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On cohomologically complete intersections. (English) Zbl 1157.13012

Let \(R\) be a noetherian local ring with maximal ideal \(\mathfrak m\) and \(I \subset R\) an ideal of height \(c\). We say that \(I\) is set-theoretic complete intersection if \(I\) is generated by \(c\) elements up to radical. In the present paper, the authors introduced a new notion which is weaker than one of set-theoretic complete intersection. That is, we say that \(I\) is cohomologically complete intersection if \(H_I^k(R) = 0\) for all \(k > c\). The main theorem of this paper is a characterization of cohomologically complete intersection ideals. Assume that \(R\) is Gorenstein. Then the authors showed that \(I\) is cohomologically complete intersection if and only if the Bass numbers of \(H_I^c(R)\) satisfy the equation \[ \mu^i(\mathfrak p, H_I^c(R)) = \delta_{i, \dim R_{\mathfrak p} - c} \quad \text{for any \(\mathfrak p \in V(I)\)}. \] That is, we can discriminate whether \(I\) is cohomologically complete intersection or not by seeing \(H_I^c(R)\) only.

MSC:

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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