Cohomology

Abstract: We show that if Z is a local complete intersection subvariety of a smooth complex variety X, of pure codimension r, then Z has k-rational singularities if and only if α ˜(Z)>k+r, where α ˜(Z) is the minimal exponent of Z. We also characterize this condition in terms of the Hodge filtration on the intersection complex Hodge module of Z. Furthermore, we show that if Z has k-rational singularities, then the Hodge filtration on the local cohomology sheaf ℋ Z r (𝒪 X ) is generated at level dim(X)-⌈α ˜(Z)⌉-1 and, assuming that k≥1 and Z is singular, of dimension d, that ℋ k (Ω ̲ Z d-k )≠0. All these results have been known for hypersurfaces in smooth varieties.

Abstract: A rink-type roller skate is provided with a plastic sole plate. To mount a toe stop on the skate, a novel bushing is embedded in the sole plate. The bushing has relatively small diameter ends and a large diameter midportion which is aggressively surfaced. Axial and rotational forces are transmitted to the sole plate through the bushing, while movement of the bushing in the plate and plate cracking are inhibited.

Abstract: CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5. Obstructions are cocycles 6. Additivity and integrability of the square 7. Restricted deformation theories and their cohomology theories 8. Rigidity of fields in the commutative theory CHAPTER II. The parameter space 1. The set of structure constants as parameter space for the deformation theory 2. Central algebras and an example justifying the choice of parameter space 3. The automorphism group as a parameter space, and examples of obstructions to derivations 4. A fiber space over the parameter space, and the upper semicontinuity theorem 5. An example of a restricted theory and the corresponding modular group CHAPTER III. The deformation theory for graded and filtered rings 1. Graded, filtered, and developable rings 2. The Hochschild theory for developable rings 3. Developable rings as deformations of their associated graded rings 4. Trivial deformations and a criterion for rigidity 5. Restriction to the commutative theory 6. Deformations of power series rings