Let H be a cocommutative weak Hopf algebra and let (B,φB) left H-module algebra. In this paper, for twisted convolution invertible morphism σ:H2→B we define its obstruction θσ as Sweedler 3-cocycle with values in the center of B. We obtain that class vanish third cohomology group HφZ(B)3(H,Z(B)) if, only there exists 2-cocycle α:H2→B such H⊗B can endowed crossed product structure α keeping coho...

For a finitely generated A-module M we define the dimension filtration M = {Mi}0≤i≤d, d = dimA M, where Mi denotes the largest submodule of M of dimension ≤ i. Several properties of this filtration are investigated. In particular, in case the local ring (A,m) possesses a dualizing complex, then this filtration occurs as the filtration of a spectral sequence related to duality. Furthermore, we c...

We describe and prove uniqueness of a natural homomorphism (called the Orchard morphism) between some groups associated to finite sets.

208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 As far as I know, the notion of a quasi-morphism does not have much to do with category theory. This very natural idea underlies several interesting recent developments at the crossroads of algebra, topology, geometry, and dynamics. Other, perhaps better, names currently in use for the same concept are quasi-homomorphism and pseudo-character. Let G be ...

We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson–Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity...