nLab pure morphism

In category theory

Definition

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

$$\array{ P & \overset{g} {\longrightarrow} & Q \\ \mathllap{^u} \big\downarrow && \big\downarrow \mathrlap{^v} \\ A & \underset{f} {\longrightarrow} & B }$$

in which $P$ and $Q$ are $\kappa$-presentable objects, the morphism $u \colon P\to A$ 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 require 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 a monomorphism, 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 arrow category $C^2 = Arr(C)$;

• more generally this characterization holds in all $\kappa$-accessible categories admitting pushouts.

(Adámek-Hub-Tholen 1996).

In ring theory and for schemes

Consider unital, possibly commutative, rings and their modules.

Definition

Given such a ring $R$, a homomorphism $f \colon M\to M'$ of left $R$-modules is pure if tensoring the exact sequence of left $R$-modules

$$0 \to Ker f \longrightarrow M \overset{f} {\longrightarrow} M' \longrightarrow Coker f \to 0$$

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

Remark

Grothendieck has proved that faithfully flat morphisms of commutative schemes are of effective descent for the categories of quasicoherent $\mathcal{O}$-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 For a local case of commutative rings, Joyal & Tierney (1984) have then proved (unpublished) that the effective descent morphisms for modules are precisely the pure morphisms of rings (or dually of affine schemes) accotding to Def . The result can be extracted also from their Memoirs volume on Galois theory. Janelidze & Tholen (2004) have reproved this theorem as a corollary of a result for noncommutative rings obtained using Beck’s comonadicity theorem.

Related concepts

• pure subobject

References

General references

• The Stacks Project, 28.10. Radicial and universally injective morphisms (of schemes) tag/01S2, 34.4. Descent for universally injective morphisms, tag/08WE

Descent along pure morphisms

The following paper was the first with the result on that pure morhisms are of the effective descent but the proof has been omitted:

• Jean-Pierre Olivier, Descente par morphismes purs, C. R. Acad. Sci. Paris Sér. A-B 271 (1970) A821–A823

The result is proved (within a larger context)

• André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309 (1984) vol. 51, pages vii+71

Clean proofs are by Mesablishvili and by Janelidze,

• 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

• Bachuki Mesablishvili, Pure morphisms are effective for modules, Applied Categorical Structures 21 (2013), 801–809. arXiv, doi.

• 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 5-6 (2004) 461-477 [MR2005i:13019, doi]

Much attention on the descent along pure morphism is in a recent survey

• Yves André, Luisa Fiorot, On the canonical, fpqc, and finite topologies on affine schemes. The state of the art, in Ann. Sc. Norm. Super. Pisa Cl. Sci. arXiv:1912.04957

Ziegler spectrum and connections to model theory

The Ziegler spectrum of indecomposable pure injectives has been introduced in

• M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984) 149 – 213.

A textbook account is in

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

The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is studied in

• G. Garkusha, M. Prest, Triangulated categories and the Ziegler spectrum, Algebras & Repres. Theory 8, 499–523 (2005) doi

Other articles:

• M. Prest, P. Rothmaler, M. Ziegler, Absolutely pure and flat modules and “indiscrete” rings, J. Alg. 174:2 (1995) 349-372 doi

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

• I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74:3 (1997) 503-558 doi

Other categoey theoretic articles

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

• 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.

• Rosanna Laking, Purity in compactly generated derivators and t-structures with Grothendieck hearts, Math. Zeitschrift, doi (2019).