A not necessarily noetherian local ring O is called regular if every finitely generated ideal I O possesses finite projective dimension. In the article localizations O = Aq, q ∈ SpecA, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdimO. ...

We show that the surface X, which is a blow-up of P in n points on a line, has finitely generated Cox ring. We give explicit generators for the ring and calculate its defining ideal of relations.

For any finitely generated torsion-free graded module over a polynomial ring, there exists a homogeneous ideal fitting into an exact sequence similar to a Bourbaki sequence even though its height is not restricted to two.

We give upper bounds for the order of the elements in a characteristic set of a regular differential ideal or a radical of a finitely generated differential ideal with respect to some specific orderings. We then show how to compute characteristic sets of these ideals using algebraic methods. 2000 Mathematics Subject Classification: 12H05, 13B25, 13P10.

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed...

We show that the blow-up of P in n points on a line has finitely generated Cox ring. We give explicit generators for the ring and calculate its defining ideal of relations.

But before starting on this, let me try to explain again, in a different way, our approach. The one habit I’ve been trying to wean you of is the an over-reliance upon concrete examples to develop your understanding. The vector space R is a very concrete and familar example of a vector space over a field. To do calculations in this setting all you need to do is apply arithmetic (over and over an...