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The \(\ast\)-transforms of acyclic complexes. (English) Zbl 1328.13018
“Let \(R\) be an \(n\)-dimensional Cohen-Macaulay local ring and \(Q\) be a parameter ideal of \(R\). Suppose that an acyclic complex \((F_\bullet,\varphi_\bullet)\) is given.” Let \(M\) denote \(\text{im}\varphi_1\); so, \(F_\bullet\) is a resolution of \(F_0/M\). This paper gives a concrete procedure for finding a resolution (called \(^*F_\bullet\), the \(*\)-transform of \(F_\bullet\)) of \(F_0/(M:_{F_0}Q)\). The complex \(^*F_\bullet\) is a subcomplex of the mapping cone obtained from the Koszul complex on a generating set for \(Q\) (tensored with \(F_n\)) and \(F_\bullet\).
The procedure has already been implemented when \(n=3\) and \(F_0\) has rank one in [K. Fukumuro et al., J. Algebra 384, 84–109 (2013; Zbl 1408.13034)]. Furthermore, this procedure is used in [K. Fukumuro et al., J. Commut. Algebra 7, No. 2, 167–187 (2015; Zbl 1331.13008)] to compare the symbolic powers and the saturation of various ordinary powers of certain determinantal rings.
MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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