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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?

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    I did a quick search through EGA, and IV-19.3.2 the following is proved: A quotient of a regular local noetherian ring is a complete intersection if and only if the ideal is generated by a regular sequence. So this means that the question reduces to whether a variety is locally a complete intersection (in the classical sense) if and only if it is cut out by a regular sequence. One direction holds (if it's cut out by a regular sequence, then the dimension is what you want), but the other seems much less obvious to me.2011-07-09

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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.

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    Well, not open anymore: http://arxiv.org/abs/1109.49212014-11-25