Duality and equivalence of module categories in noncommutative geometry II: Mukai duality for holomorphic noncommutative tori

By J. Block. 2005

112 citations recorded by the source corpus.

This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a complex manifold. In this paper we enlarge this category to include objects which correspond to quasi-coherent sheaves. We then apply this framework to proving an equivalence of categories between derived categories on the noncommutative complex torus and on a holomorphic gerbe on the dual complex torus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73a4f8cfb4807b2b4df6f04667c7ae205af4a00b

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In Algebra and Number Theory

1. M. Gualtieri
2. B. Tsygan
3. R. Nest
4. B. Keller
5. A. Gorokhovsky
6. P. Bressler
7. I. Reiten
8. Pierre-Guy Plamondon

In Geometry and Topology

1. M. Gualtieri
2. B. Keller
3. B. Tsygan
4. R. Nest
5. A. Gorokhovsky
6. P. Bressler
7. I. Reiten
8. Pierre-Guy Plamondon

In Mathematical Physics

1. M. Gualtieri
2. B. Keller
3. B. Tsygan
4. R. Nest
5. A. Gorokhovsky
6. P. Bressler
7. Alexander Kuznetsov
8. I. Reiten

Citations in the corpus

• Deformation quantization of gerbes (2005, methodology): “very close to the main theorem of [P]). More precisely, we first reduce the classification problem to classifying certain Q-algebras, using the term of A. Schwarz (or curved DGAs, as they are called in [Bl]). (The link between these objects and gerbes was rather well understood for some time; for exam”
• Generalized Kähler Geometry (2010, result): “Since the Morita equivalence is an isomorphism at the level of complex Lie algebroids over M , we can also strengthen this statement to an equivalence of the DG categories of cohesive modules [5], i.”
• On differential graded categories (2006, methodology): “s of the generator. Thus, the bounded derived category of coherent sheaves is equivalent to per(A) for a dg category with one object whose endomorphism ring satisfies a strong regularity condition. In [17], J. Block describes the bounded derived category of complexes of sheaves with coherent homology”