nLab
pure morphism

In general categorical setup

Definition

Given a regular cardinal $\kappa$, a morphism $f:A\to B$ in a category $C$ is $\kappa$-pure if for every commutative square

$$\begin{array}{ccc}P& \stackrel{g}{\to }& Q\\ u\downarrow & & \downarrow v\\ A& \stackrel{f}{\to }& B\end{array}$$\array{ P & \stackrel{g}\to & Q\\ u\downarrow && \downarrow v \\ A &\stackrel{f}\to & B }

in which $P$ and $Q$ are $\kappa$-presentable objects, the morphism $u:P\to A$ factors through $g$, i.e. there is some $h:Q\to A$ with $u=h\circ g$.

Remark

Notice that the above definition does not requrie that also the morphism $v$ is factored, hence it does not express a lifting property.

Proposition

In a $\kappa$-accessible category $C$ every $\kappa$-pure morphism is monic, hence exhibits a pure subobject. In a locally $\kappa$-presentable category $\kappa$-pure morphisms are, moreover, regular monomorphisms, and in fact coincide with the $\kappa$-directed colimits of split monomorphisms in the category of arrows $C^2=\mathrm{Arr}(C)$; more generally this characterization holds in all $\kappa$-accessible categories admiting pushouts.

(Adámek, Hub, Tholen).

In ring theory and for schemes

We work with unital, possibly commutative, rings and modules. Given a ring $R$, a morphism $f:M\to M\prime$ of left $R$-modules is pure if the tensoring the exact sequence of left $R$-modules

$$0\to \mathrm{Ker}f\to M\stackrel{f}{\to }M\prime \to \mathrm{Coker}f\to 0$$0\to Ker f \to M\stackrel{f}\to M'\to Coker f\to 0

with any right $R$-module $N$ (from the left) yields an exact sequence of abelian groups.

Grothendieck has proved that faithfully flat morphisms of commutative schemes are of effective descent for the categories of quasicoherent $𝒪$-modules. But this was not entirely optimal, as there is in fact a more general class than faithfully flat morphisms which satisfy the effective descent. For a local case of commutative rings, Joyal and Tierney have then proved (unpublished) that the effective descent morphisms for modules are precisely the pure morphisms of rings (or dually of affine schemes). Janelidze and Tholen have reproved the theorem as a corollary of a result for noncommutative rings obtained using the Beck’s comonadicity theorem.

Related concepts

• pure subobject

References

• Bachuki Mesablishvili, Pure morphisms of commutative rings are effective descent morphisms for modules – a new proof, Theory and Appl. of Categories 7, 2000, No. 3, 38-42, tac

• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge Univ. Press 2003; errata pdf

• George Janelidze, Walter Tholen, Facets of descent III: monadic descent for rings and algebras, Appl. Categ. Structures 12 (2004), no. 5-6, 461–477, MR2005i:13019, doi

• Jiří Adámek, H. Hub, Walter Tholen, On pure morphisms in accessible categories, J. Pure Appl. Alg. 107, 1 (1996), pp 1-8, doi

• Michel Hébert, Purity and injectivity in accessible categories, doi

• W.W. Crawley-Boevey, Locally finitely presented additive categories, Communications in Algebra 22(5)(1994), 1641-1674.

• Mike Prest, Purity, spectra and localisation, Enc. Math. Appl. 121, Camb. Univ. Press 2011, 798 pages; publishers book page

• Christian U. Jensen, Helmut Lenzing, Model theoretic algebra: with particular emphasis on fields, rings, modules, Algebra, Logic and Applications 2, Gordon and Breach 1989.

• Ivo Herzog, Pure-injective envelopes, pdf Journal of Algebra and Its Applications 2(4) (2003), 397-402.