Deviation of a local ring
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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(t) by
P ( t ) = ∑ n ≥ 0 t n Tor n R ( k , k ) = ∏ n ≥ 0 ( 1 + t 2 n + 1 ) ε 2 n ( 1 − t 2 n + 2 ) ε 2 n + 1 . {\displaystyle P(t)=\sum _{n\geq 0}t^{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.