0.3. For the singular Q-Fanos one can try to use the same approach. The first difference is that, if X has a non-Gorenstein singularity, then locally in a neighborhood of such a point a general element of I-Kxl should have Du Val singularity. The second observation is that -Kx, restricted on S, is not a Cartier divisor any more but only a Weil divisor such that its multiple is an ample Cartier ...

We give a purely combinatorial characterization of complete Stanley-Reisner rings having a principally generated (equivalently, finitely generated) Cartier algebra.

We characterize those invertible sheaves on a noetherian scheme which are definable by Cartier divisors and correct an erroneous counterexample in the literature.

Introduction 1 0.1. Associated primes 2 0.2. Depth 2 1. Regular local rings and their cohomological properties 4 1.1. Notions of dimension 4 1.2. Dualizing functor 5 1.3. Local rings 6 1.4. Serre’s theorem on regular local rings 6 1.5. Cohomology of D when global dimension is finite 8 2. Depth 8 2.1. Property (Sk) 8 2.2. Serre’s criterion of normality 10 2.3. Cohen-Macaulay modules 11 3. Diviso...