Modules possessing projective resolutions of finite type
Let \(k\) be a commutative ring, \(G\) be a group. Let \(R\) be a strongly \(G\)-graded \(k\)-algebra, that is, a \(k\)-algebra which admits a \(k\)-module decomposition \(R=\bigoplus_{g\in G} R_g\) such that \(R_gR_h=R_{gh}\) for all \(g,h\in G\). If \(H\) is a subgroup of \(G\) then \(R_H\) denotes the subalgebra supported on \(H\). An \(R\)-module \(M\) is said to be of type \(FP_\infty\) if and only if there is a projective resolution \(P_*\to M\) by finitely generated projective modules. A module is called elementary if it is of the form \(U\otimes_{R_E} R\) where \(E\) is a finite elementary Abelian subgroup of \(G\) and \(U\) is a finitely generated \(R_E\)-module which is projective over \(R_1\). The elementary modules are certainly of type \(FP_\infty\). A module \(M\) is called polyelementary if it has a series \[ \langle 0\rangle=M_0\leq M_1\leq\cdots\leq M_n=M, \] every factor of which is elementary. Let \(P_*\to M\) be a projective resolution of a module \(M\), then the \(n\)th kernel is the kernel of the map \(P_{n-1}\to P_{n-2}\) when \(n\geq 2\), the kernel of the map \(P_0\to M\) when \(n=1\), and \(M\) itself if \(n=0\). An \(R\)-module \(N\) is said to be stably flat over \(R\) if and only if \(\widehat{\text{Ext}}_R^0(M,N)=0\) for all \(R\)-modules \(M\) of type \(FP_\infty\). Further \(LH{\mathfrak F}\) denotes the class of locally hierarchically decomposable groups. The main results of this paper are the following Theorem A. Let \(G\) belong to \(LH{\mathfrak F}\) and suppose that \(R_1\) is coherent and has finite global dimension. Let \(M\) be an \(R\)-module of type \(FP_\infty\). Then there is an integer \(n\geq 0\) such that in any projective resolution \(P_*\to M\) of finite type the \(n\)th kernel is isomorphic to a direct summand of a polyelementary module. Theorem B. Let \(G\) belong to \(LH{\mathfrak F}\) and let \(R\) be a strongly \(G\)-graded \(k\)-algebra such that \(R_1\) is coherent. Let \(N\) be an \(R\)-module. Then the following assertions are equivalent: (1) \(N\) is stable flat as an \(R\)-module; (2) \(N\) is stable flat as an \(R_E\)-module for all finite elementary Abelian subgroups \(E\) of \(G\). When \(G\) is finite, the same result holds without the assumption that \(R_1\) is coherent.
locally hierarchically decomposable groups
scientific article; zbMATH DE number 1309248
scientific article