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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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