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 $k[[t_{1},...,t_{n}]]$ consisting of $\sum a_{I}t^{I}$ such that $|a_{I}|\to 0$ as $I\to \infty$ . 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

\|f\|=\max _{I}|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 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
http://www-math.mit.edu/~kedlaya/18.727/tate-algebras.pdf

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