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A criterion for exactness of a complex of length two. (English) Zbl 1097.13513

From the introduction: Let \(R\) be a commutative ring with an identity element, \(A=(a_{ij})\) a \((m\times n)\)-matrix with entries in \(R\) and \(\varphi_A:R^n\to R^m\) the homomorphism of \(R\)-modules defined by \(\varphi_A(\underline x)=A. \underline x\). We put \(I(A)=I_s (A)\), where \(s=\text{rank}_R(A)\) and \(I_k(A)\) is the ideal generated by the \(k\times k\)-minors. This paper is motivated by the question: What makes the complex \(R^p@>\varphi_B>> R^n@>\varphi_A>> R^m\) exact?
Suppose that \(R[T]\) is an integrally closed ring and \(\text{rank}_R(A)=n-1\). First we prove that \(\text{Ker}(\varphi_A)\) is isomorphic to an ideal constructed starting from \(I_{n-1}(A)\). As a consequence a structure result is obtained for the second syzygy ideals over an integrally closed domain. The exactness is characterized in terms of \(I_{n-1}(A)\) and \(I_1(B)\). In particular, when \(A\) is an square matrix, one has a criterion for the exactness of the complex \(R^n @>\varphi_{A^*}>>R^n @>\varphi_A>> R^n\), where \(A^*\) is the matrix of cofactors of \(A\). Finally we give a sufficient condition, that in some cases is also necessary, for the exactness of the periodic complex \[ \cdots\to R^n @>\varphi_A >>R^n@>\varphi_{A^*}>>R^n@>\varphi_A>>R^n\to\cdots. \]

MSC:

13D25 Complexes (MSC2000)
13D02 Syzygies, resolutions, complexes and commutative rings
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References:

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