Syzygy Theorems via Comparison of Order Ideals on a Hypersurface

We introduce a weak order ideal property that suffices for establishing the Evans-Griffith Syzygy Theorem. We study this weak order ideal property in settings that allow for comparison between homological algebra over a local ring $R$ versus a hypersurface ring $R/(x^n)$. Consequently we solve some relevant cases of the Evans-Griffith syzygy conjecture over local rings of unramified mixed characteristic $p$, with the case of syzygies of prime ideals of Cohen-Macaulay local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by $p^s$, $s>0$, that have finite projective dimension over a hypersurface ring.Comment: To appear in JPA

Optimal control in infinite horizon problems: a Sobolev spaces approach

In this paper, we make use of the Sobolev space W1,1 (R+,Rn) toderive at once the Pontryagin conditions for the standard optimalgrowth model in continuous time, including a necessary and sufficienttransversality condition. An application to the Ramsey model is given.We use an order ideal argument to solve the problem inherent to thefact that L1 spaces have natural positive cones with no interior points.Optimal control, Sobolev spaces, Transversality conditions, Order ideal

Optimal control in infinite horizon problems : a Sobolev space approach

In this paper, we make use of the Sobolev space W1,1(R+, Rn) to derive at once the Pontryagin conditions for the standard optimal growth model in continuous time, including a necessary and sufficient transversality condition. An application to the Ramsey model is given. We use an order ideal argument to solve the problem inherent to the fact that L1 spaces have natural positive cones with no interior points.Optimal control, Sobolev spaces, transversality conditions, order ideal.

When is Every Order Ideal a Ring Ideal?

A lattice-ordered ring R is called an OIRI-ring if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those f-rings R such that R/I is contained in an f-ring with an identity element that is a strong order unit for some nil l-ideal I of R. In particular, if P(R) denotes the set of nilpotent elements of the f-ring R, then R is an OIRI-ring if and only if R/P(R) is contained in an f-ring with an identity element that is a strong order unit

Pattern Avoidance and the Bruhat Order

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length.Comment: 18 pages, 7 figure