nLab Notes on geometric Langlands

This entry is about the collection of notes

• Dennis Gaitsgory:

  
  

  Notes on geometric Langlands

  
  

  [web]

on the categorical geometric Langlands conjecture, which is formulated as an equivalence of stable (infinity,1)-categories of

• D-modules on the derived stack of G-bundles on a curve $X$,

and

• quasi-coherent sheaves on the derived stack of $G^\vee$-equivariant local systems on $X$,

where $X$ is a smooth complete curve over a field of characteristic zero, and $G$ is a reductive group and $G^\vee$ is its Langlands dual. (More precisely, the latter category should be replaced by some appropriate category of ind-coherent sheaves.)

See also

• D. Gaitsgory, N. Rozenblyum, A study in derived algebraic geometry.

• D. Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.

Contents

Foundational stuff

• Dennis Gaitsgory, Generalities on DG categories, pdf

Studies some aspects of the symmetric monoidal (infinity,1)-category of dg-categories, including colimits, limits, and dualizable objects in it, and categories of modules over it.

• N. Rozenblyum, Filtered colimits of categories, pdf.

An explicit description of filtered colimits in the (infinity,1)-category of (infinity,1)-categories.

• Dennis Gaitsgory, Functors given by kernels, adjunctions and duality, pdf.

• Dennis Gaitsgory, Stacks, pdf

A concise review of derived stacks and derived schemes in derived algebraic geometry (the version based on coconnective commutative dg-algebras).

• Dennis Gaitsgory, Quasi-coherent sheaves on stacks, pdf

A concise review of stable (infinity,1)-categories of quasi-coherent sheaves and perfect complexes on derived stacks.

• Dennis Gaitsgory, Sheaves of categories and the notion of 1-affineness, pdf.

• Dennis Gaitsgory, Ind-coherent sheaves, pdf.

On the stable (infinity,1)-category of ind-coherent sheaves on a derived stack, and the six operations in this setting.

• Dennis Gaitsgory, N. Rozenblyum, DG Indschemes, pdf.

Develops the theory of ind-schemes in derived algebraic geometry.

• Dennis Gaitsgory, N. Rozenblyum, Crystals and D-modules, pdf.

Studies crystals and D-modules in derived algebraic geometry.

• V. Drinfeld, Dennis Gaitsgory, On some finiteness questions for algebraic stacks, pdf.

…

rest of the contents to be filled in