The f-depth of an ideal on a module

Let be an ideal of a Noetherian local ring and a finitely generated -module. The f-depth of on is the least integer such that the local cohomology module is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules.

     

Remarks on Two Approaches to the Horizontal Cohomology: Compatibility Complex and the Koszul–Tate Resolution

The Koszul–Tate resolution is described in the context of the geometry of jet spaces and differential equations. The application due to Barnich, Brandt, and Henneaux of this resolution to computing the horizontal cohomology is analyzed. Relations with the Vinogradov spectral sequence are discussed.     

2-Cocycles of original deformative Schr?dinger-Virasoro algebras

Both original and twisted Schr?dinger-Virasoro algebras, and also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim at determining the 2-cocycles of original deformative Schr?dinger-Virasoro algebras. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471091, 10671027), Foundation of Shanghai Education Committee (Grant No. 06FZ029) and “One Hundred Talents Program” from University of Science and Technology of China     

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

Here we study vector bundles E on the Hirzebruch surface F _e such that their twists by a spanned, but not ample, line bundle M = _Fe (h + ef) have natural cohomology, i.e. h ⁰(F _e , E(tM)) > 0 implies h ¹(F _e , E(tM)) = 0.      

The Combinatorial Quantum Cohomology Ring of <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">B</Emphasis>

A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which—for G = SL_n( )—was proved previously in the paper [5].     

Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and 0$">. Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of .

     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

Factorization theorems for cohomological dimension

The well-known factorization theorems for covering dimension dim and compact Hausdorff spaces are here established for the cohomological dimension dim using a new characterization of dim In particular, it is proved that every mapping f: X → Y from a compact Hausdorff space X with to a compact metric space Y admits a factorization f = hg, where g: X → Z, h: Z → Y and Z is a metric compactum with . These results are applied to the well-known open problem whether . It is shown that the problem has a positive answer for compact Hausdorff spaces X if and only if it has a positive answer for metric compacta X.