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\begin{document}

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\section*{compatible localization}

There are several notions under a name of compatible localization; where localization pertains to the localization of rings or of localization of categories.

\hypertarget{pairwise_compatibility_of_a_family_of_localizations}{}\subsection*{{(Pairwise) compatibility of a family of localizations}}\label{pairwise_compatibility_of_a_family_of_localizations}

Consider a family of localizations sharing the same source ring (or source category). Two Gabriel localizations $i_{R k}: R\to R_k$, $k = 1,2$ of a ring $R$, with corresponding localization endofunctors for modules $Q_{k}:M\to M_k$ are mutually compatible if the consecutive localization of $Q_2 Q_1 R$ is naturally a ring (or equivalently $Q_1 Q_2 R$ 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 localization]]s with the inverse image functors $Q_k^*$ and direct image functors $Q_{k*}: A\to A_k$, $k = 1,2$ are mutually compatible if the endofunctors $Q_k:=Q_{k*}Q_k^*$ mutually commute as endofunctors of $A$.

According to Thomason-Trobaugh, an [[algebraic scheme]] is \textbf{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 scheme]]s 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.

\hypertarget{coaction_compatible_localizations}{}\subsection*{{Coaction compatible localizations}}\label{coaction_compatible_localizations}

Given a right $B$-comodule algebra $(E, \rho)$ for a bialgebra $H$, with coaction $\rho : E\to E\otimes H$, an Ore localization of rings (or more general affine localization) $j : E\to E'$ is $\rho$-compatible if there exists a morphism $\rho': E'\to E\otimes H$ such that the following diagram commutes:

\begin{displaymath}
\itexarray{
E & \stackrel{j}\to & E'\\
\downarrow\rho && \downarrow \rho'\\
E\otimes H &\stackrel{j\otimes H}\to & E'\otimes H
}
\end{displaymath}
\hypertarget{compatibility_of_localizations_and_endofunctors}{}\subsection*{{Compatibility of localizations and endofunctors}}\label{compatibility_of_localizations_and_endofunctors}

A localization functor $Q^*=Q^*_\Sigma : A\to B$ (not necessarily having a right adjoint) universally inverting a family $\Sigma$ of morphisms in $A$ is compatible with an endofunctor $G:A\to A$ if $G(\Sigma)\subset \Sigma$. By the universality property of the localization functor, this is equivalent to the existence of a functor $G_\Sigma\in B$ such that $Q^* G = G_\Sigma Q^*$.

If $Q^*$ is a localization having a fully faithful right adjoint $Q_*$, with counit $\epsilon$ (which is hence iso) and unit $\eta$; then the compatibility of $Q^*$ with an endofunctor $G$ is equivalent to any of the following:

(i) there exists a distributive law $l : Q_* Q^* G\to G Q_* Q^*$ where $Q_* Q^*$ is understood as (the underlying functor of) the idempotent monad induced by the adjunction $Q^*\dashv Q_*$

(ii) the natural transformation $Q^* G\eta : Q^* G\to Q^* G Q_* Q^*$ is invertible

(iii) there is a functor $G'$ and a natural isomorphism $Q^* G\cong G' Q^*$.

If (i) holds then $l$ is invertible and uniquely determined by $G$ and the monad $Q^* Q_*$. Notice that $Q_* G_\Sigma \neq G Q_*$ in general. See [[zoranskoda:distributive law for idempotent monad]] for more.

One can also consider the distributive laws $\tilde{l} : G Q_* Q^*\to Q_* Q^* G$. The inverse $l^{-1}$ of the distributive law $l: G Q_* Q^*\to Q_* Q^* G$ as above is the unique invertible example of such. It is not clear (to me at least -- [[Zoran Škoda|Zoran]]) if there are also noninvertible examples for $\tilde{l}$.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

For the pairwise compatibility of localizations see

\begin{itemize}%
\item [[Fred Van Oystaeyen]], \emph{Compatibility of kernel functors and localization functors}, Bull. Soc. Math. Belg. \textbf{28} (1976), no. 2, 131--137.


\item [[Alain Verschoren]], \emph{Compatibility and stability}, Notas de Matem\'a{}tica Mathematical Notes, 3. Universidad de Murcia, Secretariado de Publicaciones e Intercambio Cient\'i{}fico, Murcia, 1990. xii+81 pp. ISBN: 84-7684-934-6


\item J. Mulet, A. Verschoren, \emph{On compatibility. II.}, Comm. Algebra 20 (1992), no. 7, 1897--1905. \href{http://dx.doi.org/10.1080/00927879208824438}{doi}


\item M. I. Segura, D. Tarazona, A. Verschoren, \emph{On compatibility}, Comm. Algebra \textbf{17} (1989), no. 3, 677--690.



\end{itemize}
The compatibility of coactions with Ore localizations is introduced in

\begin{itemize}%
\item [[Zoran Škoda]], \emph{Localizations for construction of quantum coset spaces}, in ``Noncommutative geometry and Quantum groups'', W.Pusz, P.M. Hajac, eds. Banach Center Publications \textbf{61}, pp. 265--298, Warszawa 2003, \href{http://arxiv.org/abs/math.QA/0301090}{math.QA/0301090}.

\end{itemize}
The compatibilities between localization functors with endofunctors is formulated in

\begin{itemize}%
\item [[V. A. Lunts]], [[A. L. Rosenberg]], \emph{Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings}, MPI 1996-53 \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=3894}{pdf}

\end{itemize}
A version with distributive laws is introduced in

\begin{itemize}%
\item [[Zoran Škoda]], Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, \href{http://arxiv.org/abs/0811.4770}{arXiv:0811.4770}.

\end{itemize}
See also

\begin{itemize}%
\item [[Zoran Škoda]], \emph{Compatibility of (co)actions and localization}, \href{http://arxiv.org/abs/0902.1398}{arXiv:0902.1398} (preliminary version).

\end{itemize}
[[!redirects compatible localizations]]



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