#11457: Witt Vectors
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Reporter: tdupu | Owner: roed
Type: enhancement | Status: new
Priority: major | Milestone: sage-5.11
Component: padics | Resolution:
Keywords: witt vectors, padic, rings | Merged in:
Authors: Taylor Dupuy, David Roe | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Dependencies:
Stopgaps: |
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Comment (by tdupu):
Replying to [comment:8 darij]:
> There are at least two issues with this implementation of Witt vectors:
>
> 1) The Frobenius map on the Witt vectors doesn't just take the p-th
power of every coordinate. This is only true if p = 0 in the base ring.
darij actually means in characteristic p base ring. The correct form of
the p-frobenius has the ghost component equation
w_{n}(f_p) = w_{n+1}
See Hazewinkel (5.27).
>
> 2) The definition of Witt vector addition and multiplication through the
ghost components don't work over a ring where p isn't invertible (or at
least a non-zero divisor). This is why every time this definition is used
to actually define the Witt vectors, it is accompanied by a functoriality
condition.
>
> I fear that fixing the issues requires rewriting the methods from
scratch.
The following two suggestions have been made:
1) Implementing a division by {{{ p^n }}} function. This is just a
morphism of modules from {{{ p^{n} R }}} to {{{ R/ann(p^{n} ) }}} .
2) Implementing a lifting function. If R has the form {{{ A/p^n A }}} then
we would like a function which takes elements of A to elements in {{{
A/p^{n+1} A }}}.
>
> I am also wondering whether anything speaks against implementing Witt
vectors for general nests as well (including the nest of all positive
integers, leading to "big Witt vectors").
I'm not sure what you mean here. Could you explain more or provide a link?
Suggestions for implementations of Witt-Burnside functors are also
welcome.
--
Ticket URL: <http://trac.sagemath.org/ticket/11457#comment:12>
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