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4 - STABILITY AND FINITE FREE RESOLUTIONS
Published online by Cambridge University Press: 15 December 2009
Summary
General remarks
We are now ready to begin the deeper study of finite free resolutions. The results that we shall obtain will enable us to provide solutions to some of the problems that have been left over from Chapter 3, but in the process we shall encounter certain new difficulties of an arithmetical nature. In order to deal with these it will be necessary to develop, from first principles, a general theory of grade. This is a topic to which Chapter 5 will be devoted.
Throughout the present chapter, R will denote a non-trivial commutative ring with an identity element. Since R is commutative, there is no essential difference between left R-modules and right R-modules. If therefore E is an R-module, r ∈ R and e ∈ E, it is immaterial whether we write the product of r and e as re or as er. We shall take advantage of this fact.
Further results on matrices
In section (4.1) we shall be concerned with p × q matrix A and q × t matrix B (each with entries in R), where p, q, t denote positive integers. (Later we shall need to introduce conventions to cover situations in which the set of rows or columns, of one or other of these matrices, is empty. However it is convenient to avoid these minor complications for the time being.) Because A has q columns and B has q rows, it is possible to form the product AB. We shall be particularly concerned with cases where AB = 0.
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- Finite Free Resolutions , pp. 99 - 130Publisher: Cambridge University PressPrint publication year: 1976