Kähler differential algebras for 0-dimensional schemes

Abstract

Given a 0-dimensional scheme in a projective n-space P n over a field K, we study the Kähler differential algebra Ω R X /K of its homogeneous coordinate ring R X . Using explicit presentations of the modules Ω m R X /K of Kähler differential m-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of P 1 , fat point schemes, and subschemes of P 2 supported on a conic.

Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

“…As a consequence of Proposition 2.10, we obtain the following explicit description for the module of Kähler differential (n + 1)-forms of R W /K (see also [9,Corollary 2.3]).…”

“…Also, the above lower bound for HP Ω n+1 R W /K (z) is attained for a fat point scheme whose support is contained in a hyperplane (see [9, Proposition 5.1]) and the upper bound for the regularity index of Ω k R W /K is sharp as well (see [9,Example 4.3]). Based on the isomorphism of graded R W -modules Ω n+1 R W /K ∼ = (S/∂I W )(−n − 1), we obtain from Propositions 2.6 and 3.1 the following consequence.…”

“…In [5], De Dominicis and the first author initiated a careful examination of the structure and the Hilbert function of Ω 1 R X /K for a set of points X (i.e., for the case m 1 = • • • = m s = 1). Then, in [8], the present authors started the study of the modules of Kähler differential k-forms for fat point schemes, and in [9] this work was continued.…”

“…A major open question in [9] is a formula for the Hilbert polynomial of Ω n+1 R W /K , i.e., for the value of the Hilbert function in large degrees, which is given as a conjecture there. Our main result here is to prove this formula.…”

“…The research underlying this paper and the calculation of the examples were greatly aided by an implementation of the relevant objects and functions in the computer algebra system ApCoCoA (see [2]). Unless explicitly stated otherwise, we adhere to the definitions and notation introduced in the books [10] and [11], as well as in our previous papers [8] and [9].…”