Minimal resolutions and the homology of matching and chessboard complexes (Q1575098)

The authors investigate the graded minimal free resolutions of certain modules over two types of rings: (i) the Segre product of \(K[X_1, \dots,X_m]\) and \(K[Y_1,\dots, Y_n]\), and (ii) the \(d\)-th Veronese subalgebra of \(K[X_1, \dots,X_m]\). Both these rings are affine semigroup rings. The structure of their minimal free resolutions with respect to the natural representation as a residue class ring is determined by simplicial complexes that are isomorphic to matching and chessboard complexes (with \(d=2\) in case (ii)). On the other hand, these rings are naturally presented as residue class rings of polynomial rings modulo the 2-minors of generic (or, in case (ii), \(d=2\), symmetric) matrices of indeterminates, and their minimal free resolutions have been determined in characteristic 0 by Lascoux, Pragacz, Weyman and others by methods of representation theory. The authors extend these results to certain naturally defined modules over the mentioned rings, characterize the Cohen-Macaulay ones among them, and transfer the information available for ideals of minors to chessboard and matching complexes. [The result on the Cohen-Macaulay property of Segre\((m,n,r)\) has previously been proved: \textit{A. Guerrieri}, Proc. Am. Math. Soc. 127, No. 3, 657-663 (1999; Zbl 0915.13008)].