Deviation of a local ring - Alchetron, the free social encyclopedia

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

Definition

The deviations ε_n 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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Sabih Azhar

Irwan Prayitno