# Deviation of a local ring

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

The zeroth deviation ε_{0} is the embedding dimension of *R* (the dimension of its tangent space). The first deviation ε_{1} vanishes exactly when the ring *R* is a regular local ring, in which case all the higher deviations also vanish. The second deviation ε_{2} vanishes exactly when the ring *R* is a complete intersection ring, in which case all the higher deviations vanish.

## References

- Gulliksen, T. H. (1971), "A homological characterization of local complete intersections",
*Compositio Mathematica*,**23**: 251–255, ISSN 0010-437X, MR 0301008

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