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The associated graded ring and associated graded module of an ideal-adic filtration
Definition
Let be a commutative ring, let be an ideal, and let be an -module. The associated graded ring of the -adic filtration is Multiplication is induced by multiplication in :
The associated graded module is viewed as a graded -module by
Depends on
Used by
- Regular centers have projective-bundle exceptional divisors Corollary
- Multiplicity of a hypersurface equation at a rational point Definition
- The Hilbert-Samuel function and eventual Hilbert-Samuel polynomial of a finite local module Definition
- The Rees algebra of an ideal and the Rees module of a filtered module Definition
- The scheme-theoretic tangent cone at a point Definition
- associated graded polynomial map singular kernel Example
- dvrs as regular local rings Example
- The associated graded ring of a regular local ring and of a cusp local ring can be computed explicitly Example
- The Hilbert-Samuel multiplicity of a plane-curve singularity is read from its associated graded ring Example
- All initial forms define the tangent cone Lemma
- Associated graded algebra of an ideal generated by a regular sequence Lemma
- associated graded polynomial surjection Lemma
- shifted adic koszul filtration euler comparison Lemma
- The module-relative Hilbert–Samuel polynomial exists without a dimension theorem Lemma
- The exceptional divisor is the projectivized normal cone Theorem
- The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form Theorem
Dependency tree · one level
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Sources
- Stacks Project, Section 10.57: Graded modules (standard reference, not scraped)
- Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, §20 (standard reference, not scraped)