The global dimension of rings of differential operators on projective spaces.

*(English)*Zbl 0821.16032The paper under review concerns the global dimension of the rings of global sections \(D^ \lambda = \Gamma (\mathbb{P}^ n, {\mathcal D}^ \lambda)\) of the sheaves of twisted differential operators \({\mathcal D}^ \lambda\) on \(\mathbb{P}^ n\), which are indexed by \(\mathbb{C}\). It turns out that the answer for \(\text{gldim }D^ \lambda\) can be \(n\), \(2n\), or \(\infty\). These cases arise when \(\lambda \in \mathbb{C} \setminus \mathbb{Z}\), \(n \in \mathbb{Z} \setminus\{ -1, \dots, -n-1\}\), \(n \in \{-1, \dots, -n-1\}\).

For the finite cases the main technique used is the Beilinson-Bernstein localisation theorem in combination with the spectral sequence \[ H^ q(\mathbb{P}^ n, \text{Ext}^ p({\mathcal F}, {\mathcal G})) \implies \text{Ext}^ n({\mathcal F}, {\mathcal G}). \] In the infinite case an ad hoc method is used to produce a module of infinite projective dimension. A complete answer for the global dimension of rings of global sections of twisted differential operators on flag varieties (equivalently, minimal primitive factors of enveloping algebras of semisimple Lie algebras) is known. In detail, suppose that \(\mathfrak g\) is a semisimple Lie algebra with Cartan subalgebra \(\mathfrak h\) and let \(\lambda \in {\mathfrak h}^*\). Associated to \(\lambda\), there is a sheaf of twisted differential operators on the flag variety, \(X\), denoted by \({\mathcal D}^ \lambda\). One writes \(D^{\lambda} := U({\mathfrak g}) \text{ann }M (\lambda) \cong \Gamma (X, {\mathcal D}^ \lambda)\). There is no loss of generality in assuming that \(\lambda\) is antidominant. One says that \(\lambda\) is regular if its stabiliser in the Weyl group is trivial. T. J. Hodges and S. P. Smith [J. Lond. Math. Soc., II. Ser. 32, 411-418 (1985; Zbl 0588.17009)] showed that if \(\lambda\) is regular then \(\text{gldim }D^ \lambda\) is finite (in fact they found an upper bound and, appealing to an earlier result of Levasseur, pointed out that it is attained). A. Joseph and J. T. Stafford [Proc. Lond. Math. Soc., III. Ser. 49, 361-384 (1984; Zbl 0543.17004)] showed that \(\text{gldim }D^ \lambda\) is infinite, if \(\lambda\) is singular. H. Hecht and D. Miličić [Proc. Am. Math. Soc. 108, 249-254 (1990; Zbl 0714.22011)] have also obtained this latter result by showing that the localisation functor \({\mathcal D}^ \lambda \otimes \underline{\phantom m}\) has infinite cohomological dimension.

There do not appear to be analogous general results, for the global dimension of the rings of global sections of twisted differential operators on complete homogeneous spaces, in the literature.

For the finite cases the main technique used is the Beilinson-Bernstein localisation theorem in combination with the spectral sequence \[ H^ q(\mathbb{P}^ n, \text{Ext}^ p({\mathcal F}, {\mathcal G})) \implies \text{Ext}^ n({\mathcal F}, {\mathcal G}). \] In the infinite case an ad hoc method is used to produce a module of infinite projective dimension. A complete answer for the global dimension of rings of global sections of twisted differential operators on flag varieties (equivalently, minimal primitive factors of enveloping algebras of semisimple Lie algebras) is known. In detail, suppose that \(\mathfrak g\) is a semisimple Lie algebra with Cartan subalgebra \(\mathfrak h\) and let \(\lambda \in {\mathfrak h}^*\). Associated to \(\lambda\), there is a sheaf of twisted differential operators on the flag variety, \(X\), denoted by \({\mathcal D}^ \lambda\). One writes \(D^{\lambda} := U({\mathfrak g}) \text{ann }M (\lambda) \cong \Gamma (X, {\mathcal D}^ \lambda)\). There is no loss of generality in assuming that \(\lambda\) is antidominant. One says that \(\lambda\) is regular if its stabiliser in the Weyl group is trivial. T. J. Hodges and S. P. Smith [J. Lond. Math. Soc., II. Ser. 32, 411-418 (1985; Zbl 0588.17009)] showed that if \(\lambda\) is regular then \(\text{gldim }D^ \lambda\) is finite (in fact they found an upper bound and, appealing to an earlier result of Levasseur, pointed out that it is attained). A. Joseph and J. T. Stafford [Proc. Lond. Math. Soc., III. Ser. 49, 361-384 (1984; Zbl 0543.17004)] showed that \(\text{gldim }D^ \lambda\) is infinite, if \(\lambda\) is singular. H. Hecht and D. Miličić [Proc. Am. Math. Soc. 108, 249-254 (1990; Zbl 0714.22011)] have also obtained this latter result by showing that the localisation functor \({\mathcal D}^ \lambda \otimes \underline{\phantom m}\) has infinite cohomological dimension.

There do not appear to be analogous general results, for the global dimension of the rings of global sections of twisted differential operators on complete homogeneous spaces, in the literature.

Reviewer: M.P.Holland (Sheffield)

##### MSC:

16S32 | Rings of differential operators (associative algebraic aspects) |

17B35 | Universal enveloping (super)algebras |

16E10 | Homological dimension in associative algebras |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

14H25 | Arithmetic ground fields for curves |

13N10 | Commutative rings of differential operators and their modules |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |