known as the left big finitistic projective dimension of A, is finite. Here pdM denotes the projective dimension of M . Unfortunately, this number is not known to be finite even if A is a finite dimensional algebra over a field, where, indeed, its finiteness is a celebrated conjecture. On the other hand, for such an algebra, finite flat certainly implies finite projective dimension, simply beca...

We extend the theory of generalized divisors so as to work on any scheme X satisfying the condition S2 of Serre. We define a generalized notion of Gorenstein biliaison for schemes in projective space. With this we give a new proof of the theorem of Gaeta, that standard determinantal schemes are in the Gorenstein biliaison class of a complete intersection.

We investigate the Cohomological Crepant Resolution Conjecture for reduced Gorenstein weighted projective spaces. Using toric methods, we prove this conjecture in some new cases. As an intermediate step, we show that weighted projective spaces are toric Deligne-Mumford stacks. We also describe a combinatorial model for the orbifold cohomology of weighted projective spaces.

For Gorenstein quotient spaces C d =G, a direct generalization of the classical McKay correspondence in dimensions d 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstei...

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Goren...

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f -vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f -vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f -vectors of flag homology spheres and the f -vectors of Gorenstein* order comp...

We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that o...

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f -vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f -vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f -vectors of flag homology spheres and the f -vectors of Gorenstein* order comp...

For Gorenstein quotient spaces C/G, a direct generalization of the classical McKay correspondence in dimensions d ≥ 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein...