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Deformation of Algebraic Schemes

Edoardo Sernesi
Springer Verlag
Publication Date: 
Number of Pages: 
Grundlehren der mathematischen Wissenschaften 334
[Reviewed by
Fernando Q. Gouvêa
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One of the goals of Springer's Grundlehren series is to provide reliable and thorough accounts of certain portions of mathematics. This volume by Edoardo Sernesi does just that, and hence fits the series well.

The idea of considering various kinds of deformations of complex structures goes back to Kodaira and Spencer (see, for example, Kodaira's Complex Manifolds and Deformation of Complex Structures). From the point of view of algebraic geometry, however, one needs a theory that works over any field, or at least any algebraically closed field, so one needs to replace analytic techniques with algebraic ones. In the case of deformation theory, the main ideas go back to Grothendieck (and perhaps Lefschetz before him). This is Sernesi's point of view. He argues in the introduction that an account of the basics of "classical [algebraic] deformation theory" was missing from the literature, and this book is his attempt to fill that gap.

Sernesi points out that the methods of Kodaira and Spencer have two parts: a formal construction, then a verification of convergence. From the algebraic point of view, the formal (or infinitesimal) part requires working over Artin rings and complete Noetherian local rings, while the convergence part is the problem of "algebraization." Most of Sernesi's book focuses on the first part of this program. The first chapter gives a fairly general introduction to deformations, and the second chapter develops the theory of formal deformations, with a brief account of the main algebraization theorems towards the end. Then he goes on to study several important examples in detail.

This is a book with quite serious pre-requisites; the reader is advised to keep at hand a copy of Grothendieck's Fondements de la Géométrie Algébrique (maybe also a copy of Fundamental Algebraic Geometry: Grothendieck's FGA Explained) and of SGA 1. I presume mentioning "EGA" isn't even necessary. So this is a book for algebraic geometers; for them, it'll prove to be a useful resource and reference.

Fernando Q. Gouvêa likes his functors representable.