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On a conjecture of Vasconcelos via Sylvester forms. (English) Zbl 1356.13001
Let \(R=k[x_{1},\dots,x_{n}]\) denote a polynomial ring over a field \(k\). In 2013 W. Vasconcelos formulated the conjecture (stated in [J. Hong, A. Simis and W.V. Vasconcelos, J. Commut. Algebra 5, No. 2, 231–267 (2013; Zbl 1274.13015)]) that the Rees algebra of an Artinian almost complete intersection \(I\subset R\) generated by monomials is almost Cohen-Macaulay. The authors consider the uniform monomial ideal, \(I:=(x_{1}^{a},\dots,x_{n}^{a},(x_{1},\dots,x_{n})^{n})\subset R\) for a given integers \(0<b<a\).
This work emphasizes the structure of the presentation ideal of Rees algebra of \(I\). It is known that the presentation ideal of Rees algebra of \(I\) is generated by binomials. They identification of these binomial generators as iterated Sylvester forms. They state that the above generators can be ordered in a such a way as to describe the Rees presentation ideal \(\mathcal{I}\) of \(I\) by a finite series of subideals of which any two consecutive ones have a monomial colon ideal. By inducting on the length of this series, they use mapping cones iteratively culminating with \(\mathcal{I}\) itself. As a consequence, the Rees algebra of \(I\) will be almost Cohen-Macaulay, thus establishing the conjecture of Vasconcelos.
MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C14 Cohen-Macaulay modules
13D02 Syzygies, resolutions, complexes and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14E07 Birational automorphisms, Cremona group and generalizations
14M07 Low codimension problems in algebraic geometry
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