Tate algebra

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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[[t1,...,tn]]displaystyle k[[t_1,...,t_n]]displaystyle k[[t_1,...,t_n]] consisting of ∑aItIdisplaystyle sum a_It^Idisplaystyle sum a_It^I such that |aI|→0a_Ia_I as |I|→∞II. The maximal spectrum of R is then a rigid-analytic space.


Define the Gauss norm of f=∑aItIdisplaystyle f=sum a_It^Idisplaystyle f=sum a_It^I in R by


‖f‖=maxI|aI|=max _I=max _I

This makes R a Banach k-algebra.


With this norm, any ideal Idisplaystyle II of Tndisplaystyle T_nT_n is closed and Tn/Idisplaystyle T_n/Idisplaystyle T_n/I is a finite field extension of ground field Kdisplaystyle KK.



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



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