commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

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

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

is from being

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

Definition

The deviations ε_''n'' of a

local ring In mathematics, more specifically in 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 algebraic varieties or manifolds, or of ...

''R'' with

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

''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 are ...

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 any Noetherian local ring with unique maxi ...

, 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 commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections. Informally, they can be thought of roughly as the local rings that can be defined using the "min ...

, in which case all the higher deviations vanish.

References

* Commutative algebra {{commutative-algebra-stub