Nakai’s conjecture for varieties smoothed by normalization

Abstract

Abstract. The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi's extension of Nakai's conjecture to arbitrary characteristic. This gives a new proof of Nakai's conjecture for curves and Stanley-Reisner rings.

A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions

“…When S/I is a regular ring, D(S/I) has a similar structure to the Weyl algebra, that is, D(S/I) is the (S/I)-algebra generated by derivations of S/I (see [4,Corollary 15.5.6]). However, as Traves in [13] proved, D(S/I) is not generated by derivations when S/I is a reduced algebra (including the case when I = QS). To observe generators and further structures of D(S/QS), Holm in [2] proved that D(S/QS) = m≥0 D (m) (A )/QD (m) (S).…”

Differential calculus over N-graded commutative rings

“…One has conjectured that this is true if and only if V is smooth [33]. This conjecture has been proved for algebraic curves [32] and, more generally, for varieties with smooth normalization [49].…”

One-dimensional infinitesimal-birational duality through differential operators

“…It implies the Zariski-Lipman conjecture [34], [3], [22]. Nakai's conjecture and its variants has been verified only for few classes of commutative algebras in [41], [24], [25], [42], [52], [35], [50], [49]. In general, the determination of the structure of differential operators of a given nonsmooth commutative algebra is a hard problem and one expects some pathology caused by singularities.…”