The present paper is devoted to a scheme-theoretic analog 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 prove that a variety over a field possesses the index function on nonzero elements of its coordinate ring iff it is an algebraic curve. In this case, the index is obtained by means of the multiplicity function from its normalization if the ground field is algebraically closed.
Algebraic variety, index of an operator tuple, integral extension, Dedekind extension, Taylor spectrum, Koszul homology groups of a variety
"Operator index of a nonsingular algebraic curve,"
Turkish Journal of Mathematics: Vol. 47:
7, Article 11.
Available at: https://journals.tubitak.gov.tr/math/vol47/iss7/11