commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...

, the deviations of a local ring ''R'' are certain invariants ε_''i''(''R'') that measure how far the

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is from being regular.

Definition

The deviations ε_''n'' of a

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...

''R'' with residue field ''k'' are non-negative integers defined in terms of its Poincaré series ''P''(''t'') by :

P(t)=\sum_t^n \operatorname^R_n(k,k) = \prod_ \frac.

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

* Commutative algebra {{abstract-algebra-stub