Tate algebra
In rigid analysis, a branch of mathematics, the Tate algebra over a complete ultrametric field k, named for John Tate, is the subring [math]\displaystyle{ R }[/math] of the formal power series ring [math]\displaystyle{ kt 1, ..., t n }[/math] consisting of [math]\displaystyle{ \sum a_I t^I }[/math] such that [math]\displaystyle{ |a_I| \to 0 }[/math] as [math]\displaystyle{ |I| \to \infty }[/math]. In other words, [math]\displaystyle{ R }[/math] is the subring of formal power series [math]\displaystyle{ kt 1, ..., t n }[/math] which converge on [math]\displaystyle{ (\text{Val}_k)^n }[/math], where [math]\displaystyle{ \text{Val}_k := \{ t\in k \text{ }| \text{ } |t| \leq 1 \} }[/math] is the valuation ring of [math]\displaystyle{ k }[/math].
The maximal spectrum of R is then a rigid-analytic space.
Define the Gauss norm of [math]\displaystyle{ f = \sum a_I t^I }[/math] in R by
- [math]\displaystyle{ \|f\| = \max_I |a_I| }[/math]
This makes R a Banach k-algebra.
With this norm, any ideal [math]\displaystyle{ I }[/math] of [math]\displaystyle{ T_n }[/math] is closed and [math]\displaystyle{ T_n/I }[/math] is a finite field extension of the ground field [math]\displaystyle{ k }[/math].
References
- Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold (1984), Non-archimedean analysis, Chapter 5: Springer
External links
- http://math.stanford.edu/~conrad/papers/aws.pdf
- https://web.archive.org/web/20060916051553/http://www-math.mit.edu/~kedlaya//18.727/tate-algebras.pdf
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