nLab Tate vector space

Definition

Given vector subspaces $V_0$ and $V_1$ of a vector space $V$, we write $V_0\prec V_1$ if $V_0/(V_0\cap V_1)$ is finite-dimensional. We write $V_0\sim V_1$ and say $V_0$ and $V_1$ are commensurable if $V_0\prec V_1$ and $V_1\prec V_0$.

A Tate vector space is a complete Hausdorff topological vector space $V$ that admits a basis of neighborhoods of 0 whose elements are mutually commensurable vector subspaces of $V$.

Duality

A vector subspace $W$ of a Tate vector space $V$ is bounded if for every open vector subspace $U\subset V$ we have $W\prec U$.

The dual of a Tate vector space $V$ is the dual vector space $Hom(V,\mathbf{C})$ equipped with a topology generated by the basis of neighborhoods of 0 whose elements are orthogonal complements to bounded subspaces of $V$.

Properties

Tate vector spaces form a pre-abelian category.

References

• John Tate: Residues of differentials on curves: Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 1 (1968) no. 1, pp. 149-159. [doi:10.24033/asens.1162]

• Alexander Beilinson, Boris Feigin, Barry Mazur: Notes on conformal field theory (1991) [pdf]