Descent for the abelian categories may be given in two standard ways: one by using transition functors, and another to pack all the transition functors in a single comonad. We shall deal here with a family of flat localizations which are jointly a conservative family (cover by flat localization). If we satisfy conditions for one of the variants of the Beck’s comonadicity theorems we have descent.
This section is explained in (Ros:Nc-Sch) and (Skoda:nloc).
By a flat localization (functor) we will mean an exact additive functor $Q^*$ having a fully faithful right adjoint $Q_*$. The composition $Q_* Q^*$ is then often denoted by $Q$. A family $F_\lambda^*: \mathcal{A}\to \mathcal{B}_\lambda$, $\lambda\in\Lambda$ of functors with the same domain is conservative if $Q^*_\lambda(f)$ is invertible for all $\lambda$ only if $f$ is invertible. A flat cover of an abelian category $\mathcal{A}$ is by definition a conservative family of flat functors $F_{\lambda}:\mathcal{A}\to\mathcal{B}_{\lambda}$.
Localization functors typically do not mutually commute. Namely, given a family of flat localizations $Q_\lambda^*: \mathcal{A}\to\mathcal{A}_\lambda$, $\lambda\in\Lambda$, the functors $Q_{\lambda} Q_\mu = Q_{\lambda*}Q_\lambda^{*}Q_{\mu*}Q_\mu^*:\mathcal{A}\to\mathcal{A}$ and $Q_\mu Q_\lambda$ for $\lambda\neq\mu$ are in general not isomorphic. If the family is a cover then define the product category $\mathcal{A}_\Lambda = \prod_{\lambda\in\Lambda}\mathcal{A}_\lambda$ and the functor $\mathbf{Q}^*:\mathcal{A}\to\mathcal{A}_\Lambda$, $\mathbf{Q}^*:M\to(Q_\lambda^* M)_{\lambda}$ where the notation $(N_\lambda)_\lambda = (N_\lambda)_{\lambda\in\Lambda}$ denotes the ordered $\Lambda$-tuple in $\mathcal{A}_\Lambda$. If $\mathcal{A}$ has products of families of $\mathrm{card}\,\Lambda$ objects, then $\mathbf{Q}^*$ has a right adjoint $\mathbf{Q}_* :\mathcal{A}_\Lambda\to\mathcal{A}$ given by $(M^\lambda)_\lambda\mapsto \prod_\lambda Q_{\lambda *} M^\lambda$. Indeed,
The unit $\mathbf{\eta}:Id_{\mathcal{A}}\to\mathbf{Q}_*\mathbf{Q}^*:\mathcal{A}\to\mathcal{A}$ of the adjunction $\mathbf{Q}^*\dashv\mathbf{Q}_*$ is the map induced from the units $\eta^\lambda$, by the universality of Cartesian product in $\mathcal{A}$, namely $\mathbf{\eta} = (\eta^\lambda)_{\lambda\in\Lambda} : M\to\prod_\lambda Q_{\lambda*}Q^*_\lambda M$. The counit $\mathbf{\epsilon}:\mathbf{Q}^*\mathbf{Q}_*\to Id_{\mathcal{A}_\Lambda}$ has the components given by the compositions
where in the first functor the projections for the Cartesian product are used.
Denote $\Omega:= \mathbf{Q}^* \mathbf{Q}_{*} :\mathcal{A}_\Lambda\to\mathcal{A}_\Lambda$; then $\mathbf{\Omega} = (\Omega,\mathbf{\delta},\mathbf{\epsilon})$ is the comonad on $\mathcal{A}_\Lambda$ induced by the adjunction $\mathbf{Q}^*\dashv\mathbf{Q}_*$, where for each $(N^\lambda)_\lambda\in\mathcal{A}_\Lambda$ the component $\mathbf{\delta}^\lambda_{(N^\lambda)_\lambda}$ of the comultiplication $\mathbf{\delta} = \mathbf{Q}^*\mathbf{\eta}\mathbf{Q}_*$ is more explicitly the map
Again, if each $Q_\mu^*$ commutes with $\Lambda$-products then the products can be placed in front: $$(\prod_\lambda Q^*_\mu Q_{\lambda *} N^\lambda)_\mu \to (\prod_{\lambda\rho}Q_\mu^*Q_{\rho*}Q^*_\rho Q_{\lambda *} N^\lambda)_\mu$$
There is a comparison functor
under the appropriate (Beck comonadicity criteria) conditions $K_{\mathbf{\Omega}}$ is an equivalence, with the (quasi)inverse mapping sending an $\mathbf{\Omega}$-comodule $(N,\nu)\in(\mathcal{A}_\Lambda)_{\mathbf{\Omega}}$, into the equalizer of morphisms $\mathbf{\eta}_{\mathbf{Q}_*N}$ and $\mathbf{Q}_*(\nu) : \mathbf{Q}_* N\to \mathbf{Q}_*\mathbf{Q}^*\mathbf{Q}_* N$ in $\mathcal{A}$, thus identifying $\mathcal{A}$ with the Eilenberg-Moore category of comodules for the comonad $\mathbf{\Omega}$.
