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Burch ideals and Burch rings. (English) Zbl 1459.13010

The ambient assumptions are that rings are commutative and Noetherian (and mostly (regular) local), all modules are finitely generated and all subcategories are full and strict. The authors of this paper are inspired by results of Lindsay Burch and other authors. For a local ring (\(R, \mathfrak m\)), an ideal \(I\) of \(R\) is called a Burch ideal if \(\mathfrak m I\neq\mathfrak m(I:\mathfrak m)\). A Burch ring of depth zero is a local ring whose completion is a quotient of a regular quotient ring mod a Burch ideal. Under mild conditions, the class of Burch ideals contains integrally closed ideals of codepth zero, \(\mathfrak m\)-full ideals, weakly \(\mathfrak m\)-full ideals, and others. Several characterizations of these objects are given and interesting properties exhibitted, along. One of the main results is as follows: Let (\(R,\mathfrak m,k\)) be a local ring and let \(I\neq\mathfrak m\) be an ideal of \(R\). Then \(I\) is Burch if and only if \(k\) is a direct summand of the second syzygy \(\Omega^2_{R/I}k\) of \(k\) over \(R/I\). Another of the results in the paper reads as follows: Let \(R\) be a Burch ring of depth \(t\) and let \(M, N\) be finitely generated \(R\)-modules. Assume that there is an integer \(\ell\geq\max\{3, t+1\}\) such that Tor\(_i^R(M,N)=0\) for all \(\ell+t\leq i\leq\ell+2t+1\). Then \(M\) or \(N\) has a finite projective dimension.

MSC:

13C13 Other special types of modules and ideals in commutative rings
13D09 Derived categories and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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