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. Promi ...

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

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

is from being

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

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 ...

''R'' with

residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is ...

''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 This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings ...

of ''R'' (the dimension of its tangent space). The first deviation ε₁ vanishes exactly when the ring ''R'' is a

regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal id ...

, 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