[Paper Review] Deformation quantization modules I:Finiteness and duality

Masaki Kashiwara, Pierre Schapira|arXiv (Cornell University)|Feb 9, 2008

Homotopy and Cohomology in Algebraic Topology26 references16 citations

TL;DR

This paper introduces cohomological completeness for sheaves of $Z[\hbar]$-modules and applies it to deformation quantization algebroids on complex Poisson manifolds. It proves coherence of convolution products of coherent kernels under properness and constructs dualizing complexes, showing that convolution commutes with duality.

ABSTRACT

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing complexes in this framework and show that the convolution of kernels commutes with duality.

Motivation & Objective

• To define and study cohomological completeness in the derived category of sheaves of $Z[\hbar]$-modules.
• To establish coherency results for modules over deformation quantization algebroids using cohomological completeness.
• To investigate the coherence of convolution products of coherent kernels under a properness condition.
• To construct dualizing complexes in the context of deformation quantization algebroids.
• To prove that convolution of kernels commutes with duality in this framework.

Proposed method

• Introduce the notion of cohomological completeness for objects in the derived category of sheaves of $Z[\hbar]$-modules.
• Use cohomological completeness to derive coherency results for modules over $Z[\hbar]$-algebras under suitable conditions.
• Apply these results to deformation quantization algebroids on complex Poisson manifolds.
• Establish that the convolution of two coherent kernels is coherent when the underlying manifold satisfies a natural properness condition.
• Construct dualizing complexes in the derived category of modules over deformation quantization algebroids.
• Prove that the convolution operation commutes with the duality functor in this setting.

Experimental results

Research questions

• RQ1 Under what conditions is the convolution of two coherent kernels over a deformation quantization algebroid coherent?
• RQ2 How does cohomological completeness relate to coherency in the derived category of $Z[\hbar]$-modules?
• RQ3 Can dualizing complexes be constructed in the framework of deformation quantization algebroids?
• RQ4 Does the convolution of kernels preserve duality in the derived category of modules over deformation quantization algebroids?
• RQ5 What role does properness play in ensuring coherence of convolution products in deformation quantization?

Key findings

• Cohomological completeness ensures coherency of modules over $Z[\hbar]$-algebras under suitable conditions.
• Under a natural properness condition, the convolution of two coherent kernels over a deformation quantization algebroid is coherent.
• Dualizing complexes exist in the derived category of modules over deformation quantization algebroids on complex Poisson manifolds.
• The convolution of kernels commutes with the duality functor in this framework.
• The results establish a coherent duality theory for modules over deformation quantization algebroids.
• The framework provides a systematic approach to studying coherence and duality in deformation quantization using cohomological completeness.