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Published online by Cambridge University Press: 29 May 2025
The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x1, . . . , xd ] generated in degrees ≤ n and block stable ideals (a class of ideals containing the Borel fixed ones) of K[x1, . . . , xn] generated in degree d. As a byproduct we give a new proof of the characterization of Betti tables of ideals with linear resolution given by Murai.
Minimal free resolutions of modules over a polynomial ring are a classical and fascinating subject. Let P = K[x1, . . . , xn] denote the polynomial ring equipped with the standard grading in n variables over a field K. For a ℤ-graded finitely generated P-module M, we consider its minimal graded free resolution:
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