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 $R$ of the formal power series ring $kt1,...,tn$ consisting of $\sum a_I{t}^I$ such that $|a_I|\to 0$ as $|I|\to \mathrm{\infty }$. In other words, $R$ is the subring of formal power series $kt1,...,tn$ which converge on $({\text{Val}}_k{)}^n$, where ${\text{Val}}_k:=\{t\in k||t|\le 1\}$ is the valuation ring of $k$.

The maximal spectrum of R is then a rigid-analytic space.

Define the Gauss norm of $f=\sum a_I{t}^I$ in R by

$\Vert f\Vert =\underset{I}{\max }|a_I|$

This makes R a Banach k-algebra.

With this norm, any ideal $I$ of $T_n$ is closed and $T_n/I$ is a finite field extension of the ground field $k$.

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