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

$P(x)=\sum_{n\ge 0}x^n \operatorname{Tor}^R_n(k,k) = \prod_{n\ge 0} \frac{(1+t^{2n+1})^{\varepsilon_{2n}}}{(1-t^{2n+2})^{\varepsilon_{2n+1}}}.$

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.