Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by elements of the form $\mathrm{in}_{\deg}(r)$ with $r\in I$, where $\mathrm{in}_{\deg}(r)$ is the nonzero homogeneous component of $r$ with the least possible grading.

On the other hand, we may consider a decreasing $\mathbb Z$-filtration on $R$ with components $R_{\ge m}=\mathrm{span}(\{r\in R|\deg(r)\ge m\})$. This projects to a filtration on $R/I$. A simple fact is that the resulting associated graded ring is isomorphic to $R/\mathrm{in}_{\deg}(I)$. This can be made more general by considering gradings of $R$ by arbitrary totally ordered abelian semigroups in place of $\mathbb Z$.

Although this seems to be a very fundamental circumstance, it has rarely occurred to me in literature (if at all). Therefore I'm somewhat at loss on how to refer to this without reproving it each time. Is this, perhaps, named after someone or might there be some standard reference?


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