Editing Length of a module (section)

== Use in multiplicity theory==
{{Main|Intersection multiplicity}}
For the needs of [[intersection theory]], [[Jean-Pierre Serre]] introduced a general notion of the [[multiplicity (mathematics)|multiplicity]] of a point, as the length of an [[Artinian local ring]] related to this point.

The first application was a complete definition of the [[intersection multiplicity]], and, in particular, a statement of [[Bézout's theorem]] that asserts that the sum of the multiplicities of the intersection points of {{mvar|n}} [[hypersurface|algebraic hypersurface]]s in a {{mvar|n}}-dimensional [[projective space]] is either infinite or is ''exactly'' the product of the degrees of the hypersurfaces.

This definition of multiplicity is quite general, and contains as special cases most of previous notions of algebraic multiplicity.

=== 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.