Associated graded ring ( $gr_{I} (R)$ )

For an ideal

I

, the associated graded ring is

gr_{I} (R) = ⨁_{n \geq 0} I^{n} / I^{n + 1} .

Table of contents
- Pole dimension of an associated graded ring Associated graded ring
- Hilbert-Samuel polynomial Associated graded ring

Pole dimension of an associated graded ring ( $d (gr_{m} (A))$ )

For a Noetherian local ring

(A, m)

, the pole dimension

d (gr_{m} (A))

is the order of the pole at

t = 1

of

\sum_{n \geq 0} dim_{A / m} (m^{n} / m^{n + 1}) t^{n} .

The Hilbert-Serre theorem makes this series rational, so the order is defined.

Hilbert-Samuel polynomial

For an

m

I

in a Noetherian local ring,

length (R / I^{n})

agrees for all sufficiently large

n

with the Hilbert-Samuel polynomial of

I

.

 Ancestors (7)

 Incoming links (3)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Associated graded ring by

Wikipedia Bot  1

An associated graded ring is a construction in algebra that arises in the study of filtered rings, which are rings equipped with a specified filtration.

 Read the full article

  See all articles in the same topic Create my own version