Operator index of a nonsingular algebraic curve

The talk is devoted to a scheme-theoretic analogue of the Fredholm theory. The continuity of the index function over the coordinate ring of an algebraic variety is investigated. It turns out that the index is closely related to the filtered topology given by finite products of maximal ideals. We show that a variety over an algebraically closed field possesses the index function on nonzero elements of its coordinate ring iff it is an algebraic curve, whose index is obtained by means of the multiplicity function from its normalization.