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Free resolutions and sparse determinantal ideals. (English) Zbl 1273.14097
Let \(S\) denote a polynomial ring over \(K\), where \(K\) is a field or \(\mathbb{Z}\). A sparse generic matrix \(X\) is a matrix whose entries are distinct variables and zeros. Ideals generated by the maximal minors of \(X\) were studied by M. Giusti and M. Merle [Algebraic geometry, Proc. int. Conf., La Rabida/Spain 1981, Lect. Notes Math. 961, 103–118 (1982; Zbl 0509.14004)] who showed that the codimension, primeness, and Cohen- Macaulayness of such ideals depend only on the perimeter of the largest rectangle of zeros in \(X\). In the paper under review, the author studies homological invariants of these ideals and describes explicitely how to compute their minimal free resolution in terms of the arrangements of zeros of \(X\). This is done by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. The author’s technique provides minimal free resolutions in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott complex. As a consequence it is shown that sparse determinantal ideals have a linear resolution over \(\mathbb Z\), and that the projective dimension only depends on the number of columns of the matrix that are identically zero. Surprisingly all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular, the nonzero generators of \(X\) form a universal Gröbner basis.

14M12 Determinantal varieties
13D02 Syzygies, resolutions, complexes and commutative rings
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