Ideal theory

In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)

Topology determined by an ideal

Reduction theory

Local cohomology in ideal theory

Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.

Let $M$ be a module over a ring $R$ and $I$ an ideal. Then $M$ determines the sheaf ${\widetilde {M}}$ on $Y=\operatorname {Spec} (R)-V(I)$ (the restriction to Y of the sheaf associated to M). Unwinding the definition, one sees:

\Gamma _{I}(M):=\Gamma (Y,{\widetilde {M}})=\varinjlim \operatorname {Hom} (I^{n},M)

.

Here, $\Gamma _{I}(M)$ is called the ideal transform of $M$ with respect to $I$ .^[1]

References

^ Eisenbud 2005, Appendix 10B.

Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432

[1] Eisenbud 2005, Appendix 10B.

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