Ideals defined by matrices and a certain complex associated with them

Abstract

For each matrix, whose elements belong to a commutative ring with an identity element, there is defined a free complex. This complex is a generalization of the standard Koszul complex, which corresponds to the case of a matrix with only a single row. The applications are to certain ideals defined by the maximal subdeterminants of a matrix. It is found that such an ideal has finite projective dimension whenever its grade reaches a certain greatest value (depending on the dimensions of the matrix) and that, in these circumstances, the complex provides a free resolution of the correct length. For semi-regular (= MacaulayCohen) rings this leads to a theorem on unmixed ideals. In the case of arbitrary Noetherian rings, a general theorem on rank is proved.