TITLE: What is the connection between the definition of complete intersection variety and complete intersection ring?
QUESTION [7 upvotes]: An algebraic variety is called a complete intersection if its defining ideal is generated
by codimension many polynomials.
A Noetherian local ring $R$ is called a complete intersection if its completion is the factor ring of a regular local ring by a regular sequence.
What is the connection between these two definitions?

REPLY [3 votes]: I  heard it is open whether a local complete intersection (in your second sense), is actually a complete intersection. That is, if a local ring $R$ is such that $\hat R\cong T/(\underline x)$, $T$ regular, $\underline x$ regular on $T$, then must $R$ be isomorphic to $Q/(\underline y)$, $Q$ regular, $\underline y$ regular on $Q$? 
But I found this on MO which is probably more relevant to your actual question.