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Publication Detail
Syzygies and Minimal Resolutions
  • Publication Type:
    Chapter
  • Authors:
    Johnson FEA
  • Publisher:
    Imperial College Press
  • Publication date:
    2017
  • Place of publication:
    London, UK
  • Pagination:
    293, 226
  • Chapter number:
    6
  • Series:
    Lectures at LTCC
  • Book title:
    Geometry in advanced pure mathematics
Abstract
The essence of linear algebra over a field resides in the fact that every vector space is free; that is, has a spanning set of linearly independent vectors. The study of linear algebra over more general rings attempts to approximate this situation by the method of free resolutions. When a module $M$ is not free we make a first approximation to its being free by taking a surjective homomorphism $\epsilon : F_0 \rightarrow M$ where $F_0$ is free to obtain an exact sequence $$ 0 \rightarrow J_1 \rightarrow F_0 \stackrel{\epsilon}{\rightarrow} M \rightarrow 0. $$ Repeating the construction we approximate $J_1$ in turn by a free module to obtain an exact sequence $\;0 \rightarrow J_2 \rightarrow F_1 \rightarrow J_1 \rightarrow 0. \; $ Iterating and splicing we obtain a {\it free resolution of} M in the sense of Hilbert
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