Non-commutative resolutions of toric varieties

Abstract

Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring End R (A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem ofŠpenko and Van den Bergh implying that End R (A) has finite global dimension. Furthermore, we show that End R (A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime characteristic, we show that the ring of differential operators D k (R) has finite global dimension. mension. 1 This theorem is due to Roos in characteristic zero [Roo72, Cha74], and Paul Smith in characteristic p [Smi87].

Derived Functors of Differential Operators

“…If K is a field of positive characteristic p, by [12,Theorem 7.2], End R p e (R) is a Cohen-Macaulay R-module for each e. Then, by the isomorphism (1.0.1) of the introduction, and passing to direct limits, one obtains that…”

“…Questions about the ring structure of differential operators, e.g., finite generation and simplicity, have been wellstudied [3,21,27,23]. In Commutative Algebra, differential operators on singularities have found a resurgence of interest due to their connections with F -singularities, symbolic powers, and noncommutative resolutions, among other topics [25,27,6,1,12]. While most commonly one encounters the ring D R|K of K-linear differential operators from a ring R (commutative, with 1) to itself, differential operators from one module M to another N are defined; the collection of these is denoted D R|K (M, N).…”

“…where R p e ⊆ R is the subring of p e -th powers of elements of R [33, Lemma 1.4.8]. This isomorphism is quite useful in understanding rings of differential operators in positive characteristic, especially in conjunction with information on the module-theoretic structure given by the Frobenius map on R [28,27,32,12]. Conversely, via this isomorphism, properties of singularities detected by the Frobenius map can sometimes be detected by the ring of differential operators; e.g., for R and K as above, R is strongly F -regular if and only if R is simple as a left D R|K -module (R is D R|K -simple) and F -pure [25,Theorem 2.2]; see the reference for definitions.…”

Trace ideals, normalization chains, and endomorphism rings

“…Recently, there were quite a few results on construction of NC(C)Rs and their properties, see e.g. [IW10,BLvdB10,FMS19,IN18,HN17,ŠVdB17]. In particular, it is interesting which values the global dimension can assume: this should give some information about the singularities of Spec(R).…”

Non-commutative crepant resolutions of Hibi rings with small class group

“…Conic modules were well studied in [Bru,BG1,SmVdB], and it is known that the number of those is finite up to isomorphism. Furthermore, the endomorphism ring of the direct sum of all conic modules has finite global dimension (see [ŠpVdB1,Proposition 1.8], [FMS,Theorem 6.1]), but in general this endomorphism ring is not an MCM module, and hence not an NCCR. In particular, taking all conic modules is too big to allow the endomorphism ring to be an MCM module.…”