= Regular sequence
{title2=$f_1,\ldots,f_r$}
A regular sequence in a <commutative ring> $A$ is a finite list such that $(f_1,\ldots,f_r)$ is proper and $f_j$ is a <non-zero-divisor> on $A/(f_1,\ldots,f_{j-1})$ 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> $0\to R_{j-1}(-\deg f_j)\to R_{j-1}\to R_j\to0$ for the successive graded quotients.
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