Introduction 2 1. A Guide to Duality 3 1.1. Local Duality 3 1.2. Dualizing Complexes and Some Vanishing Theorems 10 1.3. Cohomological Annihilators 17 2. A Few Applications of Local Cohomology 21 2.1. On Ideal Topologies 21 2.2. On Ideal Transforms 25 2.3. Asymptotic Prime Divisors 28 2.4. The Lichtenbaum-Hartshorne Vanishing Theorem 35 2.5. Connectedness results 37 3. Local Cohomology and Syzy...

Let I denote an R+-primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic p. We give a linear degree bound for the Frobenius powers I [q] of I, q = p, in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety ProjR. This provides an inclusion bound for tight closure. In the same manner we give a line...

Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, Vasconcelos shows that both can be compared in case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained i...

Associated to a projective arrangement of hyperplanes A ⊆ Pn is the module D(A), which consists of derivations tangent to A. We study D(A) when A is a configuration of lines in P. In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regulari...

Castelnuovo–Mumford regularity is one of the most important invariants in commutative algebra. It has been investigated by various authors from different point of view. One of the interesting properties of regularity is its asymptotic behavior. In this talk first we will see some well-known facts of regularity and then I will present some known results of asymptotic behavior of quadratic square...

For a rooted tree Γ, we consider path ideals of Γ, which are ideals that are generated by all directed paths of a fixed length in Γ. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of Γ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as p...

We show that if f : X −→ Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X , then f maps X(Fq) surjectively onto Y (Fq) if and only if f maps X(Fq) injectively into Y (Fq). Surprisingly, the bounds on q for these two implications have different orders of magnitude. T...

The behaviour of Castelnuovo-Mumford regularity under “geometric” transformations is not well understood. In this paper we are concerned with examples which will shed some light on certain questions concerning this behaviour. One simple question which was open (see e.g. [R]) is: May the regularity increase if we pass to the radical or remove embedded primes? By examples, we show that this happe...