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(t) by

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

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.

References

{{Citation | last1=Gulliksen | first1=T. H. | title=A homological characterization of local complete intersections | url=http://www.numdam.org/item?id=CM_1971__23_3_251_0 |mr=0301008 | year=1971 | journal=Compositio Mathematica | issn=0010-437X | volume=23 | pages=251–255}}

Category:Commutative algebra