# Is a Gorenstein ring a quotient of a local complete intersection

The title says it all - Suppose you are given a noetherian Gorenstein local ring $(A,m,k)$ of finite Krull dimension.

Does there exist a local complete intersection ring $B$ such that $A$ is a homomorphic image of $B$?

In the answer to this question

A local ring not a quotient of a regular local ring

This paper was given

http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118785530&page=record

where there is an example of a Gorenstein ring which is not a quotient of a regular ring, but it is not clear to me if it cannot be expressed as a quotient of a local complete intersection.

• What is your definition of a local complete intersection ring? For me, every such ring is a quotient of a regular local ring. Thus the negative answer to the original question implies a negative answer to this question. – Jason Starr May 30 '14 at 17:16
• I thought that c.i means that its completion is a quotient of a regular local ring... – user51400 May 30 '14 at 17:18
• "I thought that c.i. means that its completion is a quotient of a regular local ring." I guess that you are correct, at least according to Matsumura. In algebraic geometry, there is a relative notion of local complete intersection, e.g., "LCI morphisms" or an "LCI $R$-algebra" $A$. That definition definitely does require that $A$ is Zariski locally a quotient of a smooth $R$-algebra. So the different usages are somewhat inconsistent. – Jason Starr May 30 '14 at 17:30
• And they are not equivalent: arxiv.org/abs/1109.4921 – Graham Leuschke May 30 '14 at 22:33