Reference request: associated graded algebras

Question

To every filtered algebra $A$ we can associate a graded algebra $\operatorname{gr} A$ (more details here). That graded algebra is usually a simpler object than the algebra we started with, and we can deduce some properties of $A$ from looking at $\operatorname{gr} A$.

For an example, if $\operatorname{gr} A$ is an integral domain, then $A$ is also an integral domain.

I was wondering if anyone knows a good reference where those relationships are collected. It would be great if they come with a proof, but I'll take just a list of statements too.

Answer 1

There are a number of links available if you comb through this website, e.g.,

• if $\bigcap R^n=0$ and the associated graded $\text{gr}(R)$ of a ring is a domain, then $R$ is a domain.
• a counterexample to $R$ domain implies $\text{gr}(R)$ domain.

Here is a pretty good reference: Zariskian Filtration by L. Huishi and F. van Oystaeyen.

Here is another (online version): Commutative Algebra by Keerthi Madapusi, begin on page 21.