Regular sequence

In the context of commutative algebra and algebraic geometry, a regular sequence is a fundamental concept that relates to the properties of ideals and modules over a ring.

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A

is a finite list such that

(f_{1}, \dots, f_{r})

is proper and

f_{j}

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} (- de g f_{j}) \to R_{j - 1} \to R_{j} \to 0

for the successive graded quotients.