Differential operators on Stanley-Reisner rings

Abstract

Abstract. Let k be an algebraically closed field of characteristic zero, and let R = k[x 1 , . . . , xn] be a polynomial ring. Suppose that I is an ideal in R that may be generated by monomials.We investigate the ring of differential operators D(R/I) on the ring R/I, and I R (I), the idealiser of I in R. We show that D(R/I) and I R (I) are always right Noetherian rings. If I is a square-free monomial ideal then we also identify all the two-sided ideals of I R (I).To each simplicial complex ∆ on V = {v 1 , . . . , vn} there is a corresponding square-free monomial ideal I ∆ , and the Stanley-Reisner ring associated to ∆ is defined to be k[∆] = R/I ∆ . We find necessary and sufficient conditions on ∆ for D(k[∆]) to be left Noetherian.

Differential operators, retracts, and toric face rings

“…Thus we can directly apply Theorem 2.8 to compute the ring of differential operators of a toric face ring in Proposition 3.3. As a consequence, we recover results on differential operators on Stanley-Reisner rings given by Tripp, Eriksson, and Traves in [Tri97,Eri98,Tra99] over arbitrary fields.…”

“…One obstruction to the even greater use of rings of differential operators is the notorious difficulty of computing them explicitly. In fact, there are very few classes of rings whose differential operators are systematically computed, namely polynomial rings, Stanley-Reisner rings, affine semigroup rings, and coordinate rings of curves (see, for example, [Tri97,Tra99,Eri98, CB was partially supported by NSF Grants DMS 1661962 and 2001101. LFM is partially supported by the Simons Foundation Collaboration Grant for Mathematicians.…”

An illustrated view of differential operators of a reduced quotient of an affine semigroup ring

“…In [Eri98,Tra99,Tri97], Eriksson, Traves, and Tripp separately computed the ring of differential operators of a Stanley-Reisner ring over an arbitrary field, i.e., the quotient of any polynomial ring over a field by a squarefree monomial ideal. We include here the ring of differential operators of an ordinary double point R = C[x, y] ⟨xy⟩ using the viewpoint presented by the above authors, as it exhibits behavior akin to our computations in this article.…”

Finite generation of rings of differential operators of semigroup algebras

“…However, D(R) is known to be Noetherian for some families of interesting algebras; Muhasky [17] and Smith-Stafford [29] independently proved that D(R) is Noetherian if R is an integral domain of Krull dimension one. Tripp [31] proved that the ring D(K[∆]) of differential operators of the Stanley-Reisner ring K[∆] is right Noetherian, and gave a necessary and sufficient condition for D(K[∆]) to be left Noetherian.…”

“…Generally speaking, D(R) is more apt to be right Noetherian than to be left Noetherian, and a prime of height more than one is often an obstacle for the left Noetherian property of D(R) (see for example [5], [19], [29], and [31]). We observe this phenomenon as well.…”