Non-normal \({\mathcal D}\)-affine varieties with injective normalisation (Q1891735)

This paper uses similar techniques as the author's paper reviewed above [ibid. 180-199 (1995; see the preceding review)]. The main result is the following: Let \({\mathcal H}\) be a Cohen-Macaulay monomial variety with normalisation \(\mathbb{P}^r_k (m,1, \dots, 1)\) for some \(m \geq 1\) [weighted projective space] and injective normalization map. Then the rings of differential operators on \({\mathcal H}\) and \(\mathbb{P}^r_k (m,1, \dots,1)\) are Morita-equivalent iff \(H^r ({\mathcal H}, {\mathcal O}_{\mathcal H}) = 0\) or iff \(H^i ({\mathcal H}, {\mathcal O}_{\mathcal H}) = 0\) for all \(i > 0\) or iff \({\mathcal H}\) is \({\mathcal D}\)-affine (cf. definition 2.5 of the paper under review). This result is used to construct nonnormal \({\mathcal D}\)-affine varieties with injective normalisation such that the normalization is not regular.