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Algebras of minors. (English) Zbl 1015.13004

The study of the Rees algebra of ideals in Noetherian rings has attracted the attention of many researchers in the last decades. The authors of this paper give an interesting contribution by studying the Rees algebra of determinantal ideals.
Let \(X=(x_{ij})\) be a generic \(m\times n\) matrix over a field \(K\), and let \(S\) be the polynomial ring \(K[x_{ij}]\). Let \(I_t\) be the ideal generated by the \(t\)-minors of \(X\), \({\mathcal R}_t\) the Rees algebra of \(I_t\), and \(A_t\) the special fiber of \({\mathcal R}_t\), that is, the subalgebra of \(S\) generated by the \(t\)-minors of \(X\). W. Bruns and A. Conca in: Geometric and combinatoric aspects of commutative algebra, Int. Conf. Commutative Algebra Algebraic Geometry, Messina 1999, Lect. Notes Appl. Math. 217, 67-87 (2001; Zbl 0991.13006), proved that for every \(t\), and in non-exceptional characteristic, \({\mathcal R}_t\) and \(A_t\) are Cohen-Macaulay normal domains. In the case of maximal minors of \(X\), i.e. when \(t=\min(m,n)\), the canonical class and the divisor class group of \({\mathcal R}_t\) have been determined. In the paper under review the authors determine the canonical class of \({\mathcal R}_t\) and \(A_t\), and also the divisor class group of \(A_t\), when \(t < \min(m,n)\). The main tools are the \(\gamma\)-functions and the Sagbi bases deformation by which it is possible to lift the canonical module from the initial algebras of \({\mathcal R}_t\) and \(A_t\) to the algebras \({\mathcal R}_t\) and \(A_t\) themselves.
In the last section the same techniques are applied to study the case of a generic Hankel matrix. Also in this case the canonical module and the canonical class are determined.

MSC:

13C40 Linkage, complete intersections and determinantal ideals
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics

Citations:

Zbl 0991.13006
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References:

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