Editing Length of a module (section)

=== Order of vanishing of zeros and poles ===
{{technical|section and subsections|date=May 2020}}
A special case of this general definition of a multiplicity is the order of vanishing of a non-zero algebraic function <math>f \in R(X)^*</math> on an algebraic variety. Given an [[algebraic variety]] <math>X</math> and a [[Subvariety (mathematics)|subvariety]] <math>V</math> of [[codimension]] 1<ref name=":1" /> the order of vanishing for a polynomial <math>f \in R(X)</math> is defined as<ref>{{Cite web|title=Section 31.26 (0BE0): Weil divisors—The Stacks project|url=https://stacks.math.columbia.edu/tag/0BE0|website=stacks.math.columbia.edu|access-date=2020-05-22}}</ref><math display=block>\operatorname{ord}_V(f) = \text{length}_{\mathcal{O}_{V,X}}\left( \frac{\mathcal{O}_{V,X}}{(f)} \right)</math>where <math>\mathcal{O}_{V,X}</math> is the local ring defined by the stalk of <math>\mathcal{O}_X</math> along the subvariety <math>V</math><ref name=":1" /> <sup>pages 426-227</sup>, or, equivalently, the [[Stalk of a sheaf|stalk]] of <math>\mathcal{O}_X</math> at the generic point of <math>V</math><ref>{{Cite book|last=Hartshorne|first=Robin|url=http://link.springer.com/10.1007/978-1-4757-3849-0|title=Algebraic Geometry|date=1977|publisher=Springer New York|isbn=978-1-4419-2807-8|series=Graduate Texts in Mathematics|volume=52|location=New York, NY|doi=10.1007/978-1-4757-3849-0|s2cid=197660097 }}</ref> <sup>page 22</sup>. If <math>X</math> is an [[affine variety]], and <math>V</math> is defined the by vanishing locus <math>V(f)</math>, then there is the isomorphism<math display=block>\mathcal{O}_{V,X} \cong R(X)_{(f)}</math>This idea can then be extended to [[rational function]]s <math>F = f/g</math> on the variety <math>X</math> where the order is defined as<ref name=":1" /><math display=block>\operatorname{ord}_V(F) := \operatorname{ord}_V(f) - \operatorname{ord}_V(g) </math> which is similar to defining the order of zeros and poles in [[complex analysis]].

==== 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>

==== 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.