Now consider functor $F:\mathcal{A}\to\mathcal{B}$ between two comonadic categories. Throughout we assume that the comonad is induced by a cover by flat localization functors commuting with direct products $card\,\Lambda$ objects; without loss of generality, for simplicity, the categories are identified with the Eilenberg-Moore categories for the corresponding comonad. The localization data for $\mathcal{B}$ will be denoted by primed symbols, like $\Omega'$, $Q'$ etc. while the label set for the cover will be $M$ rather than $\Lambda$.
We claim that $F$ can be locally represented by fucntors $F_{\mu\lambda} : \mathcal{A}_\lambda\to\mathcal{B}_\mu$ for all pairs $\lambda,\mu$ where these local functors have to satisfy a set of compatibility conditions to explain later. The passage from $F$ to $F_{\mu\lambda}$ is given by a simple recipe, namely $F_{\mu\lambda} = Q^{'*}_\mu F Q_{\lambda*}$, reminding the description of morphisms of manifolds in pairs of local charts.
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Let $\{ Q^*_\lambda : A \rightarrow B_\lambda\}_{\lambda \in \Lambda}$ be a small family of functors having fully faithful right adjoints $Q_{\lambda *}$. In practice, the categories $B_\lambda$ are not necessarily constructed from $A$ by a localization, but of course they are equivalent to such.
One may consider the category $B_\Lambda : = \prod_{\lambda \in \Lambda}B_\lambda$ whose objects are families $\prod_{\lambda \in \Lambda} M^\lambda$ of objects $M^\lambda$ in $B_\lambda$ and morphisms are families $\prod_{\lambda \in \Lambda}f_\lambda : \prod_{\lambda \in \Lambda} M^\lambda \rightarrow \prod_{\lambda \in \Lambda} N^\lambda$ where $f_\lambda : M^\lambda \rightarrow N^\lambda$ is a morphism in $B_\lambda$, with componentwise composition. This makes sense as the family of objects is literally a function from $\Lambda$ to the disjoint union $\coprod_\lambda {Ob}\,B_\lambda$ which is in the same Grothendieck universe.
The family of adjoint pairs $Q^*_\lambda \dashv Q_{\lambda *}$ defines an inverse image functor $\mathbf{Q}^* = \prod Q^*_\lambda : A \rightarrow A_\Lambda$ by $\mathbf{Q}^*(M) := \prod_{\lambda \in \Lambda}Q^*_\lambda(M)$ on objects and $\mathbf{Q}^*(f) := \prod_\lambda Q^*_\lambda(f)$ on morphisms. However, a direct image functor may not exist. We may naturally try $\mathbf{Q}_* : \prod'_\lambda M^\lambda \mapsto \prod'_\lambda Q_*(M^\lambda)$ where $\prod'$ is now the symbol for the Cartesian product in $A$ which may not always exist. For finite families, with $A$ abelian, these trivially exist. Let $A^\Lambda = \prod_{\lambda\in\Lambda} A$ be the power category. Assume a fixed choice of the Cartesian product for all $\Lambda$-tuples in $A$. Then $\{M^\lambda\}_\lambda \mapsto \prod'_{\lambda\in \Lambda} M^\lambda$ extends to a functor $A^\Lambda\to A$, and the universality of products implies that the projections $p'_{\nu \{M^\lambda\}_\lambda} : \prod_\lambda M^\lambda \rightarrow M^\nu$ form a natural transformation of functors $p'_\nu : \prod'_{\lambda} {Id}_A\Rightarrow {p}_{\nu}$ where $p_\nu : A^\Lambda \rightarrow A$ is the $\nu$-th formal projection $\prod_\lambda M^\lambda\to M^\nu$. The unique liftings $\mathbf{\eta}_M : M \rightarrow \mathbf{Q}_* \mathbf{Q}^* (M)$ of morphisms $\eta_{\nu M} : M \rightarrow Q_{\nu *} Q^*_\nu (M)$ in the sense that $(\forall \nu)\, \eta_{\nu M} = p'_{\nu M} \circ \eta_M$ hence form a natural transformation $\mathbf{\eta} : {Id}_A \Rightarrow \mathbf{Q}_* \mathbf{Q}^*$.
