A Cousin Complex for the Quantum Projective Space
Kobi Kremnizer, David Ssevviiri
TL;DR
The paper addresses constructing a Cousin-like complex for the quantum projective space $\mathbb{P}^n_q$ to extend Grothendieck-Cousin theory into noncommutative geometry and connect with quantum group representations. It develops a framework of graded algebras modulo torsion, defines quantum global sections and local cohomology, and uses an ample autoequivalence to pass to derived functors, building a quantum Cousin complex from a filtration by quantum subspaces. The main contribution is an explicit Cousin complex for $\mathbb{P}^n_q$ whose $q=1$ limit recovers the classical Sharp-Cousin complex, with local cohomology pieces realized as modules for the quantum group with divided powers and as comodules for the quantum Borel, pointing toward a noncommutative dual BGG resolution at all roots of unity. This approach provides new homological tools for quantum projective geometry and its connections to quantum representation theory, potentially enabling broader applications in noncommutative geometry and quantum group theory.
Abstract
Grothendieck constructed a Cousin complex for abelian sheaves on an arbitrary topological space. In a special setting, its dual called the BGG resolution is applicable in representation theory. Arkhipov proposed a complex whose dual is only suitable for representation theory of quantum groups at roots of unity of prime order. It is desirable to get one which works for quantum groups at all roots of unity. For a quantum projective space, we provide such a complex.