Composition series and length of a module

Depends on

• Submodule of a module
• Quotient module $M/N$ with scalar multiplication on additive cosets
• Simple module: a nonzero module with no proper nonzero submodule

Used by

• Cellular decomposition matrices are unitriangular for the cell order Corollary
• Module length is additive in short exact sequences Corollary
• A common component makes the intersection sum infinite Counterexample
• Contact order of two regular components at a point Definition
• Degree of a nonconstant morphism of curves Definition
• Intersection multiplicity of closed subschemes at a point Definition
• koszul euler characteristic and degree indexed multiplicity Definition
• Local intersection multiplicity of two plane curves Definition
• Rational equivalence and the Chow group of cycles Definition
• The different divisor of a generically separable morphism of curves Definition
• The Hilbert function and formal Hilbert series of a graded module with finite-length pieces Definition
• The Hilbert-Samuel function and eventual Hilbert-Samuel polynomial of a finite local module Definition
• Total length of a zero-dimensional projective scheme Definition
• A field has module length one over itself Example
• A tangent line and conic have one intersection point of local length two Example
• In dimension zero the Hilbert-Samuel polynomial is constant and equals the module length Example
• Line multiplicities at a cusp Example
• Lines through a node and its two branches Example
• Ramification indices of the power map on the projective line Example
• The ℤ-module ℤ/pᵏ has length k Example
• A point blowup drops pairwise intersection multiplicity by at least one Lemma
• A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings Lemma
• An invertible quotient of an invertible subsheaf by a torsion sheaf is a twist by an effective divisor Lemma
• Blowing up a non-regular point strictly increases the finite normalization subalgebra Lemma
• Cycles of coherent sheaves and of closed subschemes, with flat pullback Lemma
• Fibre degree of the finite locally free map to the projective line Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• Finite local length exactly when no common local branch Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• For a finite-length module, the radical is a superfluous submodule Lemma
• Global intersection length is the sum of the local multiplicities Lemma
• Hilbert series and eventual Hilbert value of a two-form plane complete intersection Lemma
• Intersection with a line is the order of vanishing of the restricted equation Lemma
• Intersection with a smooth curve is a vanishing order Lemma
• Invariance of the local intersection multiplicity Lemma
• Lengths of truncated plane local rings Lemma
• Local flatness criterion by regular parameters Lemma
• Local support and index bound for the different of a curve map Lemma
• Nonzero-form cellular heads, cell purity and shared-factor linkage Lemma
• Projective representations and twisted algebra modules Lemma

…and 14 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Arvind Nair, Algebra I, Lecture 5 (standard reference, not scraped)