Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Phenols-useful templates for the synthesis of bi-functional orthogonally protected dendron building blocks via solid phase Mitsunobu reaction

Bi-functional dendritic building blocks for convergent dendrimer growth were successfully synthesized from phenolic templates in the solid phase via a Mitsunobu reaction. Each arm of the dendron building block carries an orthogonally protected secondary amine along the arm, and a peripheral primary amine or phenol group (building block type 1) or a tertiary amine junction with orthogonally protected peripheral primary amine or carboxyl groups (building block type 2). The synthetic routes reported in this work are general and applicable for the preparation of diverse building blocks, controlling protection, arm length, and peripheral moieties. These novel dendron units can form unusual dendritic architectures by solid-phase chemistry, which may be incorporated into specific complex structures expanding the scope of dendrimer science.