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.

References

• http://math.stanford.edu/~conrad/papers/aws.pdf
• http://www-math.mit.edu/~kedlaya/18.727/tate-algebras.pdf