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kernel functor

In noncommutative ring theory, particularly in the subject of noncommutative localization of rings, a kernel functor is any left exact additive subfunctor of the identity functor on the category RMod{}_R Mod of left modules over a ring RR. There is a bijective correspondence between kernel functors and uniform filters of ideals in RR. A functor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod is idempotent if σσ=σ\sigma\sigma = \sigma and a preradical if it is additive subfunctor of the identity and σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_R Mod. A kernel functor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod is said to be an idempotent kernel functor if σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_R Mod; it is idempotent as we see by calculating

σσM=σKer(MM/σM)=Ker(σMσ(M/σM))=Ker(σMM/σM)=σM \sigma \sigma M = \sigma Ker(M\to M/\sigma M) = Ker (\sigma M\to \sigma(M/\sigma M)) = Ker(\sigma M\to M/\sigma M) = \sigma M

In the last step, we used that σ\sigma is a subfunctor of the identity, hence the compositions σMMM/σM\sigma M\hookrightarrow M\to M/\sigma M and σMσ(M/σM)M/σM\sigma M\to \sigma(M/\sigma M)\to M/\sigma M coincide.

The basic reference is

  • O. Goldman, Rings and modules of quotients, J. Algebra 13, 1969 10–47, MR245608, doi

which is clearly written from the point of view of a ring theorist. Unfortunately, it just creates another formalism in localization theory of the categories of modules over a ring for basically the same results as P. Gabriel succeeded by more categorical formulations in his thesis published 7 years earlier. Some of the methods from Goldman, and even more from Gabriel apply for more general Grothendieck categories.

  • Pascual Jara, Alain Verschoren, Conchi Vidal, Localization and sheaves: a relative point of view, Pitman Research Notes in Mathematics Series, 339. Longman, Harlow, 1995. xiv+235 pp.
  • J. L. Bueso, P. Jara, A. Verschoren, Compatibility, stability, and sheaves, Monographs and Textbooks in Pure and Applied Mathematics, 185. Marcel Dekker, Inc., New York, 1995. xiv+265 pp.
Revised on June 8, 2011 18:07:06 by Zoran Škoda (161.53.130.104)