Convex subrings of partially ordered rings

The relationship between a poring and a convex subring has been studied most successfully for porings with bounded inversion. The best conceivable results are known for real closed rings. The present paper focuses on the connections between the prime spectra and between the real spectra of a poring and a convex subring. Examples show that for arbitrary porings one may not expect a very close relationship. But with the assumption of bounded inversion or spectral compatibility the results are similar to those for real closed rings (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

On endomorphism rings of local cohomology modules

Let be a local complete ring. For an -module the canonical ring map is in general neither injective nor surjective; we show that it is bijective for every local cohomology module if for every ( an ideal of ); furthermore the same holds for the Matlis dual of such a module. As an application we prove new criteria for an ideal to be a set-theoretic complete intersection.

     

Local cohomology over homogeneous rings with one-dimensional local base ring

Let be a homogeneous Noetherian ring with local base ring and let be a finitely generated graded -module. Let be the -th local cohomology module of with respect to 0}R_n$">. If , the -modules , and are Artinian for all . As a consequence, much can be said on the asymptotic behaviour of the -modules for .

     

Quasimartingales on partially ordered sets

The concept of a quasimartingale, and therefore also of a function of bounded variation, is extended to processes with a regular partially ordered index set V and with values in a Banach space. We show that quasimartingales can be described by their associated measures, defined on an inverse limit space $\text{S × Ω}$ containing $\text{V × Ω}$, furnished with the σ-algebra $\text{P}$ of the predictable sets. With the help of this measure, a Rao-Krickeberg and a Riesz decomposition is obtained, as well as a convergence theorem for quasimartingales. For a regular quasimartingale X it is proven that the spaces ($\text{S × Ω}$, $\text{P}$) and the measures associated with X are unique up to isomorphisms. In the case V = $\text{R}$₊ⁿ we prove a duality between classical (right-) quasimartingales and left-quasimartingales.     

Faltings' theorem for the annihilation of local cohomology modules over a Gorenstein ring

In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.

     

Totally ordered sets and the prime spectra of rings

Let T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain O_v such that T is order isomorphic to two special subsets of Spec(O_v). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U?Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.     

A finiteness result for associated primes of local cohomology modules

We show that the first non-finitely generated local cohomology module of a finitely generated module over a noetherian ring with respect to an ideal has only finitely many associated primes.

     

Generalized contractions in partially ordered metric spaces

We present some fixed point results for monotone operators in a metric space endowed with a partial order using a weak generalized contraction-type assumption.     

The fractional weak discrepancy of a partially ordered set

In this paper we introduce the notion of the fractional weak discrepancy of a poset, building on previous work on weak discrepancy in [J.G. Gimbel and A.N. Trenk, On the weakness of an ordered set, SIAM J. Discrete Math. 11 (1998) 655-663; P.J. Tanenbaum, A.N. Trenk, P.C. Fishburn, Linear discrepancy and weak discrepancy of partially ordered sets, ORDER 18 (2001) 201-225; A.N. Trenk, On k-weak orders: recognition and a tolerance result, Discrete Math. 181 (1998) 223-237]. The fractional weak discrepancywd_F(P) of a poset P=(V,?) is the minimum nonnegative k for which there exists a function f:V→R satisfying (1) if a?b then f(a)+1?f(b) and (2) if a∥b then |f(a)-f(b)|?k. We formulate the fractional weak discrepancy problem as a linear program and show how its solution can also be used to calculate the (integral) weak discrepancy. We interpret the dual linear program as a circulation problem in a related directed graph and use this to give a structural characterization of the fractional weak discrepancy of a poset.     

An abstract sub-supersolution method in partially ordered spaces

We present a general framework in which sub-supersolution techniques can be applied. Specifically, we consider a normed vector space endowed with a partial order. This allows us to introduce a notion of sub- and supersolutions relative to an equation driven by operators defined from the space to its topological dual. We provide sufficient conditions to guarantee that the interval determined by an ordered pair of sub-supersolutions contains a solution of the initial problem. We also study existence of extremal positive or negative solutions and location of nodal solutions. Examples of applications to quasilinear boundary value problems are given.     

Stochastic integration on partially ordered sets

The stochastic integral is introduced with respect to a stochastic process X = (X_s)_sεV, where V is any general partially ordered set satisfying some mild regularity conditions. As important examples the stochastic integral is constructed with respect to a class of Gaussian processes having similarities to the Brownian motion on the real line, and also with respect to L²-martingales under an assumption of conditional independence on the underlying σ-fields.     

On the cohomology of incidence algebras of partially ordered sets

We show how to construct a representation of the Lyons sporadic simple group in 651 dimensions over G F(3), which was apparently not previously known. This is the smallest faithful representation in charac-teristic 3.     

A note on the injective dimension of local cohomology modules

For a Noetherian ring we call an -module cofinite if there exists an ideal of such that is -cofinite; we show that every cofinite module satisfies . As an application we study the question which local cohomology modules satisfy . There are two situations where the answer is positive. On the other hand, we present two counterexamples, the failure in these two examples coming from different reasons.

     

On the cofiniteness of local cohomology modules

In this note we show that if is an ideal of a Noetherian ring and is a finitely generated -module, then for any minimax submodule of the -module is finitely generated, whenever the modules are minimax. As a consequence, it follows that the associated primes of are finite. This generalizes the main result of Brodmann and Lashgari (2000).