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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:
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