Deviation of a local ring explained

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(t) by

P(t) = \sum _{n\ge} t ^{n} \operatorname{Tor} ^{R} _{n(k,k)} = \prod _{n\ge} (( 1+ t ^{2 n +1} ) ^{\varepsilon _{2n}})/(( 1- t ^{2 n +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.