum _{S\subseteq \{1,2,\ldots ,n\}}\phi \left(\prod _{i\in S}x_{i}\right)\psi \left(\prod _{j\notin S}x_{j}\right).$

The set I of functions $\phi$ such that $\phi (1)=0$ can easily be seen to be an ideal with respect to this ring structure. Then defining $\gamma _{m}\phi :S^{\cdot }M\to A$ such that

$(\gamma _{m}\phi )(x_{1}\cdots x_{n})=\sum _{\pi \in P_{n,m}}\prod _{S\in \pi }\phi \left(\prod _{i\in S}x_{i}\right)$

gives a divided power structure on I. Here $P_{n,m}$ denotes the set of (unordered) partitions of $\{1,2,\ldots ,n\}$ into m parts.

(Note that by definition, $(\phi ^{m})(x_{1}\cdots x_{n})$ is equal to the corresponding sum where $\pi$ ranges over ordered partitions of $\{1,2,\ldots ,n\}$ into m parts, thus making the above definition of the PD structure a natural one.)

Daniel Schepler 15:20, 21 November 2006 (UTC)