nLab 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.

(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 and $Q_1 Q_2 R$ is a ring (semi-compatible if just one of the two), 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 $Q_k^*$ 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 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.

Compatible Ore localizations

Let $S$ and $T$ be two Ore sets in a domain $R$ and $i_T:R\to T^{-1}R$ the canonical map. Then $i_T(S)$ is left Ore in $T^{-1}R$ if for every $r\in R$ , $t\in T$ and $s\in S$ , there are $s'\in S$ , $r'\in R$ and $t'\in T$ such that $s' t^{-1} r = t'^{-1}r' s$ in $T^{-1} R$ .
(This can be written as $t^{-1}r s^{-1} = s'^{-1}t'^{-1}r'$ in the module $T^{-1}S^{-1}T^{-1}R$ .) We claim that it is sufficient to have this condition satisfied for $r = 1$ . Indeed, find $\tilde{s}\in S$ and $\tilde{r}\in R$ so that $\tilde{s}r = \tilde{r} r$ . We then find $t_1\in T$ , $s_1\in S$ and $r_1\in R$ so that $s_1 t^{-1} = t^{-1}_1 r_1 \tilde{s}$ . It follows that $s_1 t^{-1} r = t_1^{-1} r_1\tilde{s} r = t_1^{-1}(r_1\tilde{r})s$ .

This could be found more readily by computing in the ring $(T\vee S)^{-1} R$ or module $T^{-1}S^{-1}T^{-1}R$ where the above condition is $t^{-1}r s^{-1} = s'^{-1}t'^{-1}r'$ . This can be interpreted also as $S^{-1} T^{-1} R\subset T^{-1}S^{-1}R$ within $(T\vee S)^{-1} R$ . Then it is easy to use $r = 1$ case by $t^{-1} r s^{-1} = t^{-1}\tilde{s}^{-1}\tilde{r} = s_1^{-1}t_1^{-1}r_1\tilde{r}$ .

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:

\array{ E & \stackrel{j}\to & E'\\ \downarrow\rho && \downarrow \rho'\\ E\otimes H &\stackrel{j\otimes H}\to & E'\otimes H }

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 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) if there are also noninvertible examples for $\tilde{l}$ .

Literature

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. doi

The compatibility of coactions with Ore localizations is introduced in

Zoran Škoda, Localizations for construction of quantum coset spaces, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications 61, pp. 265–298, Warszawa 2003, math.QA/0301090.

The compatibilities between localization functors with endofunctors is formulated in

V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf

A version with distributive laws is introduced in

Zoran Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770.