In commutative algebra, the '''deviations of a local ring''' ''R'' are certain invariants ε<sub>''i''</sub>(''R'') that measure how far the ring is from being regular.
==Definition==
The deviations ε<sub>''n''</sub> of a local ring ''R'' with residue field ''k'' are non-negative integers defined in terms of its Poincaré series ''P''(''t'') by
: <math>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}}}. </math>
The zeroth deviation ε<sub>0</sub> is the embedding dimension of ''R'' (the dimension of its tangent space). The first deviation ε<sub>1</sub> vanishes exactly when the ring ''R'' is a regular local ring, in which case all the higher deviations also vanish. The second deviation ε<sub>2</sub> 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
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