A typical example of a Tate vector space over a field k are the Laurent power series It has two characteristic features: Tate modules were introduced by Drinfeld (2006) to serve as a notion of infinite-dimensional vector bundles.
These are characterized by the property that they have a base of the topology consisting of commensurable sub-vector spaces.
This construction can be iterated and yields an exact category Ind(Pro(C)).
Braunling, Groechenig & Wolfson (2016) showed that Tate objects (for C the category of finitely generated projective R-modules, and subject to the condition that the indexing families of the Ind-Pro objects are countable) are equivalent to countably generated Tate R-modules in the sense of Drinfeld mentioned above.
The construction can therefore be iterated, which is relevant to applications in higher-dimensional class field theory,[2] which studies higher local fields such as Kapranov (2001) has introduced the so-called determinant torsor for Tate vector spaces, which extends the usual linear algebra notions of determinants and traces etc.
Clausen (2009) has applied this torsor to simultaneously prove the Riemann–Roch theorem, Weil reciprocity and the sum of residues formula.