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A Study in Derived Algebraic Geometry - Volume II: Deformations, Lie Theory and Formal Geometry (Paperback)
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A Study in Derived Algebraic Geometry - Volume II: Deformations, Lie Theory and Formal Geometry (Paperback)
Series: Mathematical Surveys and Monographs
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Derived algebraic geometry is a far-reaching generalization of
algebraic geometry. It has found numerous applications in other
parts of mathematics, most prominently in representation theory.
This volume develops deformation theory, Lie theory and the theory
of algebroids in the context of derived algebraic geometry. To that
end, it introduces the notion of inf-scheme, which is an
infinitesimal deformation of a scheme and studies ind-coherent
sheaves on such. As an application of the general theory, the
six-functor formalism for D-modules in derived geometry is
obtained. This volume consists of two parts. The first part
introduces the notion of ind-scheme and extends the theory of
ind-coherent sheaves to inf-schemes, obtaining the theory of
D-modules as an application. The second part establishes the
equivalence between formal Lie group(oids) and Lie algebr(oids) in
the category of ind-coherent sheaves. This equivalence gives a vast
generalization of the equivalence between Lie algebras and formal
moduli problems. This theory is applied to study natural
filtrations in formal derived geometry generalizing the Hodge
filtration.
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