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 $|I|\to \infty$.

In other words, $R$ is the subring of formal power series $k[[t_{1},...,t_{n}]]$ which converge on $({\text{Val}}_{k})^{n}$, where ${\text{Val}}_{k}:=\{t\in k{\text{ }}|{\text{ }}|t|\leq 1\}$ 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

$\|f\|=\max _{I}|a_{I}|$

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 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