Associated graded module
= Associated graded module
{title2=$\operatorname{gr}M$}
For an increasing <filtration of a module> $M_n$ compatible with a <filtered algebra> $A$, set $\operatorname{gr}M=\bigoplus_n M_n/M_{n-1}$. Multiplication on the quotients makes it a graded <module> over $\operatorname{gr}A$. Its cumulative homogeneous dimensions equal $\dim M_n$ for a finite-dimensional exhaustive nonnegative <filtration of a module>.