Define $\mathbf{\epsilon} \equiv \prod_\lambda \epsilon_\lambda: \prod_\lambda Q^*_\lambda Q_{\lambda *} \Rightarrow \prod_\lambda {Id}_{B_\lambda} = {Id}_{ B_\Lambda}$ componentwise. This way we obtain an adjunction $\mathbf{Q}^* \dashv \mathbf{Q}_*$. As usual, every adjunction generates a comonad on $B_\Lambda$, which will be in our case denoted by $\mathbf{G}=(G,\delta,\mathbf{\epsilon})$ where $G=\mathbf{Q}^*\mathbf{Q}_*$, the comultiplication is $\delta = \mathbf{Q}^*\mathbf{\eta}\mathbf{Q}_*$ where the counit $\mathbf\epsilon$ and the unit $\mathbf{\eta}$ are from above.
If $Q_{\lambda *}$ is faithful and exact for every $\lambda$ then $\mathbf{Q}_*$ is as well.
Consider the comonad $\mathbf{G}$ in $B_\Lambda$ associated to $\mathbf{Q}^* \dashv \mathbf{Q}_*$. We are interested in situation when the comparison functor $K_{\mathbf{G}}$ is an equivalence of categories.
Typically one needs to ensure the faithfulness of $\mathbf{Q}_*$ for what we require that the family $\{Q^*_\lambda\}_{\lambda \in \Lambda}$ is a flat cover of $A$, by that in this setup we mean that $Q^*_\lambda$ are all exact (and having right adjoint as before). Flat cover is by a definition a small flat family of functors with domain $A$ which is conservative, i.e. a morphism $f \in A$ is invertible iff $Q^*_\lambda(f)$ is invertible for each $\lambda \in \Lambda$. A flat map whose direct image functor is conservative is called almost affine. In particular, this is true for adjoint triples $f^* \dashv f_* \dashv f^!$ coming from a map $f : R \rightarrow S$ of rings. Adjoint triples where the direct image functor is conservative are called affine morphisms.
The cocycle condition for gluing morphism is equivalent to the coaction axiom for the associated comodule. The remaining requirements are made to ensure that the comparison functor is an equivalence and the other original data may be reconstructed as well. The Eilenberg-Moore category of the associated monad may be constructed directly from gluing morphisms, and it appears to be just a reformulation of the descent category.
A special case of Beck's theorem. Let $\mathbf{Q}^* \dashv \mathbf{Q}_*$ be an adjoint pair and $\mathbf{G}$ its associated comonad._If $\mathbf{Q}^*$ preserves and reflects equalizers of all parallel pairs in $\mathbf{B}$ (for which equalizers exists) and if any parallel pair mapped by $\mathbf{Q}^*$ into a pair having an equalizer in $A$ has an equalizer in $\B$, then the comparison functor $K : A \rightarrow \mathbf{G}-{Comod}$ is an equivalence of categories.
Notation standard in the localization theory of modules. We usually denote the composition $Q_\lambda := Q_{\lambda *}Q_\lambda^*$ where $Q_{\lambda *}$ is the right adjoint to the categorical localization $Q_\lambda^*$. No star means hence means that we consider a localization as an endofunctor (comp. of a localization and an embedding back of the localized category). When ring theorists say that a localization $Q_\lambda$ is flat you never know if they mean really that the composition is flat (stronger property: means that both left and right adjoints $Q_{\lambda *}$ and $Q_\lambda^*$ are flat, we say that the “geometric morphism” is biflat. If $M$ is in the nonlocalized category, then the component of the adjunction is denoted by $i_{\lambda M}:M\to Q_\lambda M$. If the domain is the category $R-Mod$ of modules over a ring $R$ and $Q_\lambda$ itself has a right adjoint (affine localization) then in fact $Q_\lambda R$ is also a ring and $i_{\lambda R}$ is a ring homomorphism (this is true for example for all perfect localizations e.g. Ore localizations which are affine and biflat).
