# Deviation of a local ring

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

${\displaystyle P(x)=\sum _{n\geq 0}x^{n}\operatorname {Tor} _{n}^{R}(k,k)=\prod _{n\geq 0}{\frac {(1+t^{2n+1})^{\varepsilon _{2n}}}{(1-t^{2n+2})^{\varepsilon _{2n+1}}}}.}$

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

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