We prove that the generic quantized coordinate ringOq(G) is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group G. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a...

We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski’s theorem on convex polytopes. Also we show that for any Cohen-Macaulay cell complex as above, although there is now generalization of the Stanley-Reisner ring of simplicial co...

Let Q(k, l) be a poset whose Hasse diagram is a regular spider with k+1 legs having the same length l (cf. Fig. 1). We show that for any n ≥ 1 the nth cartesian power of the spider poset Q(k, l) is a Macaulay poset for any k ≥ 0 and l ≥ 1. In combination with our recent results [2] this provides a complete characterization of all Macaulay posets which are cartesian powers of upper semilattices,...

In this thesis we discuss a moduli space of projective curves with a map to a given projective space P. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to P that is an isomorphism onto its image away from a finite set of closed points. We proof that CM is an algebraic space by contructing a scheme W and a representable, surject...

We prove a version of the Halphen Speciality Theorem for locally Cohen-Macaulay curves in P3. To prove the theorem, we strengthen some results of Okonek and Spindler on the spectrum of the ideal sheaf of a curve. As an application, we classify curves C having index of speciality as large as possible once we fix the degree of C and the minimum degree of a surface

This thesis presents an expository account of the use of Frobenius splitting techniques in the study of Schubert varieties. After developing the basic theory of Frobenius splitting, we show that the Schubert and Bott-Samelson varieties are split and use this to derive geometric consequences in arbitrary characteristic. The main result highlighted is that Schubert varieties are normal, Cohen-Mac...

We prove that any rank two arithmetically CohenMacaulay vector bundle on a general hypersurface of degree at least three in P must be split.