**Mathematical Surveys and Monographs**

Volume: 221;
2017;
436 pp;
Hardcover

MSC: Primary 14; 18;

Print ISBN: 978-1-4704-3570-7

Product Code: SURV/221.2

List Price: $124.00

Individual Member Price: $99.20

**Electronic ISBN: 978-1-4704-4087-9
Product Code: SURV/221.2.E**

List Price: $124.00

Individual Member Price: $99.20

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# A Study in Derived Algebraic Geometry: Volume II: Deformations, Lie Theory and Formal Geometry

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*Dennis Gaitsgory; Nick Rozenblyum*

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.

#### Readership

Graduate students and researchers interested in new trends in algebraic geometry and representation theory.