Article

# Are complete intersections complete intersections?

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

ABSTRACT 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. "
##### Article: Torsion in tensor products, and tensor powers, of modules
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ABSTRACT: For finitely generated modules M and N over a complete intersection R, the vanishing of Tor_i^R(M,N) for all i> 0 gives a tight relationship among depth properties of M, N and their tensor product. Here we concentrate on the converse and show, under mild conditions, that the tensor product of M and N being torsion-free (or satisfying higher Serre conditions) forces vanishing of Tor. Special attention is paid to the case of tensor powers of a single module.
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• "Since J = J ′ + J 2 , we can find e ∈ J 2 such that (1 − e) J ⊂ J ′ and J = ( J ′ , e). Therefore by ([9] "
##### Article: Some results on Euler class groups
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ABSTRACT: Let A be a regular domain of dimension d containing an infinite field and let n be an integer with 2n\geq d+3. For a stably free A-module P of rank n, we prove that (i) P has a unimodular element if and only if the euler class of P is zero in E^n(A) and (ii) we define Whitney class homomorphism w(P):E^s(A)\ra E^{n+s}(A), where E^s(A) denotes the sth Euler class group of A for s\geq 1.
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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]. "
##### Article: On the equations defining points
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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.
Journal of Algebra 05/2006; 299(2):679-688. DOI:10.1016/j.jalgebra.2005.09.029 · 0.60 Impact Factor