Ideals of minors in free resolutions
Let \(R\) be a commutative noetherian ring, \(M\) a finitely generated \(R\)- module, \(J = \text{Ann} M\), \(g = \text{grade} M\), and let \[ : \cdots \to F_ i @>\varphi_ i>> F_{i - 1} \to \cdots \to F_ 1 @>\varphi_ 1>> F_ 0 \to M \to 0 \] be a free resolution of \(M\). Let \(r_ i = \text{rank} \varphi_ i\). It is well-known that \(J^{r_ 1} \subseteq I_{r_ 1} (\varphi_ 1) \subseteq J\). The paper studies the extensions of these inclusions. Since the left-hand one can be sharpened to \(JI_ j (\varphi_ 1) \subseteq I_{j + 1} (\varphi_ 1)\), for all \(j < r_ 1\), it was conjectured by \textit{C. Huneke} that \(JI_ j (\varphi_ i) \subseteq I_{j + 1} (\varphi_ i)\), for all \(j < r_ i\). The authors prove that for all \(i,j \geq 0\), \((r_ i - j) JI_ j (\varphi_ i) \subseteq I_{j + 1} (\varphi_ i)\), which, in characteristic zero, gives the positive answer. Concerning the right-hand inclusion, the authors are proposing the ``idealistic syzygy conjecture'', which asserts that, if \(n_ i = \left( \begin{smallmatrix} g - 1 \\ i - 1 \end{smallmatrix} \right) - 1\), we have \(I_{r_ i - n_ i} (\varphi_ i) \subseteq J\), for \(g \geq i\). They discuss this conjecture and prove it for the special case \(R\) a regular ring and \(M = R/P\), with \(P \in \text{Spec} R\). -- In the final part, several properties and questions about the annihilators of exterior powers of syzygy modules and their connection with Huneke's conjecture are discussed.
scientific article; zbMATH DE number 706691