Abstract
In Chapter 5, we saw that to work with an R-module M, we needed not just the generators f 1,..., f t of M, but the relations they satisfy. Yet the set of relations Syz (f l,..., f t ) is an R-module in a natural way and, hence, to understand it, we need not just its generators g 1,..., g s , but the set of relations Syz (g l,...,g s ) on these generators, the so-called second syzygies. The second syzygies are again an R-module and to understand it, we again need a set of generators and relations, the third syzygies, and so on. We obtain a sequence, called a resolution, of generators and relations of successive syzygy modules of M. In this chapter, we will study resolutions and the information they encode about M. Throughout this chapter, R will denote the polynomial ring k[x l,..., x n ] or one of its localizations.
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© 1998 Springer Science+Business Media New York
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Cox, D., Little, J., O’Shea, D. (1998). Free Resolutions. In: Using Algebraic Geometry. Graduate Texts in Mathematics, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-6911-1_6
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DOI: https://doi.org/10.1007/978-1-4757-6911-1_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98492-6
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