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Deviation of a local ring

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Source updated August 12, 2023

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.

01Definition

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

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

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.

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