Linear equations over commutative rings and finite free resolutions (Q1322058)

Let \(R\) be a commutative ring with identity and \(A = (a_{ij})\) an \(m \times n\) matrix with elements in \(R\). Let \(\varphi_ A : R^ n \to R^ m\) denote the \(R\)-module homomorphism defined by \(\varphi_ A(x) = A \cdot x\). Let \(\text{rank}_ R(A) = r\), where \(r\) is the largest non- negative integer such that \({\mathcal U} (A) = {\mathcal U}_ r(A) \neq(0)\), where \({\mathcal U}_ p (A)\) is the ideal of \(R\) generated by all \(p \times p\) minors of \(A\). The author's investigations are motivated by the question: when has the system of equations \(A \cdot x = b\) a solution in \(R\)? The following two theorems are the principal results of the paper. Theorem 1.6 Let \(A\) be an \(m \times n\) matrix and assume there exists an exact sequence \[ 0 \to F_ t @> \varphi_ t >> F_{t-1} @> \varphi_{t-1} >> \cdots \to F_ 2 @> \varphi_ 2 >> R^ n @> \varphi_ A >> R^ m, \] where each \(F_ p\) is a finite free \(R\)- module. Suppose that \({\mathcal U} (A) = (g)a\) where \(g\) is a nonzero divisor in \(R\) and the polynomial grade \(\text{Gr} \{a\} \geq 2\). Then the following statements are equivalent: (a) \(\text{Gr} [{\mathcal U} (\varphi_ p)\} \geq p+1\) for all \(p \geq 2\). (b) For \(b \in R^ m\) the system of linear equations \(A \cdot x = b\) has a solution in \(R\) if and only if (i) \(\text{rank}_ R(A) = \text{rank}_ R (A | b)\) and (ii) \({\mathcal U}_ r (A | b) \subseteq (g)\). Theorem 2.5. Let \(R\) be a unique factorization domain and let \(A\) be an \(m \times n\) matrix. Assume that there exists an exact sequence \[ 0 \to F_ t @> \varphi_ t >> F_{t-1} @> \varphi_{t-1} >> \cdots F_ 2 @> \varphi_ 2 >> R^ n @> \varphi_ A >> R^ m @> C >> M \to 0, \] where each \(F_ p\) is a finite free \(R\)-module. If \(g\) is a greatest common divisor of the maximal minors of \(A\) then the following statements are equivalent: (a) For \(b \in R^ m\) the system of linear equations \(A \cdot x = b\) has a solution in \(R\) if and only if (i) \(r = \text{rank}_ R (A) = \text{rank}_ R (A | b)\) and (ii) \({\mathcal U}_ r (A | b) \subseteq (g)\). (b) There exist elements \(z_ 1\), \(z_ 2, \dots, z_ m\) of \(R\), where some \(z_ p\) can be equal to zero, such that \(M\) is contained in \(\oplus^ m_{i=1} R/(z_ i)\). (c) \(M\) is contained in \(E \oplus F\) where \(E\) is an elementary \(R\)- module and \(F\) is a finite free \(R\)-module. (d) \(\text{Gr}_ R \{{\mathcal U} (\varphi_ p)\} \geq p + 1\) for all \(p \geq 2\).