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

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

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