Abstract
Flatness is a fundamental notion of commutative algebra, introduced by Serre in Ann. Inst. Fourier Grenoble 6:1-42. xxi, 444, (1955–1956), [238].
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Notes
- 1.
Or more generally if \(M\) is freely generated by a discrete set, i.e.
with \(I\) discrete. For another generalization see Exercise 16. - 2.
Actually, this refers to a variant, with essentially the same proof, which we leave to the reader.
- 3.
Concerning the general notion of a family of sets indexed by an arbitrary set, see [MRR, p. 18]; the construction of the direct sum of an arbitrary family of \(\mathbf {A}\)-modules is explained on pages 49 et 50.
- 4.
Please note that in the case of an integral ring with explicit divisibility, a principal localization matrix is known from its only diagonal elements, which can simplify computations.
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Lombardi, H., Quitté, C. (2015). Flat Modules. In: Commutative Algebra: Constructive Methods. Algebra and Applications, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9944-7_8
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