On the Fitting ideals in free resolutions
If \(R\) is a ring and if \(\varphi : F \to G\) is a map of finitely generated free \(R\)-modules, then \(I_ i (\varphi)\) \((i \geq 0)\) is the ideal of \(R\) generated by \(i \times i\) minors of a matrix representing \(\varphi\) and the rank of \(\varphi\) is the largest number \(t\) such that \(I_ i (\varphi) \neq 0\). If \((R,m,K)\) is a \(d\)-dimensional complete Noetherian local ring containing a field with maximal ideal \(m\) and residue class field \(K = R/m\), then by the Cohen structure theorem, \(R \cong K X_ 1, \dots, X_ n/(f_ 1, \dots, f_ l)\) for some indeterminates \(X_ 1, \dots, X_ n\) and some power series \(f_ 1, \dots, f_ l \in K X_ 1, \dots, X_ n \). The Jacobian ideal of \(R\) is defined to be \(I_ k (\partial (f_ 1, \dots, f_ l)/ \partial (X_ 1, \dots, X_ n)) R\), that is, the ideal of \(R\) generated by the image of \(h \times h\) minors of the Jacobian matrix \((\partial (f_ 1, \dots, f_ l)/ \partial (X_ 1, \dots, X_ n))\), where \(h = \text{height} (f_ 1, \dots, f_ l)\). The main results of this paper are as follows. 1. If \((R,m,K)\) is a \(d\)-dimensional reduced complete Cohen-Macaulay ring containing a field and \(J\) is the Jacobian ideal of \(R\), then \(J \text{Ext}_ R^{d + 1} (M,N) = 0\) for every pair of finitely generated \(R\)- modules \(M\) and \(N\). Moreover, if \(M\) is a finitely generated \(R\)-module having a well-defined rank and \((F_ \bullet, \varphi)\) is any finitely generated free resolution of \(M\), then \((t_ j - i) JI_ i (\varphi_ j) \subseteq I_{i + 1} (\varphi_ j)\) for \(i = 0, \dots, t_ j - 1\) and \(j \geq d + 1\), where \(t_ j = \text{rank} \varphi_ j\). 2. If \(R\) is equidimensional and either \(\text{char} K = 0\) or \(K\) is perfect, then there exists an integer \(k\) such that: (a) \(J^ k \text{Ext}_ R^{d + 1} (M,N) = 0\) for every pair of finitely generated \(R\)-modules \(M\) and \(N\); and (b) if \(M\) is a finitely generated \(R\)-module having a well-defined rank and \((F_ \bullet, \varphi_ \bullet)\) is any finitely generated free resolution of \(M\), then \((t_ j - i) J^ k I_ i (\varphi_ j) \subseteq i_{j + 1} (\varphi_ j)\) for \(i=0, \dots, t_ j - 1\) and \(j \geq d + 1\), where \(t_ j = \text{rank} \varphi_ j\).
scientific article; zbMATH DE number 731838