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 ${\displaystyle k[[t_{1},...,t_{n}]]}$ consisting of ${\displaystyle \sum a_{I}t^{I}}$ such that ${\displaystyle |a_{I}|\to 0}$ as ${\displaystyle I\to \infty }$. The maximal spectrum of R is then a rigid-analytic space.

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

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

This makes R a Banach k-algebra.

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