A local complete intersection ring is a Noetherian local ring whose completion is the quotient of a regular local ring by an ideal generated by a regular sequence. Taking the completion is a minor technical complication caused by the fact that not all local rings are quotients of regular ones. For rings that are quotients of regular local rings, which covers most local rings that occur in algebraic geometry, it is not necessary to take completions in the definition. There is an alternative intrinsic definition that does not depend on embedding the ring in a regular local ring. If R is a Noetherian local ring with maximal idealm, then the dimension of m/m2 is called the embedding dimension emb dim of R. Define a graded algebraH as the homology of the Koszul complexwith respect to a minimal system of generators of m/m2; up to isomorphism this only depends on R and not on the choice of the generators of m. The dimension of H1 is denoted by ε1 and is called the first deviation of R; it vanishes if and only ifR is regular. A Noetherian local ring is called a complete intersection ring if its embedding dimension is the sum of the dimension and the first deviation: There is also a recursive characterization of local complete intersection rings that can be used as a definition, as follows. Suppose that R is a complete Noetherian local ring. If R has dimension greater than 0 and x is an element in the maximal ideal that is not a zero divisor then R is a complete intersection ring if and only if R/ is. If R has dimension 0, then showed that it is a complete intersection ring if and only if the Fitting ideal of its maximal ideal is non-zero.
Examples
Regular local rings are complete intersection rings, but the converse is not true: the ring is a 0-dimensional complete intersection ring that is not regular.
Complete intersection local rings are Gorenstein rings, but the converse is not true: the ring is a 0-dimensional Gorenstein ring that is not a complete intersection ring.
An example of a locally complete intersection ring which is not a complete intersection ring is given by which has length 3 since it is isomorphic as a vector space to.
