# nLab sheaf of ideals

An (left, right, two-sided) ideal of ring (or $k$-algebra) $R$ is a (left, right, bi) $R$-submodule of $R$ itself. One can consider similarly the notion of a presheaf of modules? over a presheaf of rings: and the presheaf of ideals is then the presheaf of submodules of a presheaf of rings.

In algebraic geometry, quasicoherent sheaves of modules are of special importance; thus also quasicoherent ideals as those ideals which are quasicoherent as sheaves of modules. An important example of a quasicoherent sheaf of ideals is the defining sheaf of ideals of a closed subscheme.

Wikipedia uses less preferrable term ideal sheaf.

Having a finitary monad $T$ in the category of sets, the sheafification functor from the category of presheaves of sets to the category of sheaves of sets, can be strictly lifted to the category of presheaves of $T$-modules (or, if you like, $T$-algebras) to the category of sheaves of $T$-modules. More generally, one can consider sheaves of finitary monads and corresponding categories of sheaves of modules and of bimodules. The discussions of sheaves of ideals extends easily to this setting.

Revised on March 6, 2013 19:35:13 by Zoran Škoda (161.53.130.104)