similar to defining the order of zeros and poles in complex analysis. Example on a projective variety For example, consider a projective surface defined by a polynomial , then the order of vanishing of a rational functionis given bywhereFor example, if and and thensince is a unit in the local ring . In the other case, is a unit, so the quotient module is isomorphic toso it has length . This can be found using the maximal proper sequence Zero and poles of an analytic function The order of vanishing is a generalization of the order of zeros and poles for meromorphic functions in complex analysis. For example, the functionhas zeros of order 2 and 1 at and a pole of order at . This kind of information can be encoded using the length of modules. For example, setting and , there is the associated local ring is and the quotient module Note that is a unit, so this is isomorphic to the quotient moduleIts length is since there is the maximal chainof submodules. More generally, using the Weierstrass factorization theorem a meromorphic function factors aswhich is a (possibly infinite) product of linear polynomials in both the numerator and denominator. See also Hilbert–Poincaré series Weil divisor Chow ring Intersection theory Weierstrass factorization theorem Serre's multiplicity conjectures Hilbert scheme - can be used to study modules on a scheme with a fixed length Krull–Schmidt theorem References External links Steven H. Weintraub, Representation Theory of Finite Groups AMS (2003) , Allen Altman, Steven Kleiman, A term of commutative algebra. The Stacks project. Length Module theory