Looking out for stable syzygy bundles

We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.Comment: This paper contains an appendix by Georg Hein: Semistability of the general syzygy bundle. The new version is quite ne

Test Elements, Analogues of Tight Closure, and Size for Ideals

We give many new results related to the theory of tight closure and its generalizations. Explicitly, we establish a series of results showing that the Jacobian ideal is contained in the test ideal for tight closures both in equal characteristic p and equal characteristic 0 for algebras essentially of finite type over power series rings (they are called semianalytic algebras). We move on to introduce and study a new closure called wepf in mixed characteristic, and prove that it is a Dietz closure satisfying the Algebra axiom. This is the first known example of a Dietz closure in mixed characteristic. This is achieved by proving that the epf closure satisfies what we call the p-colon-capturing property. We define and study the relationships with properties connected with tight closure. For example, we show that a persistent closure operation that captures colons automatically captures the plus closure, i.e., the contraction of the expansion of an ideal to the absolute integral closure of the ring. We also show that the existence of persistent closure operations between two complete local domains gives us a weakly functorial version of the existence of big Cohen-Macaulay algebras for them. We also develop a new numerical notion for ideals called size using the theory of quasilength, and show that the size of an ideal is always between its height and arithmetic rank. We show under mild conditions that the size is the same as height for one-dimensional primes in a local ring whose completion is a domain. We further study the additive property and the asymptotic additive property of quasilength.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/170044/1/zoeng_1.pd

An Improved Tight Closure Algorithm for Integer Octagonal Constraints

Integer octagonal constraints (a.k.a. ``Unit Two Variables Per Inequality'' or ``UTVPI integer constraints'') constitute an interesting class of constraints for the representation and solution of integer problems in the fields of constraint programming and formal analysis and verification of software and hardware systems, since they couple algorithms having polynomial complexity with a relatively good expressive power. The main algorithms required for the manipulation of such constraints are the satisfiability check and the computation of the inferential closure of a set of constraints. The latter is called `tight' closure to mark the difference with the (incomplete) closure algorithm that does not exploit the integrality of the variables. In this paper we present and fully justify an O(n^3) algorithm to compute the tight closure of a set of UTVPI integer constraints.Comment: 15 pages, 2 figure

Tight Closure of Finite Length Modules in Graded Rings

We look at how the equivalence of tight closure and plus closure (or Frobenius closure) in the homogeneous m-coprimary case implies the same closure equivalence in the non-homogeneous m-coprimary case in standard graded rings. Although our result does not depend upon dimension, the primary application is based on results known in dimension 2 due to the recent results of H. Brenner. We also show that unlike the Noetherian case, the injective hull of the residue field over $R^+$ or $R^\infty$ contains elements that are not killed by any power of the maximal ideal of R.Comment: 14 pages, minor revisions mad