Differential operators and Nakai's conjecture

grantor: University of TorontoThis thesis is concerned with differential operators on algebraic varieties. The main goal is to understand how the presence of singularities is reflected in the algebraic structure of the ring of differential operators. Motivated by applications to tight closure, we study varieties in arbitrary characteristic. The thesis focuses on the study of ideals stable under differential operators and applies results in this area to Nakai's conjecture and the theory of tight closure. As well, results on differential operators on étale extensions are applied to show that the ring of differential operators on a smooth variety is equal to the Hasse-Schmidt algebra. Along the way, the Hasse-Schmidt derivations are shown to extend over étale extensions. The ring of differential operators on a Stanley-Reisner ring ' R' of arbitrary characteristic is explicitly described in terms of the minimal primes of 'R'. As well, the D-module structure of 'R' is completely determined. The structure of a 'k'-algebra 'R' as a module over the ring of differential operators 'D'(' R'/'k'), the Hasse-Schmidt algebra 'HS'(' R'/'k') and the derivations algebra 'der'(' R'/'k') is investigated. In particular, the conductor of the normalization 'R'' of ' R' into 'R' is 'HS'('R'/' k')-stable and Hasse-Schmidt derivations on 'R' extend to 'R''. This is the key observation in the proof that a characteristic-free analogue of Nakai's conjecture holds for varieties whose normalization is smooth. Differential operators can also be applied to Hochster and Huneke's theory of tight closure. We show that Stanley-Reisner rings are Frobenius-split by constructing an 'explicit' splitting of Frobenius in terms of differential operators. The D-module structure of a Stanley-Reisner ring ' R' is used to easily compute the test ideal of 'R', recovering a result originally due to Cowden.Ph.D