There are several notions under a name of compatible localization; where localization pertains to the localization of rings or of localization of categories.
Consider a family of localizations sharing the same source ring (or source category). Two Gabriel localizations , of a ring , with corresponding localization endofunctors for modules are mutually compatible if the consecutive localization of is naturally a ring (or equivalently is a ring), such that the corresponding localization arrow is a morphism of rings. This is equivalent to the following more general criterium.
Two affine localizations with the inverse image functors and direct image functors , are mutually compatible if the endofunctors mutually commute as endofunctors of .
According to Thomason-Trobaugh, an algebraic scheme is semiseparated if it has an affine cover such that the intersection of any two affines in the cover is affine (semiseparated cover). For example, every separated scheme is semiseparated. Alexander Rosenberg extends this terminology to noncommutative schemes by observing that an affine cover is semiseparated iff the corresponding localization functors restricting the category of quasicoherent sheaves on the whole scheme to an element of a cover, form a compatible family of affine localizations.
Given a right -comodule algebra for a bialgebra , with coaction , an Ore localization of rings (or more general affine localization) is -compatible if there exists a morphism such that the following diagram commutes:
A localization functor (not necessarily having a right adjoint) universally inverting a family of morphisms in is compatible with an endofunctor if . By the universality property of the localization functor, this is equivalent to the existence of a functor such that .
If is a localization having a fully faithful right adjoint , with counit (which is hence iso) and unit ; then the compatibility of with an endofunctor is equivalent to any of the following:
(i) there exists a distributive law where is understood as (the underlying functor of) the idempotent monad induced by the adjunction
(ii) the natural transformation is invertible
(iii) there is a functor and a natural isomorphism .
If (i) holds then is invertible and uniquely determined by and the monad . Notice that in general. See distributive law for idempotent monad for more.
One can also consider the distributive laws . The inverse of the distributive law as above is the unique invertible example of such. It is not clear (to me at least – Zoran) if there are also noninvertible examples for .
For the pairwise compatibility of localizations see
Fred Van Oystaeyen, Compatibility of kernel functors and localization functors, Bull. Soc. Math. Belg. 28 (1976), no. 2, 131–137.
Alain Verschoren, Compatibility and stability, Notas de Matemática [Mathematical Notes], 3. Universidad de Murcia, Secretariado de Publicaciones e Intercambio Científico, Murcia, 1990. xii+81 pp. ISBN: 84-7684-934-6
J. Mulet, A. Verschoren, On compatibility. II., Comm. Algebra 20 (1992), no. 7, 1897–1905. doi
M. I. Segura, D. Tarazona, A. Verschoren, On compatibility, Comm. Algebra 17 (1989), no. 3, 677–690.
The compatibility of coactions with Ore localizations is introduced in
The compatibilities between localization functors with endofunctors is formulated in
A version with distributive laws is introduced in