Are complete intersections complete intersections?

Journal of Algebra (Impact Factor: 0.6). 09/2011; 371. DOI: 10.1016/j.jalgebra.2012.08.006
Source: arXiv


A commutative local ring is generally defined to be a complete intersection
if its completion is isomorphic to the quotient of a regular local ring by an
ideal generated by a regular sequence. It has not previously been determined
whether or not such a ring is necessarily itself the quotient of a regular ring
by an ideal generated by a regular sequence. In this article, it is shown that
if a complete intersection is a one dimensional integral domain, then it is
such a quotient. However, an example is produced of a three dimensional
complete intersection domain which is not a homomorphic image of a regular
local ring, and so the property does not hold in general.

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Available from: David A. Jorgensen, Jul 21, 2014
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    • "In this section we discuss depth properties of tensor products visà vis vanishing of Tor over complete intersections. Since the terminology in the literature is not entirely standard (as one might surmise from the title of [18]), let us lay out the definitions we will use. Throughout this section R is a local ring. "
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    • "It is a standard fact that a local complete intersection ideal of height r is a complete intersection if and only if it is generated by r elements. The following two well-known lemmas are from [1] and [11] respectively, and one can find both in Murthy's paper [10]. "
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    ABSTRACT: Let A be a commutative Noetherian ring of Krull dimension n. Let I be a local complete intersection ideal of height n. Suppose (A/I) is torsion in K0(A). It is proved that I is a set theoretic complete intersection if one of the following conditions holds: (1) n is two; (2) n is odd; (3) n⩾4 even, and A contains the field of rational numbers.
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