Tate's linear algebra and residues on curves (2018)
Let X be a projective smooth curve over an algebraically closed field k. Let K = k(X) the field of rational functions on X and O_X,p the regular local ring on p, for p in X. Since X is smooth and dim X = 1, the completion of the regular local ring in p is isomorphic to ring of power series in one variable t, for t a choice of uniformizing parameter in p. Then K_p, is isomorphic to the ring of Laurent series in one variable t. In this situation, just as in complex analysis, one could define for a differential fdg, f,g in K_p, the residue in p to be the coefficient of t^-1 in the Laurent expansion of f(t)g'(t). However, it is not obvious that such coefficient is independent of the choice of uniformizing parameter t. In his article, Residues of differentials on curves, John Tate gave a free-coordinate approach to residues. In this document we explore the concept of Tate vector spaces, topological vector spaces that abstract topological properties of k((t)). This is done in order to understand Tate's construction in terms of topological properties. In this way it is shown the invariance of the residue formula and the residue theorem