Harman Patil (Editor)

Deviation of a local ring

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In commutative algebra, the deviations of a local ring R are certain invariants εi(R) that measure how far the ring is from being regular.

Definition

The deviations εn of a local ring R with residue field k are non-negative integers defined in terms of its Poincaré series P(x) by

P ( x ) = n 0 x n Tor n R ( k , k ) = n 0 ( 1 + t 2 n + 1 ) ε 2 n ( 1 t 2 n + 2 ) ε 2 n + 1 .

The zeroth deviation ε0 is the embedding dimension of R (the dimension of its tangent space). The first deviation ε1 vanishes exactly when the ring R is a regular local ring, in which case all the higher deviations also vanish. The second deviation ε2 vanishes exactly when the ring R is a complete intersection ring, in which case all the higher deviations vanish.

References

Deviation of a local ring Wikipedia


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