Editing Length of a module (section)

==== Zero and poles of an analytic function ====
The order of vanishing is a generalization of the order of zeros and poles for [[meromorphic function]]s in [[complex analysis]]. For example, the function<math display=block>\frac{(z-1)^3(z-2)}{(z-1)(z-4i)}</math>has zeros of order 2 and 1 at <math>1, 2 \in \mathbb{C}</math> and a pole of order <math>1</math> at <math>4i \in \mathbb{C}</math>. This kind of information can be encoded using the length of modules. For example, setting <math>R(X) = \mathbb{C}[z]</math> and <math>V = V(z-1)</math>, there is the associated local ring <math>\mathcal{O}_{V,X}</math> is <math>\mathbb{C}[z]_{(z-1)}</math> and the quotient module <math display=block>\frac{\mathbb{C}[z]_{(z-1)}}{((z-4i)(z-1)^2)}</math>Note that <math>z-4i</math> is a unit, so this is isomorphic to the quotient module<math display=block>\frac{\mathbb{C}[z]_{(z-1)}}{((z-1)^2)}</math>Its length is <math>2</math> since there is the maximal chain<math display=block>(0) \subset \frac{\mathbb{C}[z]_{(z-1)}}{((z-1))} \subset {\displaystyle {\frac {\mathbb {C} [z]_{(z-1)}}{((z-1)^{2})}}}</math>of submodules.<ref>{{Cite web|title=Section 10.120 (02MB): Orders of vanishing—The Stacks project|url=https://stacks.math.columbia.edu/tag/02MB|website=stacks.math.columbia.edu|access-date=2020-05-22}}</ref> More generally, using the [[Weierstrass factorization theorem]] a meromorphic function factors as<math display=block>F = \frac{f}{g}</math>which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.