Localization functors do not commute. While the descent over topological spaces compares the modules over multiple intersections and $U\cap V=V\cap U$, for the descent over the covers by localizations the consecutive localized modules $Q_\mu Q_\nu M$ and $Q_\nu Q_\mu M$ are NOT in general isomorphic!
We denote $M_\lambda = Q_\lambda M$. Thus $M_{\mu\nu} = Q_\nu Q_\mu M$ (note order!).
Canonical maps between single and (pair-) consecutive localizations are of two distinct kinds
$i^\mu_{\mu\nu M}=(i_\nu)_{M_\mu}: M_\mu\to M_{\mu\nu}$ is given by the component of the adjunction morphism $i_\nu:Id\to Q_\nu = Q^*_\nu Q_{\nu *}$ at $M_\mu$
$i^\nu_{\mu\nu M} = Q_\nu(i_{\mu M}) : M_\nu\to M_{\mu\nu}$ is obtained by applying localization functor to the component of adjunction morphism $i_\mu$ at $M$.
Notice that while Grothendieck topologies satisfy stability exiom, we do not have pullback here at all: while intersections of open sets are pullbacks, the consecutive localizations $Q_\mu Q_\nu A$ are not pullbacks of $Q_\mu A$ and $Q_\nu A$ and instead of comparing modules at the pullbacks, the descent requires that we compare $N^\mu$ from $\mu$-localization and module $N^\mu$ from $\nu$-localization both in $\mu,\nu$ and $\nu,\mu$-consecutive localizations, for the comparison we use both kinds of the canonical maps. When I say $N^\mu$ I mean some object in $\mu$-localization, which is not necessarily of the form $Q_\mu N$ for some $N$, if it were I would call it $N_\mu$ (lower index).
Globalization lemma. (version for Gabriel filters: in (Rosen:88) p. 103) Suppose $\{Q_\lambda^* : R-{Mod} \rightarrow \mathcal{M}_\lambda \}_{\lambda \in \Lambda}$ is a finite cover of $R-{Mod}$ by affine biflat localization functors (e.g. a conservative family of Ore localizations $\{S_\lambda^{-1}R \}_{\lambda \in \Lambda}$). Then for every left $R$-module $M$ the sequence
is exact, where $\iota_{\Lambda M} : m \mapsto \prod \iota_{\lambda,M}(m)$, and
The maps $\iota_{\mu\nu}$ etc. are obtained as components of adjunctions as usual. For $\iota_{\Lambda\Lambda}$ we use combination of all such maps. Here the order matters: pairs with $\mu = \nu$ may be (trivially) skipped, but, unlike in the commutative case, we can not confine to the pairs of indices with $\mu\lt\nu$ only. Nota bene!
Proof is a direct corollary of Beck theorem. For proofs in the cases and in terms of Gabriel filters and hereditary torsion theories see (Rosen:book), pp. 23–25, and (Jara-Verschoren-Vidal, Versch:96, vOyst:assalg).
(Jara-Verschoren-Vidal) P. Jara, A. Verschoren, C. Vidal, Localization and sheaves: a relative point of view, Pitman Res. Notes in Math. 339, Longman 1995.
(Rosen:88) A. L. Rosenberg, Non-commutative affine semischemes and schemes, Seminar on supermanifolds No. 26, edited by D. Leites, Dept. of Math., Univ. of Stockholm 1988, ISSN 0348-7652.
(Rosen:book) A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MAIA 330, Kluwer 1995.
(Ros:Nc-Sch) A. L. Rosenberg, Noncommutative schemes, Comp. Math. 112 (1998), pp. 93–125.
(Skoda:nloc) Z. Škoda, Noncommutative localization in noncommutative geometry
(vOyst:assalg) F. van Oystaeyen, Algebraic geometry for associative algebras, Marcel Dekker 2000.
(vOyst-Willaert:1995) F. van Oystaeyen, L. Willaert, Grothendieck topology, coherent sheaves and Serre’s theorem, J. Pure Appl. Alg. 104 (1995), pp. 109–122.
(Versch:96) A. Verschoren, Sheaves and localization, J. Algebra 182 (1996), no. 2, pp. 341–346.