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Published online by Cambridge University Press: 06 July 2010
The theory of syzygies connects the qualitative study of algebraic varieties and commutative rings with the study of their defining equations. It started with Hilbert’s work on what we now call the Hilbert function and polynomial, and is important in our day in many new ways, from the high abstractions of derived equivalences to the explicit computations made possible by Gr¨obner bases. These lectures present some highlights of these interactions, with a focus on concrete invariants of syzygies that reflect basic invariants of algebraic sets.
These notes illustrate a few of the ways in which properties of syzygies reflect qualitative geometric properties of algebraic varieties. Chapters 1, 3 and 4 were written by David Eisenbud, and closely follow his lectures at the introductory workshop. Chapter 2 was written by Jessica Sidman, from the lecture she gave enlarging on the themes of the first lecture and providing examples. The lectures may serve as an introduction to the book The Geometry of Syzygies [Eisenbud 1995]; in particular, the book contains proofs for the many of the unproved assertions here.
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