×

A Bezout theorem for determinantal modules. (English) Zbl 0844.13007

Let \(R\) be the graded polynomial ring \(\mathbb{C} [x_1, \dots, x_n]\), where to each variable \(x_i\) is assigned a positive weight \(a_i\), not necessarily 1. Let \(F = \bigoplus^{n + k - 1}_{i = 1} R(- d_i)\) and \(G = \bigoplus^k_{i = 1} R (- e_i)\) be graded free modules over \(R\), \(\varphi\) be a matrix which represents a homogeneous \(R\)-module homomorphism \(F \to G\) of degree zero, and \(M\) be the cokernel of \(\varphi\). Assume that \(M\) is a finite dimensional vector space over \(\mathbb{C}\). In this case, \(\dim_\mathbb{C} M\) depends on \(\{d_i\}\), \(\{e_i\}\), and \(\{ a_i\}\); but not on the particular entries of \(\varphi\), or even, for that matter, on the module \(M\). (The author refers to these vector space dimension as Macaulay-Bezout numbers.) Furthermore, the Macaulay-Bezout number \(\dim_\mathbb{C} M\) is also equal to the dimension, as a vector space over \(\mathbb{C}\), of the coordinate ring of the determinantal variety which is defined by the ideal of maximal order minors of \(\varphi\). An explicit formula for \(\dim_\mathbb{C} M\) is given in this paper. In particular, if \(a_i = 1\) and \(e_i = 0\) for all \(i\), then \(\dim_\mathbb{C} M\) is equal to the \(n\)-th elementary symmetric function evaluated at \((d_1, \dots, d_{n + k - 1})\). For example, if each \(d_i\) is 1, then \(\dim_\mathbb{C} M = \dim_\mathbb{C} R/(x_1, \dots, x_n)^k = {n + k - 1 \choose n},\) as expected. A second familiar example occurs when \(k = 1\). In this case, the \(n\)-th elementary symmetric function evaluated at \((d_1, \dots, d_n)\) is the product \(\prod d_i\), which, of course, is the conclusion of the classical Bezout theorem.

MSC:

13C40 Linkage, complete intersections and determinantal ideals
14M12 Determinantal varieties
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Alexandrov, A.G. : Cohomology of a quasihomogeneous complete intersection , Math. USSR Izvestiya 26iii (1986) 437-477. · Zbl 0647.14027 · doi:10.1070/IM1986v026n03ABEH001155
[2] Damon, J. : Higher Multiplicities and Almost Free Divisors and Complete Intersections (preprint). · Zbl 0867.32015
[3] Damon, J. : Topological Triviality and Versality for Subgroups of A and K II: Sufficient Conditions and Applications , Nonlinearity 5 (1992) 373-412. · Zbl 0747.58014 · doi:10.1088/0951-7715/5/2/005
[4] Damon, J. and Mond, D. : A-codimension and the vanishing topology of discriminants , Invent. Math. 106 (1991) 217-242. · Zbl 0772.32023 · doi:10.1007/BF01243911
[5] Goryunov, V.V. : PoincarĂ© polynomial of the space of residue forms on a quasihomogeneous complete intersection , Russ. Math. Surveys 35ii (1980) 241-242. · Zbl 0462.32003 · doi:10.1070/RM1980v035n02ABEH001647
[6] Macaulay, F.S. : The algebraic theory of modular systems , Cambridge Tracts 19 (1916).
[7] Milnor, J. and Orlik, P. : Isolated Singularities defined by Weighted Homogeneous Polynomials , Topology 9 (1970) 385-393. · Zbl 0204.56503 · doi:10.1016/0040-9383(70)90061-3
[8] Northcott, D.G. : Semi-regular rings and semi-regular ideals , Quart. J. Math. Oxford, (2), 11 (1960) 81-104. · Zbl 0112.03001 · doi:10.1093/qmath/11.1.81
[9] Orlik, P. and Terao, H. : Arrangements and Milnor Fibers (to appear Math. Annalen). · Zbl 0813.32033 · doi:10.1007/BF01446627
[10] Riordan, J. : Introduction to Combinatorial Analysis , Wiley, New York, 1958. · Zbl 0078.00805
[11] Terao, H. : Generalized exponents of a free arrangement of hyperplanes and the Shephard-Todd-Brieskom formula , Invent. Math. 63 (1981) 159-179. · Zbl 0437.51002 · doi:10.1007/BF01389197
[12] Wall, C.T.C. : Weighted Homogeneous Complete Intersections (preprint). · Zbl 0867.58005
[13] Zariski, O. and Samuel, P. : Commutative Algebra, reprinted as Springer Grad . Text in Math. 28 and 29, Springer Verlag, 1975. · Zbl 0313.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.