Subspace arrangements defined by products of linear forms

We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatorial structure, and we present the examples that motivated our work. We give a construction which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. We also consider generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3.$Comment: 20 pages; AMSLatex; v.2: proof of Proposition 5.1.3 corrected; proof of Proposition 5.1.6 shortened; references added, v.3: minor corrections; final version; to appear in the Journal of the London Mathematical Societ

A negative answer to a question of Bass

In this companion paper to arXiv:0802.1928 we provide an example of an isolated surface singularity $R$ over a number field such that $K_0(R) = K_0(R[t])$ but $K_0(R) \neq K_0(R[t_1,t_2])$. This answers, negatively, a question of Bass.Comment: The paper was previously part of arXiv:0802.192

Variation of Hilbert Coefficients

For a Noetherian local ring (\RR, \m), the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an \m-primary ideal $I$ are known to code for properties of \RR, of the blowup of \spec(\RR) along $V(I)$, and even of their normalizations. We give estimations for these coefficients when $I$ is enlarged (in the case of $e_1$ in the same integral closure class) for general Noetherian local rings

Betti numbers for numerical semigroup rings

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.Comment: 22 pages; v2: updated references. To appear in Multigraded Algebra and Applications (V. Ene, E. Miller Eds.

Binomial edge ideals over an exterior algebra

We introduce the study of binomial edge ideals over an exterior algebra

Graded Syzygies

The study of free resolutions is a core and beautiful area in Commutative Algebra. The main goal of this book is to inspire the readers and develop their intuition about syzygies and Hilbert functions. Many examples are given in order to illustrate ideas and key concepts. A valuable feature of the book is the inclusion of open problems and conjectures; these provide a glimpse of exciting, and often challenging, research directions in the field. Three types of problems are presented: Conjectures, Problems, and Open-Ended Problems. The latter do not describe specific problems but point to inter