Regular sequence ($f_1,\dots ,f_r$)

A regular sequence in a commutative ring $A$ is a finite list such that $(f_1,\dots ,f_r)$ is proper and $f_j$ is a non-zero-divisor on $A/(f_1,\dots ,f_{j-1})$ for each $j$. In a graded ring with homogeneous $f_j$, injectivity can be checked on homogeneous elements because components of distinct degrees cannot cancel. It gives short exact sequences $0\to R_{j-1}(-\mathrm{deg}{f}_j)\to R_{j-1}\to R_j\to 0$ for the successive graded quotients.

The depth of a Noetherian local ring is the maximal length of an $R$-regular sequence in its maximal ideal. It is at most its Krull dimension. Quotienting by a nonzero divisor lowers depth by one. This invariant measures how many independent successive nonzero-divisor equations the local algebra supports.