Editing Length of a module (section)

==== Example on a projective variety ====
For example, consider a [[projective surface]] <math>Z(h) \subset \mathbb{P}^3</math> defined by a polynomial <math>h \in k[x_0,x_1,x_2,x_3]</math>, then the order of vanishing of a rational function<math display=block>F = \frac{f}{g}</math>is given by<math display=block>\operatorname{ord}_{Z(h)}(F) = \operatorname{ord}_{Z(h)}(f) - \operatorname{ord}_{Z(h)}(g) </math>where<math display=block>\operatorname{ord}_{Z(h)}(f) = \text{length}_{\mathcal{O}_{Z(h),\mathbb{P}^3}}\left( \frac{\mathcal{O}_{Z(h),\mathbb{P}^3}}{(f)} \right)</math>For example, if <math>h = x_0^3 + x_1^3 + x_2^3 + x_2^3</math> and <math>f = x^2 + y^2</math> and <math>g = h^2(x_0 + x_1 - x_3)</math> then<math display=block>\operatorname{ord}_{Z(h)}(f) = \text{length}_{\mathcal{O}_{Z(h),\mathbb{P}^3}}\left( \frac{\mathcal{O}_{Z(h),\mathbb{P}^3}}{(x^2 + y^2)} \right) = 0</math>since <math>x^2 + y^2</math> is a [[Unit (ring theory)|unit]] in the [[local ring]] <math>\mathcal{O}_{Z(h),\mathbb{P}^3}</math>. In the other case, <math>x_0 + x_1 - x_3</math> is a unit, so the quotient module is isomorphic to<math display=block>\frac{\mathcal{O}_{Z(h), \mathbb{P}^3}}{(h^2)}</math>so it has length <math>2</math>. This can be found using the maximal proper sequence<math display=block>(0) \subset \frac{\mathcal{O}_{Z(h), \mathbb{P}^3}}{(h)} \subset \frac{\mathcal{O}_{Z(h), \mathbb{P}^3}}{(h^2)}